Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.44.0-wmf.8 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk 4 1558 2693846 2693190 2024-12-30T08:34:44Z Joanetdelonso 2995613 Edible wild plants of Mallorca island 2693846 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} = Edible wild plants of Mallorca = == And medicinal wild plants == === Uses of wild plants === <ref>Example footnote</ref>Eat wild plants is fun and healthy # You can prove this plants in your village or country # This plants are free, you can eat no problem, without spend money # Nature is the best way for an healthy body All people can prove and experiment with wild plants 7y4ep3lvri64bxe36cc75oq9h934bmm 0 7793 2693756 2037124 2024-12-29T12:34:01Z OutsideNeutron 2995580 adds code fragment formatting to method names and references 2693756 wikitext text/x-wiki {{tertiary}} {{it}} {{mathematics}} {{Template:25%done}} [[Image:Fibonacci.jpg|thumb|right|200px|A portrait of [[w:Fibonacci|Fibonacci]], after whom [[w:Fibonacci heap|Fibonacci heaps]] are named. These heaps enable O(1) insert time, amortized O(lg n) deletemin time and amortized O(1) decreasekey time, making them asymptotically faster than the basic binary heap structure discussed in this course.]] Data structures help you organize and process your [[data]]. There are many different ways of implementing them depending on available resources and whims of the [[programmer]], but here are the general ideas behind them: == Readings == * [[Wikipedia: Data structure]] * [[/Introduction/]] * [[/Applications/]] == Choosing a data structure == The type of data structure you want to use will often be determined by how quickly you need to be able to do certain things to the data and how often, with compromises sometimes made for hardware or network restrictions. *[http://nptel.iitm.ac.in/showVideo.php?v=zWg7U0OEAoE IIT Video: ''Introduction to Data Structures''] *[http://www.youtube.com/watch?v=2wT2xOZJFx8 IIT Video: ''Basic Asymptotics and Algorithm Analysis''] == Simple data structures == === Stacks === A stack is a data structure that supports first-in-last-out access to elements, meaning the most recently added element is the first to be removed. Stacks have two main operations, namely <code>push()</code> and <code>pop()</code>. <code>push()</code> adds an element to the top of the stack<ref>{{Cite web|url=https://docs.oracle.com/javase/8/docs/api/java/util/Stack.html#push-E-|title=Stack (Java Platform SE 8 )|website=docs.oracle.com|access-date=2024-12-29}}</ref>, while <code>pop()</code> removes the element at the top of the stack<ref>{{Cite web|url=https://docs.oracle.com/javase/8/docs/api/java/util/Stack.html#pop--|title=Stack (Java Platform SE 8 )|website=docs.oracle.com|access-date=2024-12-29}}</ref>. You can think of it as a stack of plates: you can 'push' additional items onto the stack of plates or 'pop' plates from the top of the stack. Stacks are usually implemented through a linked list. *[http://nptel.iitm.ac.in/showVideo.php?v=g1USSZVWDsY IIT Video: ''Stacks''] /*needs pictures*/ === Linked lists === Think of a linked list as a series of boxes(called [[nodes]]) in a row. Each piece of information (or set of information) is put into one box with a [[pointer]] to the next box. A doubly linked list is one that also has pointers that go back the other way to the previous box. They have head and tail pointers to help you keep track of where the beginning and end are, and usually at least one pointer that moves around the inside of the structure to point at a box to help you keep your place as you look for things. /*needs pictures*/ For example, suppose you wanted to keep the names, addresses, and birthdays of your friends in a linked list. Each node would have one friend's name, address, and birthday in it, plus a pointer to the next in the list. If you want, the list can be sorted as you add friends to it, based on their name, address, or birthday in whatever way you want. If you know that some friends are more important to you and you don't want to go through the whole list to look for them every time, you can add in another variable for each person that can be used to set a sorting priority. *[http://nptel.iitm.ac.in/showVideo.php?v=PGWZUgzDMYI IIT Video: ''Queues and Linked Lists''] === Queues === A Queue is a data structure that provides first-in-first-out access to elements. The two basic operations are Enqueue() and Dequeue(). Enqueue() adds an element to the back of the Queue. Dequeue() removes an element from the front of the queue. Just like a line at the supermarket, a Queue only supports adding items to the back and removing them from the front. In addition, some implementations allow you to 'peek' at the item in front without removing it. /*needs pictures*/ == Dictionaries == *[http://nptel.iitm.ac.in/showVideo.php?v=BmayUdDaDYM IIT Video: ''Dictionaries''] === Hash tables === *[http://nptel.iitm.ac.in/showVideo.php?v=KW0UvOW0XIo IIT Video: ''Hashing''] * [[Pigeonhole Principle]] === Trees === You can think of a tree structure as a linked list with more than one outgoing pointer per node. This way, it branches out and the ends are called leaves instead of tails. The top node is called the root and the branches, like a real tree, don't merge back together. Thus, the nodes all have one incoming pointer and zero or more outgoing pointers, depending on the type of tree, its location within it, and the set of data that is shaping the tree. *[http://nptel.iitm.ac.in/showVideo.php?v=tORLeHHtazM IIT Video: ''Trees''] ==== Binary Search Trees ==== A binary tree is a specific type of tree with 0, 1, or 2 child nodes. *[http://nptel.iitm.ac.in/showVideo.php?v=bvOYfDpk940 IIT Video: ''Binary Search Trees''] *[http://nptel.iitm.ac.in/showVideo.php?v=KyMiqaA0ijM IIT Video: ''Deletion in Binary Search Trees''] ===== Traversal techniques ===== There are three main ways to process data in a tree. Recursion is usually the simplest way to perform such a task, where "traverse left" and "traverse right" below are recursive functions calls with the left and right children, respectively. '''Preorder:''' process node, traverse left, traverse right '''Inorder:''' traverse left, process node, traverse right '''Postorder:''' traverse left, traverse right, process node For example, consider the following recursive function to display the elements in a tree: Routine DisplayElements( Node ) if Node = null then return DisplayElements( Node's left ) //recursive function call with left child DisplayNode( Node's value ) //display value at current node DisplayElements( Node's right ) //recursive function call with left child End Routine This is a simple inorder traversal. *[http://nptel.iitm.ac.in/showVideo.php?v=eWeqqVpgNPg IIT Video: ''Tree Walks / Traversals''] ==== AVL Trees ==== *[http://nptel.iitm.ac.in/showVideo.php?v=mRGQylRWAsI IIT Video: ''AVL Trees 1''] *[http://nptel.iitm.ac.in/showVideo.php?v=TbvhGcf6UJU IIT Video: ''AVL Trees 2''] ==== 2-4 Trees ==== *[http://nptel.iitm.ac.in/showVideo.php?v=JZhdUb5F7oY IIT Video: ''2-4 Trees''] ==== Red-Black Trees ==== *[http://nptel.iitm.ac.in/showVideo.php?v=JRsN4Oz36QU IIT Video: ''Red-Black Trees''] *[http://nptel.iitm.ac.in/showVideo.php?v=6QOKk_pcv3U IIT Video: ''Insertion in Red-Black Trees''] ==== Disk-based data structures ==== *[http://nptel.iitm.ac.in/showVideo.php?v=VbVroFR4mq4 IIT Video: ''Disk-Based Data Structures''] == Pattern matching == *[http://nptel.iitm.ac.in/showVideo.php?v=Zj_er99KMb8 IIT Video: ''Pattern Matching Case Study''] *[http://nptel.iitm.ac.in/showVideo.php?v=uhAUk63tLRM IIT Video: ''Tries''] ==Data compression== *[http://nptel.iitm.ac.in/showVideo.php?v=5wRPin4oxCo IIT Video: ''Data Compression''] ==Priority queues== *[http://nptel.iitm.ac.in/showVideo.php?v=P4toxusBX9M IIT Video: ''Priority Queues''] *[http://nptel.iitm.ac.in/showVideo.php?v=HjPmZuOXkHQ IIT Video: ''Heaps''] ==Sorting== *[http://nptel.iitm.ac.in/showVideo.php?v=gtWw_8VvHjk IIT Video: ''Quicksort''] *[http://nptel.iitm.ac.in/showVideo.php?v=4OxBvBXon5w IIT Video: ''Why Sorting?''] *[http://nptel.iitm.ac.in/showVideo.php?v=cR4rxllyiCs IIT Video: ''More Sorting''] == Graphs == *[http://nptel.iitm.ac.in/showVideo.php?v=9zpSs845wf8 IIT Video: ''Graphs''] ===Breadth-First Search=== *[http://nptel.iitm.ac.in/showVideo.php?v=hk5rQs7TQ7E IIT Video: ''Data Structures for Graphs and Breadth-First Search''] *[http://nptel.iitm.ac.in/showVideo.php?v=r1-8p11fSPw IIT Video: ''Applications of Breadth-First Search''] ===Depth-First Search=== *[http://nptel.iitm.ac.in/showVideo.php?v=CIm6RzdoPCI IIT Video: ''Depth-First Search''] *[http://nptel.iitm.ac.in/showVideo.php?v=bmyyxNyZKzI IIT Video: ''Applications of Depth-First Search''] *[http://nptel.iitm.ac.in/showVideo.php?v=Lw5rRctO9js IIT Video: ''Depth-First Search in Directed Graphs''] *[http://nptel.iitm.ac.in/showVideo.php?v=o6YWEsLslKs IIT Video: ''Applications of Depth-First Search in Directed Graphs''] ===Minimum Spanning Trees=== *[http://nptel.iitm.ac.in/showVideo.php?v=k9jemw3SZe0 IIT Video: ''Minimum Spanning Trees''] *[http://nptel.iitm.ac.in/showVideo.php?v=kajZRdXi6fA IIT Video: ''Union Find Data Structure''] *[http://nptel.iitm.ac.in/showVideo.php?v=7FtGk9yr66A IIT Video: ''Prim's Minimum Spanning Tree Algorithm''] ===Shortest Paths=== *[http://nptel.iitm.ac.in/showVideo.php?v=ng1oliu806U IIT Video: ''Single-Source Shortest Paths 1''] *[http://nptel.iitm.ac.in/showVideo.php?v=NR0qG64gZUs IIT Video: ''Correctness of Dijkstra's Algorithm''] *[http://nptel.iitm.ac.in/showVideo.php?v=R36g6qit1bE IIT Video: ''Single-Source Shortest Paths 2''] == Dynamic allocation == Dynamic allocation asks for memory as it is needed. ==Lecture Notes== Data structures [http://www.cs.umd.edu/~mount/420/Lects/420lects.pdf lecture notes] from University of Maryland, College Park ==Assignments== ADUni: *[http://www.aduni.org/courses/algorithms/courseware/psets/Problem_Set_01.pdf Sorting and Searching Assignment], [http://www.aduni.org/courses/algorithms/courseware/psets/Problem_Set_01_Solutions.html Solutions] *[http://www.aduni.org/courses/algorithms/courseware/psets/Problem_Set_02.pdf Data Structures and Graph Algorithms Assignment], [http://www.aduni.org/courses/algorithms/courseware/psets/Problem_Set_02_Solutions.pdf Solutions] College of William and Mary: *[http://www.cs.wm.edu/~wm/CS539/hw1.pdf Stacks and Queues] *[http://www.cs.wm.edu/~wm/CS539/hw3.pdf Trees, Hash Tables, Heaps] *[http://www.cs.wm.edu/~wm/CS539/hw4.pdf Sorting] IIT Delhi: *[http://www.cse.iitd.ernet.in/~subodh/courses/CSL201/a2.html binary search, stacks and backtracking] *[http://www.cse.iitd.ernet.in/~subodh/courses/CSL201/a3.html BSTs] *[http://www.cse.iitd.ernet.in/~subodh/courses/CSL201/a4.html Red-Black trees, priority queue] More: *[http://users.cis.fiu.edu/~weiss/cop3530_spr02/assignments.html Florida International University] *[http://www.cs.huji.ac.il/~gkindler/ds02c/Default.html Hebrew University of Jerusalem] *[http://www.cs.tau.ac.il/~dannyf/ds09/ds09a.htm Tel Aviv University] *[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/assignments/ MIT] ==Exams== [http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/exams/ MIT], [http://www.cs.wm.edu/~wm/CS539/exams.html College of William and Mary] Note that these exams will cover material outside the scope of this course. ==Supplementary Links== *A short overview of basic data structures: [http://www.youtube.com/watch?v=tyJffCZcb0c Video: ''An Introduction To Trees & Data Structures''], [http://www.youtube.com/watch?v=m8tMZQmENJA Video: ''The Queue Data Structure''] *Video lecture series: **[http://www.cs.sunysb.edu/~algorith/video-lectures/ Steven Skeina's algorithms and data structures course at Stony Brook University] **[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/video-lectures/ MIT algorithms and data structures] **UC Berkeley CS 61B: Data Structures [http://www.youtube.com/view_play_list?p=4BBB74C7D2A1049C&sort_field=original&page=1 Fall 2006], [http://webcast.berkeley.edu/course_details_new.php?seriesid=2008-D-26332&semesterid=2008-D Fall 2008] (other years available on same site) **[http://www.aduni.org/courses/algorithms/index.php?view=cwArs Digita University algorithms (and data structures) course] **[http://www.youtube.com/view_play_list?p=E621E25B3BF8B9D1&sort_field=original&page=1 UNSW Data Structures and Algorithms (partial)] ==See also== * [[Computer science program]] [[Category:Computer programming]] [[Category:School of Computer Science]] [[Category:Computer science]] [[Category:Technology courses]] ast1xf5nuj1cp152krhay0536ckp8m3 0 108740 2693762 2692095 2024-12-29T14:53:02Z Lbeaumont 278565 Added Improving Social Systems 2693762 wikitext text/x-wiki ==Applied Wisdom== {{TOC right }} [[w:T._S._Eliot|T. S. Eliot]] asked: {{cquote |Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?}} This ''Applied Wisdom Curriculum'' is being designed by asking how we can best prepare ourselves to solve the great universal problems that prevent us from realizing and enjoying all that is most important in life. Knowledge has not been enough; we need the broad scope, human perspective, and good judgment of ''wisdom''. Shih-Ying Yang writes: “In the last analysis, individual actualization of conceptions of wisdom in real life, and the positive impact of these wise decisions and actions, may be the vehicle of the advance of human civilizations.”<ref> Yang, Shih-Ying. 2001. “Conceptions of Wisdom Among Taiwanese Chinese.” ''Journal of Cross-Cultural Psychology'' 32(6), November:662-680.</ref> This curriculum is based on the simple premise: If folly brings us problems, then perhaps [[wisdom]] can bring us solutions. The goal of the curriculum is to help you develop a tough mind and a tender heart. {{By|lbeaumont}} '''Pursuit of [[w:Well-being|well-being]]''' is the unifying theme for these courses, [[w:eudaimonia|eudaimonia]]. The collection of [[Wise Affirmations|wise affirmations]] can help you live more wisely each day. The [[Wise Living Toolkit]] assembles various resources that can help you live wisely. Please choose courses from this curriculum and study them in any order that suits your interests. The [[Living Wisely]] course calls on these courses in a particular sequence intended to allow each new course to build upon concepts learned from previous courses. The currently available courses are listed below in that sequence. [[File:Origins_and_progression_of_wisdom.webp|thumb|Origins and Progression of Wisdom]] * [[Wisdom for the ages]] - Practical advice for [[Living Wisely|living wisely]] * The [[Virtues]] — Attaining intrinsically valuable character traits * [[Social Skills]] — Building Relationships * [[Earning Trust]] — Relying on Another * [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves * [[Practicing Dialogue]] — Thinking Together * [[Clarifying values]] — What we find most important * [[What Matters]] — Identifying what is truly most significant to you, your family, community, nation, and world. * [[Sleep Soundly]] — Attaining essential rest and restoration * [[Stoic joy]] — Seeking tranquility. * Courses from the [[Clear_Thinking/Curriculum|Clear Thinking]] curriculum. — Become more accurate and consistent in thinking. ** [[Facing Facts]] — Embracing Reality ** [[Evaluating Evidence]] — Seeking Reality ***[[Media literacy|Media Literacy]] — Identifying reliable sources ** [[Seeking True Beliefs]] — Excellence in the Quest for Knowledge ** [[Exploring Worldviews]] — Challenging our deeply embedded assumptions ** [[Deductive Logic]] — Tools for evaluating consistency ** [[Recognizing Fallacies]] — Describing inconsistencies ** [[Thinking Scientifically]] — Reliable ways of knowing ** [[Knowing_How_You_Know|Knowing How You Know]] — Developing and applying your own Theory of Knowledge. ** [[Intellectual Honesty]] — Seeking Real Good Together ** [[Socratic Methods]] — Seeking real good by questioning beliefs ***[[Street Epistemology]] — Exploring the basis for belief **[[Exploring Social Constructs]] — Constructing Reality *[[Finding Common Ground]] — Aligning concepts with reality * [[Natural Inclusion]] — Experiencing the world ''from'' nature. * [[Beyond Theism]] — A real basis for hope * [[Global Perspective]] — Applying our Wisdom to meet the Grand Challenges * Courses from the [[Emotional Competency]] curriculum: ** [[Emotional Competency]] — Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others. ** [[Studying Emotional Competency]] — a path for studying the emotional competency material ** [[Dignity]] — Improving our world by learning to preserve dignity for all people ** [[Recognizing Emotions]] — Know how you feel **[[Forming beliefs]] — Evaluating what you accept as true ** [[Resolving Anger]] — Resolving an urgent plea for justice and action ** [[Resolving Dominance Contests]] — The classic show down **[[Confronting Tyranny]] — Resisting abusive power ** [[Overcoming Hate]] — Learning acceptance **[[Appraising Emotional Responses]] — Explaining Events ** [[What you can change and what you cannot]] — Gaining the wisdom to know the difference ** [[Attributing Blame]] — Analyzing Cause and Effect **[[Coping with Ego]] — Confronting the prime mover ** [[Apologizing]] — Expressing remorse. ** [[Forgiving]] — Choosing to overcome your desire for revenge ** [[Foregoing Revenge]] — Deescalating conflict **[[Communicating Power]] — Projecting power as we speak ** [[Earning Trust]] — Relying on Another ** [[Practicing Dialogue]] — Thinking Together ** [[Candor]] — Gaining Common Understanding ** [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased. ** [[Transcending Conflict]] — Resolving contradictory goals ** [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves **[[Finding Equanimity]] — Calm throughout the storm **[[Cherishing awe]] — Connecting with vastness **[[Alleviating Loneliness]] — Reconnecting **[[Creating Communities]] — Belonging ** [[Toward congruence]] — Attaining alignment and agreement * [[Pursuing Collective Wisdom]] — Improving collaborative decision making. * [[Grand Challenges]] — The great problems and opportunities facing humanity * Courses from the [[Possibilities/Curriculum|Possibilities curriculum]]: **[[Creating Possibilities]]—Navigating problem space **[[Unleashing Creativity]] — Welcoming new and useful ideas **[[Thinking Tools]] — Boosting Imagination **[[Problem Finding]] — Discovering the ''real'' problem **[[Solving Problems]]—Creating solutions ** [[Flourishing]] **[[Playing]]—Enjoyable Activity **[[Embracing Ambiguity]]—Keep thinking **[[Transcending Conflict]]—Resolving contradictory goals ** [[Envisioning Our Future]] — Describing your vision of our future. **[[Intentional Evolution]]—Choosing our future ** [[Evolving Governments]] — Unleashing collaboration **[[Coming Together]]—Becoming wiser together ** [[Evolving Money]]—Exchanging goods and services **[[Improving Social Systems]]—Steps Toward a Better World * [[Dignity]] — Improving our world by learning to preserve dignity for all people * [[Wisdom]] — Choosing Humanity * [[Assessing Human Rights]] — Essential protections for every person * [[Moral Reasoning]] — Knowing what to do * [[Living the Golden Rule]] — Treating others as you want to be treated ** [[Understanding the Golden Rule]] — Treat others only as you consent to being treated in the same situation. * [[Practicing Dialogue]] — Thinking Together * [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased. * [[Transcending Conflict]] — Resolving contradictory goals * [[Limits To Growth]] — Recognizing the earth is finite * [[Envisioning Our Future]] — Describing your vision of our future. ** [[A Journey to GameB]] — Life as it could be ** [[Intentional Evolution]] — Choosing our future ** [[Level 5 Research Center]] — The Next Big Thing ** [[Wisdom Research|The Wisdom and the Future Research Center]] — How can we wisely create our future? *[[Finding Courage]] — Value-based action despite temptation. * [[Doing Good]] — Take real good action. * A [[Quiet Mind]] — Controlling Discursive Thought; cultivating Pure Awareness *[[Guided Meditations]] — A selection of guided meditation scripts you may wish to practice. * [[Living Wisely]] — Enjoy seeking ''real good'' throughout your life. * [[Natural Inclusion]] — Experiencing the world ''from'' nature. ==Related Lectures and Essays== Several of the courses in this applied wisdom curriculum include lectures or assign essays to read as part of the course work. Those lectures and essays are listed here, in alphabetical order. * [[Living_Wisely/advance_no_falsehoods|Advance no Falsehoods]] * [[Embracing Ambiguity/Ambiguity breeds schisms|Ambiguity breeds schisms]] * [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]] *[[Virtues/Humility/Authentic_Humility|Authentic Humility]] *[[Virtues/Humility/Being 99.9% Ignorant|Being 99.9% Ignorant]] *[[Assessing Human Rights/Beyond Olympic Gold|Beyond Olympic Gold]] *[[Knowing How You Know/gallery/Choosing my beliefs|Choosing my beliefs]] *[[Limits To Growth/Coping with Abundance|Coping with Abundance]] * [[Knowing_How_You_Know/Divided_by_epistemology|Divided by epistemology]] * [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|Does Seeking Real Good Transcend Metamodernism? ]] * [[Finding Common Ground/Doubt and our Bayesian Brains|Doubt and our Bayesian Brains]] *[[Limits To Growth/Earth at One Billion|Earth at One Billion]] *[[Living_Wisely/Economic_Faults|Economic Faults]] *[[Understanding_Fairness/fair_enough|Fair Enough]] *[[Knowing How You Know/Friendly Persuasion|Friendly Persuasion]] *[[Practicing_Dialogue/From_Demagoguery_to_Dialogue|From Demagoguery to Dialogue]] *[[Living Wisely/Genesis of Debt|Genesis of Debt]] *[[Evolving Governments/Good Government|Good Government]] *[[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]] *[[Virtues/How can you change another person?|How can you change another person?]] * [[Understanding_Fairness/Luck,_Land,_and_Legacy|Luck, Land, and Legacy]] *[[Knowing_How_You_Know/One_World|One World]] *[[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]] *[[Wisdom Research/Pinnacles|Pinnacles]] *[[Living Wisely/Real, Good Insights|Real, Good Insights]] *[[Facing Facts/Reality is our common ground|Reality is our common ground]] * [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]] *[[Beyond Theism/Resolving a Vital Paradox|Resolving a Vital Paradox]] *[[Seeking_True_Beliefs/Science_is_like_a_living_tree|Science is like a living tree]] *[[Living Wisely/Seeking Real Good|Seeking Real Good]] *[[Problem_Finding/significance|Significance]] *[[Limits To Growth/Simply Priceless|Simply Priceless ]] *[[Virtues/Spontaneous Conflict and Deliberate Restraint|Spontaneous Conflict and Deliberate Restraint]] *[[Confronting Tyranny/The Hearing|The Hearing]] *[[Thinking Scientifically/The role and limitations of scientific reduction|The role and limitations of scientific reduction]] *[[Envisioning Our Future/The World We Want in 2075|The World We Want in 2075]] *[[Global Perspective/tobacco road|Tobacco Road]] *[[Global Perspective/Toward a Global Perspective—seeing through illusion|Toward a Global Perspective—seeing through illusion]] * [[Envisioning_Our_Future/Toward_Compassion|Toward Compassion]]—Unleashing the power of kindness *[[Beyond Theism/Transcending Dogma|Transcending Dogma]] *[[Knowing How You Know/Tyranny of Evidence|Tyranny of Evidence]] *[[Exploring_Worldviews/What_Fish_Don’t_See|What Fish Don’t See]] * [[Beyond Theism/What there is|What there is]] * [[Wisdom/Wisdom, Intelligence, and Artificial Intelligence|Wisdom, Intelligence, and Artificial Intelligence]] == Research Projects == Several research projects are associated with this Applied Wisdom curriculum. These research projects include: * [[Wisdom Research|The wisdom and the future research center]] **[[Grand challenges/Causes of Suboptimal Life Experiences|Causes of Suboptimal Life Experiences]] ** [[Living Wisely/Improving our Social Operating Systems|Improving our Social Operating Systems]] *The [[Level 5 Research Center]] is helping to shape the next big thing. ==Proposed Courses yet to be Developed== Related Courses, some still to be developed, include: * Determining ''What is'' ** [[Evaluating Evidence|Evidence]] *** [[Knowing How You Know/Tyranny of Evidence|The Tyranny of Evidence]] ** [[Introduction_to_logic|Logic and logical fallacies]] ** [[Knowing How You Know#What is a Theory of Knowledge?|Theory of Knowledge]] This is now available as the course [[Knowing_How_You_Know|Knowing How You Know]]. This course covers many of the topics listed above. ** [[Street Epistemology]] Learning to conduct genuine conversations that examine the foundations of belief. ** Using the metric system *[[Living Wisely/Seeking Real Good|Seeking Real Good]] ** [[Thinking Scientifically|Scientific Method]] ** [[Beyond Theism/What there is|Taxonomy of Reality]] ** [https://www.goodreads.com/review/show/1455117057 Deep Pragmatism] ** [[Moral Reasoning|Ethics]] ** Knowing what to do -- Getting from ''is'' to ''ought''. * [http://www.wisdompage.com/2016%20Articles/Empathy%20and%20Wisdom%20Moss.pdf Developing Accurate Empathy] -- Why are they feeling that way? * Systems Analysis * Systems Design ** [[Problem Finding|Problem Seeking]] * Rational Decision making ** The [[w:Analytic_Hierarchy_Process|Analytic Hierarchy Process]] ** Using [[w:Decision matrix|decision matrices]] ** [[w:Quality_function_deployment|Quality Function Deployment]] ** Choosing Excellence! * [[Critical_Thinking_Skills|Critical thinking]] * [[w:Root_cause_analysis|Root cause analysis]] * Understanding [[w:Risk|Risk]] — Estimating likelihood and consequence. * [[Problem_solving|Problem solving]] * [[Creativity]] * [[w:Big_History|Big History]] — An integrated history of the universe from the Big Bang to the present * [[Emotional Competency]] * Marriage Excellence * [[Exploring Social Constructs|The nature of social constructs]] * Designing social constructs for greater well-being ** Debugging Social Constructs * Money Architectures — exploring implications and alternatives to national fiat currencies. * Collective Wisdom — This is now available as the course [[Pursuing Collective Wisdom]]. *Forecasting using [[w:Bayes_theorem|Bayes Theorem]] * Inner Growth * The wisdom of [[w:Ubuntu_(philosophy)|ubuntu]]. * [[w:Biomimicry|Biomimicry]] and sustainable design. * Effecting change - How change propagates, or fails to propagate, through an organization or society. ** Influencing beliefs You can help by becoming a student, improving the above list, or by developing one of these courses. ==References== <references/> [[Category:Applied Wisdom]] [[Category:Curriculum]] lpjmx8eyazj9ctnac052zpqh810orww 2693763 2693762 2024-12-29T14:54:57Z Lbeaumont 278565 Added influence and persuasion 2693763 wikitext text/x-wiki ==Applied Wisdom== {{TOC right }} [[w:T._S._Eliot|T. S. Eliot]] asked: {{cquote |Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information?}} This ''Applied Wisdom Curriculum'' is being designed by asking how we can best prepare ourselves to solve the great universal problems that prevent us from realizing and enjoying all that is most important in life. Knowledge has not been enough; we need the broad scope, human perspective, and good judgment of ''wisdom''. Shih-Ying Yang writes: “In the last analysis, individual actualization of conceptions of wisdom in real life, and the positive impact of these wise decisions and actions, may be the vehicle of the advance of human civilizations.”<ref> Yang, Shih-Ying. 2001. “Conceptions of Wisdom Among Taiwanese Chinese.” ''Journal of Cross-Cultural Psychology'' 32(6), November:662-680.</ref> This curriculum is based on the simple premise: If folly brings us problems, then perhaps [[wisdom]] can bring us solutions. The goal of the curriculum is to help you develop a tough mind and a tender heart. {{By|lbeaumont}} '''Pursuit of [[w:Well-being|well-being]]''' is the unifying theme for these courses, [[w:eudaimonia|eudaimonia]]. The collection of [[Wise Affirmations|wise affirmations]] can help you live more wisely each day. The [[Wise Living Toolkit]] assembles various resources that can help you live wisely. Please choose courses from this curriculum and study them in any order that suits your interests. The [[Living Wisely]] course calls on these courses in a particular sequence intended to allow each new course to build upon concepts learned from previous courses. The currently available courses are listed below in that sequence. [[File:Origins_and_progression_of_wisdom.webp|thumb|Origins and Progression of Wisdom]] * [[Wisdom for the ages]] - Practical advice for [[Living Wisely|living wisely]] * The [[Virtues]] — Attaining intrinsically valuable character traits * [[Social Skills]] — Building Relationships * [[Earning Trust]] — Relying on Another * [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves * [[Practicing Dialogue]] — Thinking Together * [[Clarifying values]] — What we find most important * [[What Matters]] — Identifying what is truly most significant to you, your family, community, nation, and world. * [[Sleep Soundly]] — Attaining essential rest and restoration * [[Stoic joy]] — Seeking tranquility. * Courses from the [[Clear_Thinking/Curriculum|Clear Thinking]] curriculum. — Become more accurate and consistent in thinking. ** [[Facing Facts]] — Embracing Reality ** [[Evaluating Evidence]] — Seeking Reality ***[[Media literacy|Media Literacy]] — Identifying reliable sources ** [[Seeking True Beliefs]] — Excellence in the Quest for Knowledge ** [[Exploring Worldviews]] — Challenging our deeply embedded assumptions ** [[Deductive Logic]] — Tools for evaluating consistency ** [[Recognizing Fallacies]] — Describing inconsistencies ** [[Thinking Scientifically]] — Reliable ways of knowing ** [[Knowing_How_You_Know|Knowing How You Know]] — Developing and applying your own Theory of Knowledge. ** [[Intellectual Honesty]] — Seeking Real Good Together ** [[Socratic Methods]] — Seeking real good by questioning beliefs ***[[Street Epistemology]] — Exploring the basis for belief **[[Exploring Social Constructs]] — Constructing Reality *[[Finding Common Ground]] — Aligning concepts with reality * [[Natural Inclusion]] — Experiencing the world ''from'' nature. * [[Beyond Theism]] — A real basis for hope * [[Global Perspective]] — Applying our Wisdom to meet the Grand Challenges * Courses from the [[Emotional Competency]] curriculum: ** [[Emotional Competency]] — Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others. ** [[Studying Emotional Competency]] — a path for studying the emotional competency material ** [[Dignity]] — Improving our world by learning to preserve dignity for all people ** [[Recognizing Emotions]] — Know how you feel **[[Forming beliefs]] — Evaluating what you accept as true ** [[Resolving Anger]] — Resolving an urgent plea for justice and action ** [[Resolving Dominance Contests]] — The classic show down **[[Confronting Tyranny]] — Resisting abusive power ** [[Overcoming Hate]] — Learning acceptance **[[Appraising Emotional Responses]] — Explaining Events ** [[What you can change and what you cannot]] — Gaining the wisdom to know the difference *** [[Influence and Persuasion]]—Shaping our beliefs and actions ** [[Attributing Blame]] — Analyzing Cause and Effect **[[Coping with Ego]] — Confronting the prime mover ** [[Apologizing]] — Expressing remorse. ** [[Forgiving]] — Choosing to overcome your desire for revenge ** [[Foregoing Revenge]] — Deescalating conflict **[[Communicating Power]] — Projecting power as we speak ** [[Earning Trust]] — Relying on Another ** [[Practicing Dialogue]] — Thinking Together ** [[Candor]] — Gaining Common Understanding ** [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased. ** [[Transcending Conflict]] — Resolving contradictory goals ** [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves **[[Finding Equanimity]] — Calm throughout the storm **[[Cherishing awe]] — Connecting with vastness **[[Alleviating Loneliness]] — Reconnecting **[[Creating Communities]] — Belonging ** [[Toward congruence]] — Attaining alignment and agreement * [[Pursuing Collective Wisdom]] — Improving collaborative decision making. * [[Grand Challenges]] — The great problems and opportunities facing humanity * Courses from the [[Possibilities/Curriculum|Possibilities curriculum]]: **[[Creating Possibilities]]—Navigating problem space **[[Unleashing Creativity]] — Welcoming new and useful ideas **[[Thinking Tools]] — Boosting Imagination **[[Problem Finding]] — Discovering the ''real'' problem **[[Solving Problems]]—Creating solutions ** [[Flourishing]] **[[Playing]]—Enjoyable Activity **[[Embracing Ambiguity]]—Keep thinking **[[Transcending Conflict]]—Resolving contradictory goals ** [[Envisioning Our Future]] — Describing your vision of our future. **[[Intentional Evolution]]—Choosing our future ** [[Evolving Governments]] — Unleashing collaboration **[[Coming Together]]—Becoming wiser together ** [[Evolving Money]]—Exchanging goods and services **[[Improving Social Systems]]—Steps Toward a Better World * [[Dignity]] — Improving our world by learning to preserve dignity for all people * [[Wisdom]] — Choosing Humanity * [[Assessing Human Rights]] — Essential protections for every person * [[Moral Reasoning]] — Knowing what to do * [[Living the Golden Rule]] — Treating others as you want to be treated ** [[Understanding the Golden Rule]] — Treat others only as you consent to being treated in the same situation. * [[Practicing Dialogue]] — Thinking Together * [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased. * [[Transcending Conflict]] — Resolving contradictory goals * [[Limits To Growth]] — Recognizing the earth is finite * [[Envisioning Our Future]] — Describing your vision of our future. ** [[A Journey to GameB]] — Life as it could be ** [[Intentional Evolution]] — Choosing our future ** [[Level 5 Research Center]] — The Next Big Thing ** [[Wisdom Research|The Wisdom and the Future Research Center]] — How can we wisely create our future? *[[Finding Courage]] — Value-based action despite temptation. * [[Doing Good]] — Take real good action. * A [[Quiet Mind]] — Controlling Discursive Thought; cultivating Pure Awareness *[[Guided Meditations]] — A selection of guided meditation scripts you may wish to practice. * [[Living Wisely]] — Enjoy seeking ''real good'' throughout your life. * [[Natural Inclusion]] — Experiencing the world ''from'' nature. ==Related Lectures and Essays== Several of the courses in this applied wisdom curriculum include lectures or assign essays to read as part of the course work. Those lectures and essays are listed here, in alphabetical order. * [[Living_Wisely/advance_no_falsehoods|Advance no Falsehoods]] * [[Embracing Ambiguity/Ambiguity breeds schisms|Ambiguity breeds schisms]] * [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]] *[[Virtues/Humility/Authentic_Humility|Authentic Humility]] *[[Virtues/Humility/Being 99.9% Ignorant|Being 99.9% Ignorant]] *[[Assessing Human Rights/Beyond Olympic Gold|Beyond Olympic Gold]] *[[Knowing How You Know/gallery/Choosing my beliefs|Choosing my beliefs]] *[[Limits To Growth/Coping with Abundance|Coping with Abundance]] * [[Knowing_How_You_Know/Divided_by_epistemology|Divided by epistemology]] * [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|Does Seeking Real Good Transcend Metamodernism? ]] * [[Finding Common Ground/Doubt and our Bayesian Brains|Doubt and our Bayesian Brains]] *[[Limits To Growth/Earth at One Billion|Earth at One Billion]] *[[Living_Wisely/Economic_Faults|Economic Faults]] *[[Understanding_Fairness/fair_enough|Fair Enough]] *[[Knowing How You Know/Friendly Persuasion|Friendly Persuasion]] *[[Practicing_Dialogue/From_Demagoguery_to_Dialogue|From Demagoguery to Dialogue]] *[[Living Wisely/Genesis of Debt|Genesis of Debt]] *[[Evolving Governments/Good Government|Good Government]] *[[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]] *[[Virtues/How can you change another person?|How can you change another person?]] * [[Understanding_Fairness/Luck,_Land,_and_Legacy|Luck, Land, and Legacy]] *[[Knowing_How_You_Know/One_World|One World]] *[[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]] *[[Wisdom Research/Pinnacles|Pinnacles]] *[[Living Wisely/Real, Good Insights|Real, Good Insights]] *[[Facing Facts/Reality is our common ground|Reality is our common ground]] * [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]] *[[Beyond Theism/Resolving a Vital Paradox|Resolving a Vital Paradox]] *[[Seeking_True_Beliefs/Science_is_like_a_living_tree|Science is like a living tree]] *[[Living Wisely/Seeking Real Good|Seeking Real Good]] *[[Problem_Finding/significance|Significance]] *[[Limits To Growth/Simply Priceless|Simply Priceless ]] *[[Virtues/Spontaneous Conflict and Deliberate Restraint|Spontaneous Conflict and Deliberate Restraint]] *[[Confronting Tyranny/The Hearing|The Hearing]] *[[Thinking Scientifically/The role and limitations of scientific reduction|The role and limitations of scientific reduction]] *[[Envisioning Our Future/The World We Want in 2075|The World We Want in 2075]] *[[Global Perspective/tobacco road|Tobacco Road]] *[[Global Perspective/Toward a Global Perspective—seeing through illusion|Toward a Global Perspective—seeing through illusion]] * [[Envisioning_Our_Future/Toward_Compassion|Toward Compassion]]—Unleashing the power of kindness *[[Beyond Theism/Transcending Dogma|Transcending Dogma]] *[[Knowing How You Know/Tyranny of Evidence|Tyranny of Evidence]] *[[Exploring_Worldviews/What_Fish_Don’t_See|What Fish Don’t See]] * [[Beyond Theism/What there is|What there is]] * [[Wisdom/Wisdom, Intelligence, and Artificial Intelligence|Wisdom, Intelligence, and Artificial Intelligence]] == Research Projects == Several research projects are associated with this Applied Wisdom curriculum. These research projects include: * [[Wisdom Research|The wisdom and the future research center]] **[[Grand challenges/Causes of Suboptimal Life Experiences|Causes of Suboptimal Life Experiences]] ** [[Living Wisely/Improving our Social Operating Systems|Improving our Social Operating Systems]] *The [[Level 5 Research Center]] is helping to shape the next big thing. ==Proposed Courses yet to be Developed== Related Courses, some still to be developed, include: * Determining ''What is'' ** [[Evaluating Evidence|Evidence]] *** [[Knowing How You Know/Tyranny of Evidence|The Tyranny of Evidence]] ** [[Introduction_to_logic|Logic and logical fallacies]] ** [[Knowing How You Know#What is a Theory of Knowledge?|Theory of Knowledge]] This is now available as the course [[Knowing_How_You_Know|Knowing How You Know]]. This course covers many of the topics listed above. ** [[Street Epistemology]] Learning to conduct genuine conversations that examine the foundations of belief. ** Using the metric system *[[Living Wisely/Seeking Real Good|Seeking Real Good]] ** [[Thinking Scientifically|Scientific Method]] ** [[Beyond Theism/What there is|Taxonomy of Reality]] ** [https://www.goodreads.com/review/show/1455117057 Deep Pragmatism] ** [[Moral Reasoning|Ethics]] ** Knowing what to do -- Getting from ''is'' to ''ought''. * [http://www.wisdompage.com/2016%20Articles/Empathy%20and%20Wisdom%20Moss.pdf Developing Accurate Empathy] -- Why are they feeling that way? * Systems Analysis * Systems Design ** [[Problem Finding|Problem Seeking]] * Rational Decision making ** The [[w:Analytic_Hierarchy_Process|Analytic Hierarchy Process]] ** Using [[w:Decision matrix|decision matrices]] ** [[w:Quality_function_deployment|Quality Function Deployment]] ** Choosing Excellence! * [[Critical_Thinking_Skills|Critical thinking]] * [[w:Root_cause_analysis|Root cause analysis]] * Understanding [[w:Risk|Risk]] — Estimating likelihood and consequence. * [[Problem_solving|Problem solving]] * [[Creativity]] * [[w:Big_History|Big History]] — An integrated history of the universe from the Big Bang to the present * [[Emotional Competency]] * Marriage Excellence * [[Exploring Social Constructs|The nature of social constructs]] * Designing social constructs for greater well-being ** Debugging Social Constructs * Money Architectures — exploring implications and alternatives to national fiat currencies. * Collective Wisdom — This is now available as the course [[Pursuing Collective Wisdom]]. *Forecasting using [[w:Bayes_theorem|Bayes Theorem]] * Inner Growth * The wisdom of [[w:Ubuntu_(philosophy)|ubuntu]]. * [[w:Biomimicry|Biomimicry]] and sustainable design. * Effecting change - How change propagates, or fails to propagate, through an organization or society. ** Influencing beliefs You can help by becoming a student, improving the above list, or by developing one of these courses. ==References== <references/> [[Category:Applied Wisdom]] [[Category:Curriculum]] r7wdpzggxw9cneca61h055ckkl94zdz 0 156848 2693815 1575736 2024-12-29T23:09:48Z 2A02:A03F:8952:9501:1532:F1D2:D7AE:45E7 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. 2693815 wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 2693817 2693815 2024-12-29T23:11:40Z MathXplore 2888076 Reverted edits by [[Special:Contribs/2A02:A03F:8952:9501:1532:F1D2:D7AE:45E7|2A02:A03F:8952:9501:1532:F1D2:D7AE:45E7]] ([[User talk:2A02:A03F:8952:9501:1532:F1D2:D7AE:45E7|talk]]) to last version by JackBot: unexplained content removal 1575736 wikitext text/x-wiki ==Introduction== ==Team== ===Name=== STARSWIFT * The team name refers to stars, which have an unique brightness and power capable of crossing large distances in few seconds. This logo summarizes our main objective with this project, building a fast and strong solar vehicle . ===Logo=== <gallery></gallery> https://commons.wikimedia.org/wiki/File:Logo_SSV_EE4.JPG ===Members=== {| class="wikitable" |- ! Name !! Email !! Function |- | Praveen Saragadam || r0385165@Kuleuven.be || Team Leader |- | Sanjay Kandukuri || r0456703@Kuleuvenbe || Member |- | Anjali Nedunuri|| r0451663@kuleuven.be || Member |- | Surendra Botu || r0451476@Kuleuven.be || Member |- | Anoop Nuka || r0448350@Kuleuven.be || Member |- | Chaitanya Ketha || r0451656@kuleuven.be || Member |} ==General Information== ===Plan Of Approach=== https://commons.wikimedia.org/wiki/File:Plan_Of_Approach_EE4_SSV.pdf ===Work Breakdown Structure=== https://commons.wikimedia.org/wiki/File:Work_Break_Down_Structure_Of_SSV_EE4.pdf ===Gannt Chart=== https://commons.wikimedia.org/wiki/File:Gantt_Chart_SSV_EE4.pdf ===Co-operation contract=== https://commons.wikimedia.org/wiki/File:Co-operation_contract_EE4_SSV.pdf ==Meeting Reports == ===Meeting 2 === https://commons.wikimedia.org/wiki/File:Meetings_report_final.pdf ===Meeting 3 === https://commons.wikimedia.org/wiki/File:Meeting_Report_3_EE4_SSV_final.pdf ===Meeting 4 === https://commons.wikimedia.org/wiki/File:Meeting_Report_4_EE4_SSV_Final.pdf ===Meeting 5=== [[File:Meeting Report 5.pdf|thumb|meeting report 5]] https://commons.wikimedia.org/wiki/File:Meeting_Report_5.pdf ===Meeting 6=== [[File:Meeting Report 6.pdf|thumb|Meeting report 6]] https://commons.wikimedia.org/wiki/File:Meeting_Report_6.pdf ===Meeting 7=== [[File:Meeting Report 7.pdf|thumb|Meeting report 7]] https://commons.wikimedia.org/wiki/File:Meeting_Report_7.pdf ===Meeting 8=== [[File:Meeting report 8.pdf|thumb|Meeting report 8]] https://commons.wikimedia.org/wiki/File:Meeting_report_8.pdf ===Meeting 9=== [[File:Meeting Report 9.pdf|thumb|Meeting report 9]] https://commons.wikimedia.org/wiki/File:Meeting_Report_9.pdf ===Meeting 10=== [[File:Meeting Report 10.pdf|thumb|Meeting report 10]] https://commons.wikimedia.org/wiki/File:Meeting_Report_10.pdf ===Meeting 11=== [[File:Meeting Report 11.pdf|thumb|Meeting report 11]] https://commons.wikimedia.org/wiki/File:Meeting_Report_11.pdf ===Meeting 12=== [[File:Meeting Report 12.pdf|thumb|Meeting report 12]] https://commons.wikimedia.org/wiki/File:Meeting_Report_12.pdf ===Meeting 13=== [[File:Meeting Report 13.pdf|thumb|Meeting report 13]] https://commons.wikimedia.org/wiki/File:Meeting_Report_13.pdf ===Meeting 14=== https://commons.wikimedia.org/wiki/File%3AMeeting_Report_14.pdf ===SSV Final report=== [[File:SSV FINAL REPORT.pdf|thumb|Final report]] https://commons.wikimedia.org/wiki/File:SSV_FINAL_REPORT.pdf ==Blog== ===Week 1=== In first we had a introduction class. Teams were divided.seven of us were made into a group.group members week The first meeting was held on Tuesday 11/01/2014, in Group T campus.Team discussed about the work to be done on week 1.Team leader and secretary of the team were choose. Then plan of approach, work break down structure and Gantt chart were reviewed . All different tasks were divided between team members. Discussions on different areas took place, and a brainstorming was proposed in order to produce a better project. A second meeting was scheduled for next week, the main purpose of this meeting will be the presentation of current results. ===Week 2=== The second meeting was held on Wednesday 12/01/2014, in Group T campus. During the meeting, group members presented the work done on the previous week. Break down structure and Gantt chart was completed. Discussions about the Plan of Approach, team logo and team name took place on the second half of the meeting. Another meeting was schedule for week 4. ===Week 3=== {{BookCat}} 1e0c5t6q0t6ns4jn1e3qtciqgcbk6l4 0 160684 2693840 1678139 2024-12-30T07:02:33Z 2600:1700:36C0:8240:983F:56A2:5AE2:272 everything 2693840 wikitext text/x-wiki Welcome to the Wikiversity edition of <blockquote>Anthony Cross, ''In the Land of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)'', Cambridge, UK: Open Book Publishers, 2014. {{ISBN|978-1-78374-060-4}}. http://dx.doi.org/10.11647/OBP.0042</blockquote> Our aim rfis to create a socially enhanced version of this book to make it even more comprehensive or links to discovery engines such as [https://www.worldcat.org/ Worldcat], to expand the description of the works identified, and to add any other information that you think could enrich this invaluable discovery tool. <table style="background:#ffffff; border: thin solid #aaaaaa;" class="navbox collapsible nowraplinks"> <tr><th style="background:#ACDAF2;text-align:center;width:100%;">Table of Contents</th></tr> <tr> <td style="text-align:left;width:100%;"> {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/List of Illustrations|List of Illustrations]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Preface|Preface]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Introduction|Introduction]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reigns of the First Romanovs (1613-1682)|1. Reigns of the First Romanovs (1613-1682)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reigns of Peter I (1682-1725) and Catherine I (1725-1727)|2. Reigns of Peter I (1682-1725) and Catherine I (1725-1727)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reigns of Peter II (1727-1730), Anna Ivanovna (1730-1740), Ivan VI (1740-1741), and Elizabeth (1741-1762)|3. Reigns of Peter II (1727-1730), Anna Ivanovna (1730-1740), Ivan VI (1740-1741), and Elizabeth (1741-1762)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reigns of Peter III (1762) and Catherine II (1762-1796)|4. Reigns of Peter III (1762) and Catherine II (1762-1796)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Paul I (1796-1801)|5. Reign of Paul I (1796-1801)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander I (1801-1825)|6. Reign of Alexander I (1801-1825)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Nicholas I (1825-1855)|7. Reign of Nicholas I (1825-1855)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/The Crimean War (28 March 1854-27 April 1856)|8. The Crimean War (28 March 1854-27 April 1856)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander II (1855-1881)|9. Reign of Alexander II (1855-1881)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Alexander III (1881-1894)|10. Reign of Alexander III (1881-1894)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Reign of Nicholas II (1894-1917)|11. Reign of Nicholas II (1894-1917)]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Bibliography of Bibliographies|Bibliography of Bibliographies]] {{#if:{{{1}}}|*| * }}[[In the Lands of the Romanovs: An Annotated Bibliography of First-hand English-language Accounts of the Russian Empire (1613-1917)/Index of Authors|Index of Authors]] </td> </tr> </table> The text of this book is licensed under a [http://creativecommons.org/licenses/by/4.0 Creative Commons Attribution 4.0 International] license (CC BY 4.0). {{CourseCat}} amy9zxid748n31grgubyb5xcirk86iq 0 225675 2693852 2268758 2024-12-30T11:47:00Z Nicolefox56 2995620 2693852 wikitext text/x-wiki This page is dedicated to share informations and organize my first visit in Ghana as exploratory phase of my Phd thesis [[The digital revolution experienced by the global South|''The digital revolution experienced by the global South'']]. Welcome to every one to organize with me my Ghana venue on [[Talk:The digital revolution experienced by the global South/First visit in Ghana|this talking page]]. [[User:Lionel Scheepmans|Lionel Scheepmans]] == Visa application and bureaucracy == === Embassy opening hours === From 1st to 31st August, 2017 Submission of Visa & Passport Applications 10.00 am to 12.00 noon Collection/Pickup of Passports 1.30 pm to 2.00 pm === Visa application === [https://www.ghanaembassy.be/index.php/consular/visa-application Visa application Web page on Ghana Brussels embassy webpage]is the only way to [https://www.francevisit.co.uk/visa-documents/ submit a visa application] to Brussels ambassy. An analogical submission into the office was not possible on Friday August 18th 2017. The [https://www.ghanaembassy.be/index.php/component/edocman/visa-application-form/download?Itemid=0 Visa application form], don't work with open source software but only with acrobat reader. The pdf file ask for a password when it's open with Xreader. The pdf file open normally with Chrome browser, gaps can be filed, but sending the application is not possible. On Sunday August 20th acrobat reader was not installable on linux due to a bug. My only solution was to use friend's computer. The visa application is supported by a [https://companiesuk.co/management-consultancy-activities-other-than-financial-management/avlos-limited British private limited company named AVLOS LIMITED]. I've receive my visa august 28th 2017. I was one hour late but, after a short discussion and apologize, the women was very nice to give me my passport for not coming back the next day. === Yellow fever === A yellow fever certificate is asked for the entrance in Ghana. [http://www.who.int/ith/annex7-ihr.pdf?ua=1 Since 2005, the certificate is valid for the life]. == Fly connexion == '''Arrival''' {| class="wikitable" |Date |Time of arrival |Airport |Airline |Flight Number |- |September 1st |21h25 (9h25 PM) |Accra |TAP |TP1524 |} '''Departure''' {| class="wikitable" |Date |Time of departure |Airport |Airline |Flight Number |- |September 20th |23h55 (11h55 PM) |Accra |TAP |TP1525 |} Price for the return : €469.27 == Contacts in Ghana == === Wikimedia === ==== Groups ==== * [[m:Wikimedia Ghana User Group|Wikimedia Ghana user group]] * [[w:Category:Wikipedians in Ghana|Category:Wikipedians in Ghana]] * [https://www.facebook.com/WikimediaGhanaUG/ Facebook] ==== Persons ==== * [[m:User:Sandiooses|Sandister Tei]] * [[m:Rexford Nkansah|Rexford Nkansah]] * [[wikipedia:User:Masssly|Masssly]] === OpenStreetMap === ==== Groups ==== * [[openstreetmap:WikiProject_Ghana|WikiProject Ghana]] * [[openstreetmap:Category:Users_in_Ghana|OpenStreetMap wiki editor]] * [https://www.facebook.com/OpenStreetMapGhana Facebook] ==== Persons ==== * [[openstreetmap:User:Enock4seth|Enock Seth Nyamador]] === Open foundation west Africa === ==== Groups ==== * [https://openfoundationwestafrica.org/ Open Foundation West Africa website] ==== Persons ==== * [[m:User:Flixtey|Felix Nartey]] == Activity planing == === In Accra (first week-end and second week) === * Establish contact with Wikimedia, Open street map user groups and other interesting groups involved into the digital revolution in Ghana. * Spending one week with in an intergenerational family for sharing daily live and observing the digital revolution in urban context. * Visiting Agbogbloshie and other emblematic places of the dark side of the global digital revolution. ==== Activity planing ==== * Blogging Ghana * New Media Hub * Recommended media houses (Citi FM, Joy FM) * GhanaThink Foundation * Wikimedia Ghana User Group * Recommended businesses * Agbogbloshie Scrapyard (e-waste dumpsite) === In village (first week) === Spending one week with in an intergenerational family for sharing daily live and observing the digital revolution in rural context. ==== Village proposed ==== * Korleman * Sota * Jisonaayili (Tamale) == Hosting == === Research by Internet === ==== Bewelcome ==== [http://www.bewelcome.org/places/Ghana/GH All Ghana] ==== Couchsurfing ==== [https://www.couchsurfing.com/groups/search/1?button=&country=&latitude=&longitude=&perPage=20&region=&search_query=Ghana&search_type=group&utf8=%E2%9C%93 All Groups] [https://www.couchsurfing.com/groups/54 Bigest group] Other groups https://www.couchsurfing.com/groups/volunteers-ghana https://www.couchsurfing.com/groups/the-northern-ghana-tamale https://www.couchsurfing.com/groups/responsible-volunteering-in-ghana https://www.couchsurfing.com/groups/love-seekers-40175 ==== Airb&b ==== ==== Volunteer groups and associations ==== http://www.vpwa.org/volunteer-projects/ https://www.couchsurfing.com/groups/responsible-volunteering-in-ghana https://www.couchsurfing.com/groups/29185 === In Accra === * [http://www.bewelcome.org/places/Ghana/GH/Greater%20Accra%20Region/01 Bewelcome Accra] === In village and other cities === * [http://www.bewelcome.org/members/kingraf Bewelcome Tamale] * [https://www.couchsurfing.com/groups/sekondi-takoradi CS Sekondi - Takoradi] * r == Budget == Until now, my PhD thesis is auto-financed. That's why I have to find the cheapest way to carry out this project. Public transport, specially fly connection, will be the principal cost of this first visit of Ghana. For the rest, I didn't need fancy place for living. Just street food and a bed in a safety place. As anthropologist, my target is sharing time with people not making holidays. == Planing table == {| class="wikitable" |+Planing of my visit to Ghana day by day !September !Day !Place !Activity !Host !Info |- |01 th |Friday |Accra |Arrival |Raphael |21h25 (9h25 PM) with the fly n° TP1524 |- |02 |Saturday |Accra |Meeting with people |Mohammed | |- |03 |Sunday |Dodi island |Internet activity and Accra visit |Mohammed | |- |04 |Monday |Yapei Queen ferry |Interview on ferry to Northern Ghana |Ferry |http://ghana-net.com/volta-lake-ferry.html |- |05 |Tuesday |Yapei |Interview on ferry |Justice Hotel | |- |06 |Wednesday |Tamale |Meeting with people |Rafik | |- |07 |Thursday |Tamale |Exploring country side Digital Revolution |Rafik | |- |08 |Friday |Zongo Machery |Travel and installation |Clement |http://www.greengoldghana.com/ http://www.driveaidghana.com |- |09 |Saturday |Zongo Machery |Visiting the village and collect information |Clement | |- |10 |Sunday |Zongo Machery |Visiting the village and collect information |Clement | |- |11 |Monday |Zongo Machery |Visit damabai market and testing internet cafe |Clement | |- |12 |Tuesday |Zongo Machery |Exploring country side Digital Revolution |Clement | |- |13 |Wednesday |Zongo Machery |Exploring country side Digital Revolution |Clement | |- |14 |Thursday |Accra |Village and road to Accra |Mohammed | |- |15 |Friday |Accra |Road to Accra and Mohammed place |Mohammed | |- |16 |Saturday |Accra |Software Freedom Day 2017 |Mohammed |https://www.eventbrite.com/e/software-freedom-day-2017-tickets-36988801501 |- |17 |Sunday |Aburi |Visit aburi garden |Emmanuel |http://www.aburi.ghana-net.net/ https://en.wikipedia.org/wiki/Aburi |- |18 |Monday |Accra |Visit the Impact Hub Accra 12pm till they close at 7pm |Mohammed |http://accra.impacthub.net/ |- |19 |Tuesday |Accra |Exploring city Digital Revolution |Mohammed | |- |20 |Wednesday |Accra |Departure |Airplane |23h55 (11h55 PM) fly n° TP1525 |} * [[openstreetmap:AccraMobile3|AccraMobile3]] * [[metawiki:Global Reach/Ghana Survey|Wikimedia Ghana Survey]] [[Category:Anthropology]] [[Category:Research]] [[Category:Ghana]] mqvne2knqqwf0b2k5zi8b5oarqem1xh 0 265167 2693835 2689602 2024-12-30T00:36:45Z Scogdill 1331941 2693835 wikitext text/x-wiki == Dignitaries, Ambassadors and Ministers == See also the list of Ambassadors and Ministers in the list of [[Social Victorians/People/Invited/|people invited to social events typically hosted by royals]]. === Argentina === * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation"<ref name=":13">“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.</ref> (4, Col. 2c) === Australia === * The Chief Justice of South Australia * Mr. Justice Williams of Victoria and Mrs. Williams and Miss William (assuming Australia; might be wrong) ===Austro-Hungarian Empire=== * Count Karolyi, Ambassador, and the Countess Karolyi *'''1895 April 8, Monday:''' the ''Globe'' announced that "[[Social Victorians/People/Hadik|Count Max Hadik]], the newly-appointed Attaché at the Austro-Hungarian Embassy, has arrived in London."<ref>"Court and Personal News." The ''Globe'' 08 April 1895 Monday: 6 [of 8], Col. 4b [of 5]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18950408/055/0006.</ref> *'''1895 February 2, Friday''': Count Alexander Palffy attended the [[Social Victorians/1895 Bal Poudre Warwick Castle|February 1895 Bal Poudré at Warwick Castle]] hosted by Daisy, Countess of Warwick, identified as Count Palffy, from the "Austrailian Embassy"<ref name=":02">"The Grand Bal Poudre at Warwick Castle." ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].</ref>{{rp|6, Col. 6a}} by the Leamington Spa Courier, but as the Austro-Hungarian Ambassador by the Coventry Evening Telegraph.<ref name=":12">"Grand Bal Poudre at Warwick Castle." ''Coventry Evening Telegraph'' 01 February 1895, Friday: 3 [of 4], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000337/18950201/021/0003.</ref> *'''1895 February 15''': presented at a Levee at St. James's Palace in the Diplomatic Circle to the Prince of Wales "By the Austro-Hungarian Ambassador, Herman. Prince of Stolberg Wernigerode, Attache to the German Embassy."<ref name=":0">"Levee at St. James's Palace." London ''Evening Standard'' 16 February 1895 Saturday: 3 [of 10], Col. 4a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18950216/023/0003.</ref> *'''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy"<ref name=":13" /> (4, Col. 2c) *'''1897 August 18, Wednesday''': [[Social Victorians/People/Mensdorff|Count Albert Mensdorff]] was Charge d’Affaires for Austria-Hungary; [[Social Victorians/People/Hadik|Count Hadik]] was Secretary of the Embassy.<blockquote>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, Charge d’Affaires for Austria-Hungary, presided, and among the leading members present were Count Hadik, Secretary of the Embassy; Capt. 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; Col. Hain, and many others. A telegram expressing the loyalty of the Austro-Hungarian colony was dispatched to His Majesty the Emperor-King.<ref>"The Austro-Hungarian Colony." The ''Globe'' 19 August 1897 Thursday: 6 [of 8], Col. 3c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970819/051/0006.</ref></blockquote>'''1898 January 21, Friday''': the ''Pall Mall Gazette'' reported that "[[Social Victorians/People/Hadik|Count Max Hadik]] and Count Alexander Palffy, secretaries of the Austro-Hungarian Embassy in London, have taken leave of the Court of St. James's upon their appointment to other diplomatic positions."<ref>"Pall Mall Gazette Office." ''Pall Mall Gazette'' 21 January 1898 Friday: 8 [of 10], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18980121/022/0008.</ref> ===Belgium=== * Baron Solvyns, Minister, and the Baroness Solvyns * M. Joostens (Belgian Legation, London), attended the [[Social Victorians/1895 Bal Poudre Warwick Castle|February 1895 Bal Poudré at Warwick Castle]] hosted by Daisy, Countess of Warwick.<ref name=":022">"The Grand Bal Poudre at Warwick Castle." ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].</ref>{{rp|6, Col. 4c}} * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Belgian Minister, the Councillor, and two Secretaries of Legation"<ref name=":13" /> (4, Col. 2c) ===Brazil=== * João Arthur Souza Corrêa, Minister Plenipotentiary (1890–1900) * Baron de Penodo, Minister, and the Baroness de Penedo === Cyprus === * Chief Justice of Cyprus and Lady Bovill ===Denmark=== * Monsieur de Falbe, Minister, and Madame de Falbe * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation"<ref name=":13" /> (4, Col. 2c) ===France=== * William Henry Waddington, Ambassador (1883–1893), and Madame Waddington and Mademoiselle King * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy"<ref name=":13" /> (4, Col. 2c) ===German=== * Count Hatzfeldt, Ambassador * Count Georg Herbert Münster, Ambassador (1873–1885), and Countess Marie Münster * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery"<ref name=":13" /> (4, Col. 2c) ===Greece=== * M. J. Gennadius, Minister === Hong Kong === * Mr. Justice Fielding Clarke of Hong Kong ===Italy=== * Count Corti, Ambassador *First Secretary of the Embassy, Count Bottaro Costa (presented Monday 6 July 1896 to the Prince of Wales at a Levee)<ref name=":1">"The Prince of Wales's Levee." London Evening ''Standard'' 07 July 1896 Tuesday: 5 [of 12], Col. 6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18960707/032/0005.</ref> *Marquis Montagliari, Attaché (presented Monday 6 July 1896 to the Prince of Wales at a Levee)<ref name=":1" /> *'''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy"<ref name=":13" /> (4, Col. 2c) ===Japan=== * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation"<ref name=":13" /> (4, Col. 2c) ===Liberia=== * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Liberian Minister"<ref name=":13" /> (4, Col. 2c) ===Mexico=== * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation"<ref name=":13" /> (4, Col. 2c) ===Netherlands=== * Count de Bylandt, Minister, and the Countess de Bylandt * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation"<ref name=":13" /> (4, Col. 2c) ===Persia=== * Prince Malcom Khan, Minister, the Princess Malcom Khan, and the Princess Sultana Malcom * Naser al-Din Shah Qajar, Shah of Persia (5 September 1848 – 1 May 1896) * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Persian Minister, and one Secretary of Legation"<ref name=":13" /> (4, Col. 2c) ===Portugal=== * M. M. d'Antas, Minister, and Madame d'Antas * Luis Maria Augusto Pinto de Soveral, [[Social Victorians/People/de Soveral|Marquês de Soveral]], Minister (1891), Minister of Foreign Affairs (1895–1897), Envoy Extraordinary and Minister Plenipotentiary to the Court of St James's (1897 on) ===Roumania [Romania]=== * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Roumanian Minister and the Councillor of the Legation"<ref name=":13" /> (4, Col. 2c) ===Russia=== * Baron Egor Egorovich Staal or Georges de Staal, Ambassador (1884–1902 ), Madame de Staal, and Mademoiselle de Staal * M. V. Boulatzell, 2nd Secretary in the Russian Embassy in London (until 29 June 1896, after that 1st Secretary in Vienna) * Baron Alexandre de Stoeckl, Attaché of the Russian Embassy (as of 1892) *'''1895 February 15''': presented at a Levee at St. James's Palace in the Diplomatic Circle to the Prince of Wales "By the Russian Ambassador, M. Romeiko Gourko, Secretary of Embassy, and Captain Prince Oukhtomsky, Naval Attaché."<ref name=":0" /> *'''1896 May 1''': His Excellency M. de Staal, the Russian Ambassador; Count Pahlen, First Secretary at the Embassy; M. Gourko, Second Secretary; Count Yermoloff, Military Attaché; M. Orloff, of the Chancellery; and M. Volborth, Russian Consul-General in London.<ref>"London Day by Day." London ''Daily Telegraph'' 02 May 1896 Saturday: 7 [of 12], Col. 2a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18960502/061/0007.</ref> *'''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Russian Ambassador [de Staal], Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy"<ref name=":13" /> (4, Col. 2c) === Siam [Thailand] === * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation"<ref name=":13" /> (4, Col. 2c) === Sierra Leone === * Chief Justice of Sierra Leone ===Spain=== * Count Casa de Valencia * Don Cipriano del Mazo, Minister *'''1895 February 15''': presented at a Levee at St. James's Palace in the Diplomatic Circle to the Prince of Wales "By the Spanish Ambassador, Don Pablo Sirera, Military Attaché."<ref name=":0" /> *'''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy"<ref name=":13" /> (4, Col. 2c) ===Sweden and Norway=== * Count Piper, Minister for Sweden and Norway * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": ""<ref name=":13" /> (4, Col. 2c) ===Turkey=== * Rustem Pasha, Ambassador * '''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy"<ref name=":13" /> (4, Col. 2c) ===United States=== ==== American Legation in London ==== * James Russell Lowell, Minister (–1884) * Henry White, Second Secretary (1883–1886) * Henry White, First Secretary (1886 – October 1893) ==== American Embassy in London (1895–) ==== * John Hay, Ambassador (1896–1898) *Henry White, First Secretary (1896 – 6 March 1905) *Joseph H. Choate, Ambassador (1898–) **William Woodward, Secretary to Choate (reign of King Edward VII)<ref>{{Cite book|url=https://books.google.com/books?id=kdYVBQAAQBAJ&pg=PT108&dq=Mary+Goelet&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwic5sXM9YuHAxXBk44IHdD6CPI4ChDoAXoECA0QAg|title=This Crazy Thing Called Love: The Golden World and Fatal Marriage of Ann and Billy Woodward|last=Braudy|first=Susan|date=2014-11-12|publisher=Knopf Doubleday Publishing Group|isbn=978-0-8041-5335-5|language=en}}</ref> * Mr. Edward J. Phelps, Minister, and Mrs. Phelps *'''1895 February 15''': presented at a Levee at St. James's Palace in the Diplomatic Circle to the Prince of Wales "By the United States Ambassador, [were] Mr. David D. Wells, Second Secretary, and Mr. John Ridgely Carter, Private Secretary."<ref name=":0" /> *'''1897 June 28, Monday''': present among the "members of the Corps Diplomatique and other foreigners of distinction": "The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador"<ref name=":13" /> (4, Col. 2c) == International Families == * [[Social Victorians/People/Gleichen|Prince and Princess Victor of Hohenlohe]], the Countess Feodore Gleichen and Count A. E. Gleichen, although closely related to [[Social Victorians/People/Queen Victoria|Queen Victoria]] and also British ultimately. Should be part of the [[Social Victorians/People/Royal Mob|Royal Mob]]? * Baroness Alphonse [[Social Victorians/People/Rothschild Family|de Rothschild]] * Baron Ferdinand and Miss Alice [[Social Victorians/People/Rothschild Family|de Rothschild]] == Dignitaries, Mostly from Outside the U.K. == * Count d'Eulenburg (Philipp Friedrich Alexander, Prince of Eulenburg) * Prince and Princess de Wagram (in 1886 Napoléon Alexandre Louis Joseph Berthier, 2nd Prince of Wagram [September 10, 1810 – February 10, 1887]) * Duc and Duchesse d'Otrante and Countesse Augusta d'Otrante (Gustave Fouché d'Otrante (1840 - 1910), <abbr>5^e</abbr> duc d'Otrante [https://fr.wikipedia.org/wiki/Duc_d%27Otrante<nowiki>], married I think to Adélaïde Fouché d'Otrante (1866 - 1943])</nowiki> * Duc de la Tremonille * Marques and Marquesa de [[Social Victorians/People/Santurce|Santurce]] and the Mesdemoiselles de Murrieta; Mr. C. de Murrieta and Mr. A. de Murrieta also attended social events * Marquis de Guadalmina * Marquise de Lavaggi and Comtesse de Lavaggi * Marquis Maffei * Marquis Niccolini * Count N. Adlerberg * Count and Countess X. de Florian * Count P. Esterhazy * Count Wilmoss Festitics * Count R. [[Social Victorians/People/Kinsky|Kinsky]] * Count T. Bolesta Koziebrodzki * Count P. Metternich * Count Pourtales * Countess de la Somaglia * Viscount de Saint Genys * Baron Both * Baron and Baronne de Brienen and the Mesdemoiselles de Brienen * Baron Plessen * [[Social Victorians/People/Ephrussi|M. and Madame M. Ephrussi]] * M. von Hengelmuller * M. A. Kroupensky * M. and Madame G. Meyer * Signor Tosti * ==Footnotes== {{reflist}} j6ej206yae7fcvz0nx5kbzc8pxfu7a4 2 274697 2693837 2692982 2024-12-30T02:15:15Z Platos Cave (physics) 2562653 2693837 wikitext text/x-wiki ==== Omega ==== The most precise of the experimentally measured constants is the [[w:Rydberg constant | Rydberg constant]] ''R'' = 10973731.568508(65) 1/m. Here ''c'' (exact), [[w:Vacuum permeability | Vacuum permeability]] μ₀ = 4π/10^7 (exact) and ''R'' (12-13 digits) are combined into a unit-less ratio; :<math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2} = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math> :<math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}} \frac{v^5}{r^9},\;u^{13}</math> :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7}</math> :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = 2^{295} \pi^{157} 3^{21} \alpha^{26} (\Omega^{15})^{15}</math>, units = 1 We can now define ''Ω'' using the geometries for (''c^*, μ₀^*, R^*'') and then solve by replacing (''c^*, μ₀^*, R^*'') with the numerical (''c, μ₀, R''). :<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math> :<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values) :<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values) :<math>\Omega = 2.007\;134\;949\;584...,\; units = 1</math> (CODATA 2022 mean values) There is a close natural number for Ω that is a square root :<math>\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)} = 2.007\;134\;9543... </math> implying that Ω can have a plus or a minus solution, and this agrees with theory (in the mass domain Ω occurs as Ω² = plus only, in the charge domain Ω occurs as Ω³ = can be plus or minus; see [[v:Sqrt_Planck_momentum | sqrt(momentum)]]). This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants). Using this Omega (from pi and e) in the above equation, we can solve α = 137.035996376 This formula includes [[w:Euler%27s_formula | Euler's_formula]] where {{mvar|i}} is the [[w:imaginary unit | imaginary unit]]. :<math display="block">e^{i x} = \cos x + i \sin x</math> Adding {{mvar|i}} :<math>\sqrt{ \left(i \pi^e e^{(1-e)}\right)}</math> Solves to 1.4192587369597 + 1.4192587369597{{mvar|i}}. The real and imaginary parts are equal and this implies a 45 degree angle in the complex plane, and this indicates that this complex number lies exactly on the line y = x in the complex plane. This is relevant to particle rotations and symmetries, [[w:Quantum_superposition | Quantum_superposition]] ... <ref>Macleod, M.J., "[https://theprogrammergod.com The Programmer God, are we in a computer simulation?]", Chapter 10 Alpha and Omega (2024)</ref> '''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale''' An orbital simulation program is described that emulates both gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the [[w:Planck_units |Planck units]] for comparison with real world orbits <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. [[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]] For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''_P (for example, a 1kg satellite would be divided into 1kg/''m''_P = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although only the mass state of the electron is mapped during transition, periodically the spiral angles converge to give an integer orbital radius, the transition steps between these radius can then be used to solve the transition frequency. And so although mapping a gravitational orbit on a 2-D plane, a radial quantization (as a function of pi and so of geometrical origin) emerges, (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). In this context it is thus not necessary to develop a separate `quantum' theory of gravity. === Theory === In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10<sup>23</sup> units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model. If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10<sup>23</sup> units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10<sup>23</sup> electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the [[v:Quantum_gravity_(Planck)#Photon_orbital_model |photon-orbital model]], in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this quantization has a geometrical origin. Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the formula <math>f/c = t \lambda_H/(n_f^2-n_i^2)</math>. In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravitational orbits that we observe are the time averaging sum of the underlying multiple gravitational orbitals. === N-body orbitals === [[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]] The simulation universe is a 4-axis hypersphere expanding in increments <ref>Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}</ref> with 3-axis (the [[v:Black-hole_(Planck) |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'. The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m<sub>P</sub>]], [[w:Planck length |Planck length l<sub>p</sub>]], [[w:Planck time |Planck time t<sub>p</sub>]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''<sub>p</sub>/''t''<sub>p</sub>) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}</math> As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation. Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref> is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence. [[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]] ==== 2 body orbits ('''x, y''' plane) ==== For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units :<math>r_s = \frac{2 l_p M}{m_P}</math> As the simulation itself is dimensionless, we can remove the dimensioned length component <math>2 l_p</math>, and as each point is analogous to 1 unit of Planck mass <math>m_P</math>, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius <math>r_s</math> = number of central points = total points - 1 (the orbiting point). After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. Key: 1. <math>r_s</math> = '''i'''; number of center mass points (the orbited object). 2. '''j_max''' = radius to mass co-efficient. 3. '''j''' = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 ). 4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 5. '''r_α''' = a radius constant, here r_α = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals. :<math>r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} </math> ==== Orbital formulas (2-D plane)==== Outer = orbiting point, inner = orbited center :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2</math>, orbital radius :<math>r_{barycenter} = \frac{r_{outer}}{j}</math>, barycenter :<math>v_{outer} = \frac{i}{j_{max} r_{\alpha}} </math>, orbiting point velocity :<math>v_{inner} = \frac{1}{j_{max} r_{\alpha}}</math>, orbited point(s) velocity :<math>t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 </math>, orbiting point period :<math>l_{outer} = 2 \pi (r_{outer} - r_{barycenter})</math>, distance travelled Simulation data: :period <math>t_{sim}</math> :length <math>l_{sim}</math> :radius <math>r_{sim} = \frac{l_{sim}}{2 \pi}</math> :velocity <math>v_{sim} = \frac{l_{sim}}{t_{sim}}</math> :barycenter <math>b_{sim} = \frac{x_{max} + x_{min}}{2}</math> For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 1) ''j''_max = i+1 = 8 :period <math>t = 74465.0516,\; t_{outer} = 74471.6125</math> :length <math>l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032</math> :radius <math>r_{sim} = 626.3951</math> :velocity <math>v_{sim} = 1/18.920137</math> :barycenter <math>b_{sim} = 89.5241,\; r_{barycenter} = 89.4929</math> 2) ''j''_max = 32*i+1 = 225 :period <math>t = 1656793370.3483,\; t_{outer} = 1656793381.3051</math> :length <math>l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385</math> :radius <math>r_{sim} = 495531.959</math> :velocity <math>v_{sim} = 1/532.128856</math> :barycenter <math>b_{sim} = 70790.283, \;r_{barycenter} = 70790.280</math> 3) Moon orbit. From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1. :<math>\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663</math> :<math>r_{earth-moon}</math> = 384400km :<math>M_{earth}</math> = 0.597378 10²⁵kg Solving <math>j_{max}</math> :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}</math> :<math>j_{max} = 1440443</math> Gives :<math>t_{outer} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}</math>s :<math>t_{outer} \frac{M_{earth}} {m_P } = 2371844</math>s (27.452 days) :<math>v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s</math> :<math>v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}} = 12.57m/s</math> :<math>r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km</math> ==== Gravitational coupling constant ==== In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α_G'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio; :<math>\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}</math> For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α_G then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>). :<math>{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}</math> As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α_G is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10⁴² units of Planck time (<math>t_p = 10^{-42}s</math>), this occurs about once every 3 minutes. :<math>\frac{{\alpha_G}^{-1}}{t_p}</math> Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak. If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f_e, then the point will require 10²³ electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time). :<math>f_e = \frac{m_P}{m_e} = 10^{23}</math> For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have <math>1kg/m_P = 45940509</math> particles in the point-state. The earth (A) will have <math>5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}</math> particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite; :<math>N_{orbitals} = (\frac{m_A}{m_P})(\frac{m_B}{m_P}) = 0.1261\; x10^{41}</math> Examples: :<math>i = \frac{M_{earth}}{m_P} = 0.27444 \;x10^{33}</math> (earth as the center mass) :<math>i 2 l_p = 0.00887</math> (earth Schwarzschild radius) :<math>s = \frac{1kg}{m_P} = 45940509</math> (1kg orbiting satellite) :<math>j = N_{orbitals} = i*s = 0.1261 \;x10^{41}</math> 1) 1kg satellite at earth surface orbit :<math>r_{o} = 6371000 km</math> (earth surface) :<math>j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 2289.41</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 7909.7924</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374</math> s 2) 1kg satellite at a synchronous orbit radius :<math>r_o = 42164.17 km</math> :<math>j_{max} = \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}} = 0.74256\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 5889.674</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 3074.66</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165</math> s 3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). :<math>E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J</math> (energy per orbital) :<math>N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}</math> (number of orbitals) :<math>E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg</math> 4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs. :<math>N_{sun} = \frac{M_{sun}}{m_P} </math> :<math>N_{planet} = \frac{M_{planet}}{m_P} </math> :<math>N_{orbitals} = N_{sun}N_{planet} </math> :<math>n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} </math> :<math>L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} </math> The orbital angular momentum of the planets; mercury = .9153 x10³⁹ venus = .1844 x10⁴¹ earth = .2662 x10⁴¹ mars = .3530 x10⁴⁰ jupiter = .1929 x10⁴⁴ pluto = .365 x10³⁹ Orbital angular momentum combined with orbit velocity cancels ''n_g'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly. :<math>L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} </math> [[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2³² steps they have clumped to form 1 large mass and 2 orbiting masses.]] ==== Freely moving points ==== The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted. In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2³² steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. [[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]] ==== Orbital trajectory (circular vs. straight) ==== Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit. The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other. ==== Precession ==== Can the orbital plane also rotate? semi-minor axis: <math>b = \alpha l^2 \lambda_A</math> semi-major axis: <math>a = \alpha n^2 \lambda_A</math> radius of curvature :<math>L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}</math> :<math>\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}</math> arc secs per 100 years (drift): :<math>T_{earth}</math> = 365.25 days drift = <math>\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}</math> Mercury (eccentricity = 0.205630) T = 87.9691 days a = 57909050 km (''n'' = 378.2734) b = 56671523 km (''l'' = 374.2096) drift = 42.98 Venus (eccentricity = 0.006772) T = 224.701 days a = 108208000 km (''n'' = 517.085) b = 108205519 km (''l'' = 517.079) drift = 8.6247 Earth (eccentricity = 0.0167) T = 365.25 days a = 149598000 km (''n'' = 607.989) b = 149577138 km (''l'' = 607.946) drift = 3.8388 Mars (eccentricity = 0.0934) T = 686.980 days a = 227939366 km (''n'' = 750.485) b = 226942967 km (''l'' = 748.843) drift = 1.351 [[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]] ==== Hyper-sphere orbit ==== {{main|Relativity (Planck)}} Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref>. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B¹ to B¹¹) around the ''A'' time-line (vertical expansion) axis ('''t_d''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. For object '''B''' :<math>t_d = t \sqrt{1 - v_{outer}^2}</math> For object '''A''' :<math>t_d = t \sqrt{1 - v_{inner}^2}</math> === Atomic orbitals === [[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]] In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. This means that we can treat the atomic orbital as a simple 2-body orbit with the electron as the orbiting point. Although this approach can only map the electron point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components converge reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. ==== Theory ==== {{see|Fine-structure_constant_(spiral)}} =====Hyperbolic spiral===== [[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]] A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). The spiral shape that the electron maps can be represented in Cartesian coordinates. Periodically the angles converge to give integer radius, the general form (beginning at the outer limit ranging inwards) gives; :<math>x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 < \varphi < 4\pi</math> :radius = <math>\sqrt(x^2 + y^2) r</math> :<math>\varphi = (2)\pi, \; 4r</math> (360°) :<math>\varphi = (4/3)\pi,\; 9r</math> (240°) :<math>\varphi = (1)\pi, \; 16r</math> (180°) :<math>\varphi = (4/5)\pi, \; 25r</math> (144°) :<math>\varphi = (2/3)\pi, \; 36r</math> (120°) [[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]] =====Principal quantum number '''n'''===== The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level. The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''<sup>2</sup> (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''<sup>2</sup> electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized. =====(Bohr) orbital===== The basic orbital radius has 2 components, dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and dimensioned (electron + proton wavelength); wavelength = <math>\lambda_H = \lambda_p + \lambda_e</math> radius = <math>r_{orbital} = 2\alpha n^2 (\lambda_H)</math> As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius [[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]] length = <math>l_{orbital} = \frac{1}{r_{orbital}}</math> Duration = 1 step per wavelength and so velocity velocity = <math>v_{orbital} = \frac{1}{2\alpha n}</math> Giving period of orbit period = <math>t_{orbital} = \frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha 2\alpha n^3 \lambda_H</math> As we are not mapping the wavelength component, a base (reference) orbital (''n''=1) :<math>t_{ref} = 2\pi 4\alpha^2</math> = 471964.356... The angle of rotation depends on the orbital radius :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> ===== Photon orbital model ===== The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model<ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that <math>orbital \;radius + photon = zero</math> (cancel). The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path. The photon is actually 2 photons as per the Rydberg formula (denoted initial and final). :<math>\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}</math> :<math>\lambda_{photon} = (+\lambda_i) - (+\lambda_f)</math> The wavelength of the (<math>\lambda_i</math>) photon corresponds to the wavelength of the orbital radius. The (+<math>\lambda_i</math>) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-<math>\lambda_f</math>), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+<math>\lambda_i</math>) photon wavelength, the orbital radius does not change as the 2 photons cancel each other; :<math>r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)</math> However, the (<math>\lambda_f</math>) has the longer wavelength, and so after the (<math>\lambda_i</math>) photon has been absorbed, and for the remaining duration of this (<math>\lambda_f</math>) photon wavelength, the orbital radius will be extended until the (<math>\lambda_f</math>) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial <math>n_i = 1</math> orbital to a final <math>n_f = 2</math> orbital, first the (<math>\lambda_i</math>) photon is absorbed (<math>\lambda_i + \lambda_{orbital} = zero</math> which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (<math>\lambda_f</math>) photon continues until it too is absorbed (the '''transition phase'''). :<math>t_{ref} \sim 2\pi 4\alpha^2 </math> :<math>\lambda_i = 1t_{ref}</math> :<math>\lambda_f = 4t_{ref}</math> (''n'' = 2) After the (<math>\lambda_i</math>) photon is absorbed, the (<math>\lambda_f</math>) photon still has <math>\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}</math> steps remaining until it too is absorbed. [[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]] This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of <math>r_{incr}</math> (both photon and orbital radius may be considered as constructs from multiple units of this geometry); :<math>r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}</math> In summary; the (<math>\lambda_i</math>) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step <math>r = r_{orbital}</math> WHILE (<math>\lambda_i > 0</math>) :<math>r = r + r_{incr}</math> ://<math>\lambda_i</math> photon Conversely, because of its minus term, the (<math>\lambda_i</math>) photon will simultaneously extend the orbital radius accordingly; WHILE (<math>r < 4 r_{orbital}</math>) :<math>r = r - r_{incr}</math> ://<math>\lambda_f</math> photon The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]] :<math>t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}</math> The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius; :<math>t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}</math> ==== Simulation ==== The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational and atomic orbits is only in the angle of rotation <math>\beta</math>' which has an additional <math>r_{\alpha}</math> term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately. :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}</math> [[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]] =====Spiral angle===== For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (''n''<sub>i</sub> = 1) orbital :<math>\varphi = 0, \;r_{orbital} = 2\alpha</math> For each step during transition; :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> :<math>\varphi = \varphi + \beta</math> Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and <math>n_f^2</math> at each step. :<math>r = r_{orbital} + \frac{t}{2\pi 2\alpha}</math> (number of increments ''t'' of <math>r_{incr}</math>) :<math>n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}</math> (<math>n_f^2</math> as a function of ''t'') :<math>\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}</math> (<math>\varphi</math> at any <math>n_f^2</math>) We can then re-write (<math>n_f</math> is only an integer at prescribed spiral angles); :<math>\beta = \frac{1}{{r_{orbital}}^2 n_f^3}</math> Giving integer values at these spiral angles :<math>\varphi = (2)\pi, \; r = 4 r_{orbital}</math> (360°) :<math>\varphi = (8/3)\pi,\; r = 9 r_{orbital}</math> (360+120°) :<math>\varphi = (3)\pi, \; r = 16 r_{orbital}</math> (360+180°) :<math>\varphi = (16/5)\pi, \; r = 25 r_{orbital}</math> (360+216°) :<math>\varphi = (10/3)\pi, \; r = 36 r_{orbital}</math> (360+240°) :<math>\varphi = (7/4)\pi, \; r = 49 r_{orbital}</math> :<math>\varphi = (7/2)\pi, \; r = 64 r_{orbital}</math> (360+270°) ===== Rydberg atom ===== At the ''n'' = 1 orbital, 1 complete rotation becomes (dimensionless terms measured on a 2-D plane); :<math>t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha</math> :<math>1t_{ref}</math> = 471964.3563... :<math>4t_{ref}</math> = 1887857.4255... :<math>9t_{ref}</math> = 4247679.2074... :<math>16t_{ref}</math> = 7551429.7021... ===== H atom ===== Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]). H(1s-2s) = 2466 061 413 187.035 kHz <ref>http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf</ref> H(1s-3s) = 2922 743 278 665.79 kHz <ref>https://pubmed.ncbi.nlm.nih.gov/33243883/</ref> H(1s-4s) = 3082 581 563 822.63 kHz <ref>https://codata.org/</ref> H(1s-∞s) = 3288 086 857 127.60 kHz <ref>https://codata.org/ (109678.77174307cm-1)</ref> (''n'' = ∞) R = 10973731.568157 <ref>https://codata.org/ (mean)</ref> ([[w:Rydberg constant |Rydberg constant]]) α =137.035999177 (inverse fine structure constant <ref>https://codata.org/ (mean)</ref> The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. <math>r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}</math> Dividing (dimensioned) wavelength (<math>r_{wavelength}</math>) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The <math>(n^2 - 1)</math> term gives the number of orbital wavelengths in the transition phase; :<math>h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}</math> <math>h_{(1s-2s)}</math> = 1887839.82626... <math>h_{(1s-3s)}</math> = 4247634.04874... <math>h_{(1s-4s)}</math> = 7551347.55306... ===== Simulation atom ===== The following example simulates an electron transition, the electron begins at radius <math>r = r_{orbital}</math> and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). The period <math>t_{sim}</math> and length <math>l_{sim}</math> are measured at integer <math>n^2 r</math> (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period <math>t_{sim}</math> can then be used to calculate the transition frequencies. In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''<sub>sim</sub> = period and ''l''<sub>sim</sub> = distance travelled by the electron (<math>l_{orbital} = l_{sim}</math> at ''n''=1), the radius coefficient ''r''<sub>n</sub> = radius divided by <math>r_{orbital}</math>. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter. [[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]] :<math>j_{atom} = 250</math> (atomic mass) :<math>i_{nucleus} = j_{atom} -1 = 249</math> (relative nucleus mass) :<math>r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2</math> = 2.0160965 :<math>r_{orbital} = 2 \alpha \;*\; r_{wavelength} </math> (radius) = 552.5556 :<math>t_n = \frac{t_{sim}}{r_{wavelength}}</math> :<math>l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}</math> :<math>r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}</math> :<math>r_n = \frac{r_b}{r_{orbital}}</math> {| class="wikitable" |+table 1. Electron transition (mass = 250; ''r''_n= 1 to 5) ! ''r''_n ! ''t''_sim ! ''l''_n ! angle ! ''x'', ''y'' (electron) ! ''x'', ''y'' (nucleus) ! ''x'', ''y'' (barycenter) |-1 | 1 | 471957.072 | 0.9999897 | 360° | 550.334, 0.0036 | -2.2102, -0.00002 | -0.00004, -0.00001 |- | 4 | 1887867.293 | 2.000012 | 359.952489° | 2202.8558, 0.0001 | -7.9565, -1.9475 | 0.8868, -1.9397 |- | 9 | 4247689.502 | 4.000014 | 119.92712° | -2473.180, 4296.283 | 13.558, -10.325 | 3.611, 6.901 |- | 16 | 7551439.538 | 6.000014 | 179.91669° | -8815.254, 12.818 | 25.636, 13.303 | -9.728, 13.301 |- | 25 | 11799118.905 | 8.000014 | 215.9122° | -11158.64, -8081.13 | 16.580, 39.083 | -28.118, 6.602 |} Comparison of the spiral angle at ''r''_n = 4, 9, 16 (360, 360+120, 360+180) with different mass (''m'' = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; ''m'' = 1836.15267... {| class="wikitable" |+ table 2. Spiral angle at <math>r_n</math> = 4, 9, 16 ! mass ! ''r''_n = 4 ! ''r''_n = 9 ! ''r''_n = 16 |- | ''m'' = 64 | 359.80318° | 119.70323° | 179.66239° |- | ''m'' = 128 | 359.90394° | 119.85415° | 179.83377° |- | ''m'' = 250 | 359.95249° | 119.92711° | 179.91669° |- | ''m'' = 500 | 359.97706° | 119.96501° | |- | Rydberg | 360° | 360+120° | 360+180° |} == External links == * [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]] * [[v:Physical_constant_(anomaly) | Physical constant anomalies]] * [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] * [[v:electron_(mathematical) | The mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] * [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] ==References== {{Reflist}} [[Category:Physics| ]] [[Category:Philosophy of science| ]] idgqt37fzymoglx3lda0bhmo9nwkutu 0 284602 2693772 2685382 2024-12-29T17:10:53Z Saltrabook 1417466 2693772 wikitext text/x-wiki {| class="wikitable" | colspan="6" |'''The International Type 2 Diabetes and Hypertension Research Group members:''' |- |Suyasha Koirala |Nepal |Finn Gyntelberg |Denmark |Subasha Thapa |Denmark |- |Nailet Delgado Mujica |Panama |Erik Haarløv |Denmark |Rimsky Sucre |Panama |- |Alejandro Martinez |Costa Rica |Laure Rougegrez |France |Doc Emile |Gabon |- |Augusto PML Pinto |Brazil |Dr Brigita |Ivory Coast |Maite Duque |Venezuela |- |Dr. Tam Nguyen Van |Vietnam |Dr. Tarik Ghailan |Morocco |Guido Cohen |Panama |- |Rifky Muhammad |Filippines |Jenny Mendoza |Filippines |Andra Ergle |Latvia |- |Joyce Mbogo |Kenya |Hedley Quintana |Panama | | |} 24kva9zvvy3c3v1jls8kk5u0u9l7ltf 2693773 2693772 2024-12-29T17:13:28Z Saltrabook 1417466 2693773 wikitext text/x-wiki {| class="wikitable" | colspan="6" |'''The International Type 2 Diabetes and Hypertension Research Group members:''' |- |Suyasha Koirala |Nepal |Finn Gyntelberg |Denmark |Nailet Delgado Mujica |Panama |- |Alejandro Martinez |Costa Rica |Laure Rougegrez |France |Doc Emile |Gabon |- |Augusto PML Pinto |Brazil |Dr Brigita |Ivory Coast |Maite Duque |Venezuela |- |Dr. Tam Nguyen Van |Vietnam |Dr. Tarik Ghailan |Morocco |Guido Cohen |Panama |- |Rifky Muhammad |Filippines |Jenny Mendoza |Filippines |Andra Ergle |Latvia |- |Joyce Mbogo |Kenya | | | | |} b10qfo98cu6w1j4eoypey3i08j17435 0 285384 2693826 2693713 2024-12-29T23:52:48Z Young1lim 21186 /* Linking Libraries */ 2693826 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241229.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241228.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241228.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241228.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241228.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.20241228.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]] eq8zs30kl0fo4saggb94rkgeyisu736 2693827 2693826 2024-12-29T23:54:00Z Young1lim 21186 /* Linking Libraries */ 2693827 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241230.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241228.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241228.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241228.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241228.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.20241228.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]] hi2sxaqle9o4r63o263rhz9u1fcwnk6 2693838 2693827 2024-12-30T02:16:18Z Young1lim 21186 /* Linking Libraries */ 2693838 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241230.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241230.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241228.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241228.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241228.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.20241228.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]] 2lsc28msyer8aoqzicf81gvfiyx3h1f 2693841 2693838 2024-12-30T07:21:12Z Young1lim 21186 /* Dynamic Linking - Directories and Symbolic Links */ 2693841 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241230.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241230.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241230.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241228.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241228.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.20241228.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]] oslopgsj4efgeofazcit4rxcustnhl7 2693844 2693841 2024-12-30T08:18:20Z Young1lim 21186 /* Dynamic Linking - Directories and Symbolic Links */ 2693844 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241230.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241230.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241230.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241230.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241228.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.20241228.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]] fuc0g7oqid81e09pzawzhh8c67w8n9g 2693847 2693844 2024-12-30T08:53:42Z Young1lim 21186 /* Dynamic Loading - API Functions */ 2693847 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.20241228.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241230.pdf |A.pdf]], [[Media:LIB.2B.Shared.20241230.pdf |B.pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241230.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241230.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241230.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.20241228.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]] 9i6fi72r9f6nekv6pttdd9yddigk32k 2 289273 2693802 2693574 2024-12-29T22:17:59Z Dc.samizdat 2856930 2693802 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 observed universe is 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} 5pzroiz43qjne34x05jbpks2j7ok9cu 2693812 2693802 2024-12-29T22:54:57Z Dc.samizdat 2856930 2693812 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. Objects intermediate in size between atoms and stars, such as molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow 3-sphere, with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble of galactic size. The observed universe is 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} 1sds9z3emf06e4xu350l6r22u8308y3 2693814 2693812 2024-12-29T23:03:30Z Dc.samizdat 2856930 2693814 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. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow 3-sphere, with these objects distributed on its surface. The black hole at the galaxy's center is nothing at all: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble of galactic size. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} swo1nhmqrkkojlkr9sow82prh2bxik7 2693816 2693814 2024-12-29T23:09:52Z Dc.samizdat 2856930 2693816 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are [[w:4-ball|4-balls]] of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing at all: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble of galactic size. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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. 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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}} ss0cokcw0n22qyvvum76hzm7dfeam9c 2693819 2693816 2024-12-29T23:17:46Z Dc.samizdat 2856930 2693819 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} 5quu0u2vr5uo89d7wq300cea9tc499v 2693820 2693819 2024-12-29T23:19:06Z Dc.samizdat 2856930 2693820 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble film of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} 7suyqnnsorq8sb9lkfks9g7hv80nhux 2693824 2693820 2024-12-29T23:45:36Z Dc.samizdat 2856930 2693824 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble film of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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]]. == Minkowski spacetime and Euclidean 4-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. 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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}} flhs1vvjdja7xv8nzkzlabp4w77exee 2693825 2693824 2024-12-29T23:48:21Z Dc.samizdat 2856930 /* Minkowski spacetime and Euclidean 4-space */ 2693825 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble film of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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]]. == Minkowski spacetime and Euclidean 4-space == ... == The speed of everything == ... == 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} jxmml1lpjugmultcsbku7ejza2nop1c 2693833 2693825 2024-12-30T00:26:44Z Dc.samizdat 2856930 /* The speed of everything */ 2693833 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble film of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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]]. == Minkowski spacetime and Euclidean 4-space == ... == The rate of atomic symmetry operations == ... == 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} m89c15dh5zyofod87h5nvkrcxu0t23q 2693834 2693833 2024-12-30T00:27:40Z Dc.samizdat 2856930 /* Minkowski spacetime and Euclidean 4-space */ 2693834 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 [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Atoms are [[w:4-polytope|4-polytopes]], and stars are 4-balls of atomic plasma. Objects intermediate in size between atoms and stars, including molecules, people and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in their orthogonal fourth dimension. A typical galaxy such as ours is a hollow [[w:3-sphere|3-sphere]], with these objects distributed on its surface. The black hole at the galaxy's center is nothing: the 4-ball of empty space they surround. Objects in our galaxy occupy the thin 3-dimensional manifold that is its surface, a 4-dimensional soap-bubble film of galactic size, thicker than an atom only in the interior of stars. The observed universe is a 3-sphere expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 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]]. == Minkowski spacetime and Euclidean 4-space in relativity == ... == The rate of atomic symmetry operations == ... == 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. 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A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}} * {{Cite 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}} liz4r8jmdj737xvxw5zynnvmv2ocjff 0 292769 2693764 2480478 2024-12-29T15:30:24Z 2409:40E3:4075:7D65:8000:0:0:0 2693764 wikitext text/x-wiki <!-- {{short description|Computational model used in machine learning, based on connected, hierarchical functions}} {{Machine learning|Artificial neural network}} {{Artificial intelligence}} {{Complex systems}} {{Network science}} --> [[File:Colored neural network.svg|thumb|upright=1.15|An artificial neural network is an interconnected group of nodes, inspired by a simplification of [[w:en:neuron|neuron]]s in a [[w:en:brain|brain]]. Here, each circular node represents an [[w:en:artificial neuron|artificial neuron]] and an arrow represents a connection from the output of one artificial neuron to the input of another.]] '''Artificial neural networks''' ('''ANNs'''), usually simply called '''neural networks''' ('''NNs''') or '''neural nets''',<ref>{{cite web|last=Hardesty|first=Larry|title=Explained: Neural networks|date=14 April 2017|url=https://news.mit.edu/2017/explained-neural-networks-deep-learning-0414|publisher=MIT News Office|access-date=2 June 2022}}</ref> are computing systems inspired by the [[w:en:biological neural network|biological neural network]]s that constitute animal [[w:en:brain|brain]]s.<ref>{{cite book |last1=Yang |first1=Z.R. |last2=Yang |first2=Z. |title=Comprehensive Biomedical Physics |date=2014 |publisher=Elsevier |location=Karolinska Institute, Stockholm, Sweden |isbn=978-0-444-53633-4 |page=1 |url=https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |access-date=28 July 2022 |archive-date=28 July 2022 |archive-url=https://web.archive.org/web/20220728183237/https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |url-status=live }}</ref> An ANN is based on a collection of connected units or nodes called [[w:en:artificial neuron|artificial neuron]]s, which loosely model the [[w:en:neuron|neuron]]s in a biological brain. Each connection, like the [[w:en:synapse|synapse]]s in a biological brain, can transmit a signal to other neurons. An artificial neuron receives signals then processes them and can signal neurons connected to it. The "signal" at a connection is a [[w:en:real number|real number]], and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called ''edges''. Neurons and edges typically have a ''[[w:en:Weighting|weight]]'' that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.{{toclimit|3}} ==Learning Units == * '''[[/Training/]]:''' What is the basic idea of [[machine learning]] in the context of Artificial Neural Networks. * '''[[/History/]]:''' Learn how the knowledge and application of neural networks evolved in the past. * '''[[/Models/]]:''' What basic models of representing an Artificial Neural Network (ANN) with Mathematics and/or with a corresponding computer model. ** [[/Neuron/|Model of Neuron]] ** [[/Network/|Model of a Network of Neurons]] ==Models== [[File:Neuron3.png|thumb|right|upright=1.35|Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals]] === Hyperparameter === {{Main|w:en:Hyperparameter (machine learning)}} A [[w:en:hyperparameter|hyperparameter]] is a constant [[w:en:parameter|parameter]] whose value is set before the learning process begins. The values of [[w:en:parameter|parameter]]s are derived via learning. Examples of hyperparameters include [[w:en:learning rate|learning rate]], the number of hidden layers and batch size.<ref>{{Cite web|url=https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd|title=A Walkthrough of Convolutional Neural Network – Hyperparameter Tuning|last=Lau|first=Suki|date=10 July 2017|website=Medium|access-date=23 August 2019|archive-date=4 February 2023|archive-url=https://web.archive.org/web/20230204153904/https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd?gi=95062e80a5b5|url-status=live}}</ref> The values of some hyperparameters can be dependent on those of other hyperparameters. For example, the size of some layers can depend on the overall number of layers. ===Learning=== {{No footnotes|date=August 2019|section}}{{See also|Mathematical optimization|Estimation theory|Machine learning}} Learning is the adaptation of the network to better handle a task by considering sample observations. Learning involves adjusting the weights (and optional thresholds) of the network to improve the accuracy of the result. This is done by minimizing the observed errors. Learning is complete when examining additional observations does not usefully reduce the error rate. Even after learning, the error rate typically does not reach 0. If after learning, the error rate is too high, the network typically must be redesigned. Practically this is done by defining a [[w:en:Loss function|cost function]] that is evaluated periodically during learning. As long as its output continues to decline, learning continues. The cost is frequently defined as a [[w:en:statistic|statistic]] whose value can only be approximated. The outputs are actually numbers, so when the error is low, the difference between the output (almost certainly a cat) and the correct answer (cat) is small. Learning attempts to reduce the total of the differences across the observations. Most learning models can be viewed as a straightforward application of [[w:en:Mathematical optimization|optimization]] theory and [[w:en:statistical estimation|statistical estimation]].<ref name="Zell1994ch5.2"/><ref>{{Cite book|last1=Kelleher|first1=John D. |last2=Mac Namee|first2=Brian|last3=D'Arcy|first3=Aoife |title=Fundamentals of machine learning for predictive data analytics: algorithms, worked examples, and case studies|date=2020|isbn=978-0-262-36110-1 |edition=2nd|location=Cambridge, MA|chapter=7-8|oclc=1162184998}}</ref> ==== Learning rate ==== The learning rate defines the size of the corrective steps that the model takes to adjust for errors in each observation.<ref>{{cite arXiv|last=Wei|first=Jiakai|date=2019-04-26|title=Forget the Learning Rate, Decay Loss|class=cs.LG|eprint=1905.00094}}</ref> A high learning rate shortens the training time, but with lower ultimate accuracy, while a lower learning rate takes longer, but with the potential for greater accuracy. Optimizations such as [[w:en:Quickprop|Quickprop]] are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid [[w:en:oscillation|oscillation]] inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an [[w:en:adaptive learning rate|adaptive learning rate]] that increases or decreases as appropriate.<ref>{{Cite book|title=The Improved Training Algorithm of Back Propagation Neural Network with Self-adaptive Learning Rate|last1=Li|first1=Y.|last2=Fu|first2=Y.|last3=Li|first3=H.|last4=Zhang|first4=S. W.|s2cid=10557754|date=1 June 2009|journal=2009 International Conference on Computational Intelligence and Natural Computing|isbn=978-0-7695-3645-3|volume=1|pages=73–76|doi=10.1109/CINC.2009.111}}</ref> The concept of momentum allows the balance between the gradient and the previous change to be weighted such that the weight adjustment depends to some degree on the previous change. A momentum close to 0 emphasizes the gradient, while a value close to 1 emphasizes the last change. ====Cost function==== While it is possible to define a cost function [[w:en:ad hoc|ad hoc]], frequently the choice is determined by the function's desirable properties (such as [[w:en:Convex function|convexity]]) or because it arises from the model (e.g. in a probabilistic model the model's [[w:en:posterior probability|posterior probability]] can be used as an inverse cost). ====Backpropagation==== {{Main|w:en:Backpropagation}} Backpropagation is a method used to adjust the connection weights to compensate for each error found during learning. The error amount is effectively divided among the connections. Technically, backprop calculates the [[w:en:gradient|gradient]] (the derivative) of the [[w:en:loss function|cost function]] associated with a given state with respect to the weights. The weight updates can be done via [[w:en:stochastic gradient descent|stochastic gradient descent]] or other methods, such as [[w:en:Extreme Learning Machines|Extreme Learning Machines]],<ref>{{cite journal|last1=Huang|first1=Guang-Bin|last2=Zhu |first2=Qin-Yu|last3=Siew|first3=Chee-Kheong|year=2006|title=Extreme learning machine: theory and applications|journal=Neurocomputing|volume=70|issue=1 |pages=489–501|doi=10.1016/j.neucom.2005.12.126 |citeseerx=10.1.1.217.3692|s2cid=116858 }}</ref> "No-prop" networks,<ref>{{cite journal|year=2013|title=The no-prop algorithm: A new learning algorithm for multilayer neural networks |journal=Neural Networks|volume=37 |pages=182–188|doi=10.1016/j.neunet.2012.09.020|pmid=23140797|last1=Widrow|first1=Bernard|display-authors=etal}}</ref> training without backtracking,<ref>{{cite arXiv|eprint=1507.07680|first1=Yann |last1=Ollivier|first2=Guillaume|last2=Charpiat|title=Training recurrent networks without backtracking |year=2015|class=cs.NE}}</ref> "weightless" networks,<ref name="RBMTRAIN">{{Cite journal |last=Hinton |first=G. E. |date=2010 |title=A Practical Guide to Training Restricted Boltzmann Machines |url=https://www.researchgate.net/publication/221166159 |journal=Tech. Rep. UTML TR 2010-003 |access-date=27 June 2017 |archive-date=9 May 2021 |archive-url=https://web.archive.org/web/20210509123211/https://www.researchgate.net/publication/221166159_A_brief_introduction_to_Weightless_Neural_Systems |url-status=live }}</ref><ref>ESANN. 2009.{{full citation needed|date=June 2022}}</ref> and [[w:en:Holographic associative memory|non-connectionist neural networks]].{{citation needed|date=June 2022}} ===Learning paradigms=== {{No footnotes|date=August 2019|section}} Machine learning is commonly separated into three main learning paradigms, [[w:en:supervised learning|supervised learning]], [[w:en:unsupervised learning|unsupervised learning]] and [[w:en:reinforcement learning|reinforcement learning]].<ref>{{cite book|url=https://www.wolfram.com/language/introduction-machine-learning/|title=Introduction to Machine Learning|first1=Etienne|publisher=Wolfram Media Inc|year=2021|isbn=978-1-579550-48-6|page=9|last1=Bernard}}</ref> Each corresponds to a particular learning task. ==== Supervised learning ==== [[w:en:Supervised learning|Supervised learning]] uses a set of paired inputs and desired outputs. The learning task is to produce the desired output for each input. In this case, the cost function is related to eliminating incorrect deductions.<ref>{{Cite journal|last1=Ojha|first1=Varun Kumar|last2=Abraham|first2=Ajith|last3=Snášel|first3=Václav|date=1 April 2017|title=Metaheuristic design of feedforward neural networks: A review of two decades of research|journal=Engineering Applications of Artificial Intelligence|volume=60|pages=97–116|doi=10.1016/j.engappai.2017.01.013|arxiv=1705.05584|bibcode=2017arXiv170505584O|s2cid=27910748}}</ref> A commonly used cost is the [[w:en:mean-squared error|mean-squared error]], which tries to minimize the average squared error between the network's output and the desired output. Tasks suited for supervised learning are [[w:en:pattern recognition|pattern recognition]] (also known as classification) and [[w:en:Regression analysis|regression]] (also known as function approximation). Supervised learning is also applicable to sequential data (e.g., for hand writing, speech and [[w:en:gesture recognition|gesture recognition]]). This can be thought of as learning with a "teacher", in the form of a function that provides continuous feedback on the quality of solutions obtained thus far. ====Unsupervised learning==== In [[w:en:unsupervised learning|unsupervised learning]], input data is given along with the cost function, some function of the data <math>\textstyle x</math> and the network's output. The cost function is dependent on the task (the model domain) and any ''[[w:en:A priori and a posteriori|a priori]]'' assumptions (the implicit properties of the model, its parameters and the observed variables). As a trivial example, consider the model <math>\textstyle f(x) = a</math> where <math>\textstyle a</math> is a constant and the cost <math>\textstyle C=E[(x - f(x))^2]</math>. Minimizing this cost produces a value of <math>\textstyle a</math> that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in [[w:en:Data compression|compression]] it could be related to the [[w:en:mutual information|mutual information]] between <math>\textstyle x</math> and <math>\textstyle f(x)</math>, whereas in statistical modeling, it could be related to the [[w:en:posterior probability|posterior probability]] of the model given the data (note that in both of those examples, those quantities would be maximized rather than minimized). Tasks that fall within the paradigm of unsupervised learning are in general [[w:en:Approximation|estimation]] problems; the applications include [[w:en:Data clustering|clustering]], the estimation of [[w:en:statistical distributions|statistical distributions]], [[w:en:Data compression|compression]] and [[w:en:Bayesian spam filtering|filtering]]. ====Reinforcement learning==== {{Main|w:en:Reinforcement learning}} {{See also|w:en:Stochastic control}} In applications such as playing video games, an actor takes a string of actions, receiving a generally unpredictable response from the environment after each one. The goal is to win the game, i.e., generate the most positive (lowest cost) responses. In [[w:en:reinforcement learning|reinforcement learning]], the aim is to weight the network (devise a policy) to perform actions that minimize long-term (expected cumulative) cost. At each point in time the agent performs an action and the environment generates an observation and an [[w:en:instant|instant]]aneous cost, according to some (usually unknown) rules. The rules and the long-term cost usually only can be estimated. At any juncture, the agent decides whether to explore new actions to uncover their costs or to exploit prior learning to proceed more quickly. Formally the environment is modeled as a [[w:en:Markov decision process|Markov decision process]] (MDP) with states <math>\textstyle {s_1,...,s_n}\in S </math> and actions <math>\textstyle {a_1,...,a_m} \in A</math>. Because the state transitions are not known, probability distributions are used instead: the instantaneous cost distribution <math>\textstyle P(c_t|s_t)</math>, the observation distribution <math>\textstyle P(x_t|s_t)</math> and the transition distribution <math>\textstyle P(s_{t+1}|s_t, a_t)</math>, while a policy is defined as the conditional distribution over actions given the observations. Taken together, the two define a [[w:en:Markov chain|Markov chain]] (MC). The aim is to discover the lowest-cost MC. ANNs serve as the learning component in such applications.<ref>{{cite conference | author = Dominic, S. | author2 = Das, R. | author3 = Whitley, D. | author4 = Anderson, C. | date = July 1991 | title = Genetic reinforcement learning for neural networks | pages = 71–76 | conference = IJCNN-91-Seattle International Joint Conference on Neural Networks | book-title = IJCNN-91-Seattle International Joint Conference on Neural Networks | publisher = IEEE | location = Seattle, Washington, USA | doi = 10.1109/IJCNN.1991.155315 | isbn = 0-7803-0164-1 | url-access = registration | url = https://archive.org/details/ijcnn91seattlein01ieee }}</ref><ref>{{cite journal |last=Hoskins |first=J.C. |author2=Himmelblau, D.M. |title=Process control via artificial neural networks and reinforcement learning |journal=Computers & Chemical Engineering |year=1992 |volume=16 |pages=241–251 |doi=10.1016/0098-1354(92)80045-B |issue=4}}</ref> [[w:en:Dynamic programming|Dynamic programming]] coupled with ANNs (giving [[w:en:Neural oscillation|neurodynamic]] programming)<ref>{{cite book|url=https://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|title=Neuro-dynamic programming|first1=D.P.|first2=J.N.|publisher=Athena Scientific|year=1996|isbn=978-1-886529-10-6|page=512|last1=Bertsekas|last2=Tsitsiklis|access-date=17 June 2017|archive-date=29 June 2017|archive-url=https://web.archive.org/web/20170629172039/http://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|url-status=live}}</ref> has been applied to problems such as those involved in [[w:en:vehicle routing|vehicle routing]],<ref>{{cite journal |last=Secomandi |first=Nicola |title=Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands |journal=Computers & Operations Research |year=2000 |volume=27 |pages=1201–1225 |doi=10.1016/S0305-0548(99)00146-X |issue=11–12|citeseerx=10.1.1.392.4034 }}</ref> video games, [[w:en:natural resource management|natural resource management]]<ref>{{cite conference | author = de Rigo, D. | author2 = Rizzoli, A. E. | author3 = Soncini-Sessa, R. | author4 = Weber, E. | author5 = Zenesi, P. | year = 2001 | title = Neuro-dynamic programming for the efficient management of reservoir networks | conference = MODSIM 2001, International Congress on Modelling and Simulation | url = http://www.mssanz.org.au/MODSIM01/MODSIM01.htm | book-title = Proceedings of MODSIM 2001, International Congress on Modelling and Simulation | publisher = Modelling and Simulation Society of Australia and New Zealand | location = Canberra, Australia | doi = 10.5281/zenodo.7481 | isbn = 0-86740-525-2 | access-date = 29 July 2013 | archive-date = 7 August 2013 | archive-url = https://web.archive.org/web/20130807223658/http://mssanz.org.au/MODSIM01/MODSIM01.htm | url-status = live }}</ref><ref>{{cite conference| author = Damas, M. |author2=Salmeron, M. |author3=Diaz, A. |author4=Ortega, J. |author5=Prieto, A. |author6=Olivares, G.| year = 2000 | title = Genetic algorithms and neuro-dynamic programming: application to water supply networks |volume=1 |pages=7–14 | conference = 2000 Congress on Evolutionary Computation | book-title = Proceedings of 2000 Congress on Evolutionary Computation | publisher = IEEE | location = La Jolla, California, USA | doi = 10.1109/CEC.2000.870269 | isbn = 0-7803-6375-2 }}</ref> and [[w:en:medicine|medicine]]<ref>{{Cite book |last=Deng |first=Geng |author2=Ferris, M.C. |title=Neuro-dynamic programming for fractionated radiotherapy planning |year=2008 |volume=12 |pages=47–70 |doi=10.1007/978-0-387-73299-2_3|citeseerx=10.1.1.137.8288 |series=Springer Optimization and Its Applications |isbn=978-0-387-73298-5 }}</ref> because of ANNs ability to mitigate losses of accuracy even when reducing the [[w:en:discretization|discretization]] grid density for numerically approximating the solution of control problems. Tasks that fall within the paradigm of reinforcement learning are control problems, [[w:en:game|game]]s and other sequential decision making tasks. ====Self-learning==== Self-learning in neural networks was introduced in 1982 along with a neural network capable of self-learning named Crossbar Adaptive Array (CAA).<ref>Bozinovski, S. (1982). "A self-learning system using secondary reinforcement". In R. Trappl (ed.) Cybernetics and Systems Research: Proceedings of the Sixth European Meeting on Cybernetics and Systems Research. North Holland. pp. 397–402. {{ISBN|978-0-444-86488-8}}.</ref> It is a system with only one input, situation s, and only one output, action (or behavior) a. It has neither external advice input nor external reinforcement input from the environment. The CAA computes, in a crossbar fashion, both decisions about actions and emotions (feelings) about encountered situations. The system is driven by the interaction between cognition and emotion.<ref>Bozinovski, S. (2014) "[https://core.ac.uk/download/pdf/81973924.pdf Modeling mechanisms of cognition-emotion interaction in artificial neural networks, since 1981] {{Webarchive|url=https://web.archive.org/web/20190323204838/https://core.ac.uk/download/pdf/81973924.pdf |date=23 March 2019 }}." Procedia Computer Science p. 255-263</ref> Given the memory matrix, W =||w(a,s)||, the crossbar self-learning algorithm in each iteration performs the following computation: In situation s perform action a; Receive consequence situation s'; Compute emotion of being in consequence situation v(s'); Update crossbar memory w'(a,s) = w(a,s) + v(s'). The backpropagated value (secondary reinforcement) is the emotion toward the consequence situation. The CAA exists in two environments, one is behavioral environment where it behaves, and the other is genetic environment, where from it initially and only once receives initial emotions about to be encountered situations in the behavioral environment. Having received the genome vector (species vector) from the genetic environment, the CAA will learn a goal-seeking behavior, in the behavioral environment that contains both desirable and undesirable situations.<ref>{{cite journal | last1 = Bozinovski | first1 = Stevo | last2 = Bozinovska | first2 = Liljana | year = 2001 | title = Self-learning agents: A connectionist theory of emotion based on crossbar value judgment | journal = Cybernetics and Systems | volume = 32 | issue = 6| pages = 637–667 | doi = 10.1080/01969720118145 | s2cid = 8944741 }}</ref> ==== Neuroevolution ==== {{Main|w:en:Neuroevolution}} [[w:en:Neuroevolution|Neuroevolution]] can create neural network topologies and weights using [[w:en:evolutionary computation|evolutionary computation]]. It is competitive with sophisticated gradient descent approaches{{citation needed|date=July 2019}}. One advantage of neuroevolution is that it may be less prone to get caught in "dead ends".<ref>{{cite news|date=10 January 2018|title=Artificial intelligence can 'evolve' to solve problems|language=en|work=Science {{!}} AAAS|url=https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|access-date=7 February 2018|archive-date=9 December 2021|archive-url=https://web.archive.org/web/20211209231714/https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|url-status=live}}</ref> ===Stochastic neural network=== '''Stochastic neural networks''' originating from [[w:en:Spin glass#Sherrington–Kirkpatrick model|Sherrington–Kirkpatrick model]]s are a type of artificial neural network built by introducing random variations into the network, either by giving the network's [[w:en:artificial neuron|artificial neuron]]s [[w:en:Stochastic process|stochastic]] transfer functions, or by giving them stochastic weights. This makes them useful tools for [[w:en:Optimization (mathematics)|optimization]] problems, since the random fluctuations help the network escape from [[w:en:Maxima and minima|local minima]].<ref>{{citation|title=Stochastic Models of Neural Networks|volume=102|series=Frontiers in artificial intelligence and applications: Knowledge-based intelligent engineering systems|first=Claudio|last=Turchetti|publisher=IOS Press|year=2004|isbn=9781586033880}}</ref> Stochastic neural networks trained using a Bayesian approach are known as '''Bayesian neural networks'''.<ref>{{Cite journal |last1=Jospin |first1=Laurent Valentin |last2=Laga |first2=Hamid |last3=Boussaid |first3=Farid |last4=Buntine |first4=Wray |last5=Bennamoun |first5=Mohammed |date=2022 |title=Hands-On Bayesian Neural Networks—A Tutorial for Deep Learning Users |url=http://dx.doi.org/10.1109/mci.2022.3155327 |journal=IEEE Computational Intelligence Magazine |volume=17 |issue=2 |pages=29–48 |doi=10.1109/mci.2022.3155327 |arxiv=2007.06823 |s2cid=220514248 |issn=1556-603X |access-date=19 November 2022 |archive-date=4 February 2023 |archive-url=https://web.archive.org/web/20230204153904/https://ieeexplore.ieee.org/document/9756596/ |url-status=live }}</ref> ===Other=== In a [[w:en:Bayesian probability|Bayesian]] framework, a distribution over the set of allowed models is chosen to minimize the cost. [[w:en:Evolutionary methods|Evolutionary methods]],<ref>{{cite conference |author1=de Rigo, D. |author2=Castelletti, A. |author3=Rizzoli, A. E. |author4=Soncini-Sessa, R. |author5=Weber, E. |date=January 2005 |title=A selective improvement technique for fastening Neuro-Dynamic Programming in Water Resources Network Management |conference=16th IFAC World Congress |publisher=IFAC |location=Prague, Czech Republic |conference-url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Index.html |book-title=Proceedings of the 16th IFAC World Congress – IFAC-PapersOnLine |editor=Pavel Zítek |volume=16 |pages=7–12 |url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |access-date=30 December 2011 |doi=10.3182/20050703-6-CZ-1902.02172 |isbn=978-3-902661-75-3 |hdl=11311/255236 |hdl-access=free |archive-date=26 April 2012 |archive-url=https://web.archive.org/web/20120426012450/http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |url-status=live }}</ref> [[w:en:gene expression programming|gene expression programming]],<ref>{{cite book |last=Ferreira |first=C. |year=2006 |contribution=Designing Neural Networks Using Gene Expression Programming |url=http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |editor=A. Abraham |editor2=B. de Baets |editor3=M. Köppen |editor4=B. Nickolay |title=Applied Soft Computing Technologies: The Challenge of Complexity |pages=517–536 |publisher=Springer-Verlag |access-date=8 October 2012 |archive-date=19 December 2013 |archive-url=https://web.archive.org/web/20131219022806/http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |url-status=live }}</ref> [[w:en:simulated annealing|simulated annealing]],<ref>{{cite conference |author=Da, Y. |author2=Xiurun, G. |date=July 2005 |title=An improved PSO-based ANN with simulated annealing technique |volume=63 |pages=527–533 |editor=T. Villmann |book-title=New Aspects in Neurocomputing: 11th European Symposium on Artificial Neural Networks |url=http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm |publisher=Elsevier |doi=10.1016/j.neucom.2004.07.002 |access-date=30 December 2011 |archive-date=25 April 2012 |archive-url=https://web.archive.org/web/20120425233611/http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm}}</ref> [[w:en:expectation-maximization|expectation-maximization]], [[w:en:non-parametric methods|non-parametric methods]] and [[w:en:particle swarm optimization|particle swarm optimization]]<ref>{{cite conference |author=Wu, J. |author2=Chen, E. |date=May 2009 |title=A Novel Nonparametric Regression Ensemble for Rainfall Forecasting Using Particle Swarm Optimization Technique Coupled with Artificial Neural Network |series=Lecture Notes in Computer Science |volume=5553 |pages=49–58 |book-title=6th International Symposium on Neural Networks, ISNN 2009 |url=http://www2.mae.cuhk.edu.hk/~isnn2009/ |editor=Wang, H. |editor2=Shen, Y. |editor3=Huang, T. |editor4=Zeng, Z. |publisher=Springer |doi=10.1007/978-3-642-01513-7_6 |isbn=978-3-642-01215-0 |access-date=1 January 2012 |archive-date=31 December 2014 |archive-url=https://web.archive.org/web/20141231221755/http://www2.mae.cuhk.edu.hk/~isnn2009/ |url-status=dead}}</ref> are other learning algorithms. Convergent recursion is a learning algorithm for [[w:en:cerebellar model articulation controller|cerebellar model articulation controller]] (CMAC) neural networks.<ref name="Qin1">{{cite journal |author1=Ting Qin |author2=Zonghai Chen |author3=Haitao Zhang |author4=Sifu Li |author5=Wei Xiang |author6=Ming Li |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |title=A learning algorithm of CMAC based on RLS |journal=Neural Processing Letters |volume=19 |issue=1 |date=2004 |pages=49–61 |doi=10.1023/B:NEPL.0000016847.18175.60 |s2cid=6233899 |access-date=30 January 2019 |archive-date=14 April 2021 |archive-url=https://web.archive.org/web/20210414103815/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |url-status=live }}</ref><ref name="Qin2">{{cite journal |author1=Ting Qin |author2=Haitao Zhang |author3=Zonghai Chen |author4=Wei Xiang |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |title=Continuous CMAC-QRLS and its systolic array |journal=Neural Processing Letters |volume=22 |issue=1 |date=2005 |pages=1–16 |doi=10.1007/s11063-004-2694-0 |s2cid=16095286 |access-date=30 January 2019 |archive-date=18 November 2018 |archive-url=https://web.archive.org/web/20181118122850/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |url-status=live }}</ref> ==== Modes ==== {{No footnotes|date=August 2019|section}} Two modes of learning are available: [[w:en:stochastic gradient descent|stochastic]] and batch. In stochastic learning, each input creates a weight adjustment. In batch learning weights are adjusted based on a batch of inputs, accumulating errors over the batch. Stochastic learning introduces "noise" into the process, using the local gradient calculated from one data point; this reduces the chance of the network getting stuck in local minima. However, batch learning typically yields a faster, more stable descent to a local minimum, since each update is performed in the direction of the batch's average error. A common compromise is to use "mini-batches", small batches with samples in each batch selected stochastically from the entire data set. == Types == <!-- Split to [[w:en:Types of artificial neural networks|Types of artificial neural networks]] --> {{Main|w:en:Types of artificial neural networks}} ANNs have evolved into a broad family of techniques that have advanced the state of the art across multiple domains. The simplest types have one or more static components, including number of units, number of layers, unit weights and [[w:en:topology|topology]]. Dynamic types allow one or more of these to evolve via learning. The latter are much more complicated, but can shorten learning periods and produce better results. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. Some of the main breakthroughs include: [[w:en:convolutional neural network|convolutional neural network]]s that have proven particularly successful in processing visual and other two-dimensional data;<ref name="LECUN1989">{{cite journal |vauthors=LeCun Y, Boser B, Denker JS, Henderson D, Howard RE, Hubbard W, Jackel LD |title=Backpropagation Applied to Handwritten Zip Code Recognition |journal=Neural Computation |volume=1 |issue=4 |pages=541–551 |date=1989 |doi=10.1162/neco.1989.1.4.541|s2cid=41312633 }}</ref><ref name="lecun2016slides">[[w:en:Yann LeCun|Yann LeCun]] (2016). Slides on Deep Learning [https://indico.cern.ch/event/510372/ Online] {{Webarchive|url=https://web.archive.org/web/20160423021403/https://indico.cern.ch/event/510372/ |date=23 April 2016 }}</ref> long short-term memory avoid the [[w:en:vanishing gradient problem|vanishing gradient problem]]<ref name=":03">{{Cite journal |last1=Hochreiter|first1=Sepp|author-link=Sepp Hochreiter|last2=Schmidhuber|first2=Jürgen|s2cid=1915014|author-link2=Jürgen Schmidhuber|date=1 November 1997|title=Long Short-Term Memory|journal=Neural Computation|volume=9|issue=8 |pages=1735–1780 |doi=10.1162/neco.1997.9.8.1735|pmid=9377276|issn=0899-7667}}</ref> and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition,<ref name="sak2014">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf |title=Long Short-Term Memory recurrent neural network architectures for large scale acoustic modeling |last1=Sak|first1=Hasim |last2=Senior|first2=Andrew|date=2014|last3=Beaufays|first3=Francoise|archive-url=https://web.archive.org/web/20180424203806/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf|archive-date=24 April 2018|url-status=dead}}</ref><ref name="liwu2015">{{cite arXiv|last1=Li|first1=Xiangang|last2=Wu|first2=Xihong|date=15 October 2014|title=Constructing Long Short-Term Memory based Deep Recurrent Neural Networks for Large Vocabulary Speech Recognition|eprint=1410.4281 |class=cs.CL}}</ref> text-to-speech synthesis,<ref>{{Cite journal|title=TTS synthesis with bidirectional LSTM based Recurrent Neural Networks|pages=1964–1968|last1=Fan|first1=Y. |last2=Qian|first2=Y.|date=2014 |journal=Proceedings of the Annual Conference of the International Speech Communication Association, Interspeech|url=https://www.researchgate.net/publication/287741874|access-date=13 June 2017 |last3=Xie |first3=F.|last4=Soong|first4=F. K.}}</ref><ref name="scholarpedia2"/><ref name="zen2015">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|title=Unidirectional Long Short-Term Memory Recurrent Neural Network with Recurrent Output Layer for Low-Latency Speech Synthesis|last1=Zen|first1=Heiga|last2=Sak|first2=Hasim|date=2015|website=Google.com|publisher=ICASSP|pages=4470–4474|access-date=27 June 2017|archive-date=9 May 2021|archive-url=https://web.archive.org/web/20210509123113/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|url-status=live}}</ref> and photo-real talking heads;<ref name="fan2015">{{Cite journal|last1=Fan|first1=Bo|last2=Wang|first2=Lijuan|last3=Soong|first3=Frank K.|last4=Xie|first4=Lei|date=2015|title=Photo-Real Talking Head with Deep Bidirectional LSTM|url=https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|journal=Proceedings of ICASSP|access-date=27 June 2017|archive-date=1 November 2017|archive-url=https://web.archive.org/web/20171101052317/https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|url-status=live}}</ref> competitive networks such as [[w:en:generative adversarial network|generative adversarial network]]s in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game<ref name="preprint">{{Cite arXiv |eprint=1712.01815|class=cs.AI|first1=David|last1=Silver|first2=Thomas|last2=Hubert|author-link1=David Silver (programmer)|title=Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm|date=5 December 2017|first3=Julian|last3=Schrittwieser|first4=Ioannis|last4=Antonoglou |first5=Matthew|last5=Lai |first6=Arthur|last6=Guez|first7=Marc|last7=Lanctot|first8=Laurent|last8=Sifre |first9=Dharshan|last9=Kumaran|author-link9=Dharshan Kumaran|first10=Thore|last10=Graepel|first11=Timothy |last11=Lillicrap|first12=Karen |last12=Simonyan|first13=Demis|last13=Hassabis|author-link13=Demis Hassabis}}</ref> or on deceiving the opponent about the authenticity of an input.<ref name="GANnips">{{cite conference|last1=Goodfellow|first1=Ian|last2=Pouget-Abadie|first2=Jean|last3=Mirza|first3=Mehdi|last4=Xu|first4=Bing|last5=Warde-Farley|first5=David|last6=Ozair|first6=Sherjil|last7=Courville|first7=Aaron|last8=Bengio|first8=Yoshua|year=2014|title=Generative Adversarial Networks|url=https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|conference=Proceedings of the International Conference on Neural Information Processing Systems (NIPS 2014)|pages=2672–2680|access-date=20 August 2019|archive-date=22 November 2019|archive-url=https://web.archive.org/web/20191122034612/http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|url-status=live}}</ref> == Network design == {{Main|w:en:Neural architecture search}} Neural architecture search (NAS) uses machine learning to automate ANN design. Various approaches to NAS have designed networks that compare well with hand-designed systems. The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network.<ref>{{cite arXiv|last1=Zoph|first1=Barret|last2=Le|first2=Quoc V.|date=4 November 2016|title=Neural Architecture Search with Reinforcement Learning|eprint=1611.01578|class=cs.LG}}</ref> Available systems include [[w:en:Automated machine learning|AutoML]] and AutoKeras.<ref>{{cite journal |author1=Haifeng Jin |author2=Qingquan Song |author3=Xia Hu |title=Auto-keras: An efficient neural architecture search system |journal=Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining |publisher=ACM |date=2019 |arxiv=1806.10282 |url=https://autokeras.com/ |via=autokeras.com |access-date=21 August 2019 |archive-date=21 August 2019 |archive-url=https://web.archive.org/web/20190821163310/https://autokeras.com/ |url-status=live }}</ref> Design issues include deciding the number, type and connectedness of network layers, as well as the size of each and the connection type (full, pooling, ...). [[w:en:Hyperparameter|Hyperparameter]]s must also be defined as part of the design (they are not learned), governing matters such as how many neurons are in each layer, learning rate, step, stride, depth, receptive field and padding (for CNNs), etc.<ref name="abs1502.02127">{{cite arXiv|eprint=1502.02127|last1=Claesen|first1=Marc|last2=De Moor|first2=Bart |title=Hyperparameter Search in Machine Learning |date=2015|class=cs.LG }} {{bibcode|2015arXiv150202127C}}</ref> == Use == {{Unreferenced section|date=November 2020}} Using Artificial neural networks requires an understanding of their characteristics. * Choice of model: This depends on the data representation and the application. Overly complex models are slow learning. * Learning algorithm: Numerous trade-offs exist between learning algorithms. Almost any algorithm will work well with the correct [[w:en:hyperparameter|hyperparameter]]s for training on a particular data set. However, selecting and tuning an algorithm for training on unseen data requires significant experimentation. * Robustness: If the model, cost function and learning algorithm are selected appropriately, the resulting ANN can become robust. ANN capabilities fall within the following broad categories:{{Citation needed|date=June 2017}} * [[w:en:Function approximation|Function approximation]], or [[w:en:regression analysis|regression analysis]], including [[w:en:Time series#Prediction and forecasting|time series prediction]], [[w:en:fitness approximation|fitness approximation]] and modeling. * [[w:en:Statistical classification|Classification]], including [[w:en:Pattern recognition|pattern]] and sequence recognition, [[w:en:novelty detection|novelty detection]] and sequential decision making.<ref name ="TurekNeuralNet">{{cite journal|author=Turek, Fred D.|title=Introduction to Neural Net Machine Vision|url=http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|access-date=5 March 2013|journal=Vision Systems Design|date=March 2007|volume=12|number=3|archive-date=16 May 2013|archive-url=https://web.archive.org/web/20130516124148/http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|url-status=live}}</ref> * [[w:en:Data processing|Data processing]], including filtering, clustering, [[w:en:blind source separation|blind source separation]] and compression. * [[w:en:Robotics|Robotics]], including directing manipulators and [[w:en:prosthesis|prostheses]]. ==Applications== Because of their ability to reproduce and model nonlinear processes, artificial neural networks have found applications in many disciplines. Application areas include [[w:en:system identification|system identification]] and control (vehicle control, trajectory prediction,<ref>{{cite journal|last1=Zissis|first1=Dimitrios|title=A cloud based architecture capable of perceiving and predicting multiple vessel behaviour|journal=Applied Soft Computing|date=October 2015|volume=35|doi=10.1016/j.asoc.2015.07.002|pages=652–661|url=https://zenodo.org/record/848743|access-date=18 July 2019|archive-date=26 July 2020|archive-url=https://web.archive.org/web/20200726091505/https://zenodo.org/record/848743|url-status=live}}</ref> [[w:en:process control|process control]], [[w:en:natural resource management|natural resource management]]), [[w:en:quantum chemistry|quantum chemistry]],<ref name="Balabin_2009">{{Cite journal|journal=[[w:en:J. Chem. Phys.|J. Chem. Phys.]] |volume=131 |issue=7 |page=074104 |doi=10.1063/1.3206326 |title=Neural network approach to quantum-chemistry data: Accurate prediction of density functional theory energies |year=2009 |author1=Roman M. Balabin |author2=Ekaterina I. Lomakina |pmid=19708729|bibcode=2009JChPh.131g4104B}}</ref> [[w:en:general game playing|general game playing]],<ref>{{cite journal |last1=Silver |first1=David |display-authors=etal |year=2016 |title=Mastering the game of Go with deep neural networks and tree search |url=http://web.iitd.ac.in/~sumeet/Silver16.pdf |journal=Nature |volume=529 |issue=7587 |pages=484–489 |doi=10.1038/nature16961 |pmid=26819042 |bibcode=2016Natur.529..484S |s2cid=515925 |access-date=31 January 2019 |archive-date=23 November 2018 |archive-url=https://web.archive.org/web/20181123112812/http://web.iitd.ac.in/~sumeet/Silver16.pdf |url-status=live }}</ref> [[w:en:pattern recognition|pattern recognition]] (radar systems, [[w:en:Facial recognition system|face identification]], signal classification,<ref>{{cite journal|last=Sengupta |first=Nandini|author2=Sahidullah, Md|author3=Saha, Goutam|title=Lung sound classification using cepstral-based statistical features|journal=Computers in Biology and Medicine|date=August 2016|volume=75|issue=1 |pages=118–129|doi=10.1016/j.compbiomed.2016.05.013 |pmid=27286184}}</ref> [[w:en:3D reconstruction|3D reconstruction]],<ref>Choy, Christopher B., et al. "[https://arxiv.org/abs/1604.00449 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction] {{Webarchive|url=https://web.archive.org/web/20200726091721/https://arxiv.org/abs/1604.00449 |date=26 July 2020 }}." European conference on computer vision. Springer, Cham, 2016.</ref> object recognition and more), sensor data analysis,<ref>{{cite journal|last=Gessler|first=Josef|title=Sensor for food analysis applying impedance spectroscopy and artificial neural networks|journal=RiuNet UPV|date=August 2021|issue=1|pages=8–12|url=https://riunet.upv.es/handle/10251/174498|access-date=21 October 2021|archive-date=21 October 2021|archive-url=https://web.archive.org/web/20211021115443/https://riunet.upv.es/handle/10251/174498|url-status=live}}</ref> sequence recognition (gesture, speech, [[w:en:handwriting recognition|handwritten]] and printed text recognition<ref>{{Cite journal|last1=Maitra|first1=D. S. |last2=Bhattacharya|first2=U.|last3=Parui|first3=S. K.|date=August 2015|title=CNN based common approach to handwritten character recognition of multiple scripts |url=https://ieeexplore.ieee.org/document/7333916|journal=2015 13th International Conference on Document Analysis and Recognition (ICDAR)|pages=1021–1025|doi=10.1109/ICDAR.2015.7333916|isbn=978-1-4799-1805-8 |s2cid=25739012}}</ref>), [[w:en:medical diagnosis|medical diagnosis]], finance<ref>{{cite journal|last1=French |first1=Jordan |title=The time traveller's CAPM|journal=Investment Analysts Journal|volume=46|issue=2|pages=81–96 |doi=10.1080/10293523.2016.1255469|year=2016|s2cid=157962452}}</ref> (e.g. ex-ante models for specific financial long-run forecasts and [[w:en:artificial financial market|artificial financial market]]s), [[w:en:data mining|data mining]], visualization, [[w:en:machine translation|machine translation]], social network filtering<ref>{{Cite news|url=https://www.wsj.com/articles/facebook-boosts-a-i-to-block-terrorist-propaganda-1497546000 |title=Facebook Boosts A.I. to Block Terrorist Propaganda|last=Schechner|first=Sam|date=15 June 2017 |work=[[w:en:The Wall Street Journal|The Wall Street Journal]]|access-date=16 June 2017|issn=0099-9660}}</ref> and [[w:en:e-mail spam|e-mail spam]] filtering. ANNs have been used to diagnose several types of cancers<ref>{{cite journal|last=Ganesan|first=N |title=Application of Neural Networks in Diagnosing Cancer Disease Using Demographic Data |journal=International Journal of Computer Applications|volume=1|issue=26|pages=81–97 |bibcode=2010IJCA....1z..81G|year=2010|doi=10.5120/476-783|doi-access=free}}</ref><ref>{{cite journal |url=http://www.lcc.uma.es/~jja/recidiva/042.pdf|title=Artificial Neural Networks Applied to Outcome Prediction for Colorectal Cancer Patients in Separate Institutions|journal=Lancet|volume=350|issue=9076 |pages=469–72|last=Bottaci|first=Leonardo|publisher=The Lancet|pmid=9274582|year=1997|doi=10.1016/S0140-6736(96)11196-X|s2cid=18182063|access-date=2 May 2012|url-status=dead|archive-date=23 November 2018|archive-url=https://web.archive.org/web/20181123170444/http://www.lcc.uma.es/~jja/recidiva/042.pdf}}</ref> and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.<ref>{{cite journal|last1=Alizadeh|first1=Elaheh|last2=Lyons|first2=Samanthe M|last3=Castle|first3=Jordan M |last4=Prasad|first4=Ashok|date=2016|title=Measuring systematic changes in invasive cancer cell shape using Zernike moments|url=http://pubs.rsc.org/en/Content/ArticleLanding/2016/IB/C6IB00100A|journal=Integrative Biology|volume=8|issue=11 |pages=1183–1193|doi=10.1039/C6IB00100A|pmid=27735002}}</ref><ref>{{cite journal |last1=Lyons|first1=Samanthe|date=2016|title=Changes in cell shape are correlated with metastatic potential in murine|journal=Biology Open|volume=5|issue=3|pages=289–299|doi=10.1242/bio.013409|pmid=26873952 |pmc=4810736}}</ref> ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters<ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=28 August 2017|title=Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks|journal=Computer-Aided Civil and Infrastructure Engineering|volume=33|issue=6|pages=443–458|arxiv=1708.08551|doi=10.1111/mice.12359 |bibcode=2017arXiv170808551N |s2cid=36661983}}</ref><ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|access-date=14 March 2018|archive-date=9 March 2018|archive-url=https://web.archive.org/web/20180309120108/https://trid.trb.org/view/1496617|url-status=live}}</ref> and to predict foundation settlements.<ref>{{Cite journal|last1=Díaz|first1=E.|last2=Brotons|first2=V. |last3=Tomás|first3=R.|date=September 2018|title=Use of artificial neural networks to predict 3-D elastic settlement of foundations on soils with inclined bedrock|journal=Soils and Foundations|volume=58|issue=6 |pages=1414–1422 |doi=10.1016/j.sandf.2018.08.001|issn=0038-0806|hdl=10045/81208|doi-access=free}}</ref> ANNs have also been used for building black-box models in [[w:en:geoscience|geoscience]]: [[w:en:hydrology|hydrology]],<ref>{{Cite journal |first=Rao S.|last=Govindaraju |date=1 April 2000|title=Artificial Neural Networks in Hydrology. I: Preliminary Concepts|journal=Journal of Hydrologic Engineering|volume=5|issue=2|pages=115–123|doi=10.1061/(ASCE)1084-0699(2000)5:2(115)|citeseerx=<!--10.1.1.127.3861-->}}</ref><ref>{{Cite journal|first=Rao S.|last=Govindaraju|date=1 April 2000|title=Artificial Neural Networks in Hydrology. II: Hydrologic Applications|journal=Journal of Hydrologic Engineering|volume=5|issue=2 |pages=124–137 |doi=10.1061/(ASCE)1084-0699(2000)5:2(124)}}</ref> ocean modelling and [[w:en:coastal engineering|coastal engineering]],<ref>{{Cite journal|last1=Peres|first1=D. J.|last2=Iuppa|first2=C.|last3=Cavallaro|first3=L.|last4=Cancelliere |first4=A. |last5=Foti|first5=E.|date=1 October 2015|title=Significant wave height record extension by neural networks and reanalysis wind data|journal=Ocean Modelling|volume=94|pages=128–140 |doi=10.1016/j.ocemod.2015.08.002 |bibcode=2015OcMod..94..128P}}</ref><ref>{{Cite journal|last1=Dwarakish|first1=G. S.|last2=Rakshith|first2=Shetty|last3=Natesan|first3=Usha|date=2013|title=Review on Applications of Neural Network in Coastal Engineering|journal=Artificial Intelligent Systems and Machine Learning|url=http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|volume=5|issue=7|pages=324–331|access-date=5 July 2017|archive-date=15 August 2017|archive-url=https://web.archive.org/web/20170815185634/http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|url-status=live}}</ref> and [[w:en:geomorphology|geomorphology]].<ref>{{Cite journal |last1=Ermini|first1=Leonardo|last2=Catani |first2=Filippo|last3=Casagli|first3=Nicola|date=1 March 2005|title=Artificial Neural Networks applied to landslide susceptibility assessment|journal=Geomorphology|series=Geomorphological hazard and human impact in mountain environments|volume=66|issue=1|pages=327–343|doi=10.1016/j.geomorph.2004.09.025 |bibcode=2005Geomo..66..327E}}</ref> ANNs have been employed in [[w:en:Computer security|cybersecurity]], with the objective to discriminate between legitimate activities and malicious ones. For example, machine learning has been used for classifying Android malware,<ref>{{Cite journal|last1=Nix|first1=R.|last2=Zhang |first2=J.|date=May 2017 |title=Classification of Android apps and malware using deep neural networks |journal=2017 International Joint Conference on Neural Networks (IJCNN)|pages=1871–1878|s2cid=8838479 |doi=10.1109/IJCNN.2017.7966078|isbn=978-1-5090-6182-2}}</ref> for identifying domains belonging to threat actors and for detecting URLs posing a security risk.<ref>{{Cite web|title=Detecting Malicious URLs |website=The systems and networking group at UCSD |url=http://www.sysnet.ucsd.edu/projects/url/|access-date=15 February 2019|url-status=dead|archive-date=14 July 2019|archive-url=https://web.archive.org/web/20190714201955/http://www.sysnet.ucsd.edu/projects/url/}}</ref> Research is underway on ANN systems designed for penetration testing, for detecting botnets,<ref>{{Citation |last1=Homayoun|first1=Sajad|title=BoTShark: A Deep Learning Approach for Botnet Traffic Detection |date=2018|work=Cyber Threat Intelligence|pages=137–153|editor-last=Dehghantanha|editor-first=Ali |series=Advances in Information Security|publisher=Springer International Publishing|doi=10.1007/978-3-319-73951-9_7|isbn=978-3-319-73951-9|last2=Ahmadzadeh |first2=Marzieh|last3=Hashemi|first3=Sattar |last4=Dehghantanha|first4=Ali|last5=Khayami|first5=Raouf|editor2-last=Conti|editor2-first=Mauro|editor3-last=Dargahi|editor3-first=Tooska}}</ref> credit cards frauds<ref>{{Cite journal |last1=Ghosh|last2=Reilly |name-list-style=and|s2cid=13260377 |date=January 1994|title=Credit card fraud detection with a neural-network|journal=1994 Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences |volume=3|pages=621–630|doi=10.1109/HICSS.1994.323314|isbn=978-0-8186-5090-1}}</ref> and network intrusions. ANNs have been proposed as a tool to solve [[w:en:partial differential equation|partial differential equation]]s in physics<ref>{{Cite web|last=Ananthaswamy|first=Anil|date=19 April 2021|title=Latest Neural Nets Solve World's Hardest Equations Faster Than Ever Before|url=https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/|access-date=12 May 2021|website=Quanta Magazine|language=en}}</ref><ref>{{Cite web|title=AI has cracked a key mathematical puzzle for understanding our world|url=https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/|access-date=19 November 2020|website=MIT Technology Review|language=en}}</ref><ref>{{Cite web|title=Caltech Open-Sources AI for Solving Partial Differential Equations|url=https://www.infoq.com/news/2020/12/caltech-ai-pde/|access-date=20 January 2021|website=InfoQ|language=en|archive-date=25 January 2021|archive-url=https://web.archive.org/web/20210125233952/https://www.infoq.com/news/2020/12/caltech-ai-pde/|url-status=live}}</ref> and simulate the properties of many-body [[w:en:open quantum system|open quantum system]]s.<ref>{{cite journal |last1=Nagy |first1=Alexandra |title=Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems |journal=[[w:en:Physical Review Letters|Physical Review Letters]] |volume=122 |issue=25 |pages=250501 |date=28 June 2019 |doi=10.1103/PhysRevLett.122.250501 |pmid=31347886 |bibcode=2019PhRvL.122y0501N |arxiv=1902.09483 |s2cid=119074378 }}</ref><ref>{{Cite journal|last1=Yoshioka|first1=Nobuyuki|last2=Hamazaki|first2=Ryusuke|date=28 June 2019|title=Constructing neural stationary states for open quantum many-body systems|journal=Physical Review B|volume=99|issue=21 |pages=214306|doi=10.1103/PhysRevB.99.214306|bibcode=2019PhRvB..99u4306Y|arxiv=1902.07006|s2cid=119470636}}</ref><ref>{{Cite journal|last1=Hartmann|first1=Michael J.|last2=Carleo|first2=Giuseppe |date=28 June 2019|title=Neural-Network Approach to Dissipative Quantum Many-Body Dynamics|journal=Physical Review Letters|volume=122|issue=25|pages=250502|doi=10.1103/PhysRevLett.122.250502|pmid=31347862 |bibcode=2019PhRvL.122y0502H|arxiv=1902.05131|s2cid=119357494}}</ref><ref>{{Cite journal|last1=Vicentini |first1=Filippo|last2=Biella|first2=Alberto|last3=Regnault|first3=Nicolas|last4=Ciuti|first4=Cristiano|date=28 June 2019 |title=Variational Neural-Network Ansatz for Steady States in Open Quantum Systems |journal=Physical Review Letters|volume=122|issue=25|pages=250503|doi=10.1103/PhysRevLett.122.250503 |pmid=31347877 |bibcode=2019PhRvL.122y0503V |arxiv=1902.10104|s2cid=119504484}}</ref> In brain research ANNs have studied short-term behavior of [[w:en:biological neuron models|individual neurons]],<ref>{{cite journal |author=Forrest MD |title=Simulation of alcohol action upon a detailed Purkinje neuron model and a simpler surrogate model that runs >400 times faster |journal=BMC Neuroscience |volume=16 |issue=27 |pages=27 |date=April 2015 |doi=10.1186/s12868-015-0162-6 |pmid=25928094 |pmc=4417229}}</ref> the dynamics of neural circuitry arise from interactions between individual neurons and how behavior can arise from abstract neural modules that represent complete subsystems. Studies considered long-and short-term plasticity of neural systems and their relation to learning and memory from the individual neuron to the system level. ==Theoretical properties== ===Computational power=== The [[w:en:multilayer perceptron|multilayer perceptron]] is a [[w:en:UTM theorem|universal function]] approximator, as proven by the [[w:en:universal approximation theorem|universal approximation theorem]]. However, the proof is not constructive regarding the number of neurons required, the network topology, the weights and the learning parameters. A specific recurrent architecture with [[w:en:Rational number|rational]]-valued weights (as opposed to full precision [[w:en:real number|real number]]-valued weights) has the power of a [[w:en:Universal Turing Machine|universal Turing machine]],<ref>{{Cite journal| title = Turing computability with neural nets | url = http://www.math.rutgers.edu/~sontag/FTPDIR/aml-turing.pdf | year = 1991 | journal = Appl. Math. Lett. | pages = 77–80 | volume = 4 | issue = 6 | last1 = Siegelmann | first1 = H.T. | last2 = Sontag | first2 = E.D. | doi = 10.1016/0893-9659(91)90080-F }}</ref> using a finite number of neurons and standard linear connections. Further, the use of [[w:en:Irrational number|irrational]] values for weights results in a machine with [[w:en:Hypercomputation|super-Turing]] power.<ref>{{cite journal |last1=Balcázar |first1=José |title=Computational Power of Neural Networks: A Kolmogorov Complexity Characterization |journal=IEEE Transactions on Information Theory|date=Jul 1997 |volume=43 |issue=4 |pages=1175–1183 |doi=10.1109/18.605580 |citeseerx=10.1.1.411.7782 }}</ref> ===Capacity=== A model's "capacity" property corresponds to its ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity. Two notions of capacity are known by the community. The information capacity and the VC Dimension. The information capacity of a perceptron is intensively discussed in Sir David MacKay's book<ref name="auto">{{cite book| last=MacKay| first=David, J.C.| author-link=David J.C. MacKay| year=2003| publisher=[[w:en:Cambridge University Press|Cambridge University Press]]| isbn=978-0-521-64298-9| title=Information Theory, Inference, and Learning Algorithms| url=http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| access-date=11 June 2016| archive-date=19 October 2016| archive-url=https://web.archive.org/web/20161019163258/http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| url-status=live}}</ref> which summarizes work by Thomas Cover.<ref>{{cite journal|last=Cover|first=Thomas|author-link=Thomas M. Cover|year=1965|publisher=[[w:en:IEEE|IEEE]]|url=http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|title=Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition|journal=IEEE Transactions on Electronic Computers|issue=3|pages=326–334|volume=EC-14|doi=10.1109/PGEC.1965.264137|access-date=10 March 2020|archive-date=5 March 2016|archive-url=https://web.archive.org/web/20160305031348/http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|url-status=live}}</ref> The capacity of a network of standard neurons (not convolutional) can be derived by four rules<ref>{{cite journal| last=Gerald | first=Friedland| author-link=Gerald Friedland|year=2019|publisher=[[w:en:Association for Computing Machinery|ACM]]|title=Reproducibility and Experimental Design for Machine Learning on Audio and Multimedia Data|journal=MM '19: Proceedings of the 27th ACM International Conference on Multimedia| pages=2709–2710| doi=10.1145/3343031.3350545| isbn=978-1-4503-6889-6| s2cid=204837170}}</ref> that derive from understanding a neuron as an [[w:en:ADALINE|electrical element]]. The information capacity captures the functions modelable by the network given any data as input. The second notion, is the [[w:en:VC dimension|VC dimension]]. VC Dimension uses the principles of [[w:en:measure theory|measure theory]] and finds the maximum capacity under the best possible circumstances. This is, given input data in a specific form. As noted in,<ref name="auto"/> the VC Dimension for arbitrary inputs is half the information capacity of a Perceptron. The VC Dimension for arbitrary points is sometimes referred to as Memory Capacity.<ref>{{cite web| url=http://tfmeter.icsi.berkeley.edu/| title=Stop tinkering, start measuring! Predictable experimental design of Neural Network experiments| website=The Tensorflow Meter| access-date=10 March 2020| archive-date=18 April 2022| archive-url=https://web.archive.org/web/20220418025904/http://tfmeter.icsi.berkeley.edu/| url-status=dead}}</ref> ===Convergence=== Models may not consistently converge on a single solution, firstly because local minima may exist, depending on the cost function and the model. Secondly, the optimization method used might not guarantee to converge when it begins far from any local minimum. Thirdly, for sufficiently large data or parameters, some methods become impractical. Another issue worthy to mention is that training may cross some [[w:en:Saddle point|Saddle point]] which may lead the convergence to the wrong direction. The convergence behavior of certain types of ANN architectures are more understood than others. When the width of network approaches to infinity, the ANN is well described by its first order Taylor expansion throughout training, and so inherits the convergence behavior of [[w:en:Linear model|affine models]].<ref>{{Cite journal|last1=Lee|first1=Jaehoon|last2=Xiao|first2=Lechao|last3=Schoenholz|first3=Samuel S.|last4=Bahri |first4=Yasaman|last5=Novak |first5=Roman|last6=Sohl-Dickstein|first6=Jascha|last7=Pennington |first7=Jeffrey|title=Wide neural networks of any depth evolve as linear models under gradient descent |journal=Journal of Statistical Mechanics: Theory and Experiment|year=2020|volume=2020|issue=12|page=124002 |doi=10.1088/1742-5468/abc62b|arxiv=1902.06720|bibcode=2020JSMTE2020l4002L|s2cid=62841516}}</ref><ref>{{cite conference |conference=32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montreal, Canada |author1=Arthur Jacot |author2=Franck Gabriel |author3=Clement Hongler |date=2018 |url=https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |title=Neural Tangent Kernel: Convergence and Generalization in Neural Networks |access-date=4 June 2022 |archive-date=22 June 2022 |archive-url=https://web.archive.org/web/20220622033100/https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |url-status=live }}</ref> Another example is when parameters are small, it is observed that ANNs often fits target functions from low to high frequencies. This behavior is referred to as the spectral bias, or frequency principle, of neural networks.<ref>{{cite book |vauthors=Xu ZJ, Zhang Y, Xiao Y |date=2019 |veditors=Gedeon T, Wong K, Lee M |title=Neural Information Processing. ICONIP 2019. |series=Lecture Notes in Computer Science |volume=11953 |publisher=Springer, Cham |doi=10.1007/978-3-030-36708-4_22 |chapter=Training Behavior of Deep Neural Network in Frequency Domain |pages=264–274 |arxiv=1807.01251 |isbn=978-3-030-36707-7 |s2cid=49562099 }}</ref><ref>{{cite journal |author1=Nasim Rahaman |author2=Aristide Baratin |author3=Devansh Arpit |author4=Felix Draxler |author5=Min Lin |author6=Fred Hamprecht |author7=Yoshua Bengio |author8=Aaron Courville |journal=Proceedings of the 36th International Conference on Machine Learning |volume=97 |pages=5301–5310 |date=2019 |title=On the Spectral Bias of Neural Networks |arxiv=1806.08734 |url=http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |access-date=4 June 2022 |archive-date=22 October 2022 |archive-url=https://web.archive.org/web/20221022155951/http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |url-status=live }}</ref><ref>{{cite journal |arxiv=1901.06523 |author1=Zhi-Qin John Xu |author2=Yaoyu Zhang |author3=Tao Luo |author4=Yanyang Xiao |author5=Zheng Ma |title=Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks|journal=Communications in Computational Physics |year=2020 |volume=28 |issue=5 |pages=1746–1767 |doi=10.4208/cicp.OA-2020-0085 |bibcode=2020CCoPh..28.1746X |s2cid=58981616 }}</ref><ref>{{cite arXiv |eprint=1906.09235 |author1=Tao Luo |author2=Zheng Ma |author3=Zhi-Qin John Xu |author4=Yaoyu Zhang |date=2019 |title=Theory of the Frequency Principle for General Deep Neural Networks|class=cs.LG }}</ref> This phenomenon is the opposite to the behavior of some well studied iterative numerical schemes such as [[w:en:Jacobi method|Jacobi method]]. Deeper neural networks have been observed to be more biased towards low frequency functions.<ref>{{Cite journal|last1=Xu|first1=Zhiqin John|last2=Zhou|first2=Hanxu|date=2021-05-18|title=Deep Frequency Principle Towards Understanding Why Deeper Learning Is Faster|url=https://ojs.aaai.org/index.php/AAAI/article/view/17261|journal=Proceedings of the AAAI Conference on Artificial Intelligence|volume=35|issue=12|pages=10541–10550|doi=10.1609/aaai.v35i12.17261|arxiv=2007.14313|s2cid=220831156|issn=2374-3468|access-date=5 October 2021|archive-date=5 October 2021|archive-url=https://web.archive.org/web/20211005142300/https://ojs.aaai.org/index.php/AAAI/article/view/17261|url-status=live}}</ref> ===Generalization and statistics=== {{No footnotes|date=August 2019|section}} Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. Two approaches address over-training. The first is to use [[w:en:cross-validation (statistics)|cross-validation]] and similar techniques to check for the presence of over-training and to select [[w:en:hyperparameters|hyperparameters]] to minimize the generalization error. The second is to use some form of ''[[w:en:regularization (mathematics)|regularization]]''. This concept emerges in a probabilistic (Bayesian) framework, where regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting. [[File:Synapse deployment.jpg|thumb|right|upright=1.15|Confidence analysis of a neural network]] Supervised neural networks that use a [[w:en:mean squared error|mean squared error]] (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the [[w:en:confidence interval|confidence interval]] of network output, assuming a [[w:en:normal distribution|normal distribution]]. A confidence analysis made this way is statistically valid as long as the output [[w:en:probability distribution|probability distribution]] stays the same and the network is not modified. By assigning a [[w:en:softmax activation function|softmax activation function]], a generalization of the [[w:en:logistic function|logistic function]], on the output layer of the neural network (or a softmax component in a component-based network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is useful in classification as it gives a certainty measure on classifications. The softmax activation function is: :<math>y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}</math> <section end="theory" /> ==Criticism== ===Training === A common criticism of neural networks, particularly in robotics, is that they require too much training for real-world operation.<ref>{{Cite journal |last=Parisi |first=German I. |last2=Kemker |first2=Ronald |last3=Part |first3=Jose L. |last4=Kanan |first4=Christopher |last5=Wermter |first5=Stefan |date=2019-05-01 |title=Continual lifelong learning with neural networks: A review |url=https://www.sciencedirect.com/science/article/pii/S0893608019300231 |journal=Neural Networks |language=en |volume=113 |pages=54–71 |doi=10.1016/j.neunet.2019.01.012 |issn=0893-6080}}</ref> Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example, grouping examples in so-called mini-batches and/or introducing a recursive least squares algorithm for [[w:en:cerebellar model articulation controller|CMAC]].<ref name="Qin1"/> ===Theory=== A central claim{{cn|date=January 2023}} of ANNs is that they embody new and powerful general principles for processing information. These principles are ill-defined. It is often claimed{{by whom?|date=January 2023}} that they are [[w:en:Emergent properties|emergent]] from the network itself. This allows simple statistical association (the basic function of artificial neural networks) to be described as learning or recognition. In 1997, [[w:en:Alexander Dewdney|Alexander Dewdney]] commented that, as a result, artificial neural networks have a "something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are. No human hand (or mind) intervenes; solutions are found as if by magic; and no one, it seems, has learned anything".<ref>{{cite book|url={{google books |plainurl=y |id=KcHaAAAAMAAJ|page=82}}|title=Yes, we have no neutrons: an eye-opening tour through the twists and turns of bad science|last=Dewdney|first=A. K.|date=1 April 1997|publisher=Wiley|isbn=978-0-471-10806-1|pages=82}}</ref> One response to Dewdney is that neural networks handle many complex and diverse tasks, ranging from autonomously flying aircraft<ref>[http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html NASA – Dryden Flight Research Center – News Room: News Releases: NASA NEURAL NETWORK PROJECT PASSES MILESTONE] {{Webarchive|url=https://web.archive.org/web/20100402065100/http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html |date=2 April 2010 }}. Nasa.gov. Retrieved on 20 November 2013.</ref> to detecting credit card fraud to mastering the game of [[w:en:Go (game)|Go]]. Technology writer Roger Bridgman commented: {{blockquote|Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be "an opaque, unreadable table...valueless as a scientific resource". In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers. An unreadable table that a useful machine could read would still be well worth having.<ref>{{Cite web |url=http://members.fortunecity.com/templarseries/popper.html |title=Roger Bridgman's defence of neural networks |access-date=12 July 2010 |archive-url=https://web.archive.org/web/20120319163352/http://members.fortunecity.com/templarseries/popper.html |archive-date=19 March 2012 |url-status=dead}}</ref> }} Biological brains use both shallow and deep circuits as reported by brain anatomy,<ref name="VanEssen1991">D. J. Felleman and D. C. Van Essen, "[https://archive.today/20150120022056/http://cercor.oxfordjournals.org/content/1/1/1.1.full.pdf+html Distributed hierarchical processing in the primate cerebral cortex]," ''Cerebral Cortex'', 1, pp. 1–47, 1991.</ref> displaying a wide variety of invariance. Weng<ref name="Weng2012">J. Weng, "[https://www.amazon.com/Natural-Artificial-Intelligence-Introduction-Computational/dp/0985875720 Natural and Artificial Intelligence: Introduction to Computational Brain-Mind]," BMI Press, {{ISBN|978-0-9858757-2-5}}, 2012.</ref> argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies. ===Hardware=== Large and effective neural networks require considerable computing resources.<ref name=":0">{{cite journal|last1=Edwards|first1=Chris|s2cid=11026540|title=Growing pains for deep learning|journal=Communications of the ACM|date=25 June 2015|volume=58|issue=7|pages=14–16|doi=10.1145/2771283}}</ref> While the brain has hardware tailored to the task of processing signals through a [[w:en:Graph (discrete mathematics)|graph]] of neurons, simulating even a simplified neuron on [[w:en:von Neumann architecture|von Neumann architecture]] may consume vast amounts of [[w:en:Random-access memory|memory]] and storage. Furthermore, the designer often needs to transmit signals through many of these connections and their associated neurons{{snd}} which require enormous [[w:en:Central processing unit|CPU]] power and time. [[w:en:Jürgen Schmidhuber|Schmidhuber]] noted that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by [[w:en:General-purpose computing on graphics processing units|GPGPUs]] (on [[w:en:Graphics processing unit|GPUs]]), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.<ref name="SCHIDHUB2" /> The use of accelerators such as [[w:en:Field-programmable gate array|FPGA]]s and GPUs can reduce training times from months to days.{{r|:0}} [[w:en:Neuromorphic engineering|Neuromorphic engineering]] or a [[w:en:physical neural network|physical neural network]] addresses the hardware difficulty directly, by constructing non-von-Neumann chips to directly implement neural networks in circuitry. Another type of chip optimized for neural network processing is called a [[w:en:Tensor Processing Unit|Tensor Processing Unit]], or TPU.<ref>{{cite news |url=https://www.wired.com/2016/05/google-tpu-custom-chips/ |author=Cade Metz |newspaper=Wired |date=18 May 2016 |title=Google Built Its Very Own Chips to Power Its AI Bots |access-date=5 March 2017 |archive-date=13 January 2018 |archive-url=https://web.archive.org/web/20180113150305/https://www.wired.com/2016/05/google-tpu-custom-chips/ |url-status=live }}</ref> ===Practical counterexamples === Analyzing what has been learned by an ANN is much easier than analyzing what has been learned by a biological neural network. Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful. For example, local vs. non-local learning and shallow vs. deep architecture.<ref>{{Cite web|title=Scaling Learning Algorithms towards AI|url=http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|access-date=6 July 2022|archive-date=12 August 2022|archive-url=https://web.archive.org/web/20220812081157/http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|url-status=live}}</ref> ===Hybrid approaches=== Advocates of [[w:en:Hybrid neural network|hybrid]] models (combining neural networks and symbolic approaches) say that such a mixture can better capture the mechanisms of the human mind.<ref>{{Cite journal| last1=Tahmasebi| last2=Hezarkhani| title=A hybrid neural networks-fuzzy logic-genetic algorithm for grade estimation| year=2012| journal=Computers & Geosciences| pages=18–27 |volume=42| doi=10.1016/j.cageo.2012.02.004| pmid=25540468| pmc=4268588| bibcode=2012CG.....42...18T}}</ref> ==Gallery== <gallery widths="220"> File:Single layer ann.svg|A single-layer feedforward artificial neural network. Arrows originating from <math>\scriptstyle x_2</math> are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, <math>\scriptstyle y_q</math> would be calculated as <math>\scriptstyle y_q = K*(\sum(x_i*w_{iq})-b_q) </math>. File:Two layer ann.svg|A two-layer feedforward artificial neural network File:Artificial neural network.svg|An artificial neural network File:Ann dependency (graph).svg|An ANN dependency graph File:Single-layer feedforward artificial neural network.png|A single-layer feedforward artificial neural network with 4 inputs, 6 hidden and 2 outputs. Given position state and direction outputs wheel based control values. File:Two-layer feedforward artificial neural network.png|A two-layer feedforward artificial neural network with 8 inputs, 2x8 hidden and 2 outputs. Given position state, direction and other environment values outputs thruster based control values. File:Cmac.jpg|Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step. </gallery> == See also == {{cols|colwidth=18em}} * [[w:en:ADALINE|ADALINE]] * [[w:en:Autoencoder|Autoencoder]] * [[w:en:Bio-inspired computing|Bio-inspired computing]] * [[w:en:Blue Brain Project|Blue Brain Project]] * [[w:en:Catastrophic interference|Catastrophic interference]] * [[w:en:Cognitive architecture|Cognitive architecture]] * [[w:en:Connectionist expert system|Connectionist expert system]] * [[w:en:Connectomics|Connectomics]] * [[w:en:Large width limits of neural networks|Large width limits of neural networks]] * [[w:en:List of machine learning concepts|List of machine learning concepts]] * [[w:en:Neural gas|Neural gas]] * [[w:en:Neural network software|Neural network software]] * [[w:en:Optical neural network|Optical neural network]] * [[w:en:Parallel distributed processing|Parallel distributed processing]] * [[w:en:Philosophy of artificial intelligence|Philosophy of artificial intelligence]] * [[w:en:Recurrent neural networks|Recurrent neural networks]] * [[w:en:Spiking neural network|Spiking neural network]] * [[w:en:Tensor product network|Tensor product network]] {{colend}} ==Notes== Hello{{notelist}} ==References== {{Reflist|30em}} ==Bibliography== {{colbegin|colwidth=30em}} * {{Cite journal| author=Bhadeshia H. K. D. H. | year=1999 |title=Neural Networks in Materials Science | journal=ISIJ International | volume=39 |pages=966–979 | doi=10.2355/isijinternational.39.966 | url=http://www.msm.cam.ac.uk/phase-trans/abstracts/neural.review.pdf| issue=10}} * {{Cite book|title=Neural networks for pattern recognition|last=Bishop|first=Christopher M.|date=1995|publisher=Clarendon Press|isbn=978-0-19-853849-3|oclc=33101074 }} * {{Cite book|title=Neuro-Fuzzy-Systeme : von den Grundlagen künstlicher Neuronaler Netze zur Kopplung mit Fuzzy-Systemen|first=Christian|last=Borgelt|year=2003|publisher=Vieweg|isbn=978-3-528-25265-6|oclc=76538146}} * {{cite book|title=Mathematics of Control, Signals, and Systems|author1-link=George Cybenko|last=Cybenko|first=G.V.|publisher=Springer International|year=2006|editor-last=van Schuppen|editor-first=Jan H.|chapter=Approximation by Superpositions of a Sigmoidal function|chapter-url={{google books |plainurl=y |id=4RtVAAAAMAAJ|page=303}}|pages=303–314|title-link=Mathematics of Control, Signals, and Systems}} [https://web.archive.org/web/20110719183058/http://actcomm.dartmouth.edu/gvc/papers/approx_by_superposition.pdf PDF] * {{Cite book|title=Yes, we have no neutrons : an eye-opening tour through the twists and turns of bad science|last=Dewdney |first=A. K.|isbn=978-0-471-10806-1|oclc=35558945|year=1997|publisher=Wiley|location=New York}} * {{Cite book|title=Pattern classification|first1=Richard O.|last1=Duda|last2=Hart |first2=Peter Elliot|last3=Stork |first3=David G.|year=2001|publisher=Wiley|isbn=978-0-471-05669-0|oclc=41347061|edition=2}} * {{Cite journal | last1=Egmont-Petersen|first1=M. |last2=de Ridder |first2=D. |last3=Handels |first3=H. | year=2002 | title=Image processing with neural networks – a review | journal=Pattern Recognition | volume=35 | pages=2279–2301 | doi = 10.1016/S0031-3203(01)00178-9 | issue=10|citeseerx=10.1.1.21.5444 }} * {{cite web |last1=Fahlman |first1=S. |last2=Lebiere |first2=C |year=1991 |title=The Cascade-Correlation Learning Architecture |url=http://www.cs.iastate.edu/~honavar/fahlman.pdf |access-date=28 August 2006 |archive-date=3 May 2013 |archive-url=https://web.archive.org/web/20130503184045/http://www.cs.iastate.edu/~honavar/fahlman.pdf |url-status=dead }} **created for [[w:en:National Science Foundation|National Science Foundation]], Contract Number EET-8716324, and [[w:en:Defense Advanced Research Projects Agency|Defense Advanced Research Projects Agency]] (DOD), ARPA Order No. 4976 under Contract F33615-87-C-1499. * {{Cite book|title=An introduction to neural networks|last=Gurney |first=Kevin |year=1997|publisher=UCL Press|isbn=978-1-85728-673-1|oclc=37875698}} * {{Cite book|title=Neural networks : a comprehensive foundation|last=Haykin|first= Simon S.|year=1999|publisher=Prentice Hall|isbn=978-0-13-273350-2|oclc=38908586}} * {{Cite book|title=Introduction to the theory of neural computation|last1=Hertz |first1=J.|last3=Krogh|first3=Anders S.|first2=Richard G.|last2=Palmer|year=1991|publisher=Addison-Wesley |isbn=978-0-201-51560-2|oclc=21522159}} * {{Cite book|title=Information theory, inference, and learning algorithms|publisher=Cambridge University Press|isbn=978-0-521-64298-9|oclc=52377690|date=25 September 2003|bibcode=2003itil.book.....M}} * {{Cite book|title=Computational intelligence : a methodological introduction|first1=Rudolf|last1=Kruse|first2=Christian|last2=Borgelt|first3=F.|last3=Klawonn|first4=Christian|last4=Moewes|first5=Matthias|last5=Steinbrecher|first6=Pascal|last6=Held|year=2013|publisher=Springer|isbn=978-1-4471-5012-1|oclc=837524179}} * {{Cite book|title=Introduction to neural networks : design, theory and applications|last=Lawrence|first=Jeanette|year=1994|publisher=California Scientific Software|isbn=978-1-883157-00-5|oclc=32179420}} * {{cite book| last=MacKay | first=David, J.C.| author-link=David J.C. MacKay|year=2003|publisher=[[w:en:Cambridge University Press|Cambridge University Press]]| isbn=978-0-521-64298-9|url=http://www.inference.phy.cam.ac.uk/itprnn/book.pdf|title=Information Theory, Inference, and Learning Algorithms}} * {{Cite book|title=Signal and image processing with neural networks : a C++ sourcebook|first=Timothy|last=Masters|year=1994|publisher=J. Wiley|isbn=978-0-471-04963-0|oclc=29877717}} * {{Cite book|title=Cognitive science : integrative synchronization mechanisms in cognitive neuroarchitectures of the modern connectionism|last=Maurer|first=Harald|year=2021|publisher=CRC Press|isbn=978-1-351-04352-6|doi=10.1201/9781351043526|s2cid=242963768 }} * {{cite book|url={{google books |plainurl=y |id=m12UR8QmLqoC}}|title=Pattern Recognition and Neural Networks|last=Ripley|first=Brian D.|author-link=Brian D. Ripley|publisher=Cambridge University Press|year=2007|isbn=978-0-521-71770-0}} * {{cite journal|last1=Siegelmann |first1=H.T. |first2=Eduardo D.|last2=Sontag|s2cid=2456483 |year=1994|title=Analog computation via neural networks |journal=Theoretical Computer Science |volume= 131 |issue= 2 |pages=331–360 |doi=10.1016/0304-3975(94)90178-3}} * {{Cite book|title=Neural networks for statistical modeling|last1=Smith |first1=Murray|date=1993|publisher=Van Nostrand Reinhold|isbn=978-0-442-01310-3|oclc=27145760}} * {{Cite book|title=Advanced methods in neural computing|last=Wasserman |first=Philip D.|year=1993|publisher=Van Nostrand Reinhold|isbn=978-0-442-00461-3|oclc=27429729}} * {{cite book|last1=Wilson|first1=Halsey|title=Artificial intelligence|date=2018|publisher=Grey House Publishing|isbn=978-1-68217-867-6}} {{colend}} {{Complex systems topics}} {{Control theory}} {{Differentiable computing}} {{Neuroscience}} {{Self-driving cars and enabling technologies}} {{Authority control}} {{DEFAULTSORT:Artificial Neural Network}} [[Category:Statistics]] [[Category:Artificial neural networks| ]] [[Category:Algorithms]] [[Category:Neuroscience]] [[Category:Psychology]] == Page Information == This page was based on the following [https://en.wikipedia.org/wiki/Artficial%20neural%20network wikipedia-source page]: * [https://en.wikipedia.org/wiki/Artificial%20neural%20network Artificial neural network] https://en.wikipedia.org/wiki/Artificial%20neural%20network * Datum: 3/11/2023 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity 83710qthuzj3cvqr9vamfvqq1mowany 2693767 2693764 2024-12-29T16:40:43Z Atcovi 276019 Reverted edits by [[Special:Contributions/2409:40E3:4075:7D65:8000:0:0:0|2409:40E3:4075:7D65:8000:0:0:0]] ([[User_talk:2409:40E3:4075:7D65:8000:0:0:0|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2480477 wikitext text/x-wiki <!-- {{short description|Computational model used in machine learning, based on connected, hierarchical functions}} {{Machine learning|Artificial neural network}} {{Artificial intelligence}} {{Complex systems}} {{Network science}} --> [[File:Colored neural network.svg|thumb|upright=1.15|An artificial neural network is an interconnected group of nodes, inspired by a simplification of [[w:en:neuron|neuron]]s in a [[w:en:brain|brain]]. Here, each circular node represents an [[w:en:artificial neuron|artificial neuron]] and an arrow represents a connection from the output of one artificial neuron to the input of another.]] '''Artificial neural networks''' ('''ANNs'''), usually simply called '''neural networks''' ('''NNs''') or '''neural nets''',<ref>{{cite web|last=Hardesty|first=Larry|title=Explained: Neural networks|date=14 April 2017|url=https://news.mit.edu/2017/explained-neural-networks-deep-learning-0414|publisher=MIT News Office|access-date=2 June 2022}}</ref> are computing systems inspired by the [[w:en:biological neural network|biological neural network]]s that constitute animal [[w:en:brain|brain]]s.<ref>{{cite book |last1=Yang |first1=Z.R. |last2=Yang |first2=Z. |title=Comprehensive Biomedical Physics |date=2014 |publisher=Elsevier |location=Karolinska Institute, Stockholm, Sweden |isbn=978-0-444-53633-4 |page=1 |url=https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |access-date=28 July 2022 |archive-date=28 July 2022 |archive-url=https://web.archive.org/web/20220728183237/https://www.sciencedirect.com/topics/neuroscience/artificial-neural-network |url-status=live }}</ref> An ANN is based on a collection of connected units or nodes called [[w:en:artificial neuron|artificial neuron]]s, which loosely model the [[w:en:neuron|neuron]]s in a biological brain. Each connection, like the [[w:en:synapse|synapse]]s in a biological brain, can transmit a signal to other neurons. An artificial neuron receives signals then processes them and can signal neurons connected to it. The "signal" at a connection is a [[w:en:real number|real number]], and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called ''edges''. Neurons and edges typically have a ''[[w:en:Weighting|weight]]'' that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.{{toclimit|3}} ==Learning Units == * '''[[/Training/]]:''' What is the basic idea of [[machine learning]] in the context of Artificial Neural Networks. * '''[[/History/]]:''' Learn how the knowledge and application of neural networks evolved in the past. * '''[[/Models/]]:''' What basic models of representing an Artificial Neural Network (ANN) with Mathematics and/or with a corresponding computer model. ** [[/Neuron/|Model of Neuron]] ** [[/Network/|Model of a Network of Neurons]] ==Models== [[File:Neuron3.png|thumb|right|upright=1.35|Neuron and myelinated axon, with signal flow from inputs at dendrites to outputs at axon terminals]] === Hyperparameter === {{Main|w:en:Hyperparameter (machine learning)}} A [[w:en:hyperparameter|hyperparameter]] is a constant [[w:en:parameter|parameter]] whose value is set before the learning process begins. The values of [[w:en:parameter|parameter]]s are derived via learning. Examples of hyperparameters include [[w:en:learning rate|learning rate]], the number of hidden layers and batch size.<ref>{{Cite web|url=https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd|title=A Walkthrough of Convolutional Neural Network – Hyperparameter Tuning|last=Lau|first=Suki|date=10 July 2017|website=Medium|access-date=23 August 2019|archive-date=4 February 2023|archive-url=https://web.archive.org/web/20230204153904/https://towardsdatascience.com/a-walkthrough-of-convolutional-neural-network-7f474f91d7bd?gi=95062e80a5b5|url-status=live}}</ref> The values of some hyperparameters can be dependent on those of other hyperparameters. For example, the size of some layers can depend on the overall number of layers. ===Learning=== {{No footnotes|date=August 2019|section}}{{See also|Mathematical optimization|Estimation theory|Machine learning}} Learning is the adaptation of the network to better handle a task by considering sample observations. Learning involves adjusting the weights (and optional thresholds) of the network to improve the accuracy of the result. This is done by minimizing the observed errors. Learning is complete when examining additional observations does not usefully reduce the error rate. Even after learning, the error rate typically does not reach 0. If after learning, the error rate is too high, the network typically must be redesigned. Practically this is done by defining a [[w:en:Loss function|cost function]] that is evaluated periodically during learning. As long as its output continues to decline, learning continues. The cost is frequently defined as a [[w:en:statistic|statistic]] whose value can only be approximated. The outputs are actually numbers, so when the error is low, the difference between the output (almost certainly a cat) and the correct answer (cat) is small. Learning attempts to reduce the total of the differences across the observations. Most learning models can be viewed as a straightforward application of [[w:en:Mathematical optimization|optimization]] theory and [[w:en:statistical estimation|statistical estimation]].<ref name="Zell1994ch5.2"/><ref>{{Cite book|last1=Kelleher|first1=John D. |last2=Mac Namee|first2=Brian|last3=D'Arcy|first3=Aoife |title=Fundamentals of machine learning for predictive data analytics: algorithms, worked examples, and case studies|date=2020|isbn=978-0-262-36110-1 |edition=2nd|location=Cambridge, MA|chapter=7-8|oclc=1162184998}}</ref> ==== Learning rate ==== The learning rate defines the size of the corrective steps that the model takes to adjust for errors in each observation.<ref>{{cite arXiv|last=Wei|first=Jiakai|date=2019-04-26|title=Forget the Learning Rate, Decay Loss|class=cs.LG|eprint=1905.00094}}</ref> A high learning rate shortens the training time, but with lower ultimate accuracy, while a lower learning rate takes longer, but with the potential for greater accuracy. Optimizations such as [[w:en:Quickprop|Quickprop]] are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid [[w:en:oscillation|oscillation]] inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an [[w:en:adaptive learning rate|adaptive learning rate]] that increases or decreases as appropriate.<ref>{{Cite book|title=The Improved Training Algorithm of Back Propagation Neural Network with Self-adaptive Learning Rate|last1=Li|first1=Y.|last2=Fu|first2=Y.|last3=Li|first3=H.|last4=Zhang|first4=S. W.|s2cid=10557754|date=1 June 2009|journal=2009 International Conference on Computational Intelligence and Natural Computing|isbn=978-0-7695-3645-3|volume=1|pages=73–76|doi=10.1109/CINC.2009.111}}</ref> The concept of momentum allows the balance between the gradient and the previous change to be weighted such that the weight adjustment depends to some degree on the previous change. A momentum close to 0 emphasizes the gradient, while a value close to 1 emphasizes the last change. ====Cost function==== While it is possible to define a cost function [[w:en:ad hoc|ad hoc]], frequently the choice is determined by the function's desirable properties (such as [[w:en:Convex function|convexity]]) or because it arises from the model (e.g. in a probabilistic model the model's [[w:en:posterior probability|posterior probability]] can be used as an inverse cost). ====Backpropagation==== {{Main|w:en:Backpropagation}} Backpropagation is a method used to adjust the connection weights to compensate for each error found during learning. The error amount is effectively divided among the connections. Technically, backprop calculates the [[w:en:gradient|gradient]] (the derivative) of the [[w:en:loss function|cost function]] associated with a given state with respect to the weights. The weight updates can be done via [[w:en:stochastic gradient descent|stochastic gradient descent]] or other methods, such as [[w:en:Extreme Learning Machines|Extreme Learning Machines]],<ref>{{cite journal|last1=Huang|first1=Guang-Bin|last2=Zhu |first2=Qin-Yu|last3=Siew|first3=Chee-Kheong|year=2006|title=Extreme learning machine: theory and applications|journal=Neurocomputing|volume=70|issue=1 |pages=489–501|doi=10.1016/j.neucom.2005.12.126 |citeseerx=10.1.1.217.3692|s2cid=116858 }}</ref> "No-prop" networks,<ref>{{cite journal|year=2013|title=The no-prop algorithm: A new learning algorithm for multilayer neural networks |journal=Neural Networks|volume=37 |pages=182–188|doi=10.1016/j.neunet.2012.09.020|pmid=23140797|last1=Widrow|first1=Bernard|display-authors=etal}}</ref> training without backtracking,<ref>{{cite arXiv|eprint=1507.07680|first1=Yann |last1=Ollivier|first2=Guillaume|last2=Charpiat|title=Training recurrent networks without backtracking |year=2015|class=cs.NE}}</ref> "weightless" networks,<ref name="RBMTRAIN">{{Cite journal |last=Hinton |first=G. E. |date=2010 |title=A Practical Guide to Training Restricted Boltzmann Machines |url=https://www.researchgate.net/publication/221166159 |journal=Tech. Rep. UTML TR 2010-003 |access-date=27 June 2017 |archive-date=9 May 2021 |archive-url=https://web.archive.org/web/20210509123211/https://www.researchgate.net/publication/221166159_A_brief_introduction_to_Weightless_Neural_Systems |url-status=live }}</ref><ref>ESANN. 2009.{{full citation needed|date=June 2022}}</ref> and [[w:en:Holographic associative memory|non-connectionist neural networks]].{{citation needed|date=June 2022}} ===Learning paradigms=== {{No footnotes|date=August 2019|section}} Machine learning is commonly separated into three main learning paradigms, [[w:en:supervised learning|supervised learning]], [[w:en:unsupervised learning|unsupervised learning]] and [[w:en:reinforcement learning|reinforcement learning]].<ref>{{cite book|url=https://www.wolfram.com/language/introduction-machine-learning/|title=Introduction to Machine Learning|first1=Etienne|publisher=Wolfram Media Inc|year=2021|isbn=978-1-579550-48-6|page=9|last1=Bernard}}</ref> Each corresponds to a particular learning task. ==== Supervised learning ==== [[w:en:Supervised learning|Supervised learning]] uses a set of paired inputs and desired outputs. The learning task is to produce the desired output for each input. In this case, the cost function is related to eliminating incorrect deductions.<ref>{{Cite journal|last1=Ojha|first1=Varun Kumar|last2=Abraham|first2=Ajith|last3=Snášel|first3=Václav|date=1 April 2017|title=Metaheuristic design of feedforward neural networks: A review of two decades of research|journal=Engineering Applications of Artificial Intelligence|volume=60|pages=97–116|doi=10.1016/j.engappai.2017.01.013|arxiv=1705.05584|bibcode=2017arXiv170505584O|s2cid=27910748}}</ref> A commonly used cost is the [[w:en:mean-squared error|mean-squared error]], which tries to minimize the average squared error between the network's output and the desired output. Tasks suited for supervised learning are [[w:en:pattern recognition|pattern recognition]] (also known as classification) and [[w:en:Regression analysis|regression]] (also known as function approximation). Supervised learning is also applicable to sequential data (e.g., for hand writing, speech and [[w:en:gesture recognition|gesture recognition]]). This can be thought of as learning with a "teacher", in the form of a function that provides continuous feedback on the quality of solutions obtained thus far. ====Unsupervised learning==== In [[w:en:unsupervised learning|unsupervised learning]], input data is given along with the cost function, some function of the data <math>\textstyle x</math> and the network's output. The cost function is dependent on the task (the model domain) and any ''[[w:en:A priori and a posteriori|a priori]]'' assumptions (the implicit properties of the model, its parameters and the observed variables). As a trivial example, consider the model <math>\textstyle f(x) = a</math> where <math>\textstyle a</math> is a constant and the cost <math>\textstyle C=E[(x - f(x))^2]</math>. Minimizing this cost produces a value of <math>\textstyle a</math> that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in [[w:en:Data compression|compression]] it could be related to the [[w:en:mutual information|mutual information]] between <math>\textstyle x</math> and <math>\textstyle f(x)</math>, whereas in statistical modeling, it could be related to the [[w:en:posterior probability|posterior probability]] of the model given the data (note that in both of those examples, those quantities would be maximized rather than minimized). Tasks that fall within the paradigm of unsupervised learning are in general [[w:en:Approximation|estimation]] problems; the applications include [[w:en:Data clustering|clustering]], the estimation of [[w:en:statistical distributions|statistical distributions]], [[w:en:Data compression|compression]] and [[w:en:Bayesian spam filtering|filtering]]. ====Reinforcement learning==== {{Main|w:en:Reinforcement learning}} {{See also|w:en:Stochastic control}} In applications such as playing video games, an actor takes a string of actions, receiving a generally unpredictable response from the environment after each one. The goal is to win the game, i.e., generate the most positive (lowest cost) responses. In [[w:en:reinforcement learning|reinforcement learning]], the aim is to weight the network (devise a policy) to perform actions that minimize long-term (expected cumulative) cost. At each point in time the agent performs an action and the environment generates an observation and an [[w:en:instant|instant]]aneous cost, according to some (usually unknown) rules. The rules and the long-term cost usually only can be estimated. At any juncture, the agent decides whether to explore new actions to uncover their costs or to exploit prior learning to proceed more quickly. Formally the environment is modeled as a [[w:en:Markov decision process|Markov decision process]] (MDP) with states <math>\textstyle {s_1,...,s_n}\in S </math> and actions <math>\textstyle {a_1,...,a_m} \in A</math>. Because the state transitions are not known, probability distributions are used instead: the instantaneous cost distribution <math>\textstyle P(c_t|s_t)</math>, the observation distribution <math>\textstyle P(x_t|s_t)</math> and the transition distribution <math>\textstyle P(s_{t+1}|s_t, a_t)</math>, while a policy is defined as the conditional distribution over actions given the observations. Taken together, the two define a [[w:en:Markov chain|Markov chain]] (MC). The aim is to discover the lowest-cost MC. ANNs serve as the learning component in such applications.<ref>{{cite conference | author = Dominic, S. | author2 = Das, R. | author3 = Whitley, D. | author4 = Anderson, C. | date = July 1991 | title = Genetic reinforcement learning for neural networks | pages = 71–76 | conference = IJCNN-91-Seattle International Joint Conference on Neural Networks | book-title = IJCNN-91-Seattle International Joint Conference on Neural Networks | publisher = IEEE | location = Seattle, Washington, USA | doi = 10.1109/IJCNN.1991.155315 | isbn = 0-7803-0164-1 | url-access = registration | url = https://archive.org/details/ijcnn91seattlein01ieee }}</ref><ref>{{cite journal |last=Hoskins |first=J.C. |author2=Himmelblau, D.M. |title=Process control via artificial neural networks and reinforcement learning |journal=Computers & Chemical Engineering |year=1992 |volume=16 |pages=241–251 |doi=10.1016/0098-1354(92)80045-B |issue=4}}</ref> [[w:en:Dynamic programming|Dynamic programming]] coupled with ANNs (giving [[w:en:Neural oscillation|neurodynamic]] programming)<ref>{{cite book|url=https://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|title=Neuro-dynamic programming|first1=D.P.|first2=J.N.|publisher=Athena Scientific|year=1996|isbn=978-1-886529-10-6|page=512|last1=Bertsekas|last2=Tsitsiklis|access-date=17 June 2017|archive-date=29 June 2017|archive-url=https://web.archive.org/web/20170629172039/http://papers.nips.cc/paper/4741-deep-neural-networks-segment-neuronal-membranes-in-electron-microscopy-images|url-status=live}}</ref> has been applied to problems such as those involved in [[w:en:vehicle routing|vehicle routing]],<ref>{{cite journal |last=Secomandi |first=Nicola |title=Comparing neuro-dynamic programming algorithms for the vehicle routing problem with stochastic demands |journal=Computers & Operations Research |year=2000 |volume=27 |pages=1201–1225 |doi=10.1016/S0305-0548(99)00146-X |issue=11–12|citeseerx=10.1.1.392.4034 }}</ref> video games, [[w:en:natural resource management|natural resource management]]<ref>{{cite conference | author = de Rigo, D. | author2 = Rizzoli, A. E. | author3 = Soncini-Sessa, R. | author4 = Weber, E. | author5 = Zenesi, P. | year = 2001 | title = Neuro-dynamic programming for the efficient management of reservoir networks | conference = MODSIM 2001, International Congress on Modelling and Simulation | url = http://www.mssanz.org.au/MODSIM01/MODSIM01.htm | book-title = Proceedings of MODSIM 2001, International Congress on Modelling and Simulation | publisher = Modelling and Simulation Society of Australia and New Zealand | location = Canberra, Australia | doi = 10.5281/zenodo.7481 | isbn = 0-86740-525-2 | access-date = 29 July 2013 | archive-date = 7 August 2013 | archive-url = https://web.archive.org/web/20130807223658/http://mssanz.org.au/MODSIM01/MODSIM01.htm | url-status = live }}</ref><ref>{{cite conference| author = Damas, M. |author2=Salmeron, M. |author3=Diaz, A. |author4=Ortega, J. |author5=Prieto, A. |author6=Olivares, G.| year = 2000 | title = Genetic algorithms and neuro-dynamic programming: application to water supply networks |volume=1 |pages=7–14 | conference = 2000 Congress on Evolutionary Computation | book-title = Proceedings of 2000 Congress on Evolutionary Computation | publisher = IEEE | location = La Jolla, California, USA | doi = 10.1109/CEC.2000.870269 | isbn = 0-7803-6375-2 }}</ref> and [[w:en:medicine|medicine]]<ref>{{Cite book |last=Deng |first=Geng |author2=Ferris, M.C. |title=Neuro-dynamic programming for fractionated radiotherapy planning |year=2008 |volume=12 |pages=47–70 |doi=10.1007/978-0-387-73299-2_3|citeseerx=10.1.1.137.8288 |series=Springer Optimization and Its Applications |isbn=978-0-387-73298-5 }}</ref> because of ANNs ability to mitigate losses of accuracy even when reducing the [[w:en:discretization|discretization]] grid density for numerically approximating the solution of control problems. Tasks that fall within the paradigm of reinforcement learning are control problems, [[w:en:game|game]]s and other sequential decision making tasks. ====Self-learning==== Self-learning in neural networks was introduced in 1982 along with a neural network capable of self-learning named Crossbar Adaptive Array (CAA).<ref>Bozinovski, S. (1982). "A self-learning system using secondary reinforcement". In R. Trappl (ed.) Cybernetics and Systems Research: Proceedings of the Sixth European Meeting on Cybernetics and Systems Research. North Holland. pp. 397–402. {{ISBN|978-0-444-86488-8}}.</ref> It is a system with only one input, situation s, and only one output, action (or behavior) a. It has neither external advice input nor external reinforcement input from the environment. The CAA computes, in a crossbar fashion, both decisions about actions and emotions (feelings) about encountered situations. The system is driven by the interaction between cognition and emotion.<ref>Bozinovski, S. (2014) "[https://core.ac.uk/download/pdf/81973924.pdf Modeling mechanisms of cognition-emotion interaction in artificial neural networks, since 1981] {{Webarchive|url=https://web.archive.org/web/20190323204838/https://core.ac.uk/download/pdf/81973924.pdf |date=23 March 2019 }}." Procedia Computer Science p. 255-263</ref> Given the memory matrix, W =||w(a,s)||, the crossbar self-learning algorithm in each iteration performs the following computation: In situation s perform action a; Receive consequence situation s'; Compute emotion of being in consequence situation v(s'); Update crossbar memory w'(a,s) = w(a,s) + v(s'). The backpropagated value (secondary reinforcement) is the emotion toward the consequence situation. The CAA exists in two environments, one is behavioral environment where it behaves, and the other is genetic environment, where from it initially and only once receives initial emotions about to be encountered situations in the behavioral environment. Having received the genome vector (species vector) from the genetic environment, the CAA will learn a goal-seeking behavior, in the behavioral environment that contains both desirable and undesirable situations.<ref>{{cite journal | last1 = Bozinovski | first1 = Stevo | last2 = Bozinovska | first2 = Liljana | year = 2001 | title = Self-learning agents: A connectionist theory of emotion based on crossbar value judgment | journal = Cybernetics and Systems | volume = 32 | issue = 6| pages = 637–667 | doi = 10.1080/01969720118145 | s2cid = 8944741 }}</ref> ==== Neuroevolution ==== {{Main|w:en:Neuroevolution}} [[w:en:Neuroevolution|Neuroevolution]] can create neural network topologies and weights using [[w:en:evolutionary computation|evolutionary computation]]. It is competitive with sophisticated gradient descent approaches{{citation needed|date=July 2019}}. One advantage of neuroevolution is that it may be less prone to get caught in "dead ends".<ref>{{cite news|date=10 January 2018|title=Artificial intelligence can 'evolve' to solve problems|language=en|work=Science {{!}} AAAS|url=https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|access-date=7 February 2018|archive-date=9 December 2021|archive-url=https://web.archive.org/web/20211209231714/https://www.science.org/content/article/artificial-intelligence-can-evolve-solve-problems|url-status=live}}</ref> ===Stochastic neural network=== '''Stochastic neural networks''' originating from [[w:en:Spin glass#Sherrington–Kirkpatrick model|Sherrington–Kirkpatrick model]]s are a type of artificial neural network built by introducing random variations into the network, either by giving the network's [[w:en:artificial neuron|artificial neuron]]s [[w:en:Stochastic process|stochastic]] transfer functions, or by giving them stochastic weights. This makes them useful tools for [[w:en:Optimization (mathematics)|optimization]] problems, since the random fluctuations help the network escape from [[w:en:Maxima and minima|local minima]].<ref>{{citation|title=Stochastic Models of Neural Networks|volume=102|series=Frontiers in artificial intelligence and applications: Knowledge-based intelligent engineering systems|first=Claudio|last=Turchetti|publisher=IOS Press|year=2004|isbn=9781586033880}}</ref> Stochastic neural networks trained using a Bayesian approach are known as '''Bayesian neural networks'''.<ref>{{Cite journal |last1=Jospin |first1=Laurent Valentin |last2=Laga |first2=Hamid |last3=Boussaid |first3=Farid |last4=Buntine |first4=Wray |last5=Bennamoun |first5=Mohammed |date=2022 |title=Hands-On Bayesian Neural Networks—A Tutorial for Deep Learning Users |url=http://dx.doi.org/10.1109/mci.2022.3155327 |journal=IEEE Computational Intelligence Magazine |volume=17 |issue=2 |pages=29–48 |doi=10.1109/mci.2022.3155327 |arxiv=2007.06823 |s2cid=220514248 |issn=1556-603X |access-date=19 November 2022 |archive-date=4 February 2023 |archive-url=https://web.archive.org/web/20230204153904/https://ieeexplore.ieee.org/document/9756596/ |url-status=live }}</ref> ===Other=== In a [[w:en:Bayesian probability|Bayesian]] framework, a distribution over the set of allowed models is chosen to minimize the cost. [[w:en:Evolutionary methods|Evolutionary methods]],<ref>{{cite conference |author1=de Rigo, D. |author2=Castelletti, A. |author3=Rizzoli, A. E. |author4=Soncini-Sessa, R. |author5=Weber, E. |date=January 2005 |title=A selective improvement technique for fastening Neuro-Dynamic Programming in Water Resources Network Management |conference=16th IFAC World Congress |publisher=IFAC |location=Prague, Czech Republic |conference-url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Index.html |book-title=Proceedings of the 16th IFAC World Congress – IFAC-PapersOnLine |editor=Pavel Zítek |volume=16 |pages=7–12 |url=http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |access-date=30 December 2011 |doi=10.3182/20050703-6-CZ-1902.02172 |isbn=978-3-902661-75-3 |hdl=11311/255236 |hdl-access=free |archive-date=26 April 2012 |archive-url=https://web.archive.org/web/20120426012450/http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2005/Papers/Paper4269.html |url-status=live }}</ref> [[w:en:gene expression programming|gene expression programming]],<ref>{{cite book |last=Ferreira |first=C. |year=2006 |contribution=Designing Neural Networks Using Gene Expression Programming |url=http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |editor=A. Abraham |editor2=B. de Baets |editor3=M. Köppen |editor4=B. Nickolay |title=Applied Soft Computing Technologies: The Challenge of Complexity |pages=517–536 |publisher=Springer-Verlag |access-date=8 October 2012 |archive-date=19 December 2013 |archive-url=https://web.archive.org/web/20131219022806/http://www.gene-expression-programming.com/webpapers/Ferreira-ASCT2006.pdf |url-status=live }}</ref> [[w:en:simulated annealing|simulated annealing]],<ref>{{cite conference |author=Da, Y. |author2=Xiurun, G. |date=July 2005 |title=An improved PSO-based ANN with simulated annealing technique |volume=63 |pages=527–533 |editor=T. Villmann |book-title=New Aspects in Neurocomputing: 11th European Symposium on Artificial Neural Networks |url=http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm |publisher=Elsevier |doi=10.1016/j.neucom.2004.07.002 |access-date=30 December 2011 |archive-date=25 April 2012 |archive-url=https://web.archive.org/web/20120425233611/http://www.dice.ucl.ac.be/esann/proceedings/electronicproceedings.htm}}</ref> [[w:en:expectation-maximization|expectation-maximization]], [[w:en:non-parametric methods|non-parametric methods]] and [[w:en:particle swarm optimization|particle swarm optimization]]<ref>{{cite conference |author=Wu, J. |author2=Chen, E. |date=May 2009 |title=A Novel Nonparametric Regression Ensemble for Rainfall Forecasting Using Particle Swarm Optimization Technique Coupled with Artificial Neural Network |series=Lecture Notes in Computer Science |volume=5553 |pages=49–58 |book-title=6th International Symposium on Neural Networks, ISNN 2009 |url=http://www2.mae.cuhk.edu.hk/~isnn2009/ |editor=Wang, H. |editor2=Shen, Y. |editor3=Huang, T. |editor4=Zeng, Z. |publisher=Springer |doi=10.1007/978-3-642-01513-7_6 |isbn=978-3-642-01215-0 |access-date=1 January 2012 |archive-date=31 December 2014 |archive-url=https://web.archive.org/web/20141231221755/http://www2.mae.cuhk.edu.hk/~isnn2009/ |url-status=dead}}</ref> are other learning algorithms. Convergent recursion is a learning algorithm for [[w:en:cerebellar model articulation controller|cerebellar model articulation controller]] (CMAC) neural networks.<ref name="Qin1">{{cite journal |author1=Ting Qin |author2=Zonghai Chen |author3=Haitao Zhang |author4=Sifu Li |author5=Wei Xiang |author6=Ming Li |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |title=A learning algorithm of CMAC based on RLS |journal=Neural Processing Letters |volume=19 |issue=1 |date=2004 |pages=49–61 |doi=10.1023/B:NEPL.0000016847.18175.60 |s2cid=6233899 |access-date=30 January 2019 |archive-date=14 April 2021 |archive-url=https://web.archive.org/web/20210414103815/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_998.pdf |url-status=live }}</ref><ref name="Qin2">{{cite journal |author1=Ting Qin |author2=Haitao Zhang |author3=Zonghai Chen |author4=Wei Xiang |url=http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |title=Continuous CMAC-QRLS and its systolic array |journal=Neural Processing Letters |volume=22 |issue=1 |date=2005 |pages=1–16 |doi=10.1007/s11063-004-2694-0 |s2cid=16095286 |access-date=30 January 2019 |archive-date=18 November 2018 |archive-url=https://web.archive.org/web/20181118122850/http://www-control.eng.cam.ac.uk/Homepage/papers/cued_control_997.pdf |url-status=live }}</ref> ==== Modes ==== {{No footnotes|date=August 2019|section}} Two modes of learning are available: [[w:en:stochastic gradient descent|stochastic]] and batch. In stochastic learning, each input creates a weight adjustment. In batch learning weights are adjusted based on a batch of inputs, accumulating errors over the batch. Stochastic learning introduces "noise" into the process, using the local gradient calculated from one data point; this reduces the chance of the network getting stuck in local minima. However, batch learning typically yields a faster, more stable descent to a local minimum, since each update is performed in the direction of the batch's average error. A common compromise is to use "mini-batches", small batches with samples in each batch selected stochastically from the entire data set. == Types == <!-- Split to [[w:en:Types of artificial neural networks|Types of artificial neural networks]] --> {{Main|w:en:Types of artificial neural networks}} ANNs have evolved into a broad family of techniques that have advanced the state of the art across multiple domains. The simplest types have one or more static components, including number of units, number of layers, unit weights and [[w:en:topology|topology]]. Dynamic types allow one or more of these to evolve via learning. The latter are much more complicated, but can shorten learning periods and produce better results. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. Some of the main breakthroughs include: [[w:en:convolutional neural network|convolutional neural network]]s that have proven particularly successful in processing visual and other two-dimensional data;<ref name="LECUN1989">{{cite journal |vauthors=LeCun Y, Boser B, Denker JS, Henderson D, Howard RE, Hubbard W, Jackel LD |title=Backpropagation Applied to Handwritten Zip Code Recognition |journal=Neural Computation |volume=1 |issue=4 |pages=541–551 |date=1989 |doi=10.1162/neco.1989.1.4.541|s2cid=41312633 }}</ref><ref name="lecun2016slides">[[w:en:Yann LeCun|Yann LeCun]] (2016). Slides on Deep Learning [https://indico.cern.ch/event/510372/ Online] {{Webarchive|url=https://web.archive.org/web/20160423021403/https://indico.cern.ch/event/510372/ |date=23 April 2016 }}</ref> long short-term memory avoid the [[w:en:vanishing gradient problem|vanishing gradient problem]]<ref name=":03">{{Cite journal |last1=Hochreiter|first1=Sepp|author-link=Sepp Hochreiter|last2=Schmidhuber|first2=Jürgen|s2cid=1915014|author-link2=Jürgen Schmidhuber|date=1 November 1997|title=Long Short-Term Memory|journal=Neural Computation|volume=9|issue=8 |pages=1735–1780 |doi=10.1162/neco.1997.9.8.1735|pmid=9377276|issn=0899-7667}}</ref> and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition,<ref name="sak2014">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf |title=Long Short-Term Memory recurrent neural network architectures for large scale acoustic modeling |last1=Sak|first1=Hasim |last2=Senior|first2=Andrew|date=2014|last3=Beaufays|first3=Francoise|archive-url=https://web.archive.org/web/20180424203806/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43905.pdf|archive-date=24 April 2018|url-status=dead}}</ref><ref name="liwu2015">{{cite arXiv|last1=Li|first1=Xiangang|last2=Wu|first2=Xihong|date=15 October 2014|title=Constructing Long Short-Term Memory based Deep Recurrent Neural Networks for Large Vocabulary Speech Recognition|eprint=1410.4281 |class=cs.CL}}</ref> text-to-speech synthesis,<ref>{{Cite journal|title=TTS synthesis with bidirectional LSTM based Recurrent Neural Networks|pages=1964–1968|last1=Fan|first1=Y. |last2=Qian|first2=Y.|date=2014 |journal=Proceedings of the Annual Conference of the International Speech Communication Association, Interspeech|url=https://www.researchgate.net/publication/287741874|access-date=13 June 2017 |last3=Xie |first3=F.|last4=Soong|first4=F. K.}}</ref><ref name="scholarpedia2"/><ref name="zen2015">{{Cite web|url=https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|title=Unidirectional Long Short-Term Memory Recurrent Neural Network with Recurrent Output Layer for Low-Latency Speech Synthesis|last1=Zen|first1=Heiga|last2=Sak|first2=Hasim|date=2015|website=Google.com|publisher=ICASSP|pages=4470–4474|access-date=27 June 2017|archive-date=9 May 2021|archive-url=https://web.archive.org/web/20210509123113/https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/43266.pdf|url-status=live}}</ref> and photo-real talking heads;<ref name="fan2015">{{Cite journal|last1=Fan|first1=Bo|last2=Wang|first2=Lijuan|last3=Soong|first3=Frank K.|last4=Xie|first4=Lei|date=2015|title=Photo-Real Talking Head with Deep Bidirectional LSTM|url=https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|journal=Proceedings of ICASSP|access-date=27 June 2017|archive-date=1 November 2017|archive-url=https://web.archive.org/web/20171101052317/https://www.microsoft.com/en-us/research/wp-content/uploads/2015/04/icassp2015_fanbo_1009.pdf|url-status=live}}</ref> competitive networks such as [[w:en:generative adversarial network|generative adversarial network]]s in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game<ref name="preprint">{{Cite arXiv |eprint=1712.01815|class=cs.AI|first1=David|last1=Silver|first2=Thomas|last2=Hubert|author-link1=David Silver (programmer)|title=Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm|date=5 December 2017|first3=Julian|last3=Schrittwieser|first4=Ioannis|last4=Antonoglou |first5=Matthew|last5=Lai |first6=Arthur|last6=Guez|first7=Marc|last7=Lanctot|first8=Laurent|last8=Sifre |first9=Dharshan|last9=Kumaran|author-link9=Dharshan Kumaran|first10=Thore|last10=Graepel|first11=Timothy |last11=Lillicrap|first12=Karen |last12=Simonyan|first13=Demis|last13=Hassabis|author-link13=Demis Hassabis}}</ref> or on deceiving the opponent about the authenticity of an input.<ref name="GANnips">{{cite conference|last1=Goodfellow|first1=Ian|last2=Pouget-Abadie|first2=Jean|last3=Mirza|first3=Mehdi|last4=Xu|first4=Bing|last5=Warde-Farley|first5=David|last6=Ozair|first6=Sherjil|last7=Courville|first7=Aaron|last8=Bengio|first8=Yoshua|year=2014|title=Generative Adversarial Networks|url=https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|conference=Proceedings of the International Conference on Neural Information Processing Systems (NIPS 2014)|pages=2672–2680|access-date=20 August 2019|archive-date=22 November 2019|archive-url=https://web.archive.org/web/20191122034612/http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf|url-status=live}}</ref> == Network design == {{Main|w:en:Neural architecture search}} Neural architecture search (NAS) uses machine learning to automate ANN design. Various approaches to NAS have designed networks that compare well with hand-designed systems. The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network.<ref>{{cite arXiv|last1=Zoph|first1=Barret|last2=Le|first2=Quoc V.|date=4 November 2016|title=Neural Architecture Search with Reinforcement Learning|eprint=1611.01578|class=cs.LG}}</ref> Available systems include [[w:en:Automated machine learning|AutoML]] and AutoKeras.<ref>{{cite journal |author1=Haifeng Jin |author2=Qingquan Song |author3=Xia Hu |title=Auto-keras: An efficient neural architecture search system |journal=Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining |publisher=ACM |date=2019 |arxiv=1806.10282 |url=https://autokeras.com/ |via=autokeras.com |access-date=21 August 2019 |archive-date=21 August 2019 |archive-url=https://web.archive.org/web/20190821163310/https://autokeras.com/ |url-status=live }}</ref> Design issues include deciding the number, type and connectedness of network layers, as well as the size of each and the connection type (full, pooling, ...). [[w:en:Hyperparameter|Hyperparameter]]s must also be defined as part of the design (they are not learned), governing matters such as how many neurons are in each layer, learning rate, step, stride, depth, receptive field and padding (for CNNs), etc.<ref name="abs1502.02127">{{cite arXiv|eprint=1502.02127|last1=Claesen|first1=Marc|last2=De Moor|first2=Bart |title=Hyperparameter Search in Machine Learning |date=2015|class=cs.LG }} {{bibcode|2015arXiv150202127C}}</ref> == Use == {{Unreferenced section|date=November 2020}} Using Artificial neural networks requires an understanding of their characteristics. * Choice of model: This depends on the data representation and the application. Overly complex models are slow learning. * Learning algorithm: Numerous trade-offs exist between learning algorithms. Almost any algorithm will work well with the correct [[w:en:hyperparameter|hyperparameter]]s for training on a particular data set. However, selecting and tuning an algorithm for training on unseen data requires significant experimentation. * Robustness: If the model, cost function and learning algorithm are selected appropriately, the resulting ANN can become robust. ANN capabilities fall within the following broad categories:{{Citation needed|date=June 2017}} * [[w:en:Function approximation|Function approximation]], or [[w:en:regression analysis|regression analysis]], including [[w:en:Time series#Prediction and forecasting|time series prediction]], [[w:en:fitness approximation|fitness approximation]] and modeling. * [[w:en:Statistical classification|Classification]], including [[w:en:Pattern recognition|pattern]] and sequence recognition, [[w:en:novelty detection|novelty detection]] and sequential decision making.<ref name ="TurekNeuralNet">{{cite journal|author=Turek, Fred D.|title=Introduction to Neural Net Machine Vision|url=http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|access-date=5 March 2013|journal=Vision Systems Design|date=March 2007|volume=12|number=3|archive-date=16 May 2013|archive-url=https://web.archive.org/web/20130516124148/http://www.vision-systems.com/articles/print/volume-12/issue-3/features/introduction-to-neural-net-machine-vision.html|url-status=live}}</ref> * [[w:en:Data processing|Data processing]], including filtering, clustering, [[w:en:blind source separation|blind source separation]] and compression. * [[w:en:Robotics|Robotics]], including directing manipulators and [[w:en:prosthesis|prostheses]]. ==Applications== Because of their ability to reproduce and model nonlinear processes, artificial neural networks have found applications in many disciplines. Application areas include [[w:en:system identification|system identification]] and control (vehicle control, trajectory prediction,<ref>{{cite journal|last1=Zissis|first1=Dimitrios|title=A cloud based architecture capable of perceiving and predicting multiple vessel behaviour|journal=Applied Soft Computing|date=October 2015|volume=35|doi=10.1016/j.asoc.2015.07.002|pages=652–661|url=https://zenodo.org/record/848743|access-date=18 July 2019|archive-date=26 July 2020|archive-url=https://web.archive.org/web/20200726091505/https://zenodo.org/record/848743|url-status=live}}</ref> [[w:en:process control|process control]], [[w:en:natural resource management|natural resource management]]), [[w:en:quantum chemistry|quantum chemistry]],<ref name="Balabin_2009">{{Cite journal|journal=[[w:en:J. Chem. Phys.|J. Chem. Phys.]] |volume=131 |issue=7 |page=074104 |doi=10.1063/1.3206326 |title=Neural network approach to quantum-chemistry data: Accurate prediction of density functional theory energies |year=2009 |author1=Roman M. Balabin |author2=Ekaterina I. Lomakina |pmid=19708729|bibcode=2009JChPh.131g4104B}}</ref> [[w:en:general game playing|general game playing]],<ref>{{cite journal |last1=Silver |first1=David |display-authors=etal |year=2016 |title=Mastering the game of Go with deep neural networks and tree search |url=http://web.iitd.ac.in/~sumeet/Silver16.pdf |journal=Nature |volume=529 |issue=7587 |pages=484–489 |doi=10.1038/nature16961 |pmid=26819042 |bibcode=2016Natur.529..484S |s2cid=515925 |access-date=31 January 2019 |archive-date=23 November 2018 |archive-url=https://web.archive.org/web/20181123112812/http://web.iitd.ac.in/~sumeet/Silver16.pdf |url-status=live }}</ref> [[w:en:pattern recognition|pattern recognition]] (radar systems, [[w:en:Facial recognition system|face identification]], signal classification,<ref>{{cite journal|last=Sengupta |first=Nandini|author2=Sahidullah, Md|author3=Saha, Goutam|title=Lung sound classification using cepstral-based statistical features|journal=Computers in Biology and Medicine|date=August 2016|volume=75|issue=1 |pages=118–129|doi=10.1016/j.compbiomed.2016.05.013 |pmid=27286184}}</ref> [[w:en:3D reconstruction|3D reconstruction]],<ref>Choy, Christopher B., et al. "[https://arxiv.org/abs/1604.00449 3d-r2n2: A unified approach for single and multi-view 3d object reconstruction] {{Webarchive|url=https://web.archive.org/web/20200726091721/https://arxiv.org/abs/1604.00449 |date=26 July 2020 }}." European conference on computer vision. Springer, Cham, 2016.</ref> object recognition and more), sensor data analysis,<ref>{{cite journal|last=Gessler|first=Josef|title=Sensor for food analysis applying impedance spectroscopy and artificial neural networks|journal=RiuNet UPV|date=August 2021|issue=1|pages=8–12|url=https://riunet.upv.es/handle/10251/174498|access-date=21 October 2021|archive-date=21 October 2021|archive-url=https://web.archive.org/web/20211021115443/https://riunet.upv.es/handle/10251/174498|url-status=live}}</ref> sequence recognition (gesture, speech, [[w:en:handwriting recognition|handwritten]] and printed text recognition<ref>{{Cite journal|last1=Maitra|first1=D. S. |last2=Bhattacharya|first2=U.|last3=Parui|first3=S. K.|date=August 2015|title=CNN based common approach to handwritten character recognition of multiple scripts |url=https://ieeexplore.ieee.org/document/7333916|journal=2015 13th International Conference on Document Analysis and Recognition (ICDAR)|pages=1021–1025|doi=10.1109/ICDAR.2015.7333916|isbn=978-1-4799-1805-8 |s2cid=25739012}}</ref>), [[w:en:medical diagnosis|medical diagnosis]], finance<ref>{{cite journal|last1=French |first1=Jordan |title=The time traveller's CAPM|journal=Investment Analysts Journal|volume=46|issue=2|pages=81–96 |doi=10.1080/10293523.2016.1255469|year=2016|s2cid=157962452}}</ref> (e.g. ex-ante models for specific financial long-run forecasts and [[w:en:artificial financial market|artificial financial market]]s), [[w:en:data mining|data mining]], visualization, [[w:en:machine translation|machine translation]], social network filtering<ref>{{Cite news|url=https://www.wsj.com/articles/facebook-boosts-a-i-to-block-terrorist-propaganda-1497546000 |title=Facebook Boosts A.I. to Block Terrorist Propaganda|last=Schechner|first=Sam|date=15 June 2017 |work=[[w:en:The Wall Street Journal|The Wall Street Journal]]|access-date=16 June 2017|issn=0099-9660}}</ref> and [[w:en:e-mail spam|e-mail spam]] filtering. ANNs have been used to diagnose several types of cancers<ref>{{cite journal|last=Ganesan|first=N |title=Application of Neural Networks in Diagnosing Cancer Disease Using Demographic Data |journal=International Journal of Computer Applications|volume=1|issue=26|pages=81–97 |bibcode=2010IJCA....1z..81G|year=2010|doi=10.5120/476-783|doi-access=free}}</ref><ref>{{cite journal |url=http://www.lcc.uma.es/~jja/recidiva/042.pdf|title=Artificial Neural Networks Applied to Outcome Prediction for Colorectal Cancer Patients in Separate Institutions|journal=Lancet|volume=350|issue=9076 |pages=469–72|last=Bottaci|first=Leonardo|publisher=The Lancet|pmid=9274582|year=1997|doi=10.1016/S0140-6736(96)11196-X|s2cid=18182063|access-date=2 May 2012|url-status=dead|archive-date=23 November 2018|archive-url=https://web.archive.org/web/20181123170444/http://www.lcc.uma.es/~jja/recidiva/042.pdf}}</ref> and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.<ref>{{cite journal|last1=Alizadeh|first1=Elaheh|last2=Lyons|first2=Samanthe M|last3=Castle|first3=Jordan M |last4=Prasad|first4=Ashok|date=2016|title=Measuring systematic changes in invasive cancer cell shape using Zernike moments|url=http://pubs.rsc.org/en/Content/ArticleLanding/2016/IB/C6IB00100A|journal=Integrative Biology|volume=8|issue=11 |pages=1183–1193|doi=10.1039/C6IB00100A|pmid=27735002}}</ref><ref>{{cite journal |last1=Lyons|first1=Samanthe|date=2016|title=Changes in cell shape are correlated with metastatic potential in murine|journal=Biology Open|volume=5|issue=3|pages=289–299|doi=10.1242/bio.013409|pmid=26873952 |pmc=4810736}}</ref> ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters<ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=28 August 2017|title=Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks|journal=Computer-Aided Civil and Infrastructure Engineering|volume=33|issue=6|pages=443–458|arxiv=1708.08551|doi=10.1111/mice.12359 |bibcode=2017arXiv170808551N |s2cid=36661983}}</ref><ref>{{Cite journal|last1=Nabian|first1=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|access-date=14 March 2018|archive-date=9 March 2018|archive-url=https://web.archive.org/web/20180309120108/https://trid.trb.org/view/1496617|url-status=live}}</ref> and to predict foundation settlements.<ref>{{Cite journal|last1=Díaz|first1=E.|last2=Brotons|first2=V. |last3=Tomás|first3=R.|date=September 2018|title=Use of artificial neural networks to predict 3-D elastic settlement of foundations on soils with inclined bedrock|journal=Soils and Foundations|volume=58|issue=6 |pages=1414–1422 |doi=10.1016/j.sandf.2018.08.001|issn=0038-0806|hdl=10045/81208|doi-access=free}}</ref> ANNs have also been used for building black-box models in [[w:en:geoscience|geoscience]]: [[w:en:hydrology|hydrology]],<ref>{{Cite journal |first=Rao S.|last=Govindaraju |date=1 April 2000|title=Artificial Neural Networks in Hydrology. I: Preliminary Concepts|journal=Journal of Hydrologic Engineering|volume=5|issue=2|pages=115–123|doi=10.1061/(ASCE)1084-0699(2000)5:2(115)|citeseerx=<!--10.1.1.127.3861-->}}</ref><ref>{{Cite journal|first=Rao S.|last=Govindaraju|date=1 April 2000|title=Artificial Neural Networks in Hydrology. II: Hydrologic Applications|journal=Journal of Hydrologic Engineering|volume=5|issue=2 |pages=124–137 |doi=10.1061/(ASCE)1084-0699(2000)5:2(124)}}</ref> ocean modelling and [[w:en:coastal engineering|coastal engineering]],<ref>{{Cite journal|last1=Peres|first1=D. J.|last2=Iuppa|first2=C.|last3=Cavallaro|first3=L.|last4=Cancelliere |first4=A. |last5=Foti|first5=E.|date=1 October 2015|title=Significant wave height record extension by neural networks and reanalysis wind data|journal=Ocean Modelling|volume=94|pages=128–140 |doi=10.1016/j.ocemod.2015.08.002 |bibcode=2015OcMod..94..128P}}</ref><ref>{{Cite journal|last1=Dwarakish|first1=G. S.|last2=Rakshith|first2=Shetty|last3=Natesan|first3=Usha|date=2013|title=Review on Applications of Neural Network in Coastal Engineering|journal=Artificial Intelligent Systems and Machine Learning|url=http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|volume=5|issue=7|pages=324–331|access-date=5 July 2017|archive-date=15 August 2017|archive-url=https://web.archive.org/web/20170815185634/http://www.ciitresearch.org/dl/index.php/aiml/article/view/AIML072013007|url-status=live}}</ref> and [[w:en:geomorphology|geomorphology]].<ref>{{Cite journal |last1=Ermini|first1=Leonardo|last2=Catani |first2=Filippo|last3=Casagli|first3=Nicola|date=1 March 2005|title=Artificial Neural Networks applied to landslide susceptibility assessment|journal=Geomorphology|series=Geomorphological hazard and human impact in mountain environments|volume=66|issue=1|pages=327–343|doi=10.1016/j.geomorph.2004.09.025 |bibcode=2005Geomo..66..327E}}</ref> ANNs have been employed in [[w:en:Computer security|cybersecurity]], with the objective to discriminate between legitimate activities and malicious ones. For example, machine learning has been used for classifying Android malware,<ref>{{Cite journal|last1=Nix|first1=R.|last2=Zhang |first2=J.|date=May 2017 |title=Classification of Android apps and malware using deep neural networks |journal=2017 International Joint Conference on Neural Networks (IJCNN)|pages=1871–1878|s2cid=8838479 |doi=10.1109/IJCNN.2017.7966078|isbn=978-1-5090-6182-2}}</ref> for identifying domains belonging to threat actors and for detecting URLs posing a security risk.<ref>{{Cite web|title=Detecting Malicious URLs |website=The systems and networking group at UCSD |url=http://www.sysnet.ucsd.edu/projects/url/|access-date=15 February 2019|url-status=dead|archive-date=14 July 2019|archive-url=https://web.archive.org/web/20190714201955/http://www.sysnet.ucsd.edu/projects/url/}}</ref> Research is underway on ANN systems designed for penetration testing, for detecting botnets,<ref>{{Citation |last1=Homayoun|first1=Sajad|title=BoTShark: A Deep Learning Approach for Botnet Traffic Detection |date=2018|work=Cyber Threat Intelligence|pages=137–153|editor-last=Dehghantanha|editor-first=Ali |series=Advances in Information Security|publisher=Springer International Publishing|doi=10.1007/978-3-319-73951-9_7|isbn=978-3-319-73951-9|last2=Ahmadzadeh |first2=Marzieh|last3=Hashemi|first3=Sattar |last4=Dehghantanha|first4=Ali|last5=Khayami|first5=Raouf|editor2-last=Conti|editor2-first=Mauro|editor3-last=Dargahi|editor3-first=Tooska}}</ref> credit cards frauds<ref>{{Cite journal |last1=Ghosh|last2=Reilly |name-list-style=and|s2cid=13260377 |date=January 1994|title=Credit card fraud detection with a neural-network|journal=1994 Proceedings of the Twenty-Seventh Hawaii International Conference on System Sciences |volume=3|pages=621–630|doi=10.1109/HICSS.1994.323314|isbn=978-0-8186-5090-1}}</ref> and network intrusions. ANNs have been proposed as a tool to solve [[w:en:partial differential equation|partial differential equation]]s in physics<ref>{{Cite web|last=Ananthaswamy|first=Anil|date=19 April 2021|title=Latest Neural Nets Solve World's Hardest Equations Faster Than Ever Before|url=https://www.quantamagazine.org/new-neural-networks-solve-hardest-equations-faster-than-ever-20210419/|access-date=12 May 2021|website=Quanta Magazine|language=en}}</ref><ref>{{Cite web|title=AI has cracked a key mathematical puzzle for understanding our world|url=https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/|access-date=19 November 2020|website=MIT Technology Review|language=en}}</ref><ref>{{Cite web|title=Caltech Open-Sources AI for Solving Partial Differential Equations|url=https://www.infoq.com/news/2020/12/caltech-ai-pde/|access-date=20 January 2021|website=InfoQ|language=en|archive-date=25 January 2021|archive-url=https://web.archive.org/web/20210125233952/https://www.infoq.com/news/2020/12/caltech-ai-pde/|url-status=live}}</ref> and simulate the properties of many-body [[w:en:open quantum system|open quantum system]]s.<ref>{{cite journal |last1=Nagy |first1=Alexandra |title=Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems |journal=[[w:en:Physical Review Letters|Physical Review Letters]] |volume=122 |issue=25 |pages=250501 |date=28 June 2019 |doi=10.1103/PhysRevLett.122.250501 |pmid=31347886 |bibcode=2019PhRvL.122y0501N |arxiv=1902.09483 |s2cid=119074378 }}</ref><ref>{{Cite journal|last1=Yoshioka|first1=Nobuyuki|last2=Hamazaki|first2=Ryusuke|date=28 June 2019|title=Constructing neural stationary states for open quantum many-body systems|journal=Physical Review B|volume=99|issue=21 |pages=214306|doi=10.1103/PhysRevB.99.214306|bibcode=2019PhRvB..99u4306Y|arxiv=1902.07006|s2cid=119470636}}</ref><ref>{{Cite journal|last1=Hartmann|first1=Michael J.|last2=Carleo|first2=Giuseppe |date=28 June 2019|title=Neural-Network Approach to Dissipative Quantum Many-Body Dynamics|journal=Physical Review Letters|volume=122|issue=25|pages=250502|doi=10.1103/PhysRevLett.122.250502|pmid=31347862 |bibcode=2019PhRvL.122y0502H|arxiv=1902.05131|s2cid=119357494}}</ref><ref>{{Cite journal|last1=Vicentini |first1=Filippo|last2=Biella|first2=Alberto|last3=Regnault|first3=Nicolas|last4=Ciuti|first4=Cristiano|date=28 June 2019 |title=Variational Neural-Network Ansatz for Steady States in Open Quantum Systems |journal=Physical Review Letters|volume=122|issue=25|pages=250503|doi=10.1103/PhysRevLett.122.250503 |pmid=31347877 |bibcode=2019PhRvL.122y0503V |arxiv=1902.10104|s2cid=119504484}}</ref> In brain research ANNs have studied short-term behavior of [[w:en:biological neuron models|individual neurons]],<ref>{{cite journal |author=Forrest MD |title=Simulation of alcohol action upon a detailed Purkinje neuron model and a simpler surrogate model that runs >400 times faster |journal=BMC Neuroscience |volume=16 |issue=27 |pages=27 |date=April 2015 |doi=10.1186/s12868-015-0162-6 |pmid=25928094 |pmc=4417229}}</ref> the dynamics of neural circuitry arise from interactions between individual neurons and how behavior can arise from abstract neural modules that represent complete subsystems. Studies considered long-and short-term plasticity of neural systems and their relation to learning and memory from the individual neuron to the system level. ==Theoretical properties== ===Computational power=== The [[w:en:multilayer perceptron|multilayer perceptron]] is a [[w:en:UTM theorem|universal function]] approximator, as proven by the [[w:en:universal approximation theorem|universal approximation theorem]]. However, the proof is not constructive regarding the number of neurons required, the network topology, the weights and the learning parameters. A specific recurrent architecture with [[w:en:Rational number|rational]]-valued weights (as opposed to full precision [[w:en:real number|real number]]-valued weights) has the power of a [[w:en:Universal Turing Machine|universal Turing machine]],<ref>{{Cite journal| title = Turing computability with neural nets | url = http://www.math.rutgers.edu/~sontag/FTPDIR/aml-turing.pdf | year = 1991 | journal = Appl. Math. Lett. | pages = 77–80 | volume = 4 | issue = 6 | last1 = Siegelmann | first1 = H.T. | last2 = Sontag | first2 = E.D. | doi = 10.1016/0893-9659(91)90080-F }}</ref> using a finite number of neurons and standard linear connections. Further, the use of [[w:en:Irrational number|irrational]] values for weights results in a machine with [[w:en:Hypercomputation|super-Turing]] power.<ref>{{cite journal |last1=Balcázar |first1=José |title=Computational Power of Neural Networks: A Kolmogorov Complexity Characterization |journal=IEEE Transactions on Information Theory|date=Jul 1997 |volume=43 |issue=4 |pages=1175–1183 |doi=10.1109/18.605580 |citeseerx=10.1.1.411.7782 }}</ref> ===Capacity=== A model's "capacity" property corresponds to its ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity. Two notions of capacity are known by the community. The information capacity and the VC Dimension. The information capacity of a perceptron is intensively discussed in Sir David MacKay's book<ref name="auto">{{cite book| last=MacKay| first=David, J.C.| author-link=David J.C. MacKay| year=2003| publisher=[[w:en:Cambridge University Press|Cambridge University Press]]| isbn=978-0-521-64298-9| title=Information Theory, Inference, and Learning Algorithms| url=http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| access-date=11 June 2016| archive-date=19 October 2016| archive-url=https://web.archive.org/web/20161019163258/http://www.inference.phy.cam.ac.uk/itprnn/book.pdf| url-status=live}}</ref> which summarizes work by Thomas Cover.<ref>{{cite journal|last=Cover|first=Thomas|author-link=Thomas M. Cover|year=1965|publisher=[[w:en:IEEE|IEEE]]|url=http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|title=Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition|journal=IEEE Transactions on Electronic Computers|issue=3|pages=326–334|volume=EC-14|doi=10.1109/PGEC.1965.264137|access-date=10 March 2020|archive-date=5 March 2016|archive-url=https://web.archive.org/web/20160305031348/http://www-isl.stanford.edu/people/cover/papers/paper2.pdf|url-status=live}}</ref> The capacity of a network of standard neurons (not convolutional) can be derived by four rules<ref>{{cite journal| last=Gerald | first=Friedland| author-link=Gerald Friedland|year=2019|publisher=[[w:en:Association for Computing Machinery|ACM]]|title=Reproducibility and Experimental Design for Machine Learning on Audio and Multimedia Data|journal=MM '19: Proceedings of the 27th ACM International Conference on Multimedia| pages=2709–2710| doi=10.1145/3343031.3350545| isbn=978-1-4503-6889-6| s2cid=204837170}}</ref> that derive from understanding a neuron as an [[w:en:ADALINE|electrical element]]. The information capacity captures the functions modelable by the network given any data as input. The second notion, is the [[w:en:VC dimension|VC dimension]]. VC Dimension uses the principles of [[w:en:measure theory|measure theory]] and finds the maximum capacity under the best possible circumstances. This is, given input data in a specific form. As noted in,<ref name="auto"/> the VC Dimension for arbitrary inputs is half the information capacity of a Perceptron. The VC Dimension for arbitrary points is sometimes referred to as Memory Capacity.<ref>{{cite web| url=http://tfmeter.icsi.berkeley.edu/| title=Stop tinkering, start measuring! Predictable experimental design of Neural Network experiments| website=The Tensorflow Meter| access-date=10 March 2020| archive-date=18 April 2022| archive-url=https://web.archive.org/web/20220418025904/http://tfmeter.icsi.berkeley.edu/| url-status=dead}}</ref> ===Convergence=== Models may not consistently converge on a single solution, firstly because local minima may exist, depending on the cost function and the model. Secondly, the optimization method used might not guarantee to converge when it begins far from any local minimum. Thirdly, for sufficiently large data or parameters, some methods become impractical. Another issue worthy to mention is that training may cross some [[w:en:Saddle point|Saddle point]] which may lead the convergence to the wrong direction. The convergence behavior of certain types of ANN architectures are more understood than others. When the width of network approaches to infinity, the ANN is well described by its first order Taylor expansion throughout training, and so inherits the convergence behavior of [[w:en:Linear model|affine models]].<ref>{{Cite journal|last1=Lee|first1=Jaehoon|last2=Xiao|first2=Lechao|last3=Schoenholz|first3=Samuel S.|last4=Bahri |first4=Yasaman|last5=Novak |first5=Roman|last6=Sohl-Dickstein|first6=Jascha|last7=Pennington |first7=Jeffrey|title=Wide neural networks of any depth evolve as linear models under gradient descent |journal=Journal of Statistical Mechanics: Theory and Experiment|year=2020|volume=2020|issue=12|page=124002 |doi=10.1088/1742-5468/abc62b|arxiv=1902.06720|bibcode=2020JSMTE2020l4002L|s2cid=62841516}}</ref><ref>{{cite conference |conference=32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montreal, Canada |author1=Arthur Jacot |author2=Franck Gabriel |author3=Clement Hongler |date=2018 |url=https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |title=Neural Tangent Kernel: Convergence and Generalization in Neural Networks |access-date=4 June 2022 |archive-date=22 June 2022 |archive-url=https://web.archive.org/web/20220622033100/https://proceedings.neurips.cc/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf |url-status=live }}</ref> Another example is when parameters are small, it is observed that ANNs often fits target functions from low to high frequencies. This behavior is referred to as the spectral bias, or frequency principle, of neural networks.<ref>{{cite book |vauthors=Xu ZJ, Zhang Y, Xiao Y |date=2019 |veditors=Gedeon T, Wong K, Lee M |title=Neural Information Processing. ICONIP 2019. |series=Lecture Notes in Computer Science |volume=11953 |publisher=Springer, Cham |doi=10.1007/978-3-030-36708-4_22 |chapter=Training Behavior of Deep Neural Network in Frequency Domain |pages=264–274 |arxiv=1807.01251 |isbn=978-3-030-36707-7 |s2cid=49562099 }}</ref><ref>{{cite journal |author1=Nasim Rahaman |author2=Aristide Baratin |author3=Devansh Arpit |author4=Felix Draxler |author5=Min Lin |author6=Fred Hamprecht |author7=Yoshua Bengio |author8=Aaron Courville |journal=Proceedings of the 36th International Conference on Machine Learning |volume=97 |pages=5301–5310 |date=2019 |title=On the Spectral Bias of Neural Networks |arxiv=1806.08734 |url=http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |access-date=4 June 2022 |archive-date=22 October 2022 |archive-url=https://web.archive.org/web/20221022155951/http://proceedings.mlr.press/v97/rahaman19a/rahaman19a.pdf |url-status=live }}</ref><ref>{{cite journal |arxiv=1901.06523 |author1=Zhi-Qin John Xu |author2=Yaoyu Zhang |author3=Tao Luo |author4=Yanyang Xiao |author5=Zheng Ma |title=Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks|journal=Communications in Computational Physics |year=2020 |volume=28 |issue=5 |pages=1746–1767 |doi=10.4208/cicp.OA-2020-0085 |bibcode=2020CCoPh..28.1746X |s2cid=58981616 }}</ref><ref>{{cite arXiv |eprint=1906.09235 |author1=Tao Luo |author2=Zheng Ma |author3=Zhi-Qin John Xu |author4=Yaoyu Zhang |date=2019 |title=Theory of the Frequency Principle for General Deep Neural Networks|class=cs.LG }}</ref> This phenomenon is the opposite to the behavior of some well studied iterative numerical schemes such as [[w:en:Jacobi method|Jacobi method]]. Deeper neural networks have been observed to be more biased towards low frequency functions.<ref>{{Cite journal|last1=Xu|first1=Zhiqin John|last2=Zhou|first2=Hanxu|date=2021-05-18|title=Deep Frequency Principle Towards Understanding Why Deeper Learning Is Faster|url=https://ojs.aaai.org/index.php/AAAI/article/view/17261|journal=Proceedings of the AAAI Conference on Artificial Intelligence|volume=35|issue=12|pages=10541–10550|doi=10.1609/aaai.v35i12.17261|arxiv=2007.14313|s2cid=220831156|issn=2374-3468|access-date=5 October 2021|archive-date=5 October 2021|archive-url=https://web.archive.org/web/20211005142300/https://ojs.aaai.org/index.php/AAAI/article/view/17261|url-status=live}}</ref> ===Generalization and statistics=== {{No footnotes|date=August 2019|section}} Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. Two approaches address over-training. The first is to use [[w:en:cross-validation (statistics)|cross-validation]] and similar techniques to check for the presence of over-training and to select [[w:en:hyperparameters|hyperparameters]] to minimize the generalization error. The second is to use some form of ''[[w:en:regularization (mathematics)|regularization]]''. This concept emerges in a probabilistic (Bayesian) framework, where regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting. [[File:Synapse deployment.jpg|thumb|right|upright=1.15|Confidence analysis of a neural network]] Supervised neural networks that use a [[w:en:mean squared error|mean squared error]] (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the [[w:en:confidence interval|confidence interval]] of network output, assuming a [[w:en:normal distribution|normal distribution]]. A confidence analysis made this way is statistically valid as long as the output [[w:en:probability distribution|probability distribution]] stays the same and the network is not modified. By assigning a [[w:en:softmax activation function|softmax activation function]], a generalization of the [[w:en:logistic function|logistic function]], on the output layer of the neural network (or a softmax component in a component-based network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is useful in classification as it gives a certainty measure on classifications. The softmax activation function is: :<math>y_i=\frac{e^{x_i}}{\sum_{j=1}^c e^{x_j}}</math> <section end="theory" /> ==Criticism== ===Training === A common criticism of neural networks, particularly in robotics, is that they require too much training for real-world operation.<ref>{{Cite journal |last=Parisi |first=German I. |last2=Kemker |first2=Ronald |last3=Part |first3=Jose L. |last4=Kanan |first4=Christopher |last5=Wermter |first5=Stefan |date=2019-05-01 |title=Continual lifelong learning with neural networks: A review |url=https://www.sciencedirect.com/science/article/pii/S0893608019300231 |journal=Neural Networks |language=en |volume=113 |pages=54–71 |doi=10.1016/j.neunet.2019.01.012 |issn=0893-6080}}</ref> Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example, grouping examples in so-called mini-batches and/or introducing a recursive least squares algorithm for [[w:en:cerebellar model articulation controller|CMAC]].<ref name="Qin1"/> ===Theory=== A central claim{{cn|date=January 2023}} of ANNs is that they embody new and powerful general principles for processing information. These principles are ill-defined. It is often claimed{{by whom?|date=January 2023}} that they are [[w:en:Emergent properties|emergent]] from the network itself. This allows simple statistical association (the basic function of artificial neural networks) to be described as learning or recognition. In 1997, [[w:en:Alexander Dewdney|Alexander Dewdney]] commented that, as a result, artificial neural networks have a "something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are. No human hand (or mind) intervenes; solutions are found as if by magic; and no one, it seems, has learned anything".<ref>{{cite book|url={{google books |plainurl=y |id=KcHaAAAAMAAJ|page=82}}|title=Yes, we have no neutrons: an eye-opening tour through the twists and turns of bad science|last=Dewdney|first=A. K.|date=1 April 1997|publisher=Wiley|isbn=978-0-471-10806-1|pages=82}}</ref> One response to Dewdney is that neural networks handle many complex and diverse tasks, ranging from autonomously flying aircraft<ref>[http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html NASA – Dryden Flight Research Center – News Room: News Releases: NASA NEURAL NETWORK PROJECT PASSES MILESTONE] {{Webarchive|url=https://web.archive.org/web/20100402065100/http://www.nasa.gov/centers/dryden/news/NewsReleases/2003/03-49.html |date=2 April 2010 }}. Nasa.gov. Retrieved on 20 November 2013.</ref> to detecting credit card fraud to mastering the game of [[w:en:Go (game)|Go]]. Technology writer Roger Bridgman commented: {{blockquote|Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be "an opaque, unreadable table...valueless as a scientific resource". In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers. An unreadable table that a useful machine could read would still be well worth having.<ref>{{Cite web |url=http://members.fortunecity.com/templarseries/popper.html |title=Roger Bridgman's defence of neural networks |access-date=12 July 2010 |archive-url=https://web.archive.org/web/20120319163352/http://members.fortunecity.com/templarseries/popper.html |archive-date=19 March 2012 |url-status=dead}}</ref> }} Biological brains use both shallow and deep circuits as reported by brain anatomy,<ref name="VanEssen1991">D. J. Felleman and D. C. Van Essen, "[https://archive.today/20150120022056/http://cercor.oxfordjournals.org/content/1/1/1.1.full.pdf+html Distributed hierarchical processing in the primate cerebral cortex]," ''Cerebral Cortex'', 1, pp. 1–47, 1991.</ref> displaying a wide variety of invariance. Weng<ref name="Weng2012">J. Weng, "[https://www.amazon.com/Natural-Artificial-Intelligence-Introduction-Computational/dp/0985875720 Natural and Artificial Intelligence: Introduction to Computational Brain-Mind]," BMI Press, {{ISBN|978-0-9858757-2-5}}, 2012.</ref> argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies. ===Hardware=== Large and effective neural networks require considerable computing resources.<ref name=":0">{{cite journal|last1=Edwards|first1=Chris|s2cid=11026540|title=Growing pains for deep learning|journal=Communications of the ACM|date=25 June 2015|volume=58|issue=7|pages=14–16|doi=10.1145/2771283}}</ref> While the brain has hardware tailored to the task of processing signals through a [[w:en:Graph (discrete mathematics)|graph]] of neurons, simulating even a simplified neuron on [[w:en:von Neumann architecture|von Neumann architecture]] may consume vast amounts of [[w:en:Random-access memory|memory]] and storage. Furthermore, the designer often needs to transmit signals through many of these connections and their associated neurons{{snd}} which require enormous [[w:en:Central processing unit|CPU]] power and time. [[w:en:Jürgen Schmidhuber|Schmidhuber]] noted that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by [[w:en:General-purpose computing on graphics processing units|GPGPUs]] (on [[w:en:Graphics processing unit|GPUs]]), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.<ref name="SCHIDHUB2" /> The use of accelerators such as [[w:en:Field-programmable gate array|FPGA]]s and GPUs can reduce training times from months to days.{{r|:0}} [[w:en:Neuromorphic engineering|Neuromorphic engineering]] or a [[w:en:physical neural network|physical neural network]] addresses the hardware difficulty directly, by constructing non-von-Neumann chips to directly implement neural networks in circuitry. Another type of chip optimized for neural network processing is called a [[w:en:Tensor Processing Unit|Tensor Processing Unit]], or TPU.<ref>{{cite news |url=https://www.wired.com/2016/05/google-tpu-custom-chips/ |author=Cade Metz |newspaper=Wired |date=18 May 2016 |title=Google Built Its Very Own Chips to Power Its AI Bots |access-date=5 March 2017 |archive-date=13 January 2018 |archive-url=https://web.archive.org/web/20180113150305/https://www.wired.com/2016/05/google-tpu-custom-chips/ |url-status=live }}</ref> ===Practical counterexamples === Analyzing what has been learned by an ANN is much easier than analyzing what has been learned by a biological neural network. Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful. For example, local vs. non-local learning and shallow vs. deep architecture.<ref>{{Cite web|title=Scaling Learning Algorithms towards AI|url=http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|access-date=6 July 2022|archive-date=12 August 2022|archive-url=https://web.archive.org/web/20220812081157/http://yann.lecun.com/exdb/publis/pdf/bengio-lecun-07.pdf|url-status=live}}</ref> ===Hybrid approaches=== Advocates of [[w:en:Hybrid neural network|hybrid]] models (combining neural networks and symbolic approaches) say that such a mixture can better capture the mechanisms of the human mind.<ref>{{Cite journal| last1=Tahmasebi| last2=Hezarkhani| title=A hybrid neural networks-fuzzy logic-genetic algorithm for grade estimation| year=2012| journal=Computers & Geosciences| pages=18–27 |volume=42| doi=10.1016/j.cageo.2012.02.004| pmid=25540468| pmc=4268588| bibcode=2012CG.....42...18T}}</ref> ==Gallery== <gallery widths="220"> File:Single layer ann.svg|A single-layer feedforward artificial neural network. Arrows originating from <math>\scriptstyle x_2</math> are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, <math>\scriptstyle y_q</math> would be calculated as <math>\scriptstyle y_q = K*(\sum(x_i*w_{iq})-b_q) </math>. File:Two layer ann.svg|A two-layer feedforward artificial neural network File:Artificial neural network.svg|An artificial neural network File:Ann dependency (graph).svg|An ANN dependency graph File:Single-layer feedforward artificial neural network.png|A single-layer feedforward artificial neural network with 4 inputs, 6 hidden and 2 outputs. Given position state and direction outputs wheel based control values. File:Two-layer feedforward artificial neural network.png|A two-layer feedforward artificial neural network with 8 inputs, 2x8 hidden and 2 outputs. Given position state, direction and other environment values outputs thruster based control values. File:Cmac.jpg|Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step. </gallery> == See also == {{cols|colwidth=18em}} * [[w:en:ADALINE|ADALINE]] * [[w:en:Autoencoder|Autoencoder]] * [[w:en:Bio-inspired computing|Bio-inspired computing]] * [[w:en:Blue Brain Project|Blue Brain Project]] * [[w:en:Catastrophic interference|Catastrophic interference]] * [[w:en:Cognitive architecture|Cognitive architecture]] * [[w:en:Connectionist expert system|Connectionist expert system]] * [[w:en:Connectomics|Connectomics]] * [[w:en:Large width limits of neural networks|Large width limits of neural networks]] * [[w:en:List of machine learning concepts|List of machine learning concepts]] * [[w:en:Neural gas|Neural gas]] * [[w:en:Neural network software|Neural network software]] * [[w:en:Optical neural network|Optical neural network]] * [[w:en:Parallel distributed processing|Parallel distributed processing]] * [[w:en:Philosophy of artificial intelligence|Philosophy of artificial intelligence]] * [[w:en:Recurrent neural networks|Recurrent neural networks]] * [[w:en:Spiking neural network|Spiking neural network]] * [[w:en:Tensor product network|Tensor product network]] {{colend}} ==Notes== {{notelist}} ==References== {{Reflist|30em}} ==Bibliography== {{colbegin|colwidth=30em}} * {{Cite journal| author=Bhadeshia H. 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H. | year=1999 |title=Neural Networks in Materials Science | journal=ISIJ International | volume=39 |pages=966–979 | doi=10.2355/isijinternational.39.966 | url=http://www.msm.cam.ac.uk/phase-trans/abstracts/neural.review.pdf| issue=10}} * {{Cite book|title=Neural networks for pattern recognition|last=Bishop|first=Christopher M.|date=1995|publisher=Clarendon Press|isbn=978-0-19-853849-3|oclc=33101074 }} * {{Cite book|title=Neuro-Fuzzy-Systeme : von den Grundlagen künstlicher Neuronaler Netze zur Kopplung mit Fuzzy-Systemen|first=Christian|last=Borgelt|year=2003|publisher=Vieweg|isbn=978-3-528-25265-6|oclc=76538146}} * {{cite book|title=Mathematics of Control, Signals, and Systems|author1-link=George Cybenko|last=Cybenko|first=G.V.|publisher=Springer International|year=2006|editor-last=van Schuppen|editor-first=Jan H.|chapter=Approximation by Superpositions of a Sigmoidal function|chapter-url={{google books |plainurl=y |id=4RtVAAAAMAAJ|page=303}}|pages=303–314|title-link=Mathematics of Control, Signals, and Systems}} [https://web.archive.org/web/20110719183058/http://actcomm.dartmouth.edu/gvc/papers/approx_by_superposition.pdf PDF] * {{Cite book|title=Yes, we have no neutrons : an eye-opening tour through the twists and turns of bad science|last=Dewdney |first=A. 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Ripley|publisher=Cambridge University Press|year=2007|isbn=978-0-521-71770-0}} * {{cite journal|last1=Siegelmann |first1=H.T. |first2=Eduardo D.|last2=Sontag|s2cid=2456483 |year=1994|title=Analog computation via neural networks |journal=Theoretical Computer Science |volume= 131 |issue= 2 |pages=331–360 |doi=10.1016/0304-3975(94)90178-3}} * {{Cite book|title=Neural networks for statistical modeling|last1=Smith |first1=Murray|date=1993|publisher=Van Nostrand Reinhold|isbn=978-0-442-01310-3|oclc=27145760}} * {{Cite book|title=Advanced methods in neural computing|last=Wasserman |first=Philip D.|year=1993|publisher=Van Nostrand Reinhold|isbn=978-0-442-00461-3|oclc=27429729}} * {{cite book|last1=Wilson|first1=Halsey|title=Artificial intelligence|date=2018|publisher=Grey House Publishing|isbn=978-1-68217-867-6}} {{colend}} {{Complex systems topics}} {{Control theory}} {{Differentiable computing}} {{Neuroscience}} {{Self-driving cars and enabling technologies}} {{Authority control}} {{DEFAULTSORT:Artificial Neural Network}} [[Category:Statistics]] [[Category:Artificial neural networks| ]] [[Category:Algorithms]] [[Category:Neuroscience]] [[Category:Psychology]] == Page Information == This page was based on the following [https://en.wikipedia.org/wiki/Artficial%20neural%20network wikipedia-source page]: * [https://en.wikipedia.org/wiki/Artificial%20neural%20network Artificial neural network] https://en.wikipedia.org/wiki/Artificial%20neural%20network * Datum: 3/11/2023 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity j96nt8ybux3f2lmr30u2kefqzez94xm 0 301465 2693851 2691852 2024-12-30T10:35:39Z Alandmanson 1669821 /* Potter wasps with a narrow petiole - maximum width of Tergum 1 less than half the maximum width of Tergum 2. */ 2693851 wikitext text/x-wiki ==Potter Wasps of Africa (Eumeninae)== In 2009 and 2010, Carpenter et al. published a catalogue of African Eumeninae.<ref name=Carpenter2009a>Carpenter, J. M., Gusenleitner, J., & Madl, M. (2009). A catalogue of the Eumeninae (Hymenoptera: Vespidae) of the Ethiopian region excluding Malagasy subregion. Part I: Introduction, Key to Genera, Genera ''Aethiopicodynerus'' Gusenleitner 1997 to ''Cyrtolabulus'' van der Vecht 1969. Linzer Biologische Beiträge, 41(1), 513-638.</ref><ref name=Carpenter2010a>Carpenter, J. M., Gusenleitner, J., & Madl, M. (2010). A catalogue of the Eumeninae (Hymenoptera: Vespidae) of the Ethiopian region excluding Malagasy subregion. Part II: Genera ''Delta'' de Saussure 1885 to ''Zethus'' Fabricius 1804 and species incertae sedis. Linzer Biologische Beiträge, 42(1), 251-253.</ref><ref name=Carpenter2010b>Carpenter, J. M., Gusenleitner, J., & Madl, M. (2010). A catalogue of the Eumeninae (Hymenoptera: Vespidae) of the Ethiopian Region excluding Malagasy Subregion. Part III: Classification, additions, corrections and index. Linzer Biologische Beiträge, 42(2), 919-1004.</ref> In their introduction they wrote: "Since 1918, the date of Bequaert`s fundamental paper on the Vespidae of the Ethiopian Region, many of papers dealing with taxonomy, biology or faunistics of the Eumeninae have been published. As the authors have had different opinions concerning the definition of taxa, the taxonomy of the Ethiopian Eumeninae is in a confusing state. The aim of this catalogue is to summarize the present knowledge for a revisionary treatment in the future." The catalogue included a key to the African genera, but it has 81 steps (more than 80 genera), and is difficult to use without a specimen under a microscope. Very few people, therefore, can reliably identify many of the genera from photographs. Fortunately, since 2018, Marco Selis (https://www.researchgate.net/profile/Marco-Selis/research) has been publishing modern reviews of various African genera that include photographs of most of the species in that genus. However, the task of reviewing all African genera is huge and far from complete; there are still many genera that require review. The photographs below show some of the diversity in this subfamily. ===Potter wasps with a narrow petiole - maximum width of Tergum 1 less than half the maximum width of Tergum 2.=== <gallery mode="packed" heights="200"> Afreumenes aethiopicus Deacon 2019 iNat25934867.jpg|''Afreumenes aethiopicus'' Afreumenes nestbuilding iNat92098667.jpg|''Afreumenes'' sp. building a nest Cyrtolabulus 2019 12 24 iNat37096936.jpg|''Cyrtolabulus'' sp. Potter Wasp (Delta emarginatum) filling a cell with caterpillar ... (31187442161).jpg|''Delta emarginatum'' Eumenes sp iNat 112403541 02.jpg|''Eumenes'' sp. visiting ''Persicaria capitata'' flowers Katamenes niger iNat71941388.jpg|''Katamenes niger'' Micreumenes iNat112393033 01.jpg|''Micreumenes'' sp. Paramischocyttarus buyssoni 2024 12 18 iN 256503707 c.jpg|''Paramischocyttarus buyssoni'' Pareumenes inat144303712 01.jpg|''Pareumenes sansibaricus'' Pseudonortonia iNat66914604 01.jpg|''Pseudonortonia'' sp. building a nest Zetheumenidion iNat99241223 02.jpg|''Zetheumenidion'' sp. Zethus iNat 11093897 a.jpg|''Zethus'' sp. </gallery> ===Potter wasps with the first segment of the gaster (metasoma) broad at the posterior end - maximum width of Tergum 1 more than half the maximum width of Tergum 2.=== <gallery mode="packed" heights="200"> 1 Alastor iNat 187798822 c.jpg|''Alastor'' sp. Allepipona iNat63250777.jpg|''Allepipona'' sp. nest Ancistrocerus Jan Ebr 2021 iNat105674275.jpg|''Ancistrocerus'' sp. Antepipona iNat37097420 04.jpg|''Antepipona'' sp. building a nest Anterhynchium grayi Wynand Uys 2013 inat10826034.jpg|''Anterhynchium grayi'' Anterhynchium fallax on Jatropha gossypiifolia (6741400895).jpg|''Anterhynchium fallax'' Antodynerus spoliatus iNat91099064 01.jpg|''Antodynerus spoliatus'' visiting a fairy bellflower Chlorodynerus iNat15112212 Aug 4, 2018.jpg|''Chlorodynerus'' sp. Eumenidiopsis iNat81828042 01.jpg|''Eumenidiopsis'' sp. Gibberhynchium masariforme iNat19784538.jpg|''Gibberhynchium masariforme'' Gioiella i c riddell 2015 iNat74033114 01.jpg|''Gioiella'' sp. Knemodynerus iNat 73013421 a.jpg|''Knemodynerus'' sp. Red-marked Pachodynerus - Pachodynerus erynnis, Dinner Island Ranch Wildlife Management Area, Florida.jpg|''Pachodynerus erynnis'' Paravespa iNat12988529.jpg|''Paravespa'' sp. Proepipona iNat35158982 04.jpg|''Proepipona'' sp. Rhynchalastor iNat 11014978 a.jpg|''Rhynchalastor'' sp. Rhynchium cf marginellum iNat110386778 01.jpg|''Rhynchium'' sp. Stellepipona iNat33000367.jpg|''Stellepipona'' sp. Stroudia iNat 21598854 b.jpg|''Stroudia'' sp. Synagris cf maxillosa male iNat147325729 03.jpg|''Synagris'' cf ''maxillosa'' Tricarinodynerus guerinii Paapi 2021 iNat102548983.jpg|''Tricarinodynerus guerinii'' </gallery> ==References== {{reflist}} [[Category:African Arthropods|Eumeninae]] b19i5xrcfpxteestwhtkjsjn5wf62tl 0 301480 2693850 2693668 2024-12-30T10:17:02Z Private lecturer (celestial) 2975755 /* Will science allow us to gain all the magic of heaven and do without it? */ difficult to falsify, but natural science works through falsification of false theories 2693850 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 [[w:Consensus_democracy|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. An easy explanation would be that theories about the [[w:otherworld|otherworld]], for instance, are difficult to falsify, but natural science works through falsification of false theories, thus is not helpful in explaining the otherworld. === 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]] bsdlauwcqqwhrfb8y67cquomrwb4as1 3 304127 2693823 2683322 2024-12-29T23:40:27Z Dc.samizdat 2856930 /* The dimpled fabric of spacetime */ 2693823 wikitext text/x-wiki == Galilean relativity in a space of four orthogonal dimensions == The paragraph ending in "That is all of physics" came to me on my mother's 101st birthday, just over a year after she died, like an example of rebirth in the spring. I say rebirth because it is probably not original, and even if it is, life must necessarily forget it someday, and just as necessarily rediscover it again someday afterwards. It seems to me to be as close an example to ''immortal life'' as I have encountered. That state, oxymoronic in nearly all senses I fear, may be said to exist where life discovers an immortal principle, and so (like the 11-cells) many lives distinct but not disjoint combine to give existence to an immortal object, where none could exist at all in singularity. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:44, 14 March 2024 (UTC) == Galaxies == What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 16:55, 10 November 2024 (UTC) == The dimpled fabric of spacetime == 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. 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 a 3-sphere expanding radially at velocity <math>c</math>. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:15, 10 November 2024 (UTC) pqbvq6ja2r6fzx1j5q9ho2t2azdihle 0 307962 2693836 2692728 2024-12-30T00:36:59Z Scogdill 1331941 2693836 wikitext text/x-wiki =Event= On Monday 28 June 1897, Queen Victoria hosted a garden party at Buckingham Palace, inviting between 5,000 and 6,000 people. This party was the final official event of the London Diamond Jubilee celebrations. The Queen released to the press the names of people invited, which means the newspapers could print some or all of this list. The very long article in the London ''Morning Post'', for example, prints what may be the comprehensive list of those invited, although two columns are illegible in some places. The original newspaper account seems to have been published by the ''Court Circular'', and then the popular newspapers reprinted pieces of that story, many adding contextualizing paragraphs of their own. Some of these later reports are quite long, perhaps 5 or more full columns. Sometimes the newspapers included short descriptions of the women's dresses, suggesting that for the list of people invited, the source was the ''Court Circular'', but the parts of the stories devoted to context, history or fashion might have been written by a reporter present at the event. ==Logistics== * 28 June 1897, Monday, in the gardens at Buckingham Palace, hosted by Queen Victoria. * Between 5,000 and 6,000 guests were invited. * Many visitors from the empire who were in town for the Jubilee celebrations were invited to this garden party. * The weather was fine, having improved since the day before. * The garden party was held in the grounds around Buckingham Palace, and the Palace itself was open and available for guests to visit:<blockquote>Great preparations had been made in the splendid grounds adjoining the Royal Palace for the party, the whole scene presenting a fascinating appearance. The beautifully-kept grounds were partially covered with tents and marquees for the convenience of the many guests, and the lovely lake was really in the hands of the Queen’s bargemen, who had charge of the many boats which had been placed on the extensive ornamental waters for the use of guests. There was also plenty of music, several regimental bands being in attendance, while for those who wished to become acquainted with the valuable pictures and works art which are to be found at the Royal residence, all the State and reception rooms of the Palace were thrown open.<ref name=":2">“The Queen’s Garden Party. Brilliant Scene at Buckingham Palace.” ''Globe'' 29 June 1897, Tuesday: 6 [of 8], Col. 3a–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970629/050/0006. Print p. 6.</ref> (6, Col. 3a)</blockquote> *The streets around the entrances to Buckingham Palace were lined with spectators beginning hours before the Queen was to arrive:<blockquote>Although the Garden Party was not timed to commence until after five o’clock, the Mall from Marlborough House to Buckingham Palace was well lined by two o’clock, and an hour afterwards large crowds, for the most part composed of ladies, had taken up their positions. This was also the case along Constitution-hill, where the assembly which had gathered to witness the Queen’s arrival at the Palace from Windsor nad [sic] to a large extent remained. The heat was somewhat oppressive, but the trees along the Mall and the Green Park afforded welcome shelter. Many ladies had evidently come prepared for a long wait, as they had provided themselves with the now familiar camp stool, which is always prominent on these occasions. On the other hand, the police were waging war against the men who frequent such places with stools and forms, and as soon as any of them put in an appearance they were quickly pounced upon by the officers, who at once proceeded to destroy the intended stands before the eyes of the helpless owners. Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.<ref name=":2" /> (6, Col. 3a)</blockquote> ==Related Events== This garden party was the culminating event of the official celebrations for Queen Victoria's Diamond Jubilee, and more specific events led up to it: # Trip from Windsor to Paddington Station Queen Victoria and a large retinue traveled by train from Windsor to Paddington Station the day before, preceded on an earlier train by "the royal equipages sent from Buckingham Palace for the use of the Queen and her suite," which were<blockquote>First came the splendied semi-state landau in which the Queen made her now famous journey on June 22d. It was preceded by scarlet-coated outriders, and horsed by four magnificent bays driven by postilions in navy blue and white uniforms. Two similar carriages followed, and these were in turn succeeded by a number of pair-horse clarences for the conveyance of the household and suite, and several breaks and ‘buses for luggage. A captain's escort, furnished by the 2d Life Guards, and commanded by Captain Ellison, clattered along in rear of the carriages, and took up a position opposite the spot where, by prior arrangement, Her Majesty’s saloon was to be brought to a standstill. These magnificent troops, riding their great black horses, and with the sunlight dancing upon their nodding plumes, and reflected by their burnished helmets, cuirasses, and trappings, made a very fine show indeed. The escort did not carry the colour, as it did on the 21st, nor was it accompanied by the regimental trumpets.<ref name=":0">"Jubilee Festivities. The Queen Again in London. Interesting Functions. A Visit to Kensington. The Garden Party." ''North British Daily Mail'' 29 June 1897, Tuesday: 5 [of 8], Col. 3a–7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002683/18970629/083/0005. Print p. 5.</ref>{{rp|5, Col. 3b}}</blockquote> # Reception at Paddington # Visit to Kensington # Kensington to Buckingham Palace # The Garden Party # Return to Windsor by Way of Paddington === Foreign Admirals === On 29 June 1897, the day after the garden party, the ''North British Daily Mail'' reports that, after the Queen's garden party, the foreign admirals would return to Spithead for a tour around the dockyard and luncheon:<blockquote>THE FLEET AT SPITHEAD<p> The fleet at Spithead was again illuminated last night, the railway companies having duplicated the ordinary train service to bring visitors down. The Koenig Wilhelm was to have sailed on Sunday evening, but her departure has been deferred, and last night her officers gave a private dinner party aboard for the anniversary of the Queen’s Coronation. All the commissioned ships in the harbour were dressed at noon. A royal salute was fired. The [Col. 6c–7a] foreign admirals will return from their visit to London on the occasion of the Queen’s garden party to be conducted round the dockyard to-day, and they will be entertained to luncheon.<ref name=":0" />{{rp|5, Col. 6c–7a}}</blockquote> === Colonial Premiers === The day of the garden party the colonial premiers attended a meeting with Secretary of State for the Colonies, [[Social Victorians/People/Chamberlain|Joseph Chamberlain]]:<blockquote>THE COLONIAL PREMIERS The whole of the Colonial Premiers went to the Colonial Office yesterday for further conference with Mr Chamberlain, who received them in his private room, attended by Mr F. H. Wilson, legal assistant, Mr Reid and the Hon. T. Cochrane, M.P., assistant private secretaries. The conference lasted hours, and was of a strictly private and confidential character, the matters discussed involving several points of high State policy. Premiers will be entertained at Warwick Castle by the Earl and Countess of Warwick on July 15th. On the same occasion the Attorney General of Queensland will present a loving cup from Warwick, in Queensland, to the old county town of Warwick, from which it takes its name. He will be accompanied by the Colonial troopers.<ref name=":0" />{{rp|5, Col. 7a}}</blockquote> For these visitors to London during the Diamond Jubilee, the next major social event was on 15 July, at Warwick Castle, hosted by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick and Francis, 5th Earl of Warwick]], although perhaps some attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]]. == Who Was Present == In the absence of a copy of the report about the garden party in the ''Court Circular'', the newspaper account with the fullest list of names is from the ''Morning Post'', although people further down the list can be impossible to identify, and two full columns are damaged (Col. 7 on p. 4 and Col. 1 on p. 5).<ref name=":1">“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.</ref> Whenever possible, then, what is here has been amended with other newspaper reports that have names to help decipher the illegible ones in the ''Morning Post'' account. The names in the ''Morning Post'' are grouped, mostly by rank and name. If it were not for the many military men present, the women would have outnumbered the men by quite a lot. === People of Color at This Event === One purpose of a closer look at this event is to get a more precise list of names of people of color from the various countries in the empire, who were not recognized and thus not named in newspaper descriptions of other events. For example, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] was said to include a number of South Asian dignitaries, but because the Duchess did not release to the newspapers the names of those who were invited, those dignitaries went mostly unnamed in the newspaper reports, if their presence was noted at all. Besides the South Asian guests invited to this garden party, some South Asian visitors to London were spectators as well:<blockquote>Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.<ref name=":2" /> (6, Col. 3a)</blockquote>In a section on what people — mostly women — wore, the reporter for the ''Daily News'' said,<blockquote>Suffice to say, the modistes had done their best, and that their achievements excited general admiration. Here and there, however, was an Eastern beauty whose golden lace drapery, loosely enveloping a figure that owed nothing to the corset, challenged comparison, we will not say with what success, with the European model. In the almost entire absence of uniforms or Court dress, the costumes of the East Indian notables lent colour to the assemblage, while their pearls and diamonds, the wealth of Ormuz and of Ind, were not allowed to pass unobserved.<ref name=":3" /> (5, Col. 6b)</blockquote> === People Invited === # Queen Victoria, with escort and attendants ## Captain's Escort of the 2nd Life Guards ## The Duchess of Buccleuch, Mistress of the Robes ## The Dowager Lady Churchill, Lady in Waiting ## The Hon. Harriet Phipps, Woman of the Bedchamber ## Maids of Honour in Waiting ### The Hon. Mary Hughes ### The Hon. Aline Majendie ## the Earl of Kintore, Lord in Waiting ## Captain Drummond, Groom in Waiting ## Equerries in Waiting ### Major-General Sir John M'Neill, V.C. ### Lieutenant Colonel Davidson, M.V O. [sic] #Grand Duke and Grand Duchess Serge of Russia #Princess Henry of Battenberg, with attendants ##Miss Minnie Cochrane ##Colonel John Clerk, C.S.I., C.V.O. #Her Imperial Majesty the Empress Frederic, attended by ##the Dowager Lady Ampthill ##Lord Harris ##Colonel S. Waller ##Princess Hatzfeldt Trachenberg ##Count Seckendorff ##Baron and Baroness Reischach #Their Royal Highnesses the Prince and Princess of Wales, with attendants ##Lady Suffield, Lady in Waiting ##Miss Knollys, Woman of the Bedchamber ##Lord Colville of Culross, K.T., G.C.V.O., Chamberlain to the Princess of Wales ##The Earl of Gosford, K.P., Lord in Waiting ##General Sir D. Probyn, G.C.V.O., K.C.B., K.C.S.I., V.C, Comptroller ##Sir Francis Knollys, K.C.M.G., C.B., Groom in Waiting ##Major-General Stanley Clarke, C.M.G., Equerry in Waiting #Princess Victoria of Wales #Their Royal Highnesses Prince and Princess Charles of Denmark #Their Royal Highnesses the Grand Duke and Grand Duchess of Mecklenburg-Strelitz, attended by ##Lady Caroline Cust ##Mr. Hugo Erskine Wemyss ##Count Reventlow Criminil ##Baron von der Wense #Their Royal Highnesses Prince and Princess Christian, attended by ##Baroness von und zu Egloffstein ##Colonel the Hon. Charles Eliot #Her Highness Princess Victoria #His Highness Prince Christian Victor #His Highness Prince Albert of Schleswig-Holstein #Her Royal Highness Princess Louise Marchioness of Lorne and the Marquis of Lorne, attended by ##Lady Sophia Macnamara ##[[Social Victorians/People/Arthur Collins|Colonel Arthur Collins]], M.V.O. #Their Royal Highnesses Prince and Princess Henry of Prussia, attended by ##Admiral of the Fleet Sir Edmund Commerell ##Baron and Baroness Seckendorff ##Count Hahn ##Captain Muller #Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, attended by ##The Hon. Mrs. Monson ##His Excellency Herr von Schön ##Captain the Hon. D. J. Mouson [sic, s/b Monson?], M.V.O. ##Mr. A. D. J. Monson ##Captain von Ruxleben #Princess Beatrice of Saxe-Coburg and Gotha #The Hereditary Prince of Saxe-Coburg and Gotha #Their Royal Highnesses the Duke and Duchess of Connaught and Strathearn, attended by ##Colonel and the Hon. Mrs. A. Egerton #Her Royal Highness the Duchess of Albany, attended by ##Sir Robert and Lady Collins ##Miss Potts #Her Royal Highness Princess Frederica of Hanover and Baron von Pawel Raminingen, attended by ##Mr. and Mrs. Charles Wood #His Royal Highness the Duke of Cambridge, attended by ##Colonel A. C. FitzGeorge, C.B. #Her Royal Highness the Duchess of Teck and his Highness the Duke of Teck, attended by ##Lady Katherine Coke ##The Hon. A. Nelson Hood #Her Royal Highness Princess Louise Duchess of Fife and the Duke of Fife #His Highness the Prince and her Royal Highness Princess Frederic Charles of Hesse, attended by ##The Hon. A. Hay ##Fraulein von Tasmund ##Baron von Kotwitz #Their Highnesses Prince and Princess Aribert of Anhalt, attended by ##Miss Deverell ##Major Evan Martin #Her Royal Highness the Hereditary Princess of Saxe-Meiningen and her Serene Highness Princess Feodore of Saxe-Meiningen, attended by ##The Hon. Aubrey FitzClarence ##Miss von Dreskan ##Baron von Roeder #His Serene Highness the Prince of Schaumburg-Lippe #Their Highnesses Prince and Princess Edward of Saxe-Weimar #Her Serene Highness Princess Victor of Hohenlohe #Countess Gleichen (x2) #Their Serene Highnesses Prince and Princess Adolphus of Teck #The Prince Francis and Prince Alexander of Teck #His Highness Prince Augustus Leopold of Saxe-Coburg #Their Serene Highnesses Prince and Princess Blucher von Wahlstatt #Their Serene Highnesses Prince and Princess Joachim Murat #Their Serene Highnesses [[Social Victorians/People/Pless|Prince and Princess Hans Henry Pless]] #Prince and Princess Loewenstein #Their Serene Highnesses the Duke and Duchess of Arenberg #Prince Victor Duleep Singh #Prince Frederick Duleep Singh #Princess Duleep Singh (x2) #ARGENTINE REPUBLIC — M. Florencio Dominguez and M. Carlos Dominguez #BADEN — Herr yon Brauer, Mr. Brook Taylor, and Baron Bohlen Halbach #BAVARIA — His Royal Highness the Prince Rupert, General Sir L. Gardiner, K.C.V.O., C.B., Major Fairholme, Lieutenant-Colonel Emile von le Bret Nucourt, and Captain Othon von Stettin #BELGIUM — His Serene Highness the Prince Charles de Ligne, Princess de Ligne, Madlle. de Ligne, Mr. C. lnnes Ker, Count de Jonghe d'Ardoye, and the Marquis d’Asshe #BOLIVIA — M. Caso, Mr. Conway Seymour, M. Pedro Suarez, Madame Suarez, and M. Adolfo Bolivian #BRAZIL — M. [[Social Victorians/People/Souza Correa|de Souza Correa]] [Corréa?] #BULGARIA — Their Royal Highnesses the Prince and Princess of Bulgaria, Colonel J. R. Slade, C.B., Madame Petrow Tchomakoff, Count Robert de Bourboulon, Lieutenant-Colonel Marcoff, Major Petrew, Captain Stoïanow, and Mr. Martin Furth #CENTRAL AMERICA (Greater Republic) — M. Medina and Miss Medina #CHILI — M. Ramon Subercasseaux and Mr. Raglan Somerset #CHINA — His Excellency Chang Yen Hoon, Colonel Mark Bell, V.C,. Mr. Liang, Mr. Jui, and Mr. Koo #COREA — His Excellency Min Young Hwan, Major A. Cavendish, Mr. Min Young Chan, Mr. Min Shangho, and Mr. von Rautenfeld #COSTA RICA — Senor Don Demetrio Iglesias, Mr. C. Alban Young, Dona Eudoxia Castro, Señorita Maria Iglesias, Don Ricardo Fernandez Guardia, and Dona Christina Castro Keith #DENMARK — His Royal Highness the Prince Waldemar, Major-General Arthur Ellis, C.S.I., M. Charles Rothe, and Captain Evers #EGYPT — Prince Mohammed Ali Pasha, Colonel Larking, Tigrane Pasha, Colonel Aziz Bey, Mr. George Smart, Said Zoulfikar Bey #ECUADOR — M. Navares, Colonel Concha #FRANCE — General Davoust, Duc d'Auerstadt, Duchesse d'Auerstadt, and Madlle. Davoust, Colonel Brabazon, Colonel Dawson, General Hagron, M. Crozier, Colonel Humbert, and Captain Riviers de Mauny #GERMANY — His Royal Highness the Prince Albert of Prussia, Prince Regent of Brunswick, Major-General Sir C. du Piat, K.C.B., Colonel Grierson, Lieutenant-General von Plessen, Colonel von Arnim, Captain Fischel, Count von der Schulenberg (Hofmarschall), Major Freiherr von Stein, Dr. Schreibe, Captain von Unzer #GREECE — M. Rangabi, Mr. R. D. Norton #GUATEMALA — Dr. Cruz, Madlles. Cruz (2), Señor Estrada #HAWAIIAN ISLANDS — Mr. S. M. Damon, Captain the Hon. H. Napier, Major Curtis P. Jaukea #HESSE — Their Royal Highnesses the Grand Duke and Grand Duchess of Hesse, Colonel the Hon. H. Byng, C.B., Baroness de Grancy, Baron Riedesel zu Eisenbach, Baron de Genadius Grancy #ITALY — Their Royal Highnesses the Crown Prince and Princess, the Earl of Clarendon, Colonel Needham, Countess Giulia Trigona, Lieutenant-General Terzaghi, Major Cavaliere Viganoni, Captain Cavaliere Merli Miglietti, Count Romnaldo Trigona, Cavaliere F. Comotto #JAPAN — His Imperial Highness the Prince Arisugawa, Mr. R. F. Synge, Captain Beaumont, R.N., Marquis Ito, Mr. S. Saito, Marquis Kido, Captain Funaki, Lieutenant-Colonel Murata, Lieutenant Kato, Mr. Nabeshima #LIBERIA — Mr. H. Hayman #LUXEMBURG — His Royal Highness the Hereditary Grand Duke of Luxemburg, Colonel H. D. Browne, Baron Ritter yon Grünstein #MECKLENBURG-SCHWERIN — His Excellency Herr D. yon Vietinghoff, Mr. Eyre A. Crowe #MEXICO — Don Antonio Mier y Celis, Mr. Arnold Royle, C.B., Don Francisco R. Gallardo, Don Eustagino dc Escaudon, and Captain Don Ponfirio Diaz #MONTENEGRO — His Highness the Prince Danilo, Major the Hon. C Harbord, Colonel Djurcovitch, and Captain Pejanovitch #NETHERLANDS — Count van Lynden, Countess van Lynden, Mr. Horace West, and Count W. de Bylandt #PARAGUAY — M. E. Machain and Madame Machain #PERSIA — His Imperial Highness the Prince Amir Khan, General Sir Thomas Gordon, K.C.I.E., C.B., C.S.I.[,] Mr. Harry Churchill, General Karim Khan, Mirza Ahmad Khan, Mirza Ohaness Khan, Mirza Mohamad Ali Khan #PERU — Senor Canevaro, Duchesse de Zoagli Canevaro, Dr. Don A. N. Puente, Don Alfredo Elster, and Don Carlos von der Heyde #PORTUGAL — His Royal Highness the Duke of Oporto, Major the Hon. H. C Legge, M.V.O., Colonel Duval Telles, Captain Moreira de Sà, Major d'Albuquerque, and Lieutenant Jose de Melie[?] #ROME — Right Rev. Monsignore Sambucetti, [[Social Victorians/People/Stonor|Hon. Harry Stonor]], Right Rev. Monsignore Belmont, the Right Rev. Monsignore de Vaz, Marchesi and Marchesa Muccioli, of the Noble Guard #ROUMANIA — General Pancovici, Colonel G. P. Georgescu #RUSSIA — Their Imperial Highnesses the Grand Duke Serge and Grand Duchess Feodrowna, the Grand Duke Cyril, Lord Churchill, Lieutenant-Colonel Waters, Countess Olsouffiew, Princess Youssoupoff, Princess Lobanoff de Rostow, General Stépanoff, Colonel Gadon, and Prince Youssoupoff, Colonel Clements, Mr. Alexander Gordon Ross, and Sub-Lieutenant N. Coubé (A.D.C. to Grand Duke Cyril) #SAXE-COBURG — His Royal Highness the Prince Philip of Saxe-Coburg, Captain Walter Campbell, and Herr von Schön #SAXE-WEIMAR — His Highness the Prince Hermann of Saxe-Weimar, Mr. Frederick Campbell, and Count Zeppelin #SAXONY — His Royal Highness the Prince Frederick Augustus, Duke of Saxony, Colonel Howard, Freiherr yon Reitzenstern, First Lieutenant von Metzsch, and Baron von Oppell #SERVIA — M. Mijatovich and Madame Mijatovich #SIAM — His Royal Highness the Crown Prince and the Prince Mahit of Siam, Colonel E. H. Sartorius, V.C., Lieutenant-Colonel Rajavallabha, Lieutenant-Colonel C. Vernon Hume, Colonel Indaraty, Surgeon-Major Yarr #SPAIN — Duke of Sotomayor, Captain the Hon. A. Greville, Señor José Caro, Señor Alfonso Merry del Val, and Señor Benitez al Villar #SWEDEN AND NORWAY — His Royal Highness the Prince Eugène of Sweden and Norway, Captain G. L. Holford, Count G. Gyldenstolpe, Captain Roeder, Captain Baron Cederstrom #TURKEY — Munir Pasha, Major Surtees, Brigadier-General Nassir Pasha, Captain Enver Bey, Colonel Gordon Ponsonby #UNITED STATES — His Excellency the Hon. Whitelaw Reid, Mrs. Whitelaw Reid, Colonel Hallam Parr, Major-General Nelson A. Miles, Mrs. Nelson Miles, Rear-Admiral Joseph N. Miller, Captain M. P. Maus, Mr. Ogden Mills, Mrs. Ogden Mills, Mr. G. Creighton Webb, Mr. Erskine Hewett, Commander W. H. Emory, Lieutenant Philip Andrews, Lieutenant T. S. Rogers #URUGUAY — Dr. Alberto Nin, Madlle. Nin, Don Alfonso Saenz de Zumaran, Don Luis Posadas, Colonel C. Robido #WURTEMBURG— His Royal Highness the Duke Albert of Wurtemburg, Colonel C. Swaine, Lieutenant-General von Bilfinger, First Lieutenant Count von Degenfeld- Schonburg; five officers of the Queen's German Regiment: Major C. R. Burn (in attendance), Lieutenant-Colonel von Falkenhayn, Major von Arnim, First Lieutenant Baron von Moeller-Lilienstern, First Lieutenant von Gerlach, Second Lieutenant von Studnitz #"Native Princes, and gentlemen and ladies accompanying them"<ref name=":1" /> (4, Col. 2b) ##His Highness the Raja of Kaparthala ##His Highness the Thakur Sahib of Morvi, K.C.I.E. ##His Highness the Thakur Sahib of Gondal, C.I., and her Highness the Maharani of Gondal, C.I. ##Colonel Maharaj Dhiraz ##Sir Pratab Singh, K.C.S.I. ##Thakur Hari Singh[,?] ##Kunwar Dhokal Singh ##Rajah Ajit Singh of Khetri, attended by ##Rajkumar Unmaid Singh of Shahpura, attended by ###Colonel Trevor (in attendance upon the Rajah Ajit Singh of Khetri and the Rajkumar Unmaid Singh of Shahpura) ##Bijey Singh ##Sir Jamaetjee Jejeebhoy, Bart., C.S.I., Miss Jejeebhoy, Mr. Jejeebhoy ##Mr. and Mrs. Powrala ##Major J. G. Turner and Mrs. Turner ##Mr. A. R. Wood and Mrs. Wood #The "officers of the Imperial Service Troops, with British officers and ladies"<ref name=":1" /> (4, Col. 2b) ##Captain Mir Hashim Ali Khan Hyderabad-Resaldar ##Major Sunayat Singh, Kashmir ##Commandant Abdul Ganny, Gwalior ##Commandant Gooind, Rao Matkar, Indore ##Commandant Mirza Kurim Beg, Bhopal ##Rai Bahadur Dhunpat Rai, Jeypore ##Commandant Nand Singh, Patiala ##Commandant Rai Bahadur Thakur Dip Sing, Bikanir ##Commandant Chatru Singh, Bhartpur ##Resaldar Abdul Majid Khan, Babawalpur ##Commandant Daud Khan, Ulwar ##Commandant Nazir Khan, Rampur ##Risalda-Major Didar Singh, Sindi ##Risaldar-Major Kishan Singh, Nabha ##Risaldar Hara Singh, Karpurthala ##Risaldar Dhan Singhi, Bhavnagar ##Colonel H. Melliss, C.S.I., and Mrs. Melliss ##Major F. H. R. Drummond and Mrs. Drummond ##Captain F. Angelo ##Lieutenant H. Coape-Smith ##Captain G. F. Chenevix-Trench #The "officers of Native Cavalry Corps with British officers and ladies"<ref name=":1" /> (4, Col. 2b) ##Risaldar-Major Baha-ud-din-Khan ##Sardar Bahadur, A.D.C. to Viceroy ##Risaldar-Major Sayyid Abdul Aziz ##Risaldar-Major Khan Bahadur ##Risaldar-Major Izzat Khan ##Risaldar-Major Hukam Singh ##Risaldar-Major Sher Singh ##Risaldar-Major Husain Khan ##Risaldar-Major Mangal Singh ##Risaldar-Major Kesar Singh ##Risaldar- Major Faiz Khan ##Risaldar-Major Muhammad Umar Khan ##Risaldar-Major Ali Mahomed Khan ##Risaldar-Major Mihrab Ali Khan ##Risaldar Kaddam Khan ##Risaldar Jahanzir Khan ##Risaldar Nadir Khan ##Risaldar Mir Haidar Shah Khan ##Risaldar Makbul Khan ##Risaldar Net Ram ##Ressaidar Gurdatt Singh ##Subadar Muhammed Beg Junadar ##Abdul Karin Khan ##Lieutenant-Colonel J. C. H. Gordon and Mrs. Gordon ##Major A. Phayre and Mrs. Phayre ##Captain C. F. Campbell ##Captain P. Melville, in attendance on his Highness Thakur Sahib of Morvi ##Captain M'Cartney Filgate, in attendance on their Highnesses the Thakur Sahib and Maharani of Gondal ##Mr. Nowroz ##M. Parveez ##Sir M. Mansherjee Bhownaggree, M.P. ##Mr. Percy Armytage and Mrs. Armytage ##Mr. Frank Cook, C.I.E., and Mrs. Frank Cook #The "commanding officers of Colonial contingents, with the ladies accompanying them"<ref name=":1" /> (4, Col. 2b) ##Colonel the Hon. M. and Mrs. Aylmer, Canada ##Colonel and Mrs. Lassetter, New South Wales ##Major Reay, Victoria ##Colonel Pitt, New Zealand ##Major and Miss King, Queensland ##Lieutenant and Mrs. Phillips, Cape of Good Hope ##Lieutenant-Colonel Rowell, South Australia ##Major Strickland, Western Australia ##Captain Shepstone, Natal ##Major and Miss Reeves, Ceylon ##Mr. Badeley, Hong Kong ##Colonel Walker, C.M.G., and Mrs. Walker, Straits Settlements ##Captain Lucie Smith, Jamaica ##Lieutenant-Colonel E. B. M'lnnis, C.M.G., and Mrs. M'lnnis, British Guiana ##Major Rooks, Trinidad ##Captain Bernard, Malta ##Captain Kershaw, Cyprus ##Captain and Mrs. Middlemist, Gold Coast ##Inspector Hook, Lagos ##Captain Blakeney, Sierra Leone ##Lieutenant Festing, Royal Niger Company ##Captain Flint, British North Borneo Company ##The Hon. M. Gifford, Rhodesian Horse ##The following British officers attached: Lieutenant-Colonel Boulton, Lieutenant-Colonel Prior, Lieutenant-Colonel Tucker, Lieutenant-Colonel Domville, Lieutenant-Colonel Gibson, and Lieutenant-Colonel Tyrwhitt #The "gentlemen representing the various races in the Island of Ceylon"<ref name=":1" /> (4, Col. 2c) ##Maha Mudaliyar don Solomon Dias Bandaranaihe ##The Hon. Alexander Dealius Sonewiratne ##M. E. Rowland Goonoratne ##M. Charles de Soysa Dessanayaka ##Panabokko Jikiri Banda ##Nugawela Kuia Banda ##Kobbokeduwe Loku Banda ##M. E. S. W. Senathi rajah [sic] and Mrs. Senathi ##M. J. H. de Saram and Miss de Saram ##M. P. Ramanathan ##M. Saunders and Miss Saunders #The "members of the Corps Diplomatique and other foreigners of distinction"<ref name=":1" /> (4, Col. 2c) ##The Russian Ambassador, Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy ##The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery ##The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy ##The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy ##The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy ##The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy ##The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy ##The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador ##The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation ##The Persian Minister, and one Secretary of Legation ##The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation ##The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation ##The Liberian Minister ##The Roumanian Minister and the Councillor of the Legation ##The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation ##The Belgian Minister, the Councillor, and two Secretaries of Legation ##The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation ##The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation ##The Minister for Sweden and Norway, Countess Lewenhaupt, and the Attaché of Legation ##The Chinese Minister, Lady Macartney, the English Secretary, three Secretaries, and four Attachés of Legation ##The Portuguese Minister, Madlle. de Quilinan, three Secretaries, and one Attaché of Legation ##The Swiss Minister, Madame Bourcart, Madame de Salis, the Secretary, and the Attaché of Legation ##The Haytian Chargé d’Affaires ##The Chargé d’Affaires of Greece, Madame Metaxas, and the Attaché ##The Chargé d’Affaires of Chile and Madame Bascunan ##Two Secretaries and one Attaché of the Brazilian Legation ##Count E. van Rosen ##Mr. Hippolyte de Aranjo ##Vice-Admiral Montt ##Mr. Pinto, Mrs. Pinto ##Mr. and Mrs. Scaramanga ##Vicomte de Galard ##Dr. Arnold, and Madlle. von Rappoport ##Mrs. John Meiggs, Miss Meiggs ##Miss Margaret Butler ##Mrs. Henry Morgan ##Hon. Chauncey Depew ##Mr. and Mrs. James Taylor ##Mr. and Mrs. Charles Marshall ##Mr. and Mrs. Edmund Bayliss ##Mrs. Colgate ##Miss Furniss ##Miss Wells ##Miss Harris ##Hon. Levi P. Morton, Mrs. Morton, and the Misses Morton ##The Bishop of Illinois and Mrs. Leonard, Miss Leonard ##The Bishop of Albany and Mrs. Doane ##The Bishop of New York and Mrs. Potter ##the Bishop of Minnesota and Mrs. Whipple ##Mr. and Mrs. Walter Burns ##Mrs. Douglas Grant ##Miss Scott ##Mrs. Grace, Miss Margarita Grace ##Mrs. Wentworth ##Miss van Wart ##M. Valentin de Courcel ##Madame la Marquise de Talleyrand Perigord ##Comte Boson de Perigord ##Vicomte d'Espenilles ##Madame and Madlle. Thierry Delanoue ##Madlle. de la Cherè ##M. Cellerier ##M. and Madame Delawarre ##Madame Evelina Fenzi ##Count A. Zannini ##M. and Madame Jules Cottran ##Chevalier E. Mazzuechi ##Signor A. Tedeschi ##Signor A. Mariotti ##Captain Lucian von Ziegler ##Chevalier Lieutenant von Barry ##Baron Georg Rothschild ##Privy Councillor Count Berchtold ##Baron G. E. Levi, Baroness Levi ##Commander E. Philipson, Mrs. E. Philipson ##The Duke and Duchess of San Germano Calabritto ##The Marquis of San Vito ##Donna Lidia Serramezzana ##Donna Margherita Chigi ##Marchioness Vitelleschi ##Chevalier Elia ##Count de Franqueville ##Count Urbain Chevrau ##M. Marcel Fonquier ##M. Baudon de Mony, Madame Baudon de Mony ##Duchess de Rohan ##Marquis de Lastorgrie, Marchioness de Lastorgrie ##Count de Boisgelin, Countess L. de Boisgelin ##M. Stern, Madame Stern, Madlle. Stern ##Count Charles du Luart ##General de Saucy ##M. E. Seydoux ##Count Jean de Madre ##M. de Monbrison ##Baron de la Chevrelière ##Count de la Villestreux, Countess de la Villestreux ##Count Urbain de Maille, Countess Urbain de Maille ##General Faveret de Kerbrich ##Monsieur de la Haye Jousselin ##Baronne Faveret de Kerbrich ##Colonel Matton ##M. Ferinier Didet ##Madame Ferinier Didet ##Donna Isabella Colonna, Donna Victoria Colonna ##Pom-k-Soh ##Madame Reyntiens ##Marquis de Fuente Hermosa ##Herr Rudolf Swobody ##M. Lauritz Tuxen ##Duchesse de Baiten ##M. de Marcoarti ##Comte de Heeren, Madlle. de Heeren ##Monsieur M. de Mauny Talvande ##Senor Don Nicolas Campero ##Lieutenant Charny ##Lieutenant Sanders ##Madame and Madlle. de Mouni ##Comtesse de Montsoulmin #"Foreign Admirals and Commanding Officers and Staffs"<ref name=":1" /> (4, Col. 3a / Col. 3b) ##Austrian Admiral Baron von Spaun, Commander von Ziegler, Lieutenant Retter yon Barry, Lieutenant Mitchell, R.N. (attached) ##Danish Admiral H. H. Koch, Captain Waudel, Lieutenant Middelboc, Lieutenant Majendie, R.N. (attached) ##French Admiral C. F. E. De Courthille, Captain Germinet, Commander Poidlone, Lieutenant Perdriel, Sub-Lieutenant de Caqueray, Lieutenant Phillimore, R.N. (attached) ##Italian Admiral C. E. Morin, Commander Count Prasca, Lieutenant Lunghetti, Lieutenant Count Morano, Lieutenant Henderson. R.N. (attached) ##German Admiral his Royal Highness Prince Henry of Prussia, Captain Muller, Lieutenant von Spee, Sub-Lieutenant Wittman, Lieutenant Garforth, R.N. (attached) ##Japanese Admiral H.I.H. Prince Arizugawa, Captain Miura, Commander Tsuda, Lieutenant Stewart, R.N. (attached) ##Netherlands Admiral F. K. Englebrecht, Captain de Groot, Lieutenant Baron von Hardenbrock, Lieutenant Woolcombe, R.N. (attached) ##Norwegian Rear-Admiral von Krogh, Captain Muller, Lieutenant Petersen, Lieutenant Kerr Pearse, R.N. (attached) ##Portuguese Captain Barreto de Vascomellos, Captain de Cartillo, Lieutenant Trye, R.N. (attached) ##Russian Admiral Nicholas Skrydloff, Captain Domojiroff, Lieutenant Stetsenkoff, Lieutenant Twisleton Wykeham Fiennes, R.N. (attached) ##Spanish Admiral Don Segismundo Bermijo y Merelo, Captain Don Antonio Eulate y Fery, Lieutenant Don Juan Romero, Lieutenant Don Antonio Romero, Lieutenant Fair, R.N. (attached) ##Swedish Admiral A. F. H. Klintberg, Captain Ingelman, Commander Flack, Lieutenant Alton, R.N. (attached) ##United States Admiral J. N. Miller, Lieutenaut Richmond (attached) ##Captain de Mar E. Guerra ##Captain R. S. D. Cumins #The Lord Lieutenant of Ireland and Countess Cadogan #The Right Hon. the Speaker and Mrs. Gully, Miss Gully, and Miss Shelly Gully #Cardinal Vaughan #Right Hon. the Lord Mayor and Lady Mayoress, and Misses Faudel Phillips (2) #The Gold Stick in Waiting, Silver Stick in Waiting, Silver Stick Adjutant in Waiting #Officer Commanding 1st Life Guards and five officers #Officer Commanding 2nd Life Guards and four officers #Officer Commanding Royal Horse Guards and four officers #Officer Commanding 2nd Dragoons and three officers #Field Officer in Brigade Waiting, Adjutant in Brigade Waiting #Commanding Officer Grenadier Guards #Commanding Officer Coldstream Guards #Commanding Officer Scots Guards, a Regimental Adjutant #Commanding Officer 1st, 2nd, and 3rd Battalions Grenadier Guards and three officers of each Battalion #Commanding Officer 1st and 2nd Battalions Coldstream Guards and three officers of each Battalion #Commanding Officer 1st and 2nd Battalions of Scots Guards and three officers of each Battalion #Commanding Officer Woolwich District and six officers #Commanding Officer R.H.A. Home District and two officers #Commanding Officer R.E. and four officers #Commanding Officer 2nd Battalion Lincolnshire Regiment and three officers #Commanding Officer Royal Marines (Chatham) and four officers #Commanding Officer Royal Marines (Portsmouth) and two officers #Four officers of the Honourable Corps of the Gentlemen at Arms #Archbishops — Canterbury, York, Armagh, Ontario, Rupertsland #Dukes and Duchesses ##The Duke and Duchess of [[Social Victorians/People/Argyll|Argyll]] ##The Duke and Duchess of [[Social Victorians/People/Abercorn|Abercorn]] ##The Duchess of De Baileu ##The Duke and Duchess of [[Social Victorians/People/Buccleuch|Buccleuch]] ##The Duchess of [[Social Victorians/People/Cleveland|Cleveland]] ##The Duke and Duchess of [[Social Victorians/People/Devonshire|Devonshire]] ##The Duchess of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Hamilton]] ##The Duke and Duchess of [[Social Victorians/People/Leeds|Leeds]] ##The Duke and Duchess of [[Social Victorians/People/Marlborough|Marlborough]] ##The Duke and Duchess of [[Social Victorians/People/Manchester|Manchester]] ##The Duke and Duchess of [[Social Victorians/People/Montrose|Montrose]] ##The Duke and Duchess of [[Social Victorians/People/Newcastle|Newcastle]] ##The Duke of [[Social Victorians/People/Norfolk|Norfolk]] ##The Duke of [[Social Victorians/People/Northumberland|Northumberland]] ##The Duke and Duchess of [[Social Victorians/People/Portland|Portland]] ##The Duke of [[Social Victorians/People/Richmond and Gordon|Richmond and Gordon]] ##The Duke and Duchess of [[Social Victorians/People/Roxburghe|Roxburghe]] ##The Duke and Duchess of [[Social Victorians/People/Somerset|Somerset]] ##The Duke and Duchess of [[Social Victorians/People/Sutherland|Sutherland]] ##The Duke and Duchess of St. Albans ##The Duke and Duchess of Wellington ##The Duchess of [[Social Victorians/People/Westminster|Westminster]] #Marquises and Marchionesses ##The Marquis of Abergavenny ##The Marchioness of Ailesbury ##The Marquis and Marchioness of Ailsa ##The Marquis of Anglesey ##The Marquis and Marchioness of [[Social Victorians/People/Breadalbane|Breadalbane]] ##The Marchioness of [[Social Victorians/People/Marlborough#Marchioness of Blandford|Blandford]] ##The Marquis and Marchioness of Bristol ##The Marquis of [[Social Victorians/People/Camden|Camden]] ##The Marquis and Marchioness of Conyngham ##Dowager [Marchioness of] Conyngham ##The Marchioness of Cassar de Sai[n] ##The Marquis and Marchioness of Cholmondeley ##The Marquis of D'Auerstadt ##The Marquis and Marchioness [[Social Victorians/People/Stonor|D'Hautpoul]] ##The Marquis and Marchioness of Downshire ##Dowager [Marchioness of] Downshire ##The Marquis and Marchioness of [[Social Victorians/People/Hamilton Temple Blackwood|Dufferin and Ava]] ##The Marquis and Marchioness of [[Social Victorians/People/Exeter|Exeter]] ##The Marquis and Marchioness of Granby ##The Marchioness of [[Social Victorians/People/Florence Rawdon-Hastings Chetwynd|Hastings]] ##The Marquis and Marchioness of [[Social Victorians/People/Bective|Headfort]] ##The Marquis and Marchioness of Hertford ##The Marquis and Marchioness of Huntly ##The Marquis and Marchioness of [[Social Victorians/People/Abercorn#James Hamilton, Marquess of Hamilton|Hamilton]] ##The Marquis and Marchioness of [[Social Victorians/People/Lansdowne|Lansdowne]] ##The Marquis and Marchioness of Lothian ##Dowager (Marchioness of) [[Social Victorians/People/Londonderry|Londonderry]] ##The Marquis and Marchioness of [[Social Victorians/People/Londonderry|Londonderry]] ##The Marquis and Marchioness of [[Social Victorians/People/Ormonde|Ormonde]] ##The Marchioness of [[Social Victorians/People/Queensberry|Queensberry]] ##The Marquis and Marchioness of [[Social Victorians/People/Ripon|Ripon]] ##The Marquis and Marchioness of [[Social Victorians/People/Salisbury|Salisbury]] ##The Marquis and Marchioness of [[Social Victorians/People/Tweeddale|Tweeddale]] ##Dowager (Marchioness of) [[Social Victorians/People/Tweeddale|Tweeddale]] ##John Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]] ##Lawrence, [[Social Victorians/People/Zetland|Marquess of Zetland]] and Lilian, [[Social Victorians/People/Zetland|Marchioness of Zetland]] #Earls and Countesses ##Countess of Aberdeen and Dowager Countess of Aberdeen ##Earl and Countess of Albemarle and Dowager Countess of Albemarle ##Earl and Countess of Ancaster ##Earl and Countess of Amherst ##Earl of Ava ##Earl and Countess of Antrim ##Earl and Countess of Aylesford ##Earl and Countess of Annesley ##Earl and Countess of Airlie ##Earl and Countess of Arran ##Earl of Aberdeen ##Earl and Countess of Bandon ##Countess of Bantry ##Earl and Countess of Beauchamp ##Earl and Countess of Bathurst and Dowager Countess of Bathurst ##Countess of Bective ##Earl and Countess of Belmore ##Earl of Bradford ##Countess of Bremer ##Earl and Countess of Brownlow ##Earl and Countess of Buckinghamshire ##Earl of Burford ##Earl and Countess of Cairns ##Earl and Countess of Caledon ##Earl of Camperdown ##Earl of Cardigan ##Earl and Countess of Carnarvon and Dowager Countess of Carnarvon ##Earl of Carnwath ##Earl and Countess of Carrington ##Earl and Countess of Carysfort ##Earl and Countess of Castlestuart ##Earl and Countess of Cathcart ##Earl and Countess of Cavan ##Earl and Countess of Chesterfield ##Earl and Countess of Chichester ##Dowager Countess of Clancarty ##Countess of Clanwilliam ##Earl and Countess of Compton ##Countess of Cottenham ##Earl of Courtown ##Earl and Countess of Cowper ##Earl and Countess of Cranbrook ##Earl and Countess of Craven and Dowager Countess of Craven ##Earl and Countess of Crawford ##Earl of Crewe ##Earl and Countess of Cork and Orrery ##Earl and Countess of Coventry ##Countess of Cromartie and Dowager Countess of Cromartie ##Earl and Countess of Dalkeith ##Earl and Countess of Dartmouth ##Earl and Countess of De Grey ##Dowager Countess of De La Warr ##Earl and Countess of Denbigh ##Earl and Countess of Derby ##Earl and Countess of Donoughmore ##Earl and Countess of Drogheda ##Earl of Ducie ##Earl and Countess of Dudley and Dowager Countess of Dudley ##Earl and Countess of Dundonald ##Earl and Countess of Dunmore ##Earl and Countess of Dunraven ##Earl of Durham ##Earl and Countess of Eglinton and Winton ##Earl of Eldon ##Earl and Countess of Ellesinere ##Earl and Countess of Enniskillen ##Earl and Countess of Erne ##Earl and Countess of Errol ##Earl and Countess of Essex and Dowager Countess of Erroll ##Earl of Euston ##Earl and Countess of Feversham ##Earl and Countess of Fingall ##Earl of Fortescue ##Earl and Countess of Gainsborough ##Earl and Countess of Galloway ##Earl and Countess of Glasgow ##Countess of Gosford ##Earl and Countess of Granard ##Countess of Granville ##Earl and Countess of Grey ##Countess of Grosvenor ##Countess of Guilford ##Earl and Countess of Harewood and Dowager Countess of Harewood ##Earl and Countess of Harrington ##Earl and Countess of Hopetoun ##Earl and Countess of Huntingdon ##Earl and Countess of Harrowby ##Countess of Hohenau ##Countess of Howe ##Earl and Countess of Iddesleigh ##Earl and Countess of Jersey ##Earl and Countess of Kenmare ##Earl of Kerry ##Earl and Countess of Kilmorey ##Earl of Kimberley ##Earl and Countess of Kingston ##Earl of Kinnoull ##Josephine, Countess Kinsky ##Earl and Countess of Kintore ##Countess of Leitrim ##Earl and Countess of Lanesborough ##Countess of Lathom ##Earl and Countess of Lauderdale ##Countess of Leicester ##Earl and Countess of Leven and Melville ##Earl and Countess of Lichfield ##Earl and Countess of Limerick ##Earl and Countess of Lindsay ##Earl and Countess of Lisburne ##Earl and Countess of Listowel ##Earl and Countess of Londesborough ##Earl and Countess of Longford ##Earl and Countess of Lonsdale and Dowager Countess of Lonsdale ##Earl and Countess of Loudoun ##Earl and Countess of Lovelace ##Earl and Countess of Lucan ##Countess of Lytton ##Countess of Macclesfield ##Earl and Countess of Malmesbury and Dowager Countess of Malmesbury ##Earl and Countess of Mar ##Earl and Countess of Mar and Kellie and Dowager Countess of Mar and Kellie ##Earl and Countess of Mayo and Dowager Countess of Mayo ##Countess of Meath ##Countess of Metaxas ##Earl and Countess of Mexborough ##Earl and Countess of Minto ##Earl of De Montalt ##Earl and Countess of Morley ##Earl and Countess of Morton and Dowager Countess of Morton ##Earl of Nelson ##Earl and Countess of Norbury ##Earl of Northbrook ##Earl and Countess of Northesk and Dowager Countess of Northesk ##Earl and Countess of Onslow ##Earl of Orford ##Countess of Oxford ##Earl and Countess of Pembroke ##Countess of Percy ##Earl and Countess of Portarlington ##Earl and Countess of Portsmouth ##Earl and Countess of Powis ##Earl and Countess of Radnor ##Earl and Countess of Ravensworth ##Earl and Countess of Roden ##Earl and Countess of Romney ##Lawrence, [[Social Victorians/People/Zetland|Earl of Ronaldshay]] ##Earl of Rosebery ##Earl and Countess of Rosse ##Earl and Countess of Rosslyn and Dowager Countess of Rosslyn ##Earl of Sandwich ##Earl of Scarbrough ##Earl and Countess of Selborne ##Countess of Selkirk ##Countess of Shaftesbury ##Dowager Countess of Shrewsbury and Talbot ##Earl and Countess of Spencer ##Earl and Countess of Stamford ##Earl and Countess of Stanhope ##Earl and Countess of St. Germans ##Earl of Stradbroke ##Earl of Strafford ##Earl and Countess of Suffolk and Berkshire ##Earl and Countess of Temple (of Stowe) ##Earl and Countess of Verulam ##Earl and Countess of Waldegrave ##Earl and Countess of Warwick ##Earl and Countess of Westmeath ##Earl and Countess of Wharncliffe ##Elizabeth, Dowager Countess of Wilton and Isabella, Dowager Countess of Wilton ##Earl and Countess of Winchilsea and Nottingham ##Earl and Countess of Winterton ##Earl and Countess of Yarborough and Dowager Countess of Yarborough #Viscounts<ref name=":1" /> (4, Col. 3c / Col. 4a) and Viscountesses ##Viscount and Viscountess of Boyne ##Viscountess of Cantelupe ##Viscount and Viscountess of Castlerosse ##Viscount and Viscountess of Chelsea ##Viscount and Viscountess of Chetwynd ##Viscountess of Chewton ##Viscount and Viscountess of Clifden ##Viscount and Viscountess of Cobham ##Viscount and Viscountess of Coke ##Viscount of Corry ##Viscount and Viscountess of Cranborne ##Viscount of Crichton ##Viscount and Viscountess of Cross ##Viscount of Curzon ##Viscount and Viscountess of Dalrymple ##Viscount and Viscountess of Deerhurst ##Viscount and Viscountess of De Vesci ##Viscount and Viscountess of Dillon ##Viscount of Doneraile ##Viscount and Viscountess of Duncannon ##Viscount of Dungarvan ##Viscount and Viscountess of Ebrington ##Viscount and Viscountess of Emlyn ##Viscount of Encombe ##Viscount and Viscountess of Exmouth ##Viscount and Viscountess of Falkland ##Viscount and Viscountess of Falmouth ##Viscount of Fitz Harris ##Viscount and Viscountess of Folkestone ##Viscount and Viscountess of Frankfort de Montmorency ##Viscount and Viscountess of Gage ##Viscount and Viscountess of Galway ##Viscount and Viscountess of Garnock ##Viscount and Viscountess of Gough ##Viscount of Gort ##Viscount and Viscountess of Halifax ##Viscount and Viscountess of Hardinge ##Viscount of Harrington ##Viscount and Viscountess of Hood ##Viscount and Viscountess of Kilcoursie ##Viscount and Viscountess of Knutsford ##Viscount and Viscountess of Lifford ##Viscount of Llandaff ##Viscount and Viscountess of Maitland ##Viscount and Viscountess of Marsham ##Viscount and Viscountess of Massereene and Ferrard ##Viscount and Viscountess of Melville ##Viscount and Viscountess of Midleton ##Viscount and Viscountess of Milton ##Viscount and Viscountess of Monck ##Viscount and Viscountess of Morpeth ##Dowager Viscountess of Mountmorres ##Viscount and Viscountess of Newark ##Viscount and Viscountess of Newport ##Viscount and Viscountess of Oxenbridge ##Viscount of Parker ##Viscount of Peel ##Viscount and Viscountess of Portman ##Viscount and Viscountess of Powerscourt ##Viscount and Viscountess of Raincliffe ##Viscountess of Sherbrooke ##Viscount of Sidmouth ##Viscount of St. Cyres ##Viscount of Southwell ##Viscount of Suirdale ##Viscount and Viscountess of Templetown ##Viscountess of Torrington ##Viscount and Viscountess of Trafalgar ##Viscount and Viscountess of Valentia ##Viscount of Valletort ##Viscount of Villiers ##Viscountess of Wolseley #Bishops — Auckland, Barry, Bath and Wells, British Colombia, Chichester, Durham, Ely, Exeter, Gloucester and Bristol, Gibraltar, Hereford, London, Lichfield, Lincoln, Manchester, Newcastle, Norwich, Oxford, Peterborough, Rochester, Ripon, Stepney, Southwark, St. Albans, Salisbury, Sodor and Man, Southwell, Sydney, Sierra Leone, Worcester, Winchester, Wellington #Baronesses — Burdett-Coutts, Macdonald #Lords and Ladies<ref name=":1" /> (4, Col. 4b / Col. 5a) — ##Lord and Lady Abercromby ##Lord and Lady Aberdare ##Lord Aberdour ##Lady Abinger ##Lady Alexandra Acheson ##Lady Adam ##Lady Adderley ##Lord and Lady Addington ##Lady Adye ##Lady Agnew ##Lady Alderson ##Lord and Lady Alington ##Lady Alison ##Lady Mildred Allsopp ##Lord and Lady Amherst of Hackney ##Lady Heathcoat Amory ##Lord and Lady Ampthill ##Lady Agnes Anderson ##Lady Bertha Anson ##Lady Arbuthnot ##Lady Alice Archer Houblon ##Lord Ardee ##Lord and Lady Ardilaun ##Lady Armstrong ##Lady Arnold ##Lady Arnott ##Lord and Lady Ashbourne ##Lord and Lady Ashburton and Dowager Ashburton ##Lord and Lady Ashcombe ##Lady Alice Ashley ##Lady Edith Ashley ##Lady Ashmead-Bartlett ##Lord and Lady Ashton ##Lord and Lady Ashtown ##Lady Florence Astley ##Lady Gertrude Astley-Corbett ##Lady Austin ##Lord Bagot ##Lady Bailey ##Lady Blanche Baillie ##Lady Baird ##Lady Baker ##Lord Balcarres ##Lord and Lady Balfour of Burleigh, Lady Nina Balfour and Lady Betty Balfour ##Lord Balvaird ##Lord Bangor ##Dowager Lady Barclay ##Lord and Lady Barnard ##Lady Florence Barnardiston ##Lady Constance Barne ##Lady Barran ##Lady Barrington ##Lord and Lady Basing ##Lord and Lady Bateman ##Lady Evelyn Bathurst ##Lord and Lady Battersea ##Lady Steuart Bayley ##Lady Violet Beauchamp ##Lord Osborne Beauclerk and Lady Beauclerk (2) ##Lady A. Beaumont ##Lady Bedford ##Lord and Lady Belhaven and Stenton and Dowager Belhaven and Stenton ##Lord and Lady Bellew and Dowager Bellew ##Lord and Lady Belper ##Lady Charles Beresford ##Lady William Beresford (Lilian Duchess of Marlborough) ##Lady Bergne ##Lord and Lady Bertie and Lady Elizabeth Bertie ##Lady Biddulph, Lady Elizabeth Biddulph and Lady Wilfreda Biddulph ##Lady Bigge ##Lord and Lady Bingham ##Lord and Lady Binning ##Lord Blackwood, Lord Basil Blackwood. Lady Hermione Blackwood and Lord Terence Blackwood ##Lady Bloomfield ##Lady Blythswood ##Lord and Lady Bolton ##Lady Maud Bootle-Wilbraham, Lady Bertha Bootle-Wilbraham and Lady Edith Bootle-Wilbraham ##Lord Borthwick ##Lady Margaret Boscawen ##Lord and Lady Boston ##Lady Boughey ##Lady Albreda Bourke and Lady Florence Bourke ##Lady Bowen ##Lady Bower ##Lady Muriel Boyle and Lady Boyle (2) ##Lady Mary Brabazon ##Lady Brackenbury ##Lady Braddon ##Lady Bramwell ##Lady Bramston ##Lord Brassey, Lady Idina Brassey and Lady Violet Brassey ##Lord and Lady Braye ##Lady Mary Bridgeman ##Lady Eleanor Brodie ##Lady Hilda Brodrick ##Lady De Capel Brooke and Dowager Brooke ##Lady Cunliffe Brooks ##Lord and Lady Brougham and Vaux ##Lord and Lady Ulick Browne, Lady Browne and Lady Crichton Browne ##Lady Brownlow ##Lord and Lady F. Brudenell-Bruce ##Lady Brunner ##Dowager Buchanan-Riddeil ##Lady Audrey Buller ##Lady Burdett ##Lord and Lady Burghclere ##Lord Burghley ##Lady Agnes Burne ##Lady Burrell ##Lord and Lady Burton ##Lady Butler and Lady Butler (2) ##Lord and Lady Arthur Butter ##Lady Buxton and Lady Victoria Buxton ##Lady Susan Byng ##Lord and Calthorpe ##Lady C. Cameron and Lady Margaret Cameron ##Lord and Lady Archibald Campbell and Lady A. Campbell ##Lord and Lady George Campbell ##Lady Campbell-Bannerman ##Lord and Lady Camoys ##Lord and Lady Carbery and Dowager Carbery ##Lady Carbutt ##Lady Cardon ##Lord and Lady Cardross ##Lord and Lady Carew ##Lady Carmichael ##Lord and Lady Carnegie ##Lord and Lady Castlemaine ##Lord and Lady Castletown ##Lady Eva Cathcart and Lady R. Cathcart ##Lady Frederick Cavendish, Lady Myra Cavendish, Lady Evelyn Cavendish and Lady Harriet Cavendish ##Lord Charles Cavendish-Bentinck, Lord and Lady Henry Cavendish-Bentinck, Lord William Cavendish-Bentinck, Lady Ottoline Cavendish-Bentinck ##Lord and Eustace Cecil, Lord Hugh Cecil, Lord and John Cecil, Lord and Edward Cecil, Lord and Lady Robert Cecil, Lord W. Cecil, Lady Gwendolen Cecil, Lady Florence Cecil, Lady William Cecil, Lady Louisa Cecil ##Lady Francis Cecil-Dallas ##Lady Chamberlain ##Lady Chelmsford ##Lord and Lady Chesham ##Lady Chetwode ##Lord Cheylesmore ##Lord and Lady Fitzwarine Chichester ##Lady Chitty ##Lady Cholmeley ##Lady Henry Cholmondeley ##Lady Clements (2) ##Lady Churchill, Lady Randolph Churchill, Dowager Churchill, Lady Spencer Churchill (2) ##Lord Edward Spencer-Churchill, Lady Alfred Spencer-Churchill ##Lord and Lady Churston ##Lord and Lady Clifford of Chudleigh ##Lady Marshal Clarke, Lady E. Clarke ##Lady Isabel Clayton ##Lord and Lady Clinton ##Lord and Lady Clonbrock ##Lord Cloncurry ##Lady Muriel Close ##Lady Evelyn Cobbold ##Lady Cochrane, Lady Gertrude Cochrane, Lady Adela Cochrane ##Lady Coddington ##Lady Mabel Coke ##Lord and Lady Colchester ##Lady Cole (2) ##Lady Colebrooke ##Lord and Lady Coleridge ##Lady Collins ##Lady Colomb ##Lady Colvile, Lady Colville ##Lord and Lady Colville of Culross ##Lady Jane Seymour Combe, Lady Constance Combe ##Lady Commerell ##Lord and Lady Alwyne Compton ##Lady Dowager Congleton ##Lord and Lady Connemara ##Lady Conyers ##Lady Blanche Conyngham ##Lady Cooper ##Lady Evelyn Cotterell ##Lord and Lady Cottesloe ##Lady Couch ##Lord and Lady Courtenay ##Lady Coventry (2) ##Lady Cowell ##Lady Helen Craven ##Lord and Lady Crawshaw ##Lady Evelyn Crichton, Lady Emma Crichton ##Lord Crofton ##Lady Cromer ##Lady Mary Crosse ##Lady Crossley ##Lady Mary Cuffe ##Lady Culme-Seymour ##Lady Cunliffe ##Lady Georgiana Curzon ##Lady Elizabeth Cust ##Lady Ida Dalzell ##Lady Mary Dashwood ##Lord and Lady Davey ##Lady Victoria Dawnay, Lady Evelyn Dawnay, Lady Adelaide Dawnay ##Lady Decies ##Lord and Lady De Freyne ##Lord and Lady De L’Isle and Dudley ##Lord De Manley ##Lady Mildred Denison, Lady Elinor Denison ##Lord Deramore ##Lord and Lady De Ramsey ##Lady Dering ##Lady De Ross ##Lord and Lady De Saumarez ##Lady Des Voeux ##Lady De Trafford, Lady Agnes De Trafford ##Lady De Winton ##Lord and Lady Digby ##Lady Dorchester ##Lady Dorington ##Lady Margaret Douglas, Lady Edith Douglas ##Lady H. Douglas-Hamilton ##Lady Dowell ##Lady Drummond, Lady Edith Drummond ##Lady Du Cane ##Lady Duckworth ##Lady Eva Dugdale ##Lord Dunally ##Lady Florence Duncombe, Lady Ulrica Duncombe, Lady Caroline Duncombe ##Lady Alice Dundas ##Lord and Lady Dunleath ##Lord Dunglass ##Lady Dunn ##Lord Dunsandle and Clanconal ##Lady Durand ##Lord Dynevor ##Lord Ebury ##Lady Edmonstone ##Lady Edwards, Lady J. B. Edwards, Lady Blanche Edwards ##Lady Ernestine Edgcumbe ##Lady Egerton (2) ##Lord Egerton of Tatton ##Lady Grey-Egerton ##Lord and Lady Elcho ##Lord and Lady Elibank ##Lady Ellenborough ##Lady Ellis ##Lord and Lady Elphinstone ##Lady Winifred Cary-Elwes ##Lady Engleheart ##Lord Erskine, Lady Erskine (2), Lady Horatia Erskine, Lady Erskine ##Lord and Lady Esher ##Lady Evans ##Lady Evelyn Ewart, Lady Mary Ewart ##Lady Evelyn Eyre ##Lady Fairbairn ##Lady Fairfax ##Lady Anne Fane, Lady Augusta Fane ##Lady Farquhar ##Lord and Lady Farrer ##Lady Fayrer ##Lady Louisa Feilding ##Lady Helen Munro Ferguson ##Lady Fergusson ##Lady Ffolkes ##Lady Finlay ##Lady Fisher ##Lady Dorothea Fitz-Clarence, Lady Maria Fitz-Clarence, Lady Dorothy Fitzclarence ##Lord and Lady Henry Fitz-Gerald, Lady B. Fitz Gerald, Lady M. FitzGerald, Lord Seymour Fitz-Gerald ##Lady Beatrix Fitzmaurice ##Lord and Lady F. FitzRoy, Lady C. Fitz-Roy ##Lady Mary Fitzwilliam ##Lady FitzWygram ##Lady Fletcher ##Lady Flower, Lady Flower ##Lord Foley, Lady Mary Foley ##Lady Gertrude Foljambe ##Lady Angela Forbes, Lady Forbes (2), Dowager Helen Forbes ##Lord and Lady Forester ##Lady Forrest ##Lady Susan Fortescue ##Lady Forwood ##Lady Foster ##Lady Fowler ##Lady Edith Franklin ##Lady Fremantle, Lady Fremantle ##Lady Frere ##Lady Fulton ##Lady Gardiner, Lady Lynedoch Gardiner ##Lord Garioch ##Lady Galton ##Lady Katharine Gathorne-Hardy ##Lady Garvagh ##Lord and Lady Gerard ##Lady Gilbey ##Lady Gillford ##Lady Susan Gilmour ##Lady Gipps ##Lord and Lady Glamis ##Lord and Lady Glenesk ##Lady Glyn, Lady Mary Carr Glyn ##Lady D'Arcy Godolphin-Osborne ##Lady Gordon ##Lady Margaret Ormsby Gore, Lady Constance Gore ##Lady Gore Langton (2) ##Lord Walter Gordon-Lennox, Lord Algernon Gordon-Lennox ##Lady Evelyn Goschen ##Lord R. S. Gower ##Lady Graham, Lady Margaret Graham, Lady Helen Graham ##Lady Charlotte Graham-Toler ##Lady Grant, Lady Florence Grant ##Lady Grant-Duff ##Lady Green ##Lord Greenock ##Lady Grenfell ##Lady Frances Gresley ##Lady Victoria Grey, Lady Grey ##Lady Jane Grey-Trefusis ##Lady Griffin ##Lady Helen Grimston ##Lord and Lady Arthur Grosvenor, Lady Grosvenor (2) ##Lady Gull ##Lady Haldon ##Lady Haliburton ##Lady Basil Hall ##Lady Halle ##Lord and Lady Halsbury ##Lord and Lady E. Hamilton, Lord F. Hamilton, Lady F. Douglas Hamilton, Lady Alexandra Hamilton, Lady Baillie Hamilton (2), Lady C. Hamilton, Lady Victoria Hamilton, Lady George Hamilton ##Lady Hanson ##Lady Harcourt ##Lady Cicely Hardy, Lady Hardy ##Lady Beatrice Hare ##Lord Harlech ##Lady Constance Harris, Lady Harris ##Lady Harrison, Lady Harriet Harrison ##Lady Hart ##Lady Emily Hart-Dyke ##Lady Dixon-Hartland ##Lady Hartopp ##Lord and Lady Hastings ##Lord and Lady Hatherton ##Lady Alice Havelock-Allan ##Lady Hawke ##Lord and Lady Hawkesbury ##Lady John Hay, Lady Hay ##Lady Blanche Haygarth ##Lady Hayter ##Lady Hely-Hutchinson (2) ##Lady Hemming ##Lord and Lady Heneage ##Lord and Lady Henley ##Lord Henniker ##Lady Beatrix Herbert, Lady Herbert (2) ##Lord and Lady Herries ##Lord and Lady Herschell ##Lord Francis Hervey, Lady Augustus Hervey ##Lady Hervey-Bathurst ##Lady Fermor Hesketh ##Lady Hibbert ##Lady Lucy Hicks-Beach ##Lord and Lady Arthur Hill, Lady Clement Hill, Lady Stock Hill ##Lord and Lady Hillingdon ##Lord and Lady Hindlip ##Lord and Lady Hobhouse ##Lady Norah Hodgson ##Lady Holdich ##Lady Mary Holland ##Lady Beatrix Douglas Home ##Lady Maria Hood ##Lady Hood of Avalon ##Lady Hooker ##Lady Mary Hope ##Lady Hoskins ##Lord and Lady Hotham ##Lord and Lady Hothfield ##Lady Houldsworth ##Lady Eleanor Howard, Lady Agnes Howard, Lady Howard (2), Lady Mabel Howard, Lady Rachel Howard ##Lord and Lady Howard of Glossop ##Lady Howarth ##Lady Mary Hozier ##Lady Florentia Hughes ##Lady Seager Hunt ##Lady Hunter ##Lord Hyde ##Lady Hylton ##Lord and Lady Inchiquin ##Lord Inverurie ##Lord and Lady Iveagh ##Lady Jackson ##Lord James of Hereford ##Lady Margaret Jenkins, Lady Jenkins ##Lady Jenner ##Lady Jephson ##Dowager Jessel, Lady Jessell ##Lady Jeune ##Lady Hill Johnes ##Lady Joicey ##Lady Alice Jolliffe ##Lady Burn Jones ##Lady Caroline Lister Kaye, Lady Beatrice Lister Kaye, Lady Lister Kaye ##Lady Isabella Keane ##Lady Keith-Falconer (2) ##Lord and Lady Kelvin ##Lady Kemball ##Lady Beatrice Kemp ##Lady Kennard ##Lady Kennaway ##Lady Aline Kennedy ##Lady Kennett-Barrington ##Lord Kenyon ##Lady Mabel Kenyon-Slaney ##Lord Kensington ##Lady Mary Stuart Keppel ##Lady Innes-Ker (2) ##Lady Kerr (2) ##Lord Kilmarnock ##Lady King ##Lady Florence King King ##Lady Emily Kingscote ##Lady Edith King-Tenison ##Lord and Lady Kinnaird ##Lady Kitson ##Lady Laking ##Lady Frances Lambart, Lady Ellen Lambart ##Lady Victoria Lambton ##Lady Adela Larking ##Lady Isabel Larnach ##Lady Mary Lascelles ##Lord and Lady Lawrence ##Lady Lawson ##Lord and Lady Leconfield ##Lady Elliott Lees, Lady Lees ##Lady Leese ##Lady Legard ##Lord and Lady Leigh ##Lady Henry Gordon-Lennox, Lady Walter Gordon-Lennox, Lady Algernon Gordon-Lennox, Lady Caroline Gordon-Lennox ##Lady Katharine Le Poer Trench ##Lady Constance Leslie ##Lady Susan Leslie-Melville ##Lady Lewis ##Lady Lilian Liddell ##Lady Lindley ##Lady Harriet Lindsay, Lady Jane Lindsay, Lady Jane Lindsay ##Lord and Lady Lingen ##Lord and Lady Lister ##Lady Gwendolen Little ##Lady Margaret Littleton ##Lord and Lady Llangattock ##Lady Llewelyn ##Lord and Lady Loch ##Lady Lockwood ##Lady Louise Loder ##Lady Catherine Loftus ##Lady Doreen Long ##Lady Longley ##Lady Albertha Lopes ##Lady Loraine ##Lord and Lady Lovat ##Lady Drury Lowe, Lady Lucy Drury Lowe ##Lady Lowry-Corry (2) ##Lady Mary Loyd ##Lady Lubbock ##Lord and Lady Lurgan and Dowager Lurgan ##Lady Lyall ##Lady Lyell ##Lady Mary Lygon ##Lady Lyons ##Lady Lysons ##Lady Lyttelton ##Lady Emily Lytton ##Lady MacCormac ##Lord and Lady Macdonald ##Lady Macgregor, Lady MacGregor, Lady Helen MacGregor ##Lady Mackenzie, Lady Mackenzie ##Lady Mackworth ##Lady Maclean ##Lord and Lady Macnaghten ##Lady Macpherson-Grant ##Lady Caroline Madden, Lady Madden ##Lady Louisa Magenis ##Lady Magheramorne, Dowager Magheramorne ##Lady Nora Maitland ##Lady Margaret Crichton-Maitland ##Lady Margaret Majendie ##Lord Cecil Manners, Lord Edward Manners, Lord Manners, Lady Victoria Manners, Lady Manners ##Lady Blundell Maple ##Lady Mappin ##Lady Marjoribanks ##Lady Markham ##Lady Marriott ##Lady Martin, Lady Martin ##Lady Evelyn Mason ##Lady Maude (2) ##Lady H. Maxwell, Lady Maxwell, Lady Maxwell, Lady Maxwell ##Lady Heron-Maxwell ##Lady M'Clintock ##Lady Evelyn M'Donnell ##Lady Meade (2) ##Lord and Lady Medway ##Lady Methuen ##Lady Meysey-Thompson ##Lord and Lady Middleton, Lady Middleton ##Lady Mary Milbanke ##Lady Miller ##Lady Milner ##Lady Clementina Mitford ##Lady Lady M'lver ##Lady Hilda M'Neile ##Lady Monckton ##Lord Moncreiff, Lady Scott Moncrieff ##Lady Moncreiffe ##Lord and Lady Monkswell ##Lady Monson ##Lord Charles Montagu, Lady Cecil Scott Montagu, Lady S. Montagu, Lady Agneta Montagu ##Lord Montagu of Beaulieu ##Lord and Lady Monteagle ##Lady Edith Montgomerie, Lady Sophia Montgomerie ##Lady Charlotte Montgomery ##Lady More-Molyneux ##Lord and Lady Moreton ##Lady Morgan ##Lord and Lady Morris ##Lady Blanche Morris ##Lady Mary Morrison ##Lady Moseley ##Lord and Lady Mostyn ##Lord and Lady Mowbray and Stourton, Dowager Mowbray and Stourton, Lady Mowbray ##Lord and Lady Muncaster ##Lady Anne Murray ##Lady Murray (2) ##Lady Georgiana Mure, Lady Georgiana Mure [sic] ##Lord and Lady Napier and Ettrick ##Lord and Lady Napier of Magdala and Dowager Napier of Magdala ##Lady Naylor-Leyland ##Lady Nelson ##Lord and Lady Henry Nevill ##Lord and Lady Newton ##Lord and Lady Newtown-Butler ##Lady Nicolson ##Lady Augusta Noel, Lady Agnes Noel ##Lady Norman ##Lord and Lady Norreys ##Lord and Lady North, Lady Muriel North ##Lady Northcote, Lady Northcote (2) ##Lord Norton ##Lady Elizabeth Nugent ##Lady O'Brien, Lady O'Brien [sic] ##Lady O'Hagan ##Lady Olpherts ##Lord and Lady O'Neill ##Lady Gwendoline O'Shee ##Princep [sic] Alice Packe ##Lord and Lady Berkeley Paget ##Lady Alfred Paget ##Lady Paget of Cranmore ##Lady Katherine Pakenham ##Lady Palgrave ##Lady Sophia Palmer, Lady Palmer ##Lady Evelyn Parker ##Lady Parratt ##Lady Maude Parry ##Lady Muriel Parsons ##Lord and Lady Pearson, Lady Pearson ##Lady Peel, Lady Georgiana Peel ##Lady Constance Childe-Pemberton ##Lord and Lady Penrhyn ##Lady Mary Pepys ##Lady Perceval ##Lady Percy (2) ##Lady Petre ##Dowager Lady Peyton ##Lady Phillimore ##Lady William Phipps ##Lord and Lady Pirbright ##Lord and Lady Playfair ##Lady Chichele Plowden ##Lady Anna Chandos-Pole ##Lady Pollock ##Lord and Lady Poltimore ##Lady Pontifex ##Lady Alice Portal ##Lady Powell, Lady Powell [sic] ##Lady Baden-Powell ##Lady Dickson-Poynder ##Lady Poynter ##Lord and Lady George Pratt ##Lady Priestley ##Lady Probyn ##Lady Eva Wyndham-Quin, Lady Wyndham-Quin (2) ##Lord and Lady Raglan, Dowager Raglan ##Lady Ramsay ##Lord and Lady Rathdonnell ##Lady Rathmore ##Lord and Lady Rayleigh, Dowager Rayleigh ##Lord and Lady Reay ##Lady Reid ##Lord and Lady Rendel ##Lord Rendlesham ##Lady Jane Repton ##Lord Revelstoke ##Lord and Lady Ribblesdale ##Lady Laura Ridding ##Lord and Lady Robartes ##Lady O. Roberts ##Lady Roberts of Kandahar ##Lady Robinson ##Lord and Lady Rodney ##Lord Romilly ##Lord and Lady Rookwood ##Lord and Lady Rossmore ##Lord Rowton ##Lady Roxburgh ##Lord and Lady Rothschild ##Lady Victoria Russell, Lady Arthur Russell, Lady G. Russell, Lady W. H. Russell, Lady Alexander Russell ##Lord and Lady Russell of Killowen ##Lord and Lady Ruthven ##Lady Jane Ryan ##Lady Mary Sackville ##Lady Salmon ##Lord and Lady Saltoun ##Lady Samuelson, Lady S. Samuel ##Lady Mary Saurin ##Lord and Lady Savile, Lady Marie Savile ##Lady Savory ##Lord George Scott, Lord Henry Scott, Lord Herbert Scott, Lady Sophie Scott, Lady Charles Scott, Lady Louisa Scott, Lady Scott (2) ##Lord and Lady Seaton ##Lord and Lady Settrington ##Lady Seymour, Lady Albert Seymour, Lady William Seymour, Lady Seymour (2) ##Lord and Lady Shand ##Lady Shaw ##Lady Constance Shaw-Lefevre ##Lady Octavia Shaw-Stewart, Lady Alice Shaw-Stewart ##Lady Mary Shelley ##Lord and Lady Sherborne ##Lady Shippard ##Lady Shute ##Lady Kay-Shuttleworth ##Lady Simeon ##Lady Simmons ##Lady Simpson of Windsor ##Lord and Lady Sinclair ##Lord and Lady Skelmersdale ##Lady Esther Smith, Lady Barbara Smith, Lady Smith, Lady Blanche Smith, Lady Sybil Smith, Lady Euan[-]Smith [there is a Sir Euan-Smith and perhaps a Miss Euan-smith], Lady D. Smith ##Lady Smyth ##Lady Catherine Somerset, Lady Geraldine Somerset, Lady Henry Somerset ##Lord and Lady Southampton, Dowager Southampton ##Lady Edward Spencer-Churchill ##Lady Margaret Spicer ##Lady Sprigg ##Lady Stafford ##Lord Stalbridge ##Lady Stanhope (2) ##Lord Stanmore ##Lord Stanley, Lady Alice Stanley, Lady Isobel Stanley ##Lady Stansfield ##Lord Stavordale ##Lady Stephenson ##Lady Stevenson ##Lady Helen Stewart, Lady Mary Stewart, Lady Mark Stewart, Lady Stewart, Lady Houston Stewart, Lady Stewart [sic], Lady Isabel Stewart ##Lady Stewart of Grantully ##Lady Edith St. Aubyn ##Lord and Lady St. Levan ##Lady St. Leonards ##Lord and Lady St. Oswald ##Lady Stone ##Lady Charlotte Stopford ##Lord and Lady Stratheden and Campbell ##Lady Mary Stuart-Richardson ##Lord Suffield ##Lady Sutherland ##Lady Evelyn Sutton, Lady Susan Sutton ##Lord and Lady Swansea ##Lady Swinnerton Dyer ##Lady Kathleen Swinnerton-Pilkington ##Lord and Lady E. Talbot, Lady Emma Talbot ##Lady Jane Taylor ##Lady Taylour (2) ##Lady Tatton Sykes ##Lord Herbert Vane-Tempest, Lord Henry Vane-Tempest ##Lord and Lady Templemore ##Lady Tennant ##Lord and Lady Tennyson ##Lady Tenterden ##Lord Tewkesbury ##Lord and Lady Teynham ##Lord and Lady Thring ##Lady E. Thornton ##Lady Thursby ##Lady Ulrica Thynne ##Lord and Lady Tollemache ##Lady Agnes Townshend ##Lady Mary Trefusis ##Lady Tredegar ##Lady Trevelyan, Lady Trevelyan [sic] ##Lord and Lady Trevor ##Lady Troubridge ##Lady Turner ##Lady Henrietta Turnor ##Lady Tuson ##Lord and Lady Tweedmouth ##Lady Tyler ##Lady Emily Van De Weyer ##Lady Jane Van Koughnet ##Lord and Lady Ventry ##Lady Villiers (2), Lady Edith Villiers ##Lady Howard Vincent, Lady Helen Vincent, Lady Vincent ##Lady Vivian, Lady Jane Vivian ##Lady Mary Waldegrave ##Lady F. F. Walker, Lady James Walker ##Lady Walrond ##Lady Clementine Walsh ##Lord Wandsworth ##Lady Wantage ##Lord Warksworth ##Lady Leucha Warner ##Lady Warrender ##Lord and Lady Watson ##Lady Cecilia Webb ##Lady Rose Weigall ##Lord Welby ##Lady Willes ##Lady Willis ##Lady Arthur Wellesley ##Lord and Lady Wenlock ##Lord and Lady Westbury and Dowager Westbury ##Lady Isabella Whitbread ##Lady White ##Lady Whitehead ##Lady Whiteway ##Lady Elizabeth Williamson ##Lady Williams-Wynn ##Lady Willoughby (2) ##Lord Willoughby de Broke ##Lord Willoughby de Eresby ##Lady Willshire ##Lady Wilson, Lady Sarah Gordon Wilson ##Lord and Lady Wimborne ##Lady Windeyer ##Lord and Lady Windsor ##Lady Winnington ##Lady Constance Wodehouse ##Lord and Lady Wolverton ##Lady Julia Wombwell ##Lady Wood, Lady Mary Wood ##Lady Woods ##Lord Wrottesley ##Lady Hugh Wyndham ##Lady Barbara Yeatman ##Lady Lilian Yorke ##Lord Zouche #Right Honourables ##H. H. Asquith ##E. Ashley ##A. H. Dyke Acland ##J. Atkinson ##J. B. Balfour ##Sir G. Bowen ##G. W. Balfour ##Sir Hicks-Beach ##A. J. Balfour ##James Bryce ##Sir H. Campbell-Bannerman ##A. H. Smith-Barry ##E. Carson ##H. Chaplin ##Sir J. Chitty ##Jesse Collings ##Sir R. Couch ##G. N. Curzon ##J. Chamberlain ##L. Courtney ##Sir M. Grant-Duff ##A. Akers-Douglas ##Sir W. Hart Dyke ##Sir H. Elliot ##F. Foljambe ##Sir H. Fowler ##Sir A. B. Forwood ##Sir J. Fergusson ##Herbert Gladstone ##Sir J. Gorst ##G. J. Goschen ##W. E. Gladstone ##Sir G. Grey ##C. H. Hemphill ##Charles Seale-Hayne ##R. W. Hanbury ##Lord George Hamilton ##Staveley Hill ##Sir J. T. Hibbert ##Sir W. Harcourt ##lon Hamilton ##Sir Arthur Hayter ##Sir F. Jeune ##W. L. Jackson ##Sir John Kennaway ##G. Shaw-Lefevre ##W. Lidderdale ##Sir Massey Lopes ##James Lowther ##Sir J. Lubbock ##Sir H. Lopes ##Walter Long ##Sir N. Lindley ##J. W. Mellor ##Sir G. O. Morgan ##John Morley ##Arnold Morley ##Sir J. Mowbray ##A. J. Mundella ##J. H. Macdonald ##F. Max Müller ##Sir W. Marriott ##Graham Murray (the Lord Advocate) ##Sir E. Monson ##Sir P. O'Brien ##Sir A. Otway ##Sir F. Peel ##Sir R. Paget of Cranmore ##W. J. Pirrie ##J. P. Robertson ##Sir. J. Rigby ##C. T. Ritchie ##Sir S. H. Strong ##Sir B. Saunderson ##Sir J. Stansfeld ##Sir A. Smith ##C. R. Spencer ##Sir C. Kay-Shuttleworth ##Sir R. Temple ##Sir R. Thompson ##Sir E. Thornton ##Lord Henry Thynne ##Sir G. O. Trevelyan ##C. P. Villiers ##Sir Algernon West ##Sir C. L. Wyke ##C. B. Stuart-Wortley ##S. J. Way #Honourables<ref name=":1" /> (4, Col. 5a / Col. 5b) and Honourable Ladies<ref name=":1" /> (4, Col. 5b / Col. 5c) ##Mrs. Acland ##Mrs. Alexander ##H. Allsopp, Mrs. Allsopp, George Allsopp ##Mrs. Anstruther ##Mrs. Armytage ##[Hon. Lady] Vere Annesley ##Mrs. Bagot, Mrs. Bagot [sic 2x] ##Mrs. Baillie of Dochfour ##Mrs. Balfour ##[Hon.] Coplestone and [Hon.] Mrs. Bampfylde ##John Baring, Susan Baring, Lilian Baring ##Mrs. Barker ##Mrs. Barlow ##Eric Barrington, Mrs. Barrington ##Mrs. Hamar Bass ##Misses Bateman-Hanbury (2) ##Allen B. Bathurst ##Mrs. Benyon ##[Hon. Lady] Beresford ##[Hon.] R. Chetwynd ##Arthur Chichester ##Lady Biddulph ##C. E. Bingham, Mrs. Bingham, Albert Bingham, Mrs. Bingham [sic x2] ##Lady Birkbeck ##Ivo Bligh, Mrs. Bligh ##Diana Sclater-Booth ##O. Borthwick ##J. Boscawen ##Henry Bourke, Mrs. H. Bourke, Charles Bourke, Terence Bourke, Mrs. T. Bourke, Algernon Bourke, Mrs. A. Bourke, Mrs. E. R. Bourke ##Charles Brand, Arthur Brand, Mrs. Brand, Mrs. T. Brand ##T. Brassey, Mrs. A. Brassey ##Mrs. Stapleton Bretherton ##Reginald Brett, Mrs. Brett ##Mrs. F. Bridgeman, Misses Bridgeman (2) ##Mrs. Britten ##W. St. John Brodrick, Albinia Brodrick ##Emmeline Brownlow ##Mrs. T. C. Bruce, Misses Bruce (2) ##Misses M'Clintock Bunbury (2) ##Mary Byng ##T. J. Byrnes ##Arthur Cadogan, Mrs. A. Cadogan, Mrs. C. Cadogan, Ethel Cadogan ##Mrs. Gough-Calthorpe, Rachel (Gough) Calthorpe, Misses Gough Calthorpe (2) ##Mrs. Candy ##G. H. Campbell, K. Campbell, Hugh Campbell, Mrs. H. Campbell, Mrs. Ronald Campbell, Misses Campbell (2), Mrs. J. B. Campbell, Mildred Campbell ##Mrs. Carington ##Mrs. Carpenter ##Emily Cathcart ##W. Cavendish, Mrs. W. Cavendish, Mrs. Cavendish ##Eleonora Chetwynd, Mrs. R. Chetwynd ##Mrs. A. Chichester, Hilda Chichester ##Mrs. Clowes ##T. H. Cochrane ##Audrey Coleridge ##George Colville ##Mrs. Corbett ##Mrs. H. Corry ##Caroline Courtenay ##Henry Coventry ##Osbert Craven ##Misses Cross ##Mrs. P. Crutchley ##Henry Cubitt, Mrs. Cubitt ##Hamilton Cuffe, Mrs. Otway Cuffe ##Lady Cunningham ##Montagu Curzon, Darea Curzon, Mrs. Curzon ##Hew Dalrymple ##John Dawnay, Eustace Dawnay, W. Dawnay, Mrs. Dawnay (2) ##Misses de Montmorency (2) ##Mrs. H. Dennison ##R. C. Devereux, Mrs. R. C. Devereux ##Mrs. Digby ##Conrad Dillon, Mrs. C. Dillon, Edith Dillon ##Misses Douglas-Pennant (2) ##A. Hay Drummond, Mrs. Hay Drummond, Frances Drummond, Mrs. M. Drummond ##Hubert V. Duncombe, Cecil Duncombe, Mrs. C. Duncombe ##C. T. Dundas, Mrs. C. T. Dundas, W. Dundas, Mrs. W. Dundas, Mrs. John Dundas ##Lady Du Cane ##Herbert Eaton, Mrs. H. Eaton ##F. Egerton, Mrs. A. F. Egerton, Lady Grey Egerton, Tatton Egerton, Mrs. T. Egerton ##Arthur Elliot, Mrs. Arthur Elliot, Lady Elliot, Mrs. Eliot ##Lilian Elphinstone ##Mrs. Ellis ##Muriel Erskine ##H. Escombe, Mrs. Escombe ##Mrs. Evans ##Mrs. C. Keith-Falconer ##Sir S. Ponsonby Fane ##Mrs. W. Farquhar ##Ailwyn Fellowes, Mrs. A. Fellowes ##Mrs. Ferguson of Pitfour ##Everard Fielding ##N. Fitzgerald, Mrs. N. Fitzgerald, Mrs. Fitzgerald, , Mrs. F. G. FitzGerald, Lady FitzGerald ##R. Fitzwilliam, W. H. Fitzwilliam ##Mary Forester ##Sir John Forrest ##Mrs. W. H. Forster ##Mrs. Lionel Fortescue ##Sir C. Fremantle, Mary Fremantle ##Sir Malcolm Fraser, Misses Fraser (2) ##Mrs. Charles Keith-Fraser ##Violet Gibson ##Evelyn Giffard ##Mrs. Henry Gladstone ##Lady Godley ##George Ormsby Gore ##F. Leveson-Gower ##Mrs. Gough ##Mrs. Alaric Grant ##Ronald Greville, Mrs. R. Greville, Louis Greville, Mrs. L. Greville, Sidney Greville, Mrs. A. Greville, Mrs. A. H. F. Greville ##Robert Grosvenor, Algernon Grosvenor, Mrs. A. Grosvenor, Maud Grosvenor, Elizabeth Grosvenor ##Lady Hamilton Gordon, [Hon. Lady] Nevil Gordon ##Misses Guest (2) ##Geoffrey Browne Guthrie ##Mrs. Gye ##Mrs. A. Haig ##Mrs. Halford ##, Misses Hamilton (2) ##Mrs. North Dalrymple-Hamilton ##Mrs. Hobart Hampden ##Mrs. Assheton Harbord, Mrs. C. Harbord, Judith Harbord, Bridget Harbord, Mrs. Harbord ##C. Hardinge, Mrs. C. Hardinge, A. Hardinge ##A. E. Gathorne-Hardy, Nina Gathorne-Hardy ##Misses Hawke (2) ##C. G. Hay ##Misses Heneage (2) ##Helen Henniker, Mrs. Henniker ##Robert Herbert, Sir Robert Herbert, Mrs. R. Herbert, Mrs. Herbert ##A. Holland Hibbert, Mrs. A. Holland Hibbert ##Lady Higginson ##Mrs. Hill ##Lionel Holland, Sydney Holland ##Grosvenor Hood, Dorothy Hood ##Lady Acland-Hood ##Fanny Hood of Avalon ##Mrs. Curzon Howe ##[Hon.] Evelyn Hubbard, Mrs. E. Hubbard, Alice Hubbard ##Mary Hughes ##Mrs. Meynell Ingram ##G. Jolliffe, Sydney H. Jolliffe, Mrs. Jolliffe ##Lady Johnston ##G. Keppel, Mrs. Keppel, Derek Keppel, Mrs. William Keppel ##Mrs. Alfred Ker ##Constance Kerr ##Mrs. Kingscote ##C. C. Kingston ##Lady Knollys ##Bertha Lambart ##F. W. Lambton, Mrs. Lambton ##Mary Lascelles ##Charles Laurence, Herbert Laurence ##Wilfrid Laurier ##Mrs. Lawley ##Mrs. C. Lawrence, Misses Lawrence (2), Mrs. H. Lawrence ##Mrs. Legge ##T. W. Legh, Mrs. Legh, Sybil Legh ##F. D. Leigh, Mrs. F. D. Leigh, E. Chandos Leigh, Mrs. E. C. Leigh, Cordelia Leigh ##C. Hanbury Lennox, Mrs. Hanbury Lennox ##G. W. Leslie ##R. l’Estrange ##Atholl Liddell, Mrs. A. Liddell ##Mrs. H. Gore-Lindsay ##Reginald Lister ##Henry Littleton, Misses Littleton (2) ##Misses Loch (2) ##William Lowther, Mrs. W. Lowther, L. Lowther, Mrs. L. Lowther ##Mrs. E. H. Loyd ##Mrs. Lumley ##Alfred Lyttelton, Mrs. A. Lyttelton, Misses Lyttelton (2), Mrs. Lyttelton ##Flora Macdonald, Lady Macdonald ##Mrs. Mackinnon ##Mrs. Maclagan ##Mrs. Magniac ##Mrs. Maguire ##W. Massey-Mainwaring, Mrs. Massey-Mainwaring ##Mrs. Fuller-Maitland ##Aline Majendie ##Misses Henniker Major (2) ##Mrs. Mallet ##Archibald Marjoribanks ##Misses Constable Maxwell (2) ##Mrs. M'Calmont ##Schomberg M'Donnell ##Charles Mills, Violet Mills, Mrs. Mills ##Mrs. Percy Mitford ##Maud de Moleyns ##Mrs. C. Molyneux ##Annette Monck, Mrs. Monck ##Violet Monckton ##Mrs. Monson ##John Scott Montagu ##[Hon.] Evelyn Moore ##R. Moreton, Mrs. R. Moreton ##Mrs. Mostyn, Misses Mostyn (2) ##Mrs. G. H. Murray, Alice Murray ##Lady Musgrave ##[Hon. Lady] Napier, Emilia Napier, Mrs. Scott Napier ##Mrs. Neeld ##Sir Hugh Nelson ##[Hon.] R. Nevill ##Mrs. Newdigate ##Sir H. S. Northcote ##Misses O'Brien (2) ##Mary O'Hagan ##Mrs. Okeover ##Mrs. Oliphant ##R. Terence O'Neill, Henrietta O'Neill ##Misses Palk (2) ##Cecil Parker, R. Parker, F. Parker, Mrs. F. Parker, Mrs. Parker ##Mabel Parnell ##[Hon.] C. B. Parsons, Mrs. Parsons ##Mrs. W. Paton ##[Hon.] Sydney Peel, Misses Peel (2) ##Mrs. Anderson Pelham ##E. S. Douglas-Pennant, Mrs. E. S. Douglas-Pennant ##Mrs. Heber Percy ##Albert Petre, Mrs. A. Petre ##Harriet Phipps ##Mrs. Pirie ##Thomas Playford ##Horace C. Plunkett ##[Hon.] Ashley Ponsonby, Mrs. Ponsonby, Misses Ponsonby (2) ##H. Orde Powlett, Mrs. Orde-Powlett, Myra Orde-Powlett ##E. W. B. Portman, Mrs. Portman, Mary Portman ##Mrs. Pretyman ##C. Ramsay, Mrs. C. Ramsay ##G. H. Reid ##Misses Rendel (2) ##Misses Rice (2) ##Lady White Ridley ##Mrs. Ritchie ##F. Roberts, Mrs. Phillips Roberts ##Misses Roberts (of Kandahar) (2) ##J. M. Rolls, Eleanor Rolls ##W. Rothschild, Evelina Rothschild ##W. Rowley, Mrs W. Rowley, Lady Thelluson Rowley ##A. Russell, Misses Russell (2) ##Gustavus Hamilton-Russell, Misses Hamilton Russell (2) ##the Master of Ruthven, Mrs. Ruthven ##Mrs. J. D. Ryder ##Sir Saul Samuel ##A. Saumarez, Mrs. A. Saumarez ##Mrs. E. J. Saunderson ##J. Maxwell Scott, Mrs. Maxwell Scott ##R. J. Seddon ##Mary Sidney ##Lady Simeon ##Misses Skeffington (2) ##Sir Donald Smith, Mrs. A. H. Smith, [Hon.] W. F. D. Smith ##Granville Somerset, Mrs. G. Somerset, Arthur Somerset, Mrs. A. Somerset, R. Somerset, Violet Somerset ##Mrs. C. R. Spencer ##Sir J. Gordon Sprigg ##Lyulph Stanley, F. C. Stanley, George Stanley, Mrs. E. J. Stanley, Mrs. Stanley, Mrs. V. A. Stanley, Maude Stanley ##Lady Cowell-Stepney ##Randolph Stewart, Mrs. R. Stewart, FitzRoy Stewart, Mrs. Stewart ##Mabel St. Aubyn ##Misses St. Clair (2) ##Mrs. Stirling ##Horatia Stopford ##[Hon. Lady] Alison Stourton ##Mrs. Strutt, Misses Strutt (2) ##Hilda Sugden ##Alfred Talbot, Mrs. Talbot, Mrs. R. A. J. Talbot ##Sir D. Tennant ##S. R. Thayer ##Misses Thellusson (2) ##Edward Thesiger, Mrs. E. Thesiger, Frederick Thesiger, Mrs. F. Thesiger, Mary Thesiger ##Lady Thorold ##Katharine Thring ##Misses Tollemache (2) ##R. Marsham-Townshend, Mrs. Marsham-Townshend ##Alice Hanbury-Tracy ##Charles Grey Trefusis, Misses Trefusis (2) ##Mrs. Trelawny ##Mrs Tremayne ##Mrs. W. le Poer Trench ##Charles Trevor ##George Hill-Trevor, Marcus Hill-Trevor, Mrs. Hill-Trevor, Misses Hill-Trevor (2) ##Mrs. C. W. Trotter ##Lady Tryon ##Rosamond Tufton ##Sir G. Turner ##Rev. L. Tyrwhitt ##Misses Tyssen Amherst (2) ##Misses Vereker (2) ##R. Greville-Verney, Mrs. R. G. Verney, Misses Verney (2) ##F. Villiers, Mrs. F. Villiers ##Misses Vivian (2) ##Arthur Walsh ##Mrs. P. E. Warburton ##Robert Ward, Mrs. Dudley-Ward ##Mrs. West ##Mrs. Whateley ##Sir W. Whiteway ##F. Bootle-Wilbraham ##Ella Williamson ##Tatton Willoughby ##Lady Wilson ##[Hon.] Armine Wodehouse, Mrs. Wodehouse ##Frances Wolseley ##F. Wood, Misses Wood (2) ##Mrs. G. Wrottesley, Evelyn Wrottesley ##Percy Wyndham, Mrs. P. Wyndham, Misses Wyndham (2) ##Maud Wynn ##Lois Yarde-Buller ##Alex. G. Yorke, Mrs. J. Yorke, Mrs. E. C. Yorke #Sirs<ref name=":1" /> (4, Col. 5c–6a) ##Augustus Adderley ##Edwin Arnold ##John Austin ##George Arthur ##John Heathcoat-Amory ##A. Armstrong ##Andrew Agnew ##Frederick Abel ##Henry Acland ##A. Arnold ##Alexander Arbuthnot ##John Barran ##G. Bower ##J. W. Bonser ##J. Crichton-Browne ##Joseph Bailey ##E. Ashmead-Bartlett ##Henry Barkly ##R. Beauchamp ##Raymond Burrell ##Charles Barrington ##David Baird ##Arthur Birch ##Edward Birkbeck ##W. Cunliffe Brooks ##A. de Capel Brooke ##Courtenay Boyle ##F. Burton ##F. Buxton ##Steuart Bayley ##John Bramston ##John Baker ##H. Bullard ##J. T. Brunner ##H. Bellingham ##Henry Bergne ##Thomas Boughey ##F. J. Bramwell ##E. Burne-Jones ##James Blyth ##Seymour Blane ##Henry Chamberlain ##Roderick Cameron ##Hugh Cholmeley ##John Conroy ##Edward Clarke ##C. Cameron ##E. Carbutt ##W. Coddington ##Marshal Clarke ##Reginald Cathcart ##Savile Crossley ##Edward Colebrooke ##Reginald Cust ##Charles Crosthwaite ##John Colomb ##Daniel Cooper ##F. Astley-Corbett ##Donald Currie ##Henry Cunningham ##Robert Cunliffe ##Henry Cotterell ##T. D. Gibson Carmichael ##F. Curden, ##George Dallas ##James Drummond ##Mortimer Durand ##G. Des Vieux ##Henry Dering ##J. N. Dick ##Dyce Duckworth ##T. Swinnerton Dyer ##E. Hastings Doyle ##John Dorington ##William Dunn ##Humphrey de Trafford ##Charles Dalrymple ##G. Dashwood ##Gardner ##Engleheart ##Francis Evans ##A. Edmonstone ##Whittaker Ellis ##W. H. Flower ##Horace Farquhar ##Joseph Fayrer ##H. Fletcher ##William Ffolkes ##William Fraser ##Bartle Frere ##Gerald Seymour Fitz-Gerald ##Robert Finlay ##B. Walter Foster ##Gerald FitzGerald ##R. FitzGerald ##Maurice FitzGerald ##Forrest Fulton ##William Flower ##Andrew Fairbairn ##John Gilbert ##E. T. Gourley ##Edward Grey ##W. Gull ##Walter Gilbey ##Lepel Griffin ##G. Macpherson-Grant ##Reginald Graham ##Philip Grey Egerton ##Douglas Galton ##R. Glyn ##Arthur Godley ##Charles Grant ##R. Gresley ##Alexander Acland-Hood ##T. G. Fermor Hesketh ##Arthur Haliburton ##Brydges Henniker ##F. Dixon-Hartland ##R. Hanson ##Alfred Hickman ##W. Houldsworth ##Henry Howorth ##F. Seager Hunt ##Charles Hall ##E. W. Hamilton ##Reginald Hardy ##Clement Hill ##Basil Hall ##Joseph Hooker ##Charles Hunter ##Charles Hartopp ##Victor Houlton ##Augustus Hemming ##Henry Irving ##Frederic Johnstone ##W. Jenner ##J. Jenkins ##James Joicey ##Charles Jessell ##Harry Johnston ##Edward Jenkinson ##James Hill Johnes ##John Jackson ##H. Seymour King ##James Kitson ##J. Lister-Kaye ##V. Kennett-Barrington ##George Kekewich ##John Leslie ##Thomas Dick Lander ##T. Villiers Lister ##James Linton ##Charles Lees ##Charles Legard ##Thomas Lea ##Wilfrid Lawson ##Elliott Lees ##A. C. Lyall ##J. T. D. Llewelyn ##Joseph Leese ##Leonard Lyell ##F. Laking ##Godfrey Lushington ##F. Lockwood ##Henry Longley ##George Lewis ##F. Milner ##Herbert Maxwell ##Francis Montefiore ##Graham Montgomery ##Robert Moncreiffe ##Musgrave ##Colin Scott Moncrieff ##Francis Mowatt ##Evan MacGregor ##J. G. Miller ##F. D. Maclean ##J. Blundell Maple ##Allan Mackenzie ##Lewis M'lver ##F. Mappin ##Theodore Martin ##Samuel Montagu ##William MacCormac ##Hubert Miller ##Lewis Morris ##Clements Markham ##A. C. Mackenzie ##John Monckton ##J. Stirling-Maxwell ##J. Heron Maxwell ##Kenneth Matheson ##J. S. Montefiore ##Acquin Martin ##W. Maxwell ##Oswald Moseley ##Arthur Nicolson ##Terence O'Brien ##Reginald Ogilvy ##Herbert Oakeley ##Hush Owen ##G. G. Petre ##Walter Parratt ##Frederick Pollock ##Herbert Perrott ##Douglas Powell ##Weetman Pearson ##Joseph Pease ##Francis S. Powell ##Reginald Palgrave ##W. Priestley ##E. G. Poynter ##G. S. Baden-Powell ##Charles Pontifex ##J. Dickson-Poynder ##James Paget ##C. M. Palmer ##C. Lennox Peel ##James B. Peile ##Westby Perceval ##Charles Pigott ##John Puleston ##W. Plowden ##Richard Quain ##George Russell ##C. Lister Ryan ##W. H. Russell ##J. Ramsay ##Owen Roberts ##R. T. Reid ##Charles Robinson ##J. Thellusson Rowley ##James Reid ##C. Euan-Smith ##J. Barrington Simeon ##J. B. Stone ##M. Shaw-Stewart ##Edward Sieveking ##T. H. Sanderson ##Augustus K. Stephenson ##Thomas Sutherland ##Mark Stewart ##Andrew Scoble ##Joseph Savory ##Douglas Straight ##Charles Shelley ##S. Shippard ##E. Sassoon ##A. Condie Stephen ##E. Sullivan ##Arthur Sullivan ##S. Scott ##H. Simpson ##E. Stafford ##Ernest Satow ##Tatton Sykes ##John Tyler ##Charles Tennant ##John Tenniel ##J. Thorold ##John Thursby ##Thomas Troubridge ##Charles Turner ##H. Meysey-Thompson ##W. Vincent ##Edgar Vincent ##Arthur Vicars ##W. Williams-Wynn ##James Walker ##R. Webster ##George Wombwell ##C. Rivers Wilson ##W. H. Wills ##Donald Mackenzie Wallace ##George Warrender ##F. Winnington ##James Whitehead ##Arthur Willshire ##Henry Wood ##Hugh Wyndham ##W. White ##Sidney Waterlow ##Hedworth Williamson ##Jacob Wilson ##W. Windeyer ##Albert Woods (Garter) ##Allen Young #Chairman of County Council (Dr. Collins) #Counts and Countesses ##Count Cassini ##Count and Countess De Ganay ##Count Gurowski ##Count Hohenau ##Count Theodor Bolesta Koziebrodski ##Count Leon Mniszeek ##Count and Countess Potocki ##Count and Countess Raben #Barons and Baronesses ##Baroness Emile Beaumont d'Erlanger ##Baroness De Brienen ##Baron De Onethau and Baroness D’Onethan [sic] ##Baron and Baroness Alphonse de Rothschild ##Baron Ferdinand Rothschild ##Baron and Baroness Schröder ##Baron and Baroness von Deichmann ##Baron von Heeckeren van Wassenaer ##Baroness von Hügel, Baroness Gertrud von Hügel [sic] ##Baron and Baroness Campbell von Laurentz ##Baroness Wilhelm von Rothschild #Rev. the Moderator of the General Assembly of the Church of Scotland #Deans — Christ Church, St. Paul's, Westminster, Windsor #The Provost of Eton #Master of Trinity (Mr. Butler) #The Sub-Dean of the Chapels Royal #Canons — Blundell, Dalton, Duckworth, Fleming, Hervey, Teignmouth Shore, Wilberforce #Dr. Adler (Chief Rabbi) #Dr. M'Cormick #Chaplain of the Fleet #Chaplain General #Reverend Doctors — Edmund Warre, C. J. Welldon #Reverends — Prebendary Hawkshaw, Albert Baillie, W. H. Bliss, M. Ebrington Bisset, Lord W. Cecil, Lord Charles Fitzroy, J. H. Ellison, H. Haweis, W. R. Jolly, G. J. Martin, Newton Mant, Marquis of Normanby, A. Robins. W. Gunion Rutherford, Clement Smith, Montagu Villiers #Doctors — Lennox Browne, J. V. Bridge, Barlow, Robert Farquharson, J. F. Fox, Surgeon-Major Kilkelly, John Lowe, C. H. H. Parry, G. V. Poore, Dorrien Smith, S. Wilks #Messieurs<ref name=":1" /> (4, Col. 6b–7a), Mesdames (4, Col. 7a–b) and Misses<ref name=":1" /> (4, Col. 7c – 5, Col. 1a) ##Mme Abdy ##Mr C. T. Dyke-Acland, Mme A. H. Dyke Acland, Mme Dyke Acland ##Mme Adair ##Misses Adam (2) ##Mr and Mme Adeane ##Misses Adye [?] (2) ##Mme Agar ##Mr Hamilton Aidé ##Mr John Aird, Misses Aird (2) ##Miss Akers-Douglas ##Mr Edward Alderson ##Mr George Alexander, Mme Alexander, Miss Alexander ##Miss Alison ##Mr and Mme Allhusen ##Mme Alma-Tadema ##Mr W. Ambrose ##Miss Heathcoat-Amory ##Mr R. Anderson, Miss Florence Anderson ##Mr E. H. Anson ##Mr H. T. Anstruther, Miss Rosomond Anstruther ##Mme Antrobus ##Mr Arbuthnot, Miss Arbuthnott [sic] ##Miss Archer-Houblon ##Mme Argles ##Mme Arkwright, Miss Arkwright ##Misses Armytage (2) ##Miss Arnott ##Mr and Mme Ascroft, Miss Ascroft ##Mr Arthur Ash ##Mr A. Asher ##Mme Ashton ##Mme Asquith ##Mr Astor, Mr W. Astor ##Mr B. F. Astley ##Mme Evelyn Atherley ##Mr and Mme Alfred Austin, Misses Austin (2) ##Mr and Mrs C. H. Babington ##Mr and Mrs Bagge ##Mrs Charles Bagot, Mrs J. F. Bagot, Miss Alice Bagot ##Mr James Bailey, Mrs J. Bailey, Mrs Bailey, Misses Bailey (2) ##Mrs Duncan Baillie, Misses Duncan Baillie (2) ##Mr Baillie of Dochfour ##Mr and Mrs W. A. Baillie-Hamilton ##Mr E. Bainbridge ##Mr and Mrs H. R. Baird, Mr and Mrs J. G. A. Baird, Misses Baird (2) ##Mr and Mrs Baldwin ##Mr and Mrs E. Balfour, Mr and Mrs Charles Balfour, Miss Balfour ##Mr and Mrs Banbury, Miss Banbury ##Mr and Mrs S. B. Bancroft [actor "Bancroft and his wife accepted with becoming grace the congratulations with which they were well-nigh overwhelmed"<ref name=":3" /> (5, Col. 6b)] ##Bandanaratke [?] ##Mrs Bankes ##Mr Banks ##Mr and Mrs Walter Baring, Miss Baring ##Miss Barker ##Mr J. Emmott Barlow, Mrs Barlow, Mrs Barlow [sic 2x] ##Misses Barnardiston (2) ##Miss Barne ##Mr and Mrs F. G. Barnes, Mr and Mrs Barnes, Misses Barnes (2) ##Miss Barran (2) ##Mr and Mrs J. Wolfe Barry, Mr and Mrs F. Tress Barry, Mrs A. Barry ##Misses Bartlett (2) ##Mr and Mrs D. P. Barton, Mr and Mrs Barton ##Mr Hamar Bass ##Mrs Bates, Miss Bates ##Mr and Mrs H. Bathurst, Misses Bathurst (2) ##Mr and Mrs Baxendale, Miss Baxendale ##Miss Mariot [?] Bayley ##Mr and Mrs W. W. Beach, Miss Beach ##Misses Hicks-Beach (2) ##Mr R. M. Beachcroft ##Mr and Mrs Wentworth Beaumont, Mr Wentworth B. Beaumont, Mrs Beaumont, Miss Hilda Beaumont ##Mr and Mrs Rupert Beckett, Mr E. W. Beckett ##Mr and Mrs Beer ##Mr and Mrs F. F. Begg ##Mr Charles Bell, Mr and Mrs Bell, Misses Bell (2) ##Miss Bellingham ##Mr and Mrs R. Benson, Mr and Mrs Benson ##Miss Berens ##Mr and Mrs Beresford, Miss Beresford ##Miss Berkeley, Misses Berkeley (2) ##Mr and Mrs Bertier, Miss Bertier ##Mr and Mrs Cosmo Bevan, Mr and Mrs F. Bevan, Miss Bevan ##Mr M. M. Bhownaggree ##Mr and Mrs F. Bibby ##Mr Leonard Biddulph, Mr Biddulph, Mr Victor Biddulph, Mr M. Biddulph, Mrs H. M. Biddulph, Misses Biddulph (2), Miss Biddulph, Miss Freda Biddulph ##Mr and Mrs Bigham ##Mr Bigwood ##Mrs C. Bill, Miss Bill ##Miss Birch ##Mrs Birch-Reynardson, Misses Birch-Reynardson (2) ##Mr A. Birrell, Mrs Birrell ##Mr and Mrs Bischoffsheim ##Mrs Ebrington Bissett ##Misses Blackwood (2) ##Mr and Mrs R. G. Blennerhassett ##Mrs W. H. Bliss ##Mrs Blundell, Miss Blundell ##Misses Blyth (2) ##Mr and Mrs Bolitho, Miss Bolitho ##Mr H. C. O. Bonsor, Mrs Bonsor, Miss Bonsor ##Mrs W. Borsel ##Mrs Griffith-Boscawen ##Mr and Mrs Boulnois ##Miss Bourke ##Mr W. R. Bousfield ##Mrs Bowden-Smith, Misses Bowden-Smith (2) ##Miss Bowen (2) ##Mr T. G. Bowles, Mrs Bowles ##Mr Edmund R. Boyle ##Miss Mabel Brackenbury ##Mrs Bradley, Miss Bradley ##Miss Beryl Bradford ##Miss Braddon ##Miss Bramwell ##Mr H. L. C. Brassey, Mrs H. A. Brassey, Misses Brassey (2), Misses Brassey (2) [sic 2x] ##Mr Stapleton Bretherton, Misses Stapleton Bretherton (2), Mr F. Stapleton Bretherton ##Mrs Bridge ##Mr G. Bridgman, Mr and Mrs C. G. O. Bridgeman ##Mr Brigg ##Mrs Brocklehurst ##Misses Brodie (2) ##Mr and Mrs Brookfield, Miss Brookfield ##Miss Bromley-Davenport ##Miss Brooke ##Miss Rhoda Broughton ##Mr and Mrs A. H. Brown, Miss Brown ##Mrs Browne, Misses Browne (2), Misses Browne (2) [sic 2x] ##Mrs Brownrigg, Miss Brownrigg ##Mr A. O. Bruce, Mrs A. C. Bruce [sic], Misses Bruce (2) ##Miss Brunner ##Mrs Bryce ##Mr Brymer ##Mr and Mrs Buchanan ##Mrs C. E. Buckle ##Mr Bucknill ##Miss Budgett ##Miss Mary Bulteel ##Miss Burdett ##Mr and Mrs Burges, Misses Burges (2) ##Mrs C. K. Burn ##Mr and Mrs F. C Burnand ##Miss Evelyne Burne ##Mr and Mrs W. Burns, Miss Burns ##Misses Burrell (2) ##Mr J. G. Butcher ##Mrs Butler, Mrs Butler, Miss Butler ##Mr Sydney Buxton, Mrs S. Buxton, Misses Buxton (2) ##Mr P. H. Calderon ##Mrs Calley ##Mrs Archibald Calvert, Miss Calvert ##Mr Cameron, Miss Cameron, Misses Cameron (2) ##Mr and Mrs J. D. Campbell, Mr J. A. Campbell, Miss J. A. Campbell, Mrs F. Campbell, Mrs W. Campbell, Mrs Hastings Campbell, Mrs W. Campbell [sic 2x], Mrs F. L. Campbell, Mrs D. B. O. Campbell, Miss Lilah Campbell, Miss Campbell, Miss Ronald Campbell, Misses Campbell (2) ##Miss Grace de Capell-Brooke ##Miss Carden ##Miss Carleton ##Mr and Mrs W. W. Carlile, Miss Carlisle ##Mrs Rivett Carnac ##Mrs Carnegy ##Mrs Boyd Carpenter, Misses Boyd Carpenter (2) ##Mrs Carson ##Mr and Mrs D'Oyly Carte ##Mrs Carter ##Mrs Castance ##Mr R. K. Causton, Mrs Causton, Miss Causton ##Mrs Cavaye ##Mr and Mrs C. Tyrall Cavendish, Mr Victor Cavendish, Mr Henry Cavendish, Mr Cavendish, Mrs Cavendish ##Mr and Mrs F. Cavendish-Bentinck, Mr Cavendish-Bentinck, Mrs W. G. Cavendish-Bentinck ##Mr F. Cawley ##Mr and Mrs Cayzer, Miss Cayzer ##Mr and Mrs W. M. Cazalet ##Mr F. Cazenove ##Mr Evelyn Cecil, Miss Cecil ##Mrs Chaine ##Mrs Chaloner ##Mr Austen Chamberlain, Mrs Chamberlain, Misses Chamberlain (2) ##Misses Chaning (2) ##Mr and Mrs Channing ##Mr and Mrs Cecil Chaplin, Misses Chaplin (2), Miss Edith Chaplin, Miss Chaplin ##Mrs Chapman ##Misses Chetwode (2) ##Mrs W. Chetwynd, Miss Chetwynd (2) ##Mr Childe-Pemberton ##Miss Chitty ##Miss Leila Crichton ##Miss Cholmeley (2) ##Miss Cholmondeley ##Miss Chrichton-Maitland ##Mrs H. Churchill ##Miss Spencer Churchill ##Mr J. D. Clark, Mr and Mrs Atkinson Clark, Mr Clark, Mrs B. F. Clark, Mrs G. D. Clark, Stanley Clark, Miss Clark ##Mr Purdon Clarke, Mr Ernest Clarke, Miss Clarke, Miss Stanley Clarke ##Mrs Clerk ##Mr and Mrs Henry Pelham Clinton ##Mrs Clive, Misses Clive (2) ##Mrs Close ##Mr Clough ##Mr Clowes, Misses Clowes (2) ##Mr Cobbold ##Mr T. B. Cochrane, Miss Cochrane ##Mr and Mrs W. A. Cockerell, Miss Cockerell, Miss Cockerell [sic 2x] ##Mr and Mrs D. Coghill ##Mr B. Cohen ##Mr Wentworth Cole ##Miss Colomb ##Mr and Mrs Colston ##Miss Colville ##Mr Richard Combe ##Miss Commerell, Miss Commerell [sic 2x] ##Mr and Mrs Compton ##Mr and Mrs Consett, Miss Vera Consett ##Mr and Mrs F. L. Cook, Mr Ward Cook, Miss Cook ##Mr and Mrs Kinloch Cooke, Mr Cooke, Mr and Mrs C. Kinloch Cooke ##Mr and Mrs Daniel Cooper, Mrs E. H. Cooper, Misses Cooper (2), Miss Cooper ##Mr and Mrs Cameron Corbett, Miss Corbett ##Mr and Mrs V. Seymour Corkran, Miss Corkran ##Mr and Mrs F. S. W. Cornwallis ##Mr and Mrs Cory ##Mrs Armar Corry, Mrs Clifford Corry, Miss Corry ##Mr J. R. G. Cotterell, Miss Cotterell (2) ##Mrs Stapleton Coton ##Mr and Mrs George Courroux ##Mrs Courtney ##Mr Burdett-Coutts ##Mrs Coventry ##Miss Cowell ##Miss Cowell-Stepney ##Mr and Mrs R. Cox, Mrs Cox, Miss Cox ##Mrs Crabbe, Misses Crabbe (2) ##Mrs Craik ##Mr and Mrs Crawshay ##Mrs Creignton, Miss Lucia Creighton ##Mr C. A. Cripps, Mr and Mrs Wilfrid Cripps ##Mr and Mrs Critchett ##Mr and Mrs Croombie ##Mrs A. B. Crosbie ##Mr and Mrs Shepherd Cross, Mr A. Cross, Miss Crosse ##Mr and Mrs Cruddas, Misses Cruddas (2) ##Mr and Mrs Percy Crutchley, Misses Crutchley (2) ##Miss Cuffe ##Miss Culme-Seymour ##Mrs Cuninghame ##Miss Cunliffe ##Mrs Dick-Cunynghame ##Mrs Curzon ##Misses Cust (2) ##Miss Custance ##Mrs Dalbiac ##Miss Gladys Dalgety [?] ##Mr C. B. Dalison ##Miss Dalrymple ##Mrs Dalton ##Mrs Denis Daly ##Mr and Mrs Darling ##Miss Dashwood ##Mr W. Bromley-Davenport ##Miss Davey ##Mr and Mrs Louis Davidson, Mrs Randall Davidson ##Mr W. Rees Davies, Mr Ben Davies, Mr and Mrs Vaughan Davies ##Mrs Davis ##Miss Dawnay (2) ##Mrs de Arcos ##Misses De Brienen (2) ##[Miss] La Baronne de Friesen ##Mrs R. C. de Grey Vyner ##[Miss] La Baronne Sirtema de Grovestins [?] ##Mr and Mrs J. de la Cour ##Mr and Mrs Edwin de Lisle ##Mr W. E. Denison ##Mrs Denny ##Miss De Perpigna ##Mrs de Salis ##Mr de Soria ##Mr De Trafford, Miss De Trafford ##Mr Deverell, Miss Deverell ##Mr and Mrs W. de Winton, Miss De Winton ##Mr and Mrs Gerard Dicconson ##Mr and Mrs Dicken ##Mr and Mrs C. S. Dickson, Mrs Dickson ##Mr J. K. Digby, Kenelm E. Digby, Mrs Digby, Misses Digby (2), Miss Digby ##Mr and Mrs J. Diggle ##Mr Lee Dillon, Misses Dillon (2) ##Mr and Mrs Coningsby Disraeli, Mr and Mrs R. Disraeli, Miss Disraeli ##Mrs Domvile, Miss Domvile ##Mr Greville Douglas, Mrs A. L. Douglas, Misses Douglas (2) ##Mrs Akers-Douglas ##Miss Dowell ##Mr and Mrs Doxford, Miss Doxford ##Mrs Geoffrey Drage ##Mr A. Drummond, Mr and Mrs G. Drummond, Mrs A. Hay Drummond, Mrs Lawrence Drummond, Mrs Drummond, Miss Edith Drummond, Misses Drummond (2), Miss Mary Drummond, Miss Adelizs [?] Drummond, Misses Drummond (2) [sic 2x] ##Misses Du Cane (2) ##Miss Du Chair ##Mr W. H. Dudley-Ward, Miss Sybil Dudley-Ward ##Mr F. Dugdale ##Misses Duncombe (2) ##Mrs Dundas, Miss May Dundas ##Miss Dunn ##Mrs Dunne, Miss Marion Dunne ##Mr Du Plat Taylor, Mrs G. Du Plat Taylor ##Mrs Durnford ##Mr and Mrs Thiselton Dyer ##Mrs East, Misses East (2) ##Mr F. Eaton ##Mr R. Edgcumb ##Mrs Edis, Misses Edis (2) ##Mr Bevan Edwards, Miss Bevan Edwards (2), Mr C. C. Edwards, Mrs Edwards ##Mrs Egerton, Miss Egerton (2), Miss Egerton ##Miss Grey Egerton ##Mr and Mrs M. Eliot, Misses Eliot (2) ##Miss Ellaby ##Mrs Ellicott, Miss Ellicott ##Mr and Mrs F. Elliot, Mr T. H. Elliott, Miss Gertrude Elliot ##Mr T. E. Ellis, Miss Ellis (2), Miss Evelyn Ellis ##Mrs Ellison, Miss Ellison ##Misses Elphinstone (2) ##Mr Cary-Elwes ##Mr Erskine, Miss Rachel Erskine ##Mr Maurice Euphrussi ##Mr W. H. Evans, Misses Evans (2) ##Mr H. P. Ewart, Mrs C. B. Ewart ##Mr Eyre ##Mr Cecil Fane, Mr G. H. Fane, Mr Fane ##Mr Dyafer Fakhry ##Misses Keith Falconer (2) ##Mrs Fane ##Mrs Fanshawe, Miss Fanshawe ##Mr and Mrs Fardell, Misses Fardell (2) ##Mr and Mrs Farmer, Mrs Lancelot Farmer, Miss Farmer ##Mrs Farnham ##Mr Alfred Farquhar, Mr W. Farquhar, Mr and Mrs E. Farquhar, Mrs G. M. Farquhar ##Mr J. N. Farquharson, Miss Amelia Farquharson, Miss Henrietta Farquharson ##Mr and Mrs Farquharson of Invercauld, Misses Farquharson of Invercauld (2) ##Misses Feilding (2) ##Mrs Fellowes ##Mrs Fenn ##Mrs Fenwick, Misses Fenwick (2) ##Mr and Mrs Johnson-Ferguson ##Mr Munro-Ferguson ##Misses Ferguson of Pitfour (2) ##Miss Fergusson ##Miss Dorothy Ffolkes ##Mrs Field ##Mr and Mrs Fielden, Misses Fielden (2) ##Mrs G. H. Finch, Mrs Wynne Finch, Misses Finch (2) ##Mr and Mrs Firbank ##Mr Herbert Fisher, Mr and Mrs Hayes Fisher, Misses Fisher (2) ##Mr and Mrs Fison, Miss Fison ##Miss FitzClarence (2) ##Mrs FitzGeorge, Miss Olga FitzGeorge ##Mr Fitzgerald, Mr F. G. Fitzgerald, Miss Fitz Gerald ##Mr and Mrs Almeric Fitzroy, Miss Ethel Fitz-Roy ##Mrs R. Fitzwilliam, Misses Fitzwilliam (2) ##Mr Flannery ##Mr E. Flower, Miss Flower, Miss Flower [sic 2x] ##Mrs Floyd ##Mrs H. Fludyer ##Mr H. St. George Foley ##Mrs Barrington Foote ##Mr J. S. Forbes, Mr Forbes ##Mr John Ford ##Mr H. W. Forster ##Mr and Mrs Arnold-Forster ##Mr and Mrs Bevill Fortescue ##Misses Forwood (2) ##Mr W. S. Foster, Mrs W. H. Foster, Mrs H. S. Foster, Miss Foster ##Misses Fowler (2) ##Mr Franklin ##Mrs Houston French ##Misses Frere (2) ##Mr L. Fry ##Mrs Fullerton, Misses Fullerton (2) ##Mr Gadson ##Mr Wilhelm Ganz ##Miss Gardiner, Miss Gardiner [sic 2x] ##Mrs Gardner ##Mr and Mrs Garfit [?] ##Mrs Gathorne-Hardy, Miss Gathorne-Hardy ##Mr Hamilton Gatliff ##Mr and Mrs Scott Gatty ##Mr and Mrs Sydney Gedge ##Mr Geoffrey Drage [sic; does this belong here?] ##Mr F. W. Gibbs, Misses Gibbs (2) ##Mr and Mrs Walter Gibson ##Miss Gilbey ##Mr and Mrs Tyrell Giles ##Mr W. Gillett ##Mr and Mrs Gilliat, Misses Gilliat (2) ##Mr Henry Gladstone, Mrs Gladstone, Miss Helen Gladstone ##Miss Glyn ##Misses Godley (2) ##Mrs Godson ##Mr and Mrs Goelet, Miss Goelet ##Mr Charles Gold, Miss Gold ##Mr G. P. Goldney ##Mr and Mrs S. Hoffnung Goldnung Goldsmid ##Mrs A. Goldsmid, Miss Goldsmid ##Mr Otto Goldsmidt ##Mrs Goldsworthy ##Mrs Goodden, Miss Gurrney Goodden ##Mrs Goodenough ##Mr and Mrs John Gordon, Mr and Mrs J. E. Gordon, Mrs Gordon, Mrs G. G. Gordon, Mrs S. Gordon, Mrs Gordon [sic 2x], Miss Hamilton Gordon, Misses Gordon (2) ##Mr and Mrs Frank Gore, V. Gore, Mr and Mrs S. W. Gore, Mrs Gore, Miss Gore ##Mr and Mrs Goschen, Misses Goschen (2) ##Mr and Mrs A. Gosling, Miss Gosling ##Mr and Mrs F. R. Gosset ##Misses Gough-Calthorpe (2) ##Mr E. A. Goulding ##Mr G. Leveson-Gower ##Mr F. Graham, Mr Graham, Mr H. R. Graham, Mr and Mrs C. C. Graham ##Mrs Grant, Miss Grant ##Miss Victona Grant-Duff ##Mr and Mrs Henry Graves, Miss Graves ##Mr Ernest Gray ##Mrs Green ##Mr H. D. Greene, Mr W. R. Greene ##Mrs Gregory, Miss Gregory ##Mr and Mrs W. H. Grenfell, Mrs H. Grenfell, Miss Maud Grenfell ##Mr J. A. Gretton ##Mr Howard of Greystoke ##Mr Grifflth-Boscawen ##Mr and Mrs W. H. Kendal Grimston ##Mr George Grossmith ["George Grossmith was not a little lionised by titled ladies"<ref name=":3">“The Queen’s Garden Party. Buckingham Palace Grounds. A Brilliant Scene. The Queen’s Cup of Tea.” ''Daily News'' (London) 29 June 1897, Tuesday: 5 [of 10], Col. 6a [of 7] – 6, Col. 2a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970629/021/0005. Print pp. 5–6.</ref> (5, Col. 6b)] ##Mr Montagu Guest ##Mrs Gunter, Misses Gunter (2) ##Mr Gurdon ##Mrs Gurney ##Mrs Guy-Pym ##Mr and Mrs Gye ##Mr and Mrs Carl Haag, Miss Carl Haag ##The Munshi Abdul Hafiz Karim ##Mr and Mrs Haggard ##Miss Haig ##Mr R. B. Haldane ##Mr Halford, Misses Halford (2) ##Mr and Mrs Lewis Hall, Mrs Hall, Miss Hall, Miss (Lewis) Hall ##Mr and Mrs Thomas Halsey, Misses Halsey (2) ##Mr Francis Hamilton, Mrs R. W. Hamilton, Mrs Ian Hamilton, Misses Hamilton (2), Miss Hamilton ##Mrs Hammet ##Mr and Mrs Hanbury, Miss Dora Hanbury, Mrs Hanbury ##Miss V. Hanson ##Mr L. V. Harcourt ##Mr and Mrs Hardcastle, Misses Hardcastle (2) ##Mr and Mrs Hardy, Misses Hardy (2) ##Mr Cozens-Hardy ##Mr T. Hare, Mr Augustus Hare, Mr and Mrs John Hare, Mrs Marcus Hare, Mrs Marcus Hare [sic 2x], Miss Hare, Misses Hare (2), Misses Hare (2) [sic 2x] ##Mrs Harford ##Mrs Hargreaves-Rogers ##Mr C. Harrison, Miss Harrison ##Miss Hart ##Misses Hart-Dyke (2) ##Mr and Mrs Hartmann ##Mr George Harwood, Misses Harwood (2) ##Mr Hatch ##Mrs Hatton ##Mrs Haweis ##Mr and Mrs Claude Hay, Misses Hay (2), Misses Hay (2) [sic 2x] ##Mrs Arthur Heath, Mrs Heath ##Miss Louisa Heathcote ##Mr J. Henniker Heaton ##Misses Hemming (2) ##Mr Philip Henriques ##Mrs Heneage, Miss Heneage ##Mrs Henderson ##Miss (Brydges) Henniker ##Mrs Philip Henriques ##Mrs Herbert, Miss Herbert ##Mr and Mrs Hermon-Hodge ##Miss Heron Maxwell (2) ##Mr G. T. Hertslet ##Mrs Hervey, Miss Hervey ##Misses Hervey-Bathurst (2) ##Mr and Mrs Heseltine, Miss Heseltine ##Miss Hickman ##Mr H. Higgins, Mr Cecil Higgins, Mrs Higgins ##Mrs Platt-Higgins ##Miss Gladys Higginson ##Mrs Hildyard ##Mrs Staveley Hill, Miss Hill ##Misses (Stock) Hill (2) ##Mrs Hills ##Mrs Hippisley ##Mr and Mrs E. Brodie Hoare, Misses (Brodie) Hoare (2), Mr G. Hoare, Mrs S. Hoare, Misses Hoare (2) ##Mr and Mrs H. Hobhouse ##Mr R. K. Hodgson ##Mr and Mrs C. D. Hohler ##Mr R. R. Holmes, Mrs Holmes ##Mr R. Hallett Holt ##Mr Maurice Holzman ##Misses Hood (2) ##Miss Margaret Acland Hood ##Miss Hooker ##Mr and Mrs E. Hope, Mr and Mrs Adrian Hope, Misses Adrian Hope (2), Mr and Mrs James Hope, Mr Hope, Miss Mary Hope, Miss Hope ##Mr and Mrs Beresford-Hope, Miss Agnes Beresford Hope ##Mr and Mrs W. H. Hornby, Misses Hornby (2) ##Mr and Mrs Horner ##Mr and Mrs Hornyold [sic] ##Mr J. C. Horsley ##Miss Jean Hotham ##Misses Houldsworth (2) ##Mr R. P. Houston ##Mr E. S. Howard, Mr and Mrs A. C. Howard, Mr Joseph and Mrs J. Howard, Mr and Mrs H. Howard, Mrs Howard, Misses Howard (2), Miss Howard, Misses Howard (2) [sic 2x] ##Mr J. Hozier ##Mr and Mrs G. B. Hudson ##Hughes, Misses Hughes (2) ##Mr and Mrs A. C. Humphreys-Owen ##Mr and Mrs Hungerford ##Miss M. Carew Hunt ##Mrs W. G. G. Hutchinson, Miss Hutchinson ##Mr and Mrs G. M. Hutton, Mr John Hutton, Mr A. E. Hutton, Mrs G. Hutton ##Mrs Inglefield ##Mr and Mrs Wootton Isaacson ##Mrs Jackson, Miss Grace Jackson ##Mr Jacobs ##Mrs Jacoby ##Mr and Mrs W. James, Mr Arthur and Mrs A. James, Miss Helena James ##Mrs J. E. Jameson, Misses Jameson (2) ##Mr and Mrs Jebb ##Mr and Mrs A. F. Jeffreys ##Mr E. R. Jenkins, Missess Jenkins (2) ##Mrs Jenkinson ##Miss Jenner ##Mr and Mrs H. C. Jervoise, Miss Jervoise ##Mrs Jessel, Miss Jessel ##Mrs Cotton-Jodrell, Miss Cotton-Jodrell ##Mr and Mrs J. H. Johnstone, Mrs G. Johnstone, Miss Johnstone, Miss S. L. Johnstone ##Mrs Joicey, Miss Joicey ##Misses Jolliffe (2) ##Mr and Mrs Atherley-Jones ##Mr and Mrs Brynmor-Jones ##Mrs Inigo Jones ##Mr Philip Burne Jones ##Mrs Pryce Jones ##Mr Henry Joslin ##Mr and Mrs Kearley ##Mrs Keeley ##Misses Keith-Falconer (2) ##Miss Kemball ##Mr George Kemp ##Mr C. Kempe ##Mr and Mrs A. Kennard, Mrs Hegan Kennard, Miss Kennard, Misses Kennard (2) ##Misses Kennaway (2) ##Mrs Kennedy, Miss Kennedy ##Mrs Kennion [?] ##Mrs Kennison ##Mr and Mrs W. Kenny, Miss Ethel Kenny ##Mr J. Kenyon ##Mrs Colin Keppel, Miss Keppel ##Misses Ker (2) ##Misses Kerr (2), Miss Nona Kerr ##Mrs Kilkelly ##Mr and Mrs Kimber, Miss Kimber ##Mr King King, Miss King King ##Mr Nigel Kingscote, Mr T. Kingscote ##Mrs Kingston ##Mrs Kitching ##Misses Kitson (2) ##Mr Lees Knowles and Mrs Knowles ##Mr Knowles ##Mr and Mrs Kuhe ##Mrs A. P. Lake ##Misses Lambart (2) ##Miss Aline Lambton ##Mr Landon ##Mrs Lane ##Mrs Langenbach ##Miss Larking ##Mrs Lascelles ##Mr W. F. Laurence ##Mr Edwin Laurence ##Misses Laurie (2) ##Mrs Laurier ##Mr and Mrs E. Law ##Mrs E. Lawrence, Misses Lawrence (2) ##Mrs Lawrie ##Mr J. Grant Lawson, Miss J. Lawson ##Mr and Mrs Lecky ##Mrs Hanning Lee, Misses Hanning Lee (2) ##Miss C. Lees ##Miss Leese ##Miss Violet Leigh ##Mr and Mrs S. Leighton, Misses Leighton (2) ##Mrs Leslie ##Mr L’Estrange, Miss l’Estrange ##Mr Letchworth ##Mrs Lewis, Misses Lewis (2) ##Mrs Naylor Leyland ##Mrs Liddell, Miss Liddell, Misses Liddell (2) ##Mrs Lidderdale, Misses Lidderdale (2) ##Misses Lindley (2) ##Mr Henry Gore Lindsay, Miss Gore Lindsay, Mr H. B. Lindsay, Mr W. A. Lindsay, Miss Lindsay, Miss Lindsay [sic 2x] ##Mr Leonard Lindsey ##Misses Linton (2) ##Miss Lister ##Mr Cecil Lister-Kaye ##Miss Llewelyn ##Mr E. Lloyd, Mrs Lloyd, Misses Lloyd (2) ##Miss Alice Loch, Miss Emily Loch ##Mrs Lockhart ##Mrs Lockwood, Miss Lockwood ##Mr Loder ##Miss Loftus ##Mr Heathcote Long and Mrs Long ##Mr H. T. Lopes, Misses Lopes (2) ##Mr Drury Lowe and Mrs Lowe, Miss Drury Lowe ##Mr and Mrs J. W. Lowther, Miss Aimee Lowther ##Mr E. H. Loyd, Mr and Mrs A. K. Loyd ##Mr and Mrs H Lubbock, Miss Lubbock ##Mr Reginald Lucas, Mrs Lucas, Mrs F. A. Lucas ##Mrs Lucas-Shadwell, Miss Lucas-Shadwell ##Mrs Luck ##Mr and Mrs Fairfax Lucy ##Mr H. Luttrell, Mr W. C. F. Luttrell, Mrs Luttrell, Miss Luttrell ##Miss Lyall ##Misses Lyell (2) ##Mr and Mrs Lyon ##Miss Lyson ##Misses Lyte (2) ##Mr W. G. E. Macartney ##Mr and Mrs J. C. Macdona ##Miss Macdonald ##Mr Alpin Macgregor, Misses MacGregor(2), Miss Macgregor ##Mr and Mrs Muir Mackenzie, Miss Margaret Muir MacKenzie, Miss Mackenzie ##Mr Mackinnon ##Misses Mackworth (2) ##Mr and Mrs J. Maclean ##Mr and Mrs J. W. Maclure, Miss Maclure ##Mr and Mrs Frederick Macmillan ##Miss Macnaghten ##Miss Macpherson-Grant ##Mr Madden, Miss Madden, Misses Madden (2) ##Misses Magniac (2) ##Mr R. Maguire ##Mr Fuller Maitland ##Mr lan Z. Malcolm ##Mrs Malet, Miss Malet ##Mr B. Mallet ##Mr and Mrs G. Manners ##Mrs Newton Mant ##Mr and Mrs Marjoribanks, Mrs Majoribanks ##Mr T. C. March ##Mrs Markham, Misses Markham (2) ##Mr and Mrs H. H. Marks ##Mrs Marshall ##Mr and Mrs W. A. M’Arthur ##Mr Martin, Mrs R. B. Martin ##Miss Martyn ##Mr Mason ##Miss Massey-Mainwaring ##Mr C. Maud ##Mr and Mrs F. W. Maude, Mrs C. Maude, Miss Constance Maude ##Mrs Maurice ##Misses Maxwell (2) ##Mr and Mrs Maxwell-Lyte ##Mrs May ##Mr H. L. B. MCalmont, Mrs J. M'Calmont ##Miss M'Clintock ##Mrs J. M'Donald ##Mrs Meeking, Miss Meeking ##Mrs Mellor, Misses Mellor (2) ##Mr and Mrs Beresford Melville ##Mr and Mrs T. G. Menzies, Miss Menzies ##Mr and Mrs M'Ewan ##Mr and Mrs P. C. Milbank, Miss May Milbank ##Mr F. Bingham Mildmay, Mr and Mrs Bingham Mildmay, Miss Beatrice Mildmay ##Mr and Mrs Arundel St. John Mildmay, Miss St. John Mildmay ##Mrs Napier Miles ##Mrs Millett ##Mr and Mrs A. Milman, Miss Lena Milman, Misses Milman (2) ##Mr and Mrs Milvain ##Mrs Victor Milward, Miss Milward ##Mr A. B. F. Mitford, Miss Mitford ##Mr and Mrs M’Laren ##Mrs M'Neile ##Mr and Mrs C. M’Neill, Mrs M’Neill ##Mrs W. C. F. Molyneux ##Mrs G. Moncrieff ##Mr Monk, Misses Monk (2) ##Mr E. P. Monckton ##Mr Ronald Moncrieffe ##Miss Cicely Monson ##Mr V. Montagu, Misses Montagu (2) ##Mrs Montefiore ##Mr Montgomerie ##Mrs Montgomery, Miss Montgomery ##Mr Moon ##Miss Mary Moore ##Mrs Moorhouse ##Mrs Manvers Moorson ##Mr R. J. More ##Miss More-Molyneux ##Miss Evelyn Moreton ##Mr Charles Morley ##Mr and Mrs Morrell ##Mrs Ashurst Morris, Misses Morris (2), Miss Ethel Morris ##Mr Hugh Morrison ##Misses Moseley (2) ##Mrs Mostyn ##Mr and Mrs Mount, Misses Mount (2) ##Misses Mowatt (2) ##Miss Mowbray ##Mrs Max Muller ##Miss Mundella ##Mr and Mrs Campbell Munro, Miss Campbell Munro ##Mr and Mrs Muntz, Miss Muntz ##Mr and Mrs Murdoch, Miss Murdoch ##Mr and Mrs W. J. Mure, Mr Mure, Miss Mure ##Mr C. J. Murray, Mr G. H. Murray, Mrs Graham Murray, Miss Graham Murray, Mrs J. Murray, Mrs Wyndham Murray, Miss Murray ##Mr W. H. Myers ##Mr M. Myther ##Misses Nelson (2) ##Misses Nevill (2), Miss Nevill ##Mrs F. Neville, Miss Neville ##Mrs Nevul [?] ##Mr F. A. Newdigate ##Mrs Newhouse ##Mrs H. F. Nicholson ##Mr and Mrs Nicol, Miss Nicol ##Mr G. and Mrs Noel, Misses Noel (2) ##Mrs Nugent ##Mr T. W. Nussey ##Mrs Oakley, Miss Oakley ##Misses O’Brien (2) ##Mr J. C. O'Dowd ##Miss Ogilvy ##Miss V. A. Okeover ##Mrs H. H. Oldham ##Mr and Mrs M. Oldroyd ##Mr and Mrs Arthur Oliphant ##Misses Olpherts [?] (2) ##Mr and Mrs Oppenheim, Miss Linda Oppenheim ##Mr and Mrs Charles Orde ##Mr C. L. Orr-Ewing ##Miss Alina O'Shee ##Mr and Mrs R. A. Oswald, Mr and Mrs Oswald ##Miss Phoebe Otway ##Miss Humphreys Owen ##Mr Hussey Packe, Misses Packe (2) ##Mrs A. Paget, Mrs Paget, Miss Alice Paget, Miss Paget ##Misses Paget of Cranmore (2) ##Mrs Pakenham, Mrs Pakenham ##Miss Palgrave ##Mrs Dampier Palmer, Miss Palmer ##Mr Paoli ##Miss Parker ##Mr E. and Mrs Parkes ##Mrs H. Parr ##Miss Parry ##Mr Paton ##Mr and Mrs J. L. Pattison, Miss Pattison ##Mr J. Balfour Paul ##Mr J. M. Paulton ##Mr Walter Peace, Miss Peace ##Mrs Peacocke [?] ##Mr Godfrey and Mrs G. Pearse ##Mr Joseph and Mrs J. Pease, Mr Arthur and Mrs A. Pease, Mr A. E. Pease, Miss Pease, Miss Pease ##Mr A. Peckover, Misses Peekover [?] ##Mr Archibald Peel, Mr Algernon Peel, Mrs A. Peel, Misses Peel (2), Miss Cecilia Peel ##Mrs Aldrich Pelham, Miss Anderson Pelham ##Misses Pelly (2) ##Mr J. and Mrs Pender ##Mr J. Penn, Miss Penn ##Mr Pennefather ##Mr Heber Percy ##Miss Pereira ##Mr and Mrs Perks ##Mrs Perowne, Miss Perowne ##Mr Henry Petre ##Mrs Peyton, Miss Peyton ##Mrs N. G. Philips ##Mr and Mrs Phellps, Miss Pollock Phellps ##Mr and Mrs Lort Phillips, Miss [?ort] Phillips ##Mr B. Faudel-Phillips, Mr L. Faudel-Phillips ##Mrs Phillpotts ##Mr and Mrs Constantine Phipps, Mr Charles Phipps, Miss Phipps, Mr and Mrs Wilton Phipps, Miss Wilton Phipps ##Mrs Pipon ##Mrs Pirie [?] ##Mrs Pitman ##Mrs Fox Pitt ##Mr Platt-Higgins ##Mrs Poe ##Miss Pole, Miss Chandos Pole ##Mr and Mrs Pollock ##Mr and Mrs John Ponsonby, Miss Julia Ponsonby, Miss Ponsonby ##Mr and Mrs Wyndham Portal ##Mrs F. Post, Miss Post ##Mr and Mrs Powell, Miss Baden Powell, Miss Powell (2) ##Mrs Powlett, Miss Powlett, Miss Orde Powlett ##Mr Herbert Praed ##Mr and Mrs Price, Mr and Mrs Montagu Price ##Miss Priestley ##Mr H. W. and Mrs Primrose ##Mrs Upton Prior ##Mr and Mrs Leslie Probyn, Miss Charlotte Probyn ##Mr Roland and Mrs R. Protheroe ##Mr Provand ##Mr A. V. Pryor ##Mr Guy Pym ##Miss Quain ##Mr and Mrs Quilter, Misses Quilter (2) ##Mr Henry and Mrs H. Raikes, Mrs Raikes, Miss Lucy Raikes ##Mr Pandeli Ralli ##Mr and Mrs Alexander Ramsay, Miss Ramsay ##Mr and Mrs J. Rankin, Miss Rankin ##Mr Read ##Mr and Mrs Rebow, Miss Rebow ##Mr G. A. Redford and Mrs Redford ##Miss K. Reiss ##Mr and Mrs J. Rennell Rodd, Mrs Rennell Rodd ##Mr and Mrs Renshaw ##Mr J. A. Rentoul ##Mr Repton ##Mrs I[?]. Ricardo, Misses Ricardo (2), Miss Ricardo ##Mrs Rice, Miss Beatrix Rice ##Mr H. C. Richards ##Mr T. Richardson, Mr and Mrs Richardson ##Mrs Riddel ##Miss Rigby ##Alderman and Sheriff Ritchie, Misses Ritchie (2) ##Mr A. T. Phillips Roberts ##Mr Forbes Robertson, Mr Edmund Robertson, Mrs Robertson, Miss Sibyl Robertson ##Mrs Robins, Miss Robins ##Mr Brooke Robinson, Mrs Robinson ##Miss Frances Rod ##Sheriff Hargreaves Rogers ##Mrs Rolfe ##[[Social Victorians/People/Fanny Ronalds|Mrs Ronalds]] ##Mrs James Ronand ##Mrs Carl Rosa ##Mrs Harcourt Rose ##Mr and Mrs Leopold Rothschild, Mr Alfred Rothschild ##Mr James Round, Misses Round (2) ##Mr Bowen Rowlands ##Mr and Mrs Royds, Misses Royds (2) ##Mrs Arnold Royle [? Royce?], Mrs G. Royle ##Mr Hugo von Ruffer ##Mr G. W. E. Russell, Mr H. J. H. Russell, Mr T. W. RusseII, Mrs J. C. Russell, Mrs F. Russell, Miss Russell, Misses Russell (2), Misses Russell (2) ##Misses Russell of Killowen (2) ##Mrs W. W. Russon[?] ##Mr John Rutherford and Mrs Rutherford ##Miss Jane Ryan ##Mr G. L. Ryder, Mr J. D. Ryder ##Mrs Salmon [?] ##Mrs Salmond, Misses Salmond (2) ##Mr and Mrs Salting ##Mr and Mrs H. S. Samuel ##Mr Albert and Mrs A. Sandeman, Mrs Sandeman, Misses Sandeman (2) ##Miss [Sanderson?] ##Mrs Sandford ##Mr and Mrs Sant, Miss Sant ##Misses Sar[?] (2) ##Misses Sartorius (2) ##Mr and Mrs R. Sassoon, Mr and Mrs Arthur Sassoon, Misses Sassoon (2) ##Miss Saumarez Smith ##Miss Truda Saunderson ##Miss Saurin ##Mrs Graves Sawle ##Mr and Mrs Scaramanga ##Mr Leo Schuster ##Mr P. L. Sclater, Miss P. L. Sclater ##Mrs Scobell ##Mr J. Murray Scott, Mrs Scott, Miss Maxwell Scott ##Mrs Seddon, Misses Seddon (2) ##Mr and Mrs C. H. Seely ##Mr and Mrs Senhouse ##Mrs Sergison [?] ##Mr and Mrs H. Seton-Karr ##Mrs Settle, Mrs Settle, ##Miss Lily Severn ##Mr Horace and Mrs H. Seymour, Mrs L. Seymour, Miss M. Seymour, Misses Seymour (2), Miss Mabel Seymour ##Mr Lucas Shadwell ##Mr W. E. T. Sharpe ##Mr H. H. Shaw, Mr C. E. Shaw, Mr and Mrs Shaw, Miss Shaw ##Mr Michael Shaw-Stewart, Miss Shaw-Stewart ##Miss Shelley ##Mr and Mrs Shelley-Bontein ##Mrs Edgar Shephard ##Miss [Sheppart?] ##Mrs Brinsley Sheridan ##Miss Maud S[?]hey ##Mrs Teignmouth [?] Shore ##Miss Shute ##Misses Kay Shuttleworth [?] (2) ##Mr W. Sidebottom, Mr and Mrs T. H. Sidebottom ##Mr and Mrs Louis Sinclair ##Mr and Mrs T. Skewes-Cox ##Mrs Bridgman Simpson ##Mr and Mrs Skefflngton Smyth ##Mrs Slade, Mrs Frederick Slade ##Mrs P. L. Slater ##Mrs Hawley Smart ##Mr and Mrs Abel Smith, Miss Abel Smith, Mr and Mrs J. P. Smith, Mr A. H. Smith, Mr and Mrs H. C. Smith, Mr and Mrs T. Smith, Mr Smith, Mr G. D. Smith, Mr and Mrs Dudley Smith, Miss Dudley Smith, Mrs Graham Smith, Mrs C. Smith, Miss Smith, Miss Dorrien Smith, Smith (2), Miss [Euan?]-Smith [there is a Sir Euan-Smith and a Lady Euan[-]Smith], Miss Smith (Clement), Miss Rachel Smith ##Mrs Smith-Barry ##Mr Philip Somers-Cocks ##Mr H. Somerset ##Mr Augustus Spalding ##Mr and Mrs E. B. Sparke, Miss Sparke ##Miss Sparkes ##Miss Ruby Spencer Churchill ##Mr and Mrs A. Spicer ##Misses Sprigg (2) ##Miss Stafford ##Mr and Mrs H. M. Stanley, Mr E. J. Stanley, Miss Magaret Stanley, Miss [Stanley?], Miss Evelyn Stanley ##Mrs Starkie ##Miss Evelyn Starling[?] ##Mrs St. Clair ##Mr and Mrs Leslie Stephen, Miss [?] Stephen ['''could this have been Virginia Woolf?'''] ##Misses Stephenson (2) ##Mrs Sterling, Miss R. Sterling ##Mr and Mrs J. Stern, Misses Stern (2) ##Mr and Mrs Stevenson, Mrs Stevenson, Miss Stevenson ##Mr and Mrs Steward ##Mr C. J. Stewart, Mrs A. C. Stewart, Mrs Stewart, Miss Stewart, Misses Stewart (2), Miss Nita Houston Stewart, Miss Hilda Stewart ##Mr Stibbert ##Mr and Mrs J. H. Stock ##Miss Dora Stone ##Mrs Stopford, Misses Stopford (2) ##Mr and Mrs Eames Storey ##Mr and Mrs E. Strachey ##Mr and Mrs J. Sturgis ##Mrs Napier Sturt ##Mrs Sullivan, Miss [Sullivan?] ##Mrs Surtees ##Mr F. Sutton ##Mrs Swaine, Miss Swaine ##Mr and Mrs J. A. Swettenham ##Miss Swinburne ##Mr Christopher Sykes ##Mrs R. F. Synge, Mrs Synge ##Miss [T??s ?] ##Mr Alma-Tadema ##Miss Satyendra Bala Tagore ##Miss Mary Talbot, Miss Talbot, Miss Muriel [?Talbot??] ##Mr and Mrs Tarleton ##Mr and Mrs F. Taylor, Mr John Taylor, Mrs J. W. Taylor, Mrs Taylor, Mrs Brook Taylor, Miss Taylor, Miss Ella Taylor, Misses Taylor (2) ##Mrs Temple, Miss Temple ##Mr H. J. Tennant ##Miss Ellen Terry ##Mr and Mrs Montagu Tharp ##Miss Thesiger ##Mr Abel Thomas, Mrs H. Thomas, Miss Ethel Thomas ##Mrs Thomson, Mrs Anstruther Thomson, Mrs C. F. Anstruther Thomson, Misses Thomson (2) ##Mr Montagu and Mrs M. Thorold ##Mr P. Thornton, Miss [?Thornton??], Misses Thornton (2) ##Miss Thorold ##Miss [?Thursby?? there are a Sir John Thursby and a Lady Thursby] ##Mrs Thwaites ##Mr and Mrs J. C. Thynne, Mr and Mrs C. E. Thynne, Mr F. J. Thynne, Mrs R. T. Thynne, Miss Rachel Thynne, Miss Thynne, Misses Thynne (2), Miss Thynne, Misses "Thynne, I. C. (2)" ##Mr and Mrs Edward Tighe ##Mrs Tillard ##Mrs Tillbrook ##Mr W. H. Wilson-Todd ##Mr and Mrs H. Graham Toler ##Mr H. F. Tollemache ##Mr W. E. M. Tomlinson, Miss Tomlinson ##Mr A. M. Torrance ##Mr and Mrs Tosti ##Mr and Mrs Christopher Tower ##Mr and Mrs Beerbohm Tree ##Miss Adela[?] Trefusis ##Mr H. D. Trelawny, Miss Trelawny [2?] ##Mr Tremayne, Misses Tremayne (2) ##Mr A. J. R. Trendell ##Mr and Mrs C. E. Tritton, Misses Tritton (2) ##Mr and Mrs C. W. Trotter ##Mr and Mrs Tudway ##Mr and Mrs Dan Tupper, Mrs Tupper, Miss G. le M. Tupper ##Miss Turner ##Mr Algernon Turnor ##Miss E. Tuson ##Mr and Mrs A. Ure ##Mr T. Usborne ##Mr and Mrs T. Usher, Miss Usher ##Mr and Mrs Val Prinsep ##Miss Van [der Byl?] ##Mr Van De Weyer, Misses Van de Weyer (2) ##Mr L. Van Loon ##Mr and Mrs Chas van Raalte ##Mrs Vance ##Mrs Edmund Vaughan ##Mr and Mrs Venning, Miss Venning ##Mr and Mrs Hope Vere ##Mrs Verschoyle, Miss Verschoyle ##Mr and Mrs F. E. Villiers, Mrs E. Villiers, Mrs Villiers, Miss Dorothy Villiers, Miss Freda Villiers, Miss Villiers ##Mr Graham Vivian, Mrs R. Vivian, Misses [Vivian?] (2) ##Mr and Mrs Von André ##Miss Hilda von Deichmann ##Mr R. C. de Grey Vyner ##Mr Charles Waldstein ##Mr F. Walker, Miss F. Walker, Miss Smart Walker, Miss Walker ##Mrs Wallis ##Mr and Mrs Spencer Walpole, Miss Maud Walpole ##Misses Walrond (2) ##Mr and Mrs Walter, Misses [Walter??] (2) ##Mr and Mrs Walton ##Mr Wanklyn ##Mr Piers Egerton Warburton ##Mr and Mrs J. Humphrey Ward, Mrs C. [?] E. Ward, Miss Ward ##Mr and Mrs Lee Warner, Mrs Warner ##Mr A. F. Warr ##Mrs Warre ##Misses Warren (2) ##Misses Warrender (2) ##Mrs Watson, Miss Watson ##Mrs S. J. Way ##Mr Thomas Wayman ##Mr Godfrey Webb ##Mr and Mrs R. G. Webster, Misses Webster (2) ##Mr Weigall, Miss Rachel Weigall ##Mr and Mrs John Welby, Mr and Mrs J. Welby ##Mr W. West ##Mr and Mrs Cornwallis West, Miss Cornwallis West ##Mr and Mrs Sackville West ##Miss Evelyn Wellesley ##Mrs Wells ##Mrs F. Charteris Wemyss ##Misses [Van De Weyer?] (2) ##Mrs Weywan ##Mrs Wharton ##Mr Whateley ##Mrs Whatley ##Mrs Whatman, Misses Whatman (2) ##Misses Whitehead (2) ##Mr and Mrs G. Whiteley, Mrs Herbert Whiteley, Mrs H. Whiteley ##Mr Whitbread ##Mrs Wickham ##Mrs Wilberforce, Misses Wilberforce (2), Miss Wilberforce ##Miss Wilbraham ##Mr and Mrs Hwfa Williams, Mr Powell Williams, Mrs Williams, Mrs Ellis Williams, Miss Williams ##Misses Wills (2) ##Mrs Eardley Wilmot, Miss Eardley-Wilmot ##Mr and Mrs A. S. Wilson, Mr G. Fleetwood Wilson, Miss Fleetwood Wilson, Mr F. W. Wilson, Mr J. W. Wilson, Mr A. Wilson, Mr and Mrs John Wilson, Mr and Mrs C. H. Wilson, Mrs Wilson, Miss [?hend?] Wilson, Miss Wilson ##Mr and Mrs Wingfleld ##Mr Wodehouse, Mrs E. F. Wodehouse ##Mr S. Wombwell, Miss Wombwell ##Mrs Wood, Mrs Charles Wood, Misses Wood (2) ##Miss de la Wood[?] ##Mr W. Woodall ##Mrs J. Woodford, Miss Wood[ford?] ##Mr H. C. Woods, M.D. ##Misses [Workham?] (2) ##Mrs Stuart Wortley ##Mrs Wray ##Mr Wylie ##Mrs Williams Wynn ##Mr and Mrs Wynne ##Mr D’Arcy Wyvill, Misses [Wyvill?] (2) ##Deputy Inspector-General and Mrs Charles Wyndham, Mr Wyndham, Misses Wynd[ham?] (2) ##Miss Yeatman ##Mr and Mrs Yerburgh ##Mr H. Yorke #Admirals of the Fleet [initial large caps, rest sm caps] — Earl of Clanwilliam, Lord John [Hay?], the Hon. Sir H. Keppel #Admirals — H. G. Andoe, C. E. Buckle, Sir F. Bedford, Britten, the Hon. W. Carpenter, H. F. Cleveland, Sir H. Chads, Close, [?], Carr, E. J. Church, Sir W. Dowell, R. G. Douglas, A. L. [?], C. E. Domvile, A. T. Dale, D’Eyncourt, Field, Sir A. [Farquhar?], Fitzgerald, Fellowes, Fanshawe, Sir H. Fairfax, Sir [?] Fisher, C. J. Fane, Fullerton, the Hon. Sir E. Fremantle, [?] FitzGeorge, Woods Pasha, Sir W. Hunt-Grubbe, Sir Anthony [?] Hoskin, Lord Hood of Avalon, Sir Leopold Heath, Sir [?] [F.?] Hotham, Sir Algernon Heneage, R. H. Hamond, the Right Hon. Sir [J.?] Hay, St. G. C. D’Arcy Irvine, Jones, Kennedy, Sir A. [?s], A. P. Lake, R. M. Lloyd, Sir L. Loraine, A. H. Markham, [Sir?] R. More-Molyneux, Sir F. L. M'Clintock, Sir R. Macdonald, [the?] Hon. V. Montagu, Nicholson, Noel, Marquis of Northampton, Sir E. Ommaney [?], Sir Augustus Phillimore, A. T. Powlett, [?], [?. ?.] Rowley, Sir F. Richards, Lord Charles Scott, [? St.? John?], W. H. C. St. Clair, Bowden Smith, Sulivan, E. H. Sey[mour?], H. Stephenson, Sir Nowell Salmon, Sir W. Houston [Stewart?], Sir M. [Cuhne?]-Seymour, E. W. Turnour, E. W. Van[?] Wharton, Sir G. Willes, the Hon. W. J. Ward #Captain, R.N. — W. A. D. Acland, C. J. Barlow, F. R. Board[?], H. Bainbridge, Hon. T. Brand, Bickford, Lord Charles [B?ford?], B. F. Clark, Colville, Carter, Hon. S. Cecil Colville, [?ford?], A. G. Douglas, Sir C. Domville, Hon. A. Hay Dru[?], [?] [W.?] [?] Gordon, Hammet, Hon. Curzon Howe, Hender[?], [?] Ingles, Jellicoe, Jephson, Johnstone, Jeffreys, H. C. [?], Hon. A. Littleton, Hon. Hedworth Lambton, Moore, May, [? Net?], Poe, Pipon [?], Aldrich Pelham, Alfred Paget, [Bi.idcl?], Rolleston, John Sinclair, Bridgeman Simpson, [?], Van Koughnet [?], Burges Watson, Eardley-Wilmot, [?ham, Winsloe, Hon. J. Yorke #[Lieutenants???] — Anson, G. R. Bethell, Blair, Bayley, Cave[?], [?] Cave,Hon. Cecil Cadogan, de Salis, Fraser, Floyd, Hon. [?] [F?], Alaric Grant, Morgan, Moore, Marescaux, [?] Stuart, Tupper, Wells, Williams, G. J. S. Warrender #[Lieutenants?] R.N. — Alton, Murray Aynsley, Boyle, Bather, [?], [R. F.?] Boyle, Chaytor, Sir Charles Cust, G. W. Davy, [?] Wyndham-Fiennes, Fair, Godfrey Faussett, Garforth, [L?]ord Clifford, Hopkinson, Henderson, Keyes, Keppel, [?] Lloyd, Majendie, Mitchell, Morant, Kerr-Pearse, [?] Richmond, Rae, Stewart, Hon. Victor Stanley, [?] [Calta?]-Seymoar, Trye, Thring, Hon. Cyril Ward, W[?], R. E. Wemyss, Woolcombe #[Captain?] Trinity House, Sir J. Sydney Webbe #[Field?] Marshall — Sir F. P. Haines, Sir Lintorn Simmons, Sir [?] Stewart, Lord Roberts of Kandahar, Viscount Wolseley #[Generals?] —Sir J. Ardagh, Sir A. Alison, Sir H. J. Alderson, [?n] Annesley, J. Alleyne, Sir J. M. Adye, Sir C. G. [Arbuth?]not, Sir H. Havelock-Allan, R. Bateson, Sir W. F. [B?er, Sir H. Brackenbury, H. M. Bengough, the Right Hon. [?] Buller, Sir Owen Tador-Burne, H. J. Buchanan, Sir C. H. [Brown?low], Sir S. Browne, Sir M. Biddulph, Viscount Bridport, [?. O.?] Barnard, E. F. Chapman, Lord Clarina, C. F. Clery, the Hon. S. Gough-Calthorpe, E. H. Clive, Godfrey Clerk, Lord [Ch?]sford, the Hon. Sir Andrew Clarke, Sir E. Du Cane, Crutchley [?], Lord de Ros, Sir John Donelly, J. H. Dunne, Sir Martin Dillon, Sir Collingwood Dickson, Sir H. de Bathe, Davis, Sir F. de Winton, Sir T. Dennehy, Sir H. Ewart, Sir J. B. Edwards, C. B. Ewart, Cecil East, Arthur French, Sir T. Fitz-Wygram, the Hon. Sir P. Feilding, Sir T. E. Gallwey, Sir T. Goldsmid, Sir R. Gipps, Sir R. Grant, Sir F. W. Grenfell, Coleridge Grove, Goldsworthy, J. J. H. Gordon, Sir E. A. Holdich, Sir E. W. Higginson, Sir R. J. Hay, Sir R. Harrison, Julian Hall, Earl Howe, the Hon. W. Home, J. Jameson, Sir Arnold Kemball, Kelly-Kenay, Lord Mark Kerr, F. T. Lloyd, Sir D. Lysons, Sir Drury Lowe, G. Luck, J. W. Laurie, F. Marshall, the Hon. R. Monck, Crichton Maitland, Sir J. M'Neill, Montgomery, the Hon. S. Mostyn, G. Moncrieff, E. Markham, Sir W. A. Mackinnon, Bryan Milman [?], H. M’Calmont [?], M'Donnell, W. C. F. Molyneux, Lord [Methuen?], J. F. Maurice, Sir F. Middleton, O. H. Nicolls, Sir E. [?] Newdegate, Sir H. N[orman?], Sir W. Olpherts, F. Peyton [?], G. [?] Upton Prior, T. H. Pakenham, G. W. T. Rich, Lord [?der] Russell, Robinson, Rowlands, J. C. Russell, F. [Russell?], A. C. Stewart, Sir Henry Smyth, Sterling, Sir C. [?] Shute, N. Stevenson, Swaine, Lord William Seymour, [?] [Sahmond?], Sir Frederick Stephenson, Sir John Stokes, Sir R. [?], Sir H. B. Tuson, the Hon. R. A. J. Talbot, G. le M. [Tupper?], Taylor, Hon. C. Thesiger, R. T. Thynne, Upperton, [?]H. Utterson, Sir J. Watson, Sir C. W. Wilson, Sir F. F. Walker, Sir Evelyn Wood, Sir C. Warren, Albert Williams, the Hon. G. Wrottesley, Sir G. H. Willis, Sir H. Wilmot #Colonels — Armytage, Arkwright, Pat Boyle, Burges, the Hon. [?] Byng, H. B. H. Blundell, M. S. Brownrigg, Sir E. Bradford, Sir A. [Blyge? Bigge?], the Hon. F. Bridgeman, Brassey, Lord William Beresford, St. John Barne, N. Barnardiston, Lord Blythswood, [?] Cunynghame, F. H. Custance, Clayton, Sir Henry Colville, [?] Carnac [?], Cavaye, Seymour Corkran, the Hon. Charles [?], W. Campbell, Chaloner, Archibald Calvert, the Hon. [?] Campbell, the Hon. Wenman C. Coke, the Hon. W. [?ton], the Hon. Sir W. Colville, Chaine, A. B. Crosbie, [T.?] [R?] Crosse, Lord Edward Pelham Clinton, the Hon. Henry [C?hton], E. H. Cooper, the Hon. H. Corry, John Clerk, Lord Dorchestcr, C. R. Dease, the Hon, Lewis Dawnay, [the?] Hon. H. Denison, Denny, Dalbiac, A. Davidson, the Hon. Cathbert Edwards, the Right Hon. Sir F. Edwards, [?son], R. Edis, the Hon. Charles Edgecumbe, Aubone Fife, [?], Wynne Finch, Ferguson of Pitfour, Forster, Lancelot [?r] H. Frudyer, Barrington Foote, Goldsmid, Gore, Grenfell, [?n], C. G. Gordon, R. Gunter, Alan Gardner, Hon. G. Gough, [?] [?iton], the Hon. A. Hood, the Earl of Home, Lord Claud [Hamilton?], Harford, Herbert, the Earl of Haddington, Haygarth, G. Hatton [?], Hillyard, Arthur Haig, Sir E. Stock Hill, R. Hennell, Archer Houblon [?], the Hon. Cospatrick Home, the Hon. C. Gathorne-Hardy, Johnstone, Cotton-Jodrell, Hegan, [H?nard], Sir N. Kingscote, H. A. Lascelles, the Hon. Heneage [L?], Hanning Lee, F. A. Lucas, the Hon. H. Lyttelton, Lockwood, L. V. Loyd, C. W. Long, Ronald Lane, Lucas, J. Leslie, the Hon. Caryl [?]Molyneux, John Murray, Sir A. W. Mackworth, J. M'Calmont [?], Milward, the Hon. F. C. Morgan, J. J. Mellor, Meeking, Manvers [?], Moorsom, H. Malet, the Earl of Mount Edgecumbe, the [Earl?] of March, Wyndham Murray, Sir V. Majendie, the Hon. G. [Napper?], H. H. Oldham, L. J. Oliphant, A. Paget, Dampier Palmer, [Earl?] Percy, George Paget, C. D. Patterson, Arthur Peel, [Birch?] [Richardson?], the Hon. F. W. Stopford, Sir W. G. Stirling, E J. [Sanderson?], T. M. Sandys, H. Smith, J. F. Sandeman, Renyon-[Surrey?], C. E Stewart, E. H. Sartorius, the Hon. Walter [Stewart?], L. Seymour, Settle, Stevenson, Starkie, C. H. Seafe, the Hon. Sir W. P. Talbot, J. Du Plat[?] Taylor, H. Thomas, A. W. [T?], the Hon. W. Ie Poer Trench, H. P. Vance, Sir C. E. Howard Vincent, M.P.; R. Vivian, A. P. Vivian, E. Villiers, the Duke of Westminster, the Earl of Wemyss, Lord Wantage, Ward, [Waring?], [Earle?] Welby, Lord Arthur Wellesley, Robert Williams, the Hon. H. L. Wood, Sir W. H. Walroud, F. Smart Walker, A. [Williams?] Wynn, Wardrop #Majors — Anne, Atherley, Ashton, F. H. Bowles, the Hon. [?] R. Bourke, Carnegy, H. Candy, Close, the Hon. F. Colborue, the Hon. Wenman Coke, Lawrence Drummond, Alfred [Edgecombe?], G. Egerton, E. H. Elliot, the Hon. A. Henniker, J. [H?a?h], the Hon. Assheton Harbord, the Hon. North Dalrymple [Hamilton?], Jameson, Pryce Jones, Larnach, the Hon. Osbert [Lumley?], C. Little, Marindin, the Hon. J. Scott Napier, Wyndham Quin, F. C Rasch, the Hon. A. Sidney, the Hon. J. T. St. Aubyn, Sir Edgar Sebright, Stirling, T. E. M. Swinnington-Parkington, [?.] M. Temple, Tillbrook, Anstruther Thomson, [E.?] [L.?] Woodhouse, and the Marquis of Winchester #Captains — O. Ames, J. Acland, Alan Boisragon, Bates, H. M. [Biddulph?], the Hon. Baring, Butler, the Hon. J. Byng, the Hon. [N.?] Yarde-Butler, E. W. Blunt, J. F. Bagot, the Hon. W. Bagot, Seymour Combe, W. Chetwynd, Dundas, Denis Daly, Cecil Drummond, M. Drummond, Ellison, Houston French, Gye, R. G. [Gilmour?], P. Green, W. G. Grice-Hutchinson, Ahmed Hussain, G. [L.?] Holford, Jessel, the Hon. W. Lambton, the Hon. G. H. [L?], Sir H. Naylor-Leyland, G. Lister, Matthews, A. D. Miller, [?],M. M'Neill, C. Norton, Phillpotts, N. G. Philips, Prety[man?], Duncan Pirie, Pitman, Fox Pitt, Petre, Harcourt Rose, [W.?] [J.?] Stopford, Sir Eyre Shaw, H. G. D. Shute, Spicer, the Hon. [?.] St. Aubyn, Sutton, Tillard, Webbe, Wray, and Gordon [Watson?] #Lieutenants — Baun, A. Cowell, the Hon. E. C. Lennox, F. Ponsonby, J. Ponsonby, Vandeleur, the Hon. C. Willoughby, and the Hon. C. S. H. D. Willoughby ===Entertainment=== "The Bands of the 1st Life Guards, Grenadier Guards, and Royal Artillery played a selection of music during the afternoon."<ref name=":1" /> (4, Col. 2c) ==Anthology== == Notes and Questions == # ==References== * <references /> n6uam9zok2xogjvohfeaee1ctpvfg6u 10 313705 2693775 2679356 2024-12-29T18:06:47Z Watchduck 137431 2693775 wikitext text/x-wiki <templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" /> {| class="wikitable" style="text-align: center;" |+ 4 fixed points | [[File:Venn 0000 0000.svg|40px]]<br>0<br><small style="opacity: .5;">0</small> | [[File:Venn 0001 0100.svg|40px]]<br>40<br><small style="opacity: .5;">2</small> | [[File:Venn 0001 0010.svg|40px]]<br>72<br><small style="opacity: .5;">4</small> | [[File:Venn 0000 0110.svg|40px]]<br>96<br><small style="opacity: .5;">6</small> |} {| class="wikitable sortable mentor-cycles" |+ 6 cycles of length 2 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="3"| cycle |- |class="f"| 40 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (20, 60) ||class="entry q0 g0"| 20₂ ||class="entry q0 g0"| 60₄ |- |class="f"| 40 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (92, 116) ||class="entry q0 g0"| 92₄ ||class="entry q0 g0"| 116₄ |- |class="f"| 72 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (18, 90) ||class="entry q0 g0"| 18₂ ||class="entry q0 g0"| 90₄ |- |class="f"| 72 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (58, 114) ||class="entry q0 g0"| 58₄ ||class="entry q0 g0"| 114₄ |- |class="f"| 96 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (6, 102) ||class="entry q0 g0"| 6₂ ||class="entry q0 g0"| 102₄ |- |class="f"| 96 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (46, 78) ||class="entry q0 g0"| 46₄ ||class="entry q0 g0"| 78₄ |} {| class="wikitable sortable mentor-cycles" |+ 12 cycles of length 5 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="6"| cycle |- |class="f"| 0 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 7) ||class="c"| (232, 105, 23, 104, 254) ||class="entry q0 g1"| 232₄ ||class="entry q1 g0"| 105₄ ||class="entry q1 g0"| 23₄ ||class="entry q2 g0"| 104₃ ||class="entry q2 g1"| 254₇ |- |class="f"| 0 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (4, 3, 2, 6, 7) ||class="c"| (150, 22, 129, 126, 127) ||class="entry q0 g1"| 150₄ ||class="entry q2 g0"| 22₃ ||class="entry q1 g1"| 129₂ ||class="entry q0 g0"| 126₆ ||class="entry q3 g0"| 127₇ |- |class="f"| 0 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 1, 1, 1, 1) ||class="w"| (8, 1, 5, 1, 5) ||class="c"| (255, 1, 233, 128, 151) ||class="entry q1 g1"| 255₈ ||class="entry q3 g1"| 1₁ ||class="entry q3 g1"| 233₅ ||class="entry q2 g1"| 128₁ ||class="entry q3 g1"| 151₅ |- |class="f"| 40 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (192, 65, 63, 64, 214) ||class="entry q0 g1"| 192₂ ||class="entry q1 g1"| 65₂ ||class="entry q1 g0"| 63₆ ||class="entry q2 g1"| 64₁ ||class="entry q2 g1"| 214₅ |- |class="f"| 40 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (190, 62, 169, 86, 87) ||class="entry q0 g1"| 190₆ ||class="entry q2 g0"| 62₅ ||class="entry q1 g1"| 169₄ ||class="entry q0 g0"| 86₄ ||class="entry q3 g0"| 87₅ |- |class="f"| 40 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (215, 41, 193, 168, 191) ||class="entry q1 g1"| 215₆ ||class="entry q3 g0"| 41₃ ||class="entry q3 g1"| 193₃ ||class="entry q2 g1"| 168₃ ||class="entry q3 g1"| 191₇ |- |class="f"| 72 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (160, 33, 95, 32, 182) ||class="entry q0 g1"| 160₂ ||class="entry q1 g1"| 33₂ ||class="entry q1 g0"| 95₆ ||class="entry q2 g1"| 32₁ ||class="entry q2 g1"| 182₅ |- |class="f"| 72 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (222, 94, 201, 54, 55) ||class="entry q0 g1"| 222₆ ||class="entry q2 g0"| 94₅ ||class="entry q1 g1"| 201₄ ||class="entry q0 g0"| 54₄ ||class="entry q3 g0"| 55₅ |- |class="f"| 72 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (183, 73, 161, 200, 223) ||class="entry q1 g1"| 183₆ ||class="entry q3 g0"| 73₃ ||class="entry q3 g1"| 161₃ ||class="entry q2 g1"| 200₃ ||class="entry q3 g1"| 223₇ |- |class="f"| 96 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (136, 9, 119, 8, 158) ||class="entry q0 g1"| 136₂ ||class="entry q1 g1"| 9₂ ||class="entry q1 g0"| 119₆ ||class="entry q2 g1"| 8₁ ||class="entry q2 g1"| 158₅ |- |class="f"| 96 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (246, 118, 225, 30, 31) ||class="entry q0 g1"| 246₆ ||class="entry q2 g0"| 118₅ ||class="entry q1 g1"| 225₄ ||class="entry q0 g0"| 30₄ ||class="entry q3 g0"| 31₅ |- |class="f"| 96 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (159, 97, 137, 224, 247) ||class="entry q1 g1"| 159₆ ||class="entry q3 g0"| 97₃ ||class="entry q3 g1"| 137₃ ||class="entry q2 g1"| 224₃ ||class="entry q3 g1"| 247₇ |} {| class="wikitable sortable mentor-cycles" |+ 18 cycles of length 10 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="11"| cycle |- |class="f"| 40 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (156, 53, 99, 52, 138, 180, 29, 75, 28, 162) ||class="entry q0 g1"| 156₄ ||class="entry q1 g0"| 53₄ ||class="entry q1 g0"| 99₄ ||class="entry q2 g0"| 52₃ ||class="entry q2 g1"| 138₃ ||class="entry q0 g1"| 180₄ ||class="entry q1 g0"| 29₄ ||class="entry q1 g0"| 75₄ ||class="entry q2 g0"| 28₃ ||class="entry q2 g1"| 162₃ |- |class="f"| 40 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (212, 125, 43, 124, 194, 252, 85, 3, 84, 234) ||class="entry q0 g1"| 212₄ ||class="entry q1 g0"| 125₆ ||class="entry q1 g0"| 43₄ ||class="entry q2 g0"| 124₅ ||class="entry q2 g1"| 194₃ ||class="entry q0 g1"| 252₆ ||class="entry q1 g1"| 85₄ ||class="entry q1 g1"| 3₂ ||class="entry q2 g1"| 84₃ ||class="entry q2 g1"| 234₅ |- |class="f"| 40 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (130, 42, 149, 66, 107, 170, 2, 189, 106, 67) ||class="entry q0 g1"| 130₂ ||class="entry q2 g0"| 42₃ ||class="entry q1 g1"| 149₄ ||class="entry q0 g0"| 66₂ ||class="entry q3 g0"| 107₅ ||class="entry q0 g1"| 170₄ ||class="entry q2 g1"| 2₁ ||class="entry q1 g1"| 189₆ ||class="entry q0 g0"| 106₄ ||class="entry q3 g0"| 67₃ |- |class="f"| 40 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (202, 98, 221, 10, 35, 226, 74, 245, 34, 11) ||class="entry q0 g1"| 202₄ ||class="entry q2 g0"| 98₃ ||class="entry q1 g1"| 221₆ ||class="entry q0 g1"| 10₂ ||class="entry q3 g1"| 35₃ ||class="entry q0 g1"| 226₄ ||class="entry q2 g0"| 74₃ ||class="entry q1 g1"| 245₆ ||class="entry q0 g1"| 34₂ ||class="entry q3 g1"| 11₃ |- |class="f"| 40 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (195, 21, 213, 148, 171, 235, 61, 253, 188, 131) ||class="entry q1 g1"| 195₄ ||class="entry q3 g0"| 21₃ ||class="entry q3 g1"| 213₅ ||class="entry q2 g1"| 148₃ ||class="entry q3 g1"| 171₅ ||class="entry q1 g1"| 235₆ ||class="entry q3 g0"| 61₅ ||class="entry q3 g1"| 253₇ ||class="entry q2 g1"| 188₅ ||class="entry q3 g1"| 131₃ |- |class="f"| 40 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (139, 93, 157, 220, 227, 163, 117, 181, 244, 203) ||class="entry q1 g1"| 139₄ ||class="entry q3 g0"| 93₅ ||class="entry q3 g1"| 157₅ ||class="entry q2 g1"| 220₅ ||class="entry q3 g1"| 227₅ ||class="entry q1 g1"| 163₄ ||class="entry q3 g0"| 117₅ ||class="entry q3 g1"| 181₅ ||class="entry q2 g1"| 244₅ ||class="entry q3 g1"| 203₅ |- |class="f"| 72 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (154, 83, 101, 82, 140, 210, 27, 45, 26, 196) ||class="entry q0 g1"| 154₄ ||class="entry q1 g0"| 83₄ ||class="entry q1 g0"| 101₄ ||class="entry q2 g0"| 82₃ ||class="entry q2 g1"| 140₃ ||class="entry q0 g1"| 210₄ ||class="entry q1 g0"| 27₄ ||class="entry q1 g0"| 45₄ ||class="entry q2 g0"| 26₃ ||class="entry q2 g1"| 196₃ |- |class="f"| 72 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (178, 123, 77, 122, 164, 250, 51, 5, 50, 236) ||class="entry q0 g1"| 178₄ ||class="entry q1 g0"| 123₆ ||class="entry q1 g0"| 77₄ ||class="entry q2 g0"| 122₅ ||class="entry q2 g1"| 164₃ ||class="entry q0 g1"| 250₆ ||class="entry q1 g1"| 51₄ ||class="entry q1 g1"| 5₂ ||class="entry q2 g1"| 50₃ ||class="entry q2 g1"| 236₅ |- |class="f"| 72 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (132, 76, 147, 36, 109, 204, 4, 219, 108, 37) ||class="entry q0 g1"| 132₂ ||class="entry q2 g0"| 76₃ ||class="entry q1 g1"| 147₄ ||class="entry q0 g0"| 36₂ ||class="entry q3 g0"| 109₅ ||class="entry q0 g1"| 204₄ ||class="entry q2 g1"| 4₁ ||class="entry q1 g1"| 219₆ ||class="entry q0 g0"| 108₄ ||class="entry q3 g0"| 37₃ |- |class="f"| 72 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (172, 100, 187, 12, 69, 228, 44, 243, 68, 13) ||class="entry q0 g1"| 172₄ ||class="entry q2 g0"| 100₃ ||class="entry q1 g1"| 187₆ ||class="entry q0 g1"| 12₂ ||class="entry q3 g1"| 69₃ ||class="entry q0 g1"| 228₄ ||class="entry q2 g0"| 44₃ ||class="entry q1 g1"| 243₆ ||class="entry q0 g1"| 68₂ ||class="entry q3 g1"| 13₃ |- |class="f"| 72 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (165, 19, 179, 146, 205, 237, 91, 251, 218, 133) ||class="entry q1 g1"| 165₄ ||class="entry q3 g0"| 19₃ ||class="entry q3 g1"| 179₅ ||class="entry q2 g1"| 146₃ ||class="entry q3 g1"| 205₅ ||class="entry q1 g1"| 237₆ ||class="entry q3 g0"| 91₅ ||class="entry q3 g1"| 251₇ ||class="entry q2 g1"| 218₅ ||class="entry q3 g1"| 133₃ |- |class="f"| 72 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (141, 59, 155, 186, 229, 197, 115, 211, 242, 173) ||class="entry q1 g1"| 141₄ ||class="entry q3 g0"| 59₅ ||class="entry q3 g1"| 155₅ ||class="entry q2 g1"| 186₅ ||class="entry q3 g1"| 229₅ ||class="entry q1 g1"| 197₄ ||class="entry q3 g0"| 115₅ ||class="entry q3 g1"| 211₅ ||class="entry q2 g1"| 242₅ ||class="entry q3 g1"| 173₅ |- |class="f"| 96 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (166, 71, 89, 70, 176, 198, 39, 57, 38, 208) ||class="entry q0 g1"| 166₄ ||class="entry q1 g0"| 71₄ ||class="entry q1 g0"| 89₄ ||class="entry q2 g0"| 70₃ ||class="entry q2 g1"| 176₃ ||class="entry q0 g1"| 198₄ ||class="entry q1 g0"| 39₄ ||class="entry q1 g0"| 57₄ ||class="entry q2 g0"| 38₃ ||class="entry q2 g1"| 208₃ |- |class="f"| 96 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (142, 111, 113, 110, 152, 238, 15, 17, 14, 248) ||class="entry q0 g1"| 142₄ ||class="entry q1 g0"| 111₆ ||class="entry q1 g0"| 113₄ ||class="entry q2 g0"| 110₅ ||class="entry q2 g1"| 152₃ ||class="entry q0 g1"| 238₆ ||class="entry q1 g1"| 15₄ ||class="entry q1 g1"| 17₂ ||class="entry q2 g1"| 14₃ ||class="entry q2 g1"| 248₅ |- |class="f"| 96 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (144, 112, 135, 24, 121, 240, 16, 231, 120, 25) ||class="entry q0 g1"| 144₂ ||class="entry q2 g0"| 112₃ ||class="entry q1 g1"| 135₄ ||class="entry q0 g0"| 24₂ ||class="entry q3 g0"| 121₅ ||class="entry q0 g1"| 240₄ ||class="entry q2 g1"| 16₁ ||class="entry q1 g1"| 231₆ ||class="entry q0 g0"| 120₄ ||class="entry q3 g0"| 25₃ |- |class="f"| 96 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (184, 88, 175, 48, 81, 216, 56, 207, 80, 49) ||class="entry q0 g1"| 184₄ ||class="entry q2 g0"| 88₃ ||class="entry q1 g1"| 175₆ ||class="entry q0 g1"| 48₂ ||class="entry q3 g1"| 81₃ ||class="entry q0 g1"| 216₄ ||class="entry q2 g0"| 56₃ ||class="entry q1 g1"| 207₆ ||class="entry q0 g1"| 80₂ ||class="entry q3 g1"| 49₃ |- |class="f"| 96 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (153, 7, 143, 134, 241, 249, 103, 239, 230, 145) ||class="entry q1 g1"| 153₄ ||class="entry q3 g0"| 7₃ ||class="entry q3 g1"| 143₅ ||class="entry q2 g1"| 134₃ ||class="entry q3 g1"| 241₅ ||class="entry q1 g1"| 249₆ ||class="entry q3 g0"| 103₅ ||class="entry q3 g1"| 239₇ ||class="entry q2 g1"| 230₅ ||class="entry q3 g1"| 145₃ |- |class="f"| 96 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (177, 47, 167, 174, 217, 209, 79, 199, 206, 185) ||class="entry q1 g1"| 177₄ ||class="entry q3 g0"| 47₅ ||class="entry q3 g1"| 167₅ ||class="entry q2 g1"| 174₅ ||class="entry q3 g1"| 217₅ ||class="entry q1 g1"| 209₄ ||class="entry q3 g0"| 79₅ ||class="entry q3 g1"| 199₅ ||class="entry q2 g1"| 206₅ ||class="entry q3 g1"| 185₅ |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> tod334059t57d0hwkybdsovecjl71sx 14 313706 2693782 2680030 2024-12-29T18:12:51Z Watchduck 137431 2693782 wikitext text/x-wiki [[Category:Studies of Boolean functions]] [[Category:Walsh permutation]] 9ud7c9k7hney95q0uy7mtjf5t211mg9 10 313707 2693776 2679236 2024-12-29T18:07:03Z Watchduck 137431 2693776 wikitext text/x-wiki * {{tl|Mentors of Boolean functions/cycles/4-ary/fixed}} * {{tl|Mentors of Boolean functions/cycles/4-ary/length 2}} (28) * {{tl|Mentors of Boolean functions/cycles/4-ary/length 3}} (168) * {{tl|Mentors of Boolean functions/cycles/4-ary/length 4}} (48) * {{tl|Mentors of Boolean functions/cycles/4-ary/length 6}} (2636) * There are 4080 of length 12, but that exceeds the size limit. <noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> 4c7dwq2et2f17j2w7ri4kuj7kk13zsp 10 313709 2693778 2678094 2024-12-29T18:07:26Z Watchduck 137431 2693778 wikitext text/x-wiki <templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" /> {| class="wikitable sortable mentor-cycles" |+ 28 cycles of length 2 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="3"| cycle |- |class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59580, 60394) ||class="entry q2 g1"| 59580₉ ||class="entry q2 g1"| 60394₁₁ |- |class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60810, 61148) ||class="entry q2 g1"| 60810₉ ||class="entry q2 g1"| 61148₁₁ |- |class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (63906, 64244) ||class="entry q2 g1"| 63906₉ ||class="entry q2 g1"| 64244₁₁ |- |class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64660, 65474) ||class="entry q2 g1"| 64660₉ ||class="entry q2 g1"| 65474₁₁ |- |class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59610, 60908) ||class="entry q2 g1"| 59610₉ ||class="entry q2 g1"| 60908₁₁ |- |class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60300, 61114) ||class="entry q2 g1"| 60300₉ ||class="entry q2 g1"| 61114₁₁ |- |class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (63940, 64754) ||class="entry q2 g1"| 63940₉ ||class="entry q2 g1"| 64754₁₁ |- |class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64146, 65444) ||class="entry q2 g1"| 64146₉ ||class="entry q2 g1"| 65444₁₁ |- |class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (102, 1542) ||class="entry q0 g0"| 102₄ ||class="entry q0 g0"| 1542₄ |- |class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (816, 1360) ||class="entry q0 g0"| 816₄ ||class="entry q0 g0"| 1360₄ |- |class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (4472, 5912) ||class="entry q0 g0"| 4472₆ ||class="entry q0 g0"| 5912₆ |- |class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (4654, 5198) ||class="entry q0 g0"| 4654₆ ||class="entry q0 g0"| 5198₆ |- |class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59622, 63992) ||class="entry q2 g1"| 59622₉ ||class="entry q2 g1"| 63992₁₁ |- |class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60336, 64174) ||class="entry q2 g1"| 60336₉ 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||class="entry q2 g1"| 63902₁₁ |- |class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64200, 60854) ||class="entry q2 g1"| 64200₉ ||class="entry q2 g1"| 60854₁₁ |- |class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64680, 60374) ||class="entry q2 g1"| 64680₉ ||class="entry q2 g1"| 60374₁₁ |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> t1i6vkl312xr7wlg8013ktwf9zfoma1 10 313710 2693779 2678095 2024-12-29T18:07:41Z Watchduck 137431 2693779 wikitext text/x-wiki <templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" /> {| class="wikitable sortable mentor-cycles" |+ 168 cycles of length 3 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="4"| cycle |- |class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (6, 26208, 26214) ||class="entry q0 g0"| 6₂ ||class="entry q0 g0"| 26208₆ ||class="entry q0 g0"| 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||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1076, 20894, 20636) ||class="entry q0 g0"| 1076₄ ||class="entry q0 g0"| 20894₈ ||class="entry q0 g0"| 20636₆ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1314, 14622, 14602) ||class="entry q0 g0"| 1314₄ ||class="entry q0 g0"| 14622₈ ||class="entry q0 g0"| 14602₆ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1328, 25430, 25424) ||class="entry q0 g0"| 1328₄ ||class="entry q0 g0"| 25430₈ ||class="entry q0 g0"| 25424₆ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (1074, 14334, 14074) ||class="entry q0 g0"| 1074₄ ||class="entry q0 g0"| 14334₁₂ ||class="entry q0 g0"| 14074₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (1316, 24446, 24428) ||class="entry q0 g0"| 1316₄ ||class="entry q0 g0"| 24446₁₂ ||class="entry q0 g0"| 24428₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (2770, 14350, 14314) ||class="entry q0 g0"| 2770₆ ||class="entry q0 g0"| 14350₆ ||class="entry q0 g0"| 14314₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (3012, 20622, 24188) ||class="entry q0 g0"| 3012₆ ||class="entry q0 g0"| 20622₆ ||class="entry q0 g0"| 24188₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (14362, 3010, 14062) ||class="entry q0 g0"| 14362₆ ||class="entry q0 g0"| 3010₆ ||class="entry q0 g0"| 14062₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (14604, 25410, 24440) ||class="entry q0 g0"| 14604₆ ||class="entry q0 g0"| 25410₆ ||class="entry q0 g0"| 24440₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (20634, 25172, 14328) ||class="entry q0 g0"| 20634₆ ||class="entry q0 g0"| 25172₆ ||class="entry q0 g0"| 14328₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (20876, 2772, 24174) ||class="entry q0 g0"| 20876₆ ||class="entry q0 g0"| 2772₆ ||class="entry q0 g0"| 24174₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25170, 20888, 14076) ||class="entry q0 g0"| 25170₆ ||class="entry q0 g0"| 20888₆ ||class="entry q0 g0"| 14076₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25412, 14616, 24426) ||class="entry q0 g0"| 25412₆ ||class="entry q0 g0"| 14616₆ ||class="entry q0 g0"| 24426₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (27826, 24168, 14316) ||class="entry q0 g0"| 27826₈ ||class="entry q0 g0"| 24168₈ ||class="entry q0 g0"| 14316₁₀ |- |class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (28068, 14056, 24186) ||class="entry q0 g0"| 28068₈ ||class="entry q0 g0"| 14056₈ ||class="entry q0 g0"| 24186₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (2176, 2448, 1904) ||class="entry q0 g0"| 2176₂ ||class="entry q0 g0"| 2448₄ ||class="entry q0 g0"| 1904₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (24576, 24582, 1638) ||class="entry q0 g0"| 24576₂ ||class="entry q0 g0"| 24582₄ ||class="entry q0 g0"| 1638₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2194, 21464, 23850) ||class="entry q0 g0"| 2194₄ ||class="entry q0 g0"| 21464₈ ||class="entry q0 g0"| 23850₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2196, 13752, 15180) ||class="entry q0 g0"| 2196₄ ||class="entry q0 g0"| 13752₈ ||class="entry q0 g0"| 15180₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2434, 23864, 21210) ||class="entry q0 g0"| 2434₄ ||class="entry q0 g0"| 23864₈ ||class="entry q0 g0"| 21210₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2436, 15192, 13500) ||class="entry q0 g0"| 2436₄ ||class="entry q0 g0"| 15192₈ ||class="entry q0 g0"| 13500₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24594, 14926, 23612) ||class="entry q0 g0"| 24594₄ ||class="entry q0 g0"| 14926₈ ||class="entry q0 g0"| 23612₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24596, 23598, 14938) ||class="entry q0 g0"| 24596₄ ||class="entry q0 g0"| 23598₈ ||class="entry q0 g0"| 14938₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24834, 13486, 21452) ||class="entry q0 g0"| 24834₄ ||class="entry q0 g0"| 13486₈ ||class="entry q0 g0"| 21452₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24836, 21198, 13738) ||class="entry q0 g0"| 24836₄ ||class="entry q0 g0"| 21198₈ ||class="entry q0 g0"| 13738₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (2182, 28656, 24854) ||class="entry q0 g0"| 2182₄ ||class="entry q0 g0"| 28656₁₀ ||class="entry q0 g0"| 24854₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (24848, 28390, 2454) ||class="entry q0 g0"| 24848₄ ||class="entry q0 g0"| 28390₁₀ ||class="entry q0 g0"| 2454₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1650, 23592, 23610) ||class="entry q0 g0"| 1650₆ ||class="entry q0 g0"| 23592₆ ||class="entry q0 g0"| 23610₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1652, 14920, 14940) ||class="entry q0 g0"| 1652₆ ||class="entry q0 g0"| 14920₆ ||class="entry q0 g0"| 14940₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1890, 21192, 21450) ||class="entry q0 g0"| 1890₆ ||class="entry q0 g0"| 21192₆ ||class="entry q0 g0"| 21450₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1892, 13480, 13740) ||class="entry q0 g0"| 1892₆ ||class="entry q0 g0"| 13480₆ ||class="entry q0 g0"| 13740₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 12) ||class="c"| (1910, 28384, 28662) ||class="entry q0 g0"| 1910₈ ||class="entry q0 g0"| 28384₈ ||class="entry q0 g0"| 28662₁₂ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (13498, 28644, 23870) ||class="entry q0 g0"| 13498₈ ||class="entry q0 g0"| 28644₁₀ ||class="entry q0 g0"| 23870₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (15178, 28404, 21470) ||class="entry q0 g0"| 15178₈ ||class="entry q0 g0"| 28404₁₀ ||class="entry q0 g0"| 21470₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (21212, 28642, 15198) ||class="entry q0 g0"| 21212₈ ||class="entry q0 g0"| 28642₁₀ ||class="entry q0 g0"| 15198₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (23852, 28402, 13758) ||class="entry q0 g0"| 23852₈ ||class="entry q0 g0"| 28402₁₀ ||class="entry q0 g0"| 13758₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 10, 6) ||class="c"| (4104, 31134, 30856) ||class="entry q0 g0"| 4104₂ ||class="entry q0 g0"| 31134₁₀ ||class="entry q0 g0"| 30856₆ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (8896, 19026, 31116) ||class="entry q0 g0"| 8896₄ ||class="entry q0 g0"| 19026₆ ||class="entry q0 g0"| 31116₈ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (11296, 17826, 30876) ||class="entry q0 g0"| 11296₄ ||class="entry q0 g0"| 17826₆ ||class="entry q0 g0"| 30876₈ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (17568, 11316, 31114) ||class="entry q0 g0"| 17568₄ ||class="entry q0 g0"| 11316₆ ||class="entry q0 g0"| 31114₈ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (19008, 9156, 30874) ||class="entry q0 g0"| 19008₄ ||class="entry q0 g0"| 9156₆ ||class="entry q0 g0"| 30874₈ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4122, 9174, 8914) ||class="entry q0 g0"| 4122₄ ||class="entry q0 g0"| 9174₈ ||class="entry q0 g0"| 8914₆ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4124, 17846, 17588) ||class="entry q0 g0"| 4124₄ ||class="entry q0 g0"| 17846₈ ||class="entry q0 g0"| 17588₆ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4362, 11574, 11554) ||class="entry q0 g0"| 4362₄ ||class="entry q0 g0"| 11574₈ ||class="entry q0 g0"| 11554₆ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4364, 19286, 19268) ||class="entry q0 g0"| 4364₄ ||class="entry q0 g0"| 19286₈ ||class="entry q0 g0"| 19268₆ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (4110, 8190, 7918) ||class="entry q0 g0"| 4110₄ ||class="entry q0 g0"| 8190₁₂ ||class="entry q0 g0"| 7918₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (4376, 30590, 30584) ||class="entry q0 g0"| 4376₄ ||class="entry q0 g0"| 30590₁₂ ||class="entry q0 g0"| 30584₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (8902, 11314, 8170) ||class="entry q0 g0"| 8902₆ ||class="entry q0 g0"| 11314₆ ||class="entry q0 g0"| 8170₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (9168, 17586, 30332) ||class="entry q0 g0"| 9168₆ ||class="entry q0 g0"| 17586₆ ||class="entry q0 g0"| 30332₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (11302, 9154, 7930) ||class="entry q0 g0"| 11302₆ ||class="entry q0 g0"| 9154₆ ||class="entry q0 g0"| 7930₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (11568, 19266, 30572) ||class="entry q0 g0"| 11568₆ ||class="entry q0 g0"| 19266₆ ||class="entry q0 g0"| 30572₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (17574, 19028, 8172) ||class="entry q0 g0"| 17574₆ ||class="entry q0 g0"| 19028₆ ||class="entry q0 g0"| 8172₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (17840, 8916, 30330) ||class="entry q0 g0"| 17840₆ ||class="entry q0 g0"| 8916₆ ||class="entry q0 g0"| 30330₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (19014, 17828, 7932) ||class="entry q0 g0"| 19014₆ ||class="entry q0 g0"| 17828₆ ||class="entry q0 g0"| 7932₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (19280, 11556, 30570) ||class="entry q0 g0"| 19280₆ ||class="entry q0 g0"| 11556₆ ||class="entry q0 g0"| 30570₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (30862, 30312, 8184) ||class="entry q0 g0"| 30862₈ ||class="entry q0 g0"| 30312₈ ||class="entry q0 g0"| 8184₁₀ |- |class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (31128, 7912, 30318) ||class="entry q0 g0"| 31128₈ ||class="entry q0 g0"| 7912₈ ||class="entry q0 g0"| 30318₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (8320, 8580, 4940) ||class="entry q0 g0"| 8320₂ ||class="entry q0 g0"| 8580₄ ||class="entry q0 g0"| 4940₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (18432, 18450, 4698) ||class="entry q0 g0"| 18432₂ ||class="entry q0 g0"| 18450₄ ||class="entry q0 g0"| 4698₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8326, 18404, 29994) ||class="entry q0 g0"| 8326₄ ||class="entry q0 g0"| 18404₈ ||class="entry q0 g0"| 29994₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8340, 7596, 12144) ||class="entry q0 g0"| 8340₄ ||class="entry q0 g0"| 7596₈ ||class="entry q0 g0"| 12144₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8578, 29996, 18150) ||class="entry q0 g0"| 8578₄ ||class="entry q0 g0"| 29996₈ ||class="entry q0 g0"| 18150₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8592, 12132, 7356) ||class="entry q0 g0"| 8592₄ ||class="entry q0 g0"| 12132₈ ||class="entry q0 g0"| 7356₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18438, 11890, 29756) ||class="entry q0 g0"| 18438₄ ||class="entry q0 g0"| 11890₈ ||class="entry q0 g0"| 29756₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18452, 29754, 11878) ||class="entry q0 g0"| 18452₄ ||class="entry q0 g0"| 29754₈ ||class="entry q0 g0"| 11878₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18690, 7354, 18416) ||class="entry q0 g0"| 18690₄ ||class="entry q0 g0"| 7354₈ ||class="entry q0 g0"| 18416₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18704, 18162, 7594) ||class="entry q0 g0"| 18704₄ ||class="entry q0 g0"| 18162₈ ||class="entry q0 g0"| 7594₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (8338, 31692, 18710) ||class="entry q0 g0"| 8338₄ ||class="entry q0 g0"| 31692₁₀ ||class="entry q0 g0"| 18710₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (18692, 31450, 8598) ||class="entry q0 g0"| 18692₄ ||class="entry q0 g0"| 31450₁₀ ||class="entry q0 g0"| 8598₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (4686, 29736, 29742) ||class="entry q0 g0"| 4686₆ ||class="entry q0 g0"| 29736₆ ||class="entry q0 g0"| 29742₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (4700, 11872, 11892) ||class="entry q0 g0"| 4700₆ ||class="entry q0 g0"| 11872₆ ||class="entry q0 g0"| 11892₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (4938, 18144, 18402) ||class="entry q0 g0"| 4938₆ ||class="entry q0 g0"| 18144₆ ||class="entry q0 g0"| 18402₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (4952, 7336, 7608) ||class="entry q0 g0"| 4952₆ ||class="entry q0 g0"| 7336₆ ||class="entry q0 g0"| 7608₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 12) ||class="c"| (4958, 31432, 31710) ||class="entry q0 g0"| 4958₈ ||class="entry q0 g0"| 31432₈ ||class="entry q0 g0"| 31710₁₂ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (7342, 31704, 30014) ||class="entry q0 g0"| 7342₈ ||class="entry q0 g0"| 31704₁₀ ||class="entry q0 g0"| 30014₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (12130, 31452, 18422) ||class="entry q0 g0"| 12130₈ ||class="entry q0 g0"| 31452₁₀ ||class="entry q0 g0"| 18422₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (18164, 31690, 12150) ||class="entry q0 g0"| 18164₈ ||class="entry q0 g0"| 31690₁₀ ||class="entry q0 g0"| 12150₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (30008, 31438, 7614) ||class="entry q0 g0"| 30008₈ ||class="entry q0 g0"| 31438₁₀ ||class="entry q0 g0"| 7614₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (10240, 10260, 5180) ||class="entry q0 g0"| 10240₂ ||class="entry q0 g0"| 10260₄ ||class="entry q0 g0"| 5180₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (16512, 16770, 5418) 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|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (19748, 5996, 19766) ||class="entry q0 g0"| 19748₆ ||class="entry q0 g0"| 5996₈ ||class="entry q0 g0"| 19766₈ |- |class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (28952, 6008, 28958) ||class="entry q0 g0"| 28952₆ ||class="entry q0 g0"| 6008₈ ||class="entry q0 g0"| 28958₈ |- |class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 14, 10) ||class="c"| (5736, 32766, 32488) ||class="entry q0 g0"| 5736₆ ||class="entry q0 g0"| 32766₁₄ ||class="entry q0 g0"| 32488₁₀ |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> 9xw2mil6kss94ikoton2phevc3y9t1v 10 313711 2693780 2678096 2024-12-29T18:07:54Z Watchduck 137431 2693780 wikitext text/x-wiki <templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" /> {| class="wikitable sortable mentor-cycles" |+ 48 cycles of length 4 !class="f"| 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||class="w"| (6, 11, 16, 1) ||class="c"| (59521, 6015, 65535, 1) ||class="entry q1 g1"| 59521₆ ||class="entry q3 g0"| 6015₁₁ ||class="entry q1 g1"| 65535₁₆ ||class="entry q3 g1"| 1₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (59624, 105, 5910, 59625) ||class="entry q0 g1"| 59624₈ ||class="entry q1 g0"| 105₄ ||class="entry q2 g0"| 5910₇ ||class="entry q3 g1"| 59625₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (61064, 1545, 4470, 61065) ||class="entry q0 g1"| 61064₈ ||class="entry q1 g0"| 1545₄ ||class="entry q2 g0"| 4470₇ ||class="entry q3 g1"| 61065₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (64160, 4641, 1374, 64161) ||class="entry q0 g1"| 64160₈ ||class="entry q1 g0"| 4641₄ ||class="entry q2 g0"| 1374₇ ||class="entry q3 g1"| 64161₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (64704, 5185, 830, 64705) ||class="entry q0 g1"| 64704₈ ||class="entry q1 g0"| 5185₄ ||class="entry q2 g0"| 830₇ ||class="entry q3 g1"| 64705₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59534, 1647, 6000, 61167) ||class="entry q0 g1"| 59534₈ ||class="entry q1 g0"| 1647₈ ||class="entry q2 g0"| 6000₇ ||class="entry q3 g1"| 61167₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59570, 4731, 5964, 64251) ||class="entry q0 g1"| 59570₈ ||class="entry q1 g0"| 4731₈ ||class="entry q2 g0"| 5964₇ ||class="entry q3 g1"| 64251₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59604, 5245, 5930, 64765) ||class="entry q0 g1"| 59604₈ ||class="entry q1 g0"| 5245₈ ||class="entry q2 g0"| 5930₇ ||class="entry q3 g1"| 64765₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (60290, 5931, 5244, 65451) ||class="entry q0 g1"| 60290₈ ||class="entry q1 g0"| 5931₈ ||class="entry q2 g0"| 5244₇ ||class="entry q3 g1"| 65451₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (60804, 5965, 4730, 65485) ||class="entry q0 g1"| 60804₈ ||class="entry q1 g0"| 5965₈ ||class="entry q2 g0"| 4730₇ ||class="entry q3 g1"| 65485₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (63888, 6001, 1646, 65521) ||class="entry q0 g1"| 63888₈ ||class="entry q1 g0"| 6001₈ ||class="entry q2 g0"| 1646₇ ||class="entry q3 g1"| 65521₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60376, 1337, 5158, 60857) ||class="entry q0 g1"| 60376₁₀ ||class="entry q1 g0"| 1337₆ ||class="entry q2 g0"| 5158₅ ||class="entry q3 g1"| 60857₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60388, 4397, 5146, 63917) ||class="entry q0 g1"| 60388₁₀ ||class="entry q1 g0"| 4397₆ ||class="entry q2 g0"| 5146₅ ||class="entry q3 g1"| 63917₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60856, 857, 4678, 60377) ||class="entry q0 g1"| 60856₁₀ ||class="entry q1 g0"| 857₆ ||class="entry q2 g0"| 4678₅ ||class="entry q3 g1"| 60377₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60898, 4427, 4636, 63947) ||class="entry q0 g1"| 60898₁₀ ||class="entry q1 g0"| 4427₆ ||class="entry q2 g0"| 4636₅ ||class="entry q3 g1"| 63947₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (61108, 4637, 4426, 64157) ||class="entry q0 g1"| 61108₁₀ ||class="entry q1 g0"| 4637₆ ||class="entry q2 g0"| 4426₅ ||class="entry q3 g1"| 64157₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (61138, 5147, 4396, 64667) ||class="entry q0 g1"| 61138₁₀ ||class="entry q1 g0"| 5147₆ ||class="entry q2 g0"| 4396₅ ||class="entry q3 g1"| 64667₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (63916, 869, 1618, 60389) ||class="entry q0 g1"| 63916₁₀ ||class="entry q1 g0"| 869₆ ||class="entry q2 g0"| 1618₅ ||class="entry q3 g1"| 60389₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (63946, 1379, 1588, 60899) ||class="entry q0 g1"| 63946₁₀ ||class="entry q1 g0"| 1379₆ ||class="entry q2 g0"| 1588₅ ||class="entry q3 g1"| 60899₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64156, 1589, 1378, 61109) ||class="entry q0 g1"| 64156₁₀ ||class="entry q1 g0"| 1589₆ ||class="entry q2 g0"| 1378₅ ||class="entry q3 g1"| 61109₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64198, 5159, 1336, 64679) ||class="entry q0 g1"| 64198₁₀ ||class="entry q1 g0"| 5159₆ ||class="entry q2 g0"| 1336₅ ||class="entry q3 g1"| 64679₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64666, 1619, 868, 61139) ||class="entry q0 g1"| 64666₁₀ ||class="entry q1 g0"| 1619₆ ||class="entry q2 g0"| 868₅ ||class="entry q3 g1"| 61139₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64678, 4679, 856, 64199) ||class="entry q0 g1"| 64678₁₀ ||class="entry q1 g0"| 4679₆ ||class="entry q2 g0"| 856₅ ||class="entry q3 g1"| 64199₁₁ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (60350, 831, 5184, 60351) ||class="entry q0 g1"| 60350₁₂ ||class="entry q1 g0"| 831₈ ||class="entry q2 g0"| 5184₃ ||class="entry q3 g1"| 60351₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (60894, 1375, 4640, 60895) ||class="entry q0 g1"| 60894₁₂ ||class="entry q1 g0"| 1375₈ ||class="entry q2 g0"| 4640₃ ||class="entry q3 g1"| 60895₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (63990, 4471, 1544, 63991) ||class="entry q0 g1"| 63990₁₂ ||class="entry q1 g0"| 4471₈ ||class="entry q2 g0"| 1544₃ ||class="entry q3 g1"| 63991₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (65430, 5911, 104, 65431) ||class="entry q0 g1"| 65430₁₂ ||class="entry q1 g0"| 5911₈ ||class="entry q2 g0"| 104₃ ||class="entry q3 g1"| 65431₁₃ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (61166, 15, 4368, 59535) ||class="entry q0 g1"| 61166₁₂ ||class="entry q1 g1"| 15₄ ||class="entry q2 g1"| 4368₃ ||class="entry q3 g1"| 59535₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (64250, 51, 1284, 59571) ||class="entry q0 g1"| 64250₁₂ ||class="entry q1 g1"| 51₄ ||class="entry q2 g1"| 1284₃ ||class="entry q3 g1"| 59571₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (64764, 85, 770, 59605) ||class="entry q0 g1"| 64764₁₂ ||class="entry q1 g1"| 85₄ ||class="entry q2 g1"| 770₃ ||class="entry q3 g1"| 59605₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65450, 771, 84, 60291) ||class="entry q0 g1"| 65450₁₂ ||class="entry q1 g1"| 771₄ ||class="entry q2 g1"| 84₃ ||class="entry q3 g1"| 60291₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65484, 1285, 50, 60805) ||class="entry q0 g1"| 65484₁₂ ||class="entry q1 g1"| 1285₄ ||class="entry q2 g1"| 50₃ ||class="entry q3 g1"| 60805₉ |- |class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65520, 4369, 14, 63889) ||class="entry q0 g1"| 65520₁₂ ||class="entry q1 g1"| 4369₄ ||class="entry q2 g1"| 14₃ ||class="entry q3 g1"| 63889₉ |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> d6vp0d4au8ea18eay9lyh0cu6txqbis 10 313712 2693781 2678097 2024-12-29T18:08:24Z Watchduck 137431 2693781 wikitext text/x-wiki <templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" /> {| class="wikitable sortable mentor-cycles" |+ 2636 cycles of length 6 !class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="7"| cycle |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (1672, 28425, 1655, 28168, 1695, 1889) ||class="entry q0 g0"| 1672₄ ||class="entry q1 g0"| 28425₈ ||class="entry q1 g0"| 1655₈ ||class="entry q0 g0"| 28168₆ ||class="entry q1 g0"| 1695₈ ||class="entry q1 g0"| 1889₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (2152, 24825, 1895, 24808, 2415, 1649) ||class="entry q0 g0"| 2152₄ ||class="entry q1 g0"| 24825₈ ||class="entry q1 g0"| 1895₈ ||class="entry q0 g0"| 24808₆ ||class="entry q1 g0"| 2415₈ ||class="entry q1 g0"| 1649₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (4768, 31521, 4703, 31264, 4791, 4937) ||class="entry q0 g0"| 4768₄ ||class="entry q1 g0"| 31521₈ ||class="entry q1 g0"| 4703₈ ||class="entry q0 g0"| 31264₆ ||class="entry q1 g0"| 4791₈ ||class="entry q1 g0"| 4937₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (5312, 32065, 5183, 31808, 5335, 5417) ||class="entry q0 g0"| 5312₄ ||class="entry q1 g0"| 32065₈ ||class="entry q1 g0"| 5183₈ ||class="entry q0 g0"| 31808₆ ||class="entry q1 g0"| 5335₈ ||class="entry q1 g0"| 5417₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (6688, 29361, 5423, 29344, 6951, 5177) ||class="entry q0 g0"| 6688₄ ||class="entry q1 g0"| 29361₈ ||class="entry q1 g0"| 5423₈ ||class="entry q0 g0"| 29344₆ ||class="entry q1 g0"| 6951₈ ||class="entry q1 g0"| 5177₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (7232, 29905, 4943, 29888, 7495, 4697) ||class="entry q0 g0"| 7232₄ ||class="entry q1 g0"| 29905₈ ||class="entry q1 g0"| 4943₈ ||class="entry q0 g0"| 29888₆ ||class="entry q1 g0"| 7495₈ ||class="entry q1 g0"| 4697₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (8296, 18669, 4955, 18664, 8571, 4685) ||class="entry q0 g0"| 8296₄ ||class="entry q1 g0"| 18669₈ ||class="entry q1 g0"| 4955₈ ||class="entry q0 g0"| 18664₆ ||class="entry q1 g0"| 8571₈ ||class="entry q1 g0"| 4685₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (9736, 20109, 5435, 20104, 10011, 5165) ||class="entry q0 g0"| 9736₄ ||class="entry q1 g0"| 20109₈ ||class="entry q1 g0"| 5435₈ ||class="entry q0 g0"| 20104₆ ||class="entry q1 g0"| 10011₈ ||class="entry q1 g0"| 5165₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (13376, 23749, 1907, 23744, 13651, 1637) ||class="entry q0 g0"| 13376₄ ||class="entry q1 g0"| 23749₈ ||class="entry q1 g0"| 1907₈ ||class="entry q0 g0"| 23744₆ ||class="entry q1 g0"| 13651₈ ||class="entry q1 g0"| 1637₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (16488, 10475, 5437, 10472, 16765, 5163) ||class="entry q0 g0"| 16488₄ ||class="entry q1 g0"| 10475₈ ||class="entry q1 g0"| 5437₈ ||class="entry q0 g0"| 10472₆ ||class="entry q1 g0"| 16765₈ ||class="entry q1 g0"| 5163₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (17928, 11915, 4957, 11912, 18205, 4683) ||class="entry q0 g0"| 17928₄ ||class="entry q1 g0"| 11915₈ ||class="entry q1 g0"| 4957₈ ||class="entry q0 g0"| 11912₆ ||class="entry q1 g0"| 18205₈ ||class="entry q1 g0"| 4683₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (21024, 15011, 1909, 15008, 21301, 1635) ||class="entry q0 g0"| 21024₄ ||class="entry q1 g0"| 15011₈ ||class="entry q1 g0"| 1909₈ ||class="entry q0 g0"| 15008₆ ||class="entry q1 g0"| 21301₈ ||class="entry q1 g0"| 1635₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (680, 703, 27351, 27176, 27433, 27585) ||class="entry q0 g0"| 680₄ ||class="entry q1 g0"| 703₈ ||class="entry q1 g0"| 27351₁₀ ||class="entry q0 g0"| 27176₆ ||class="entry q1 g0"| 27433₈ ||class="entry q1 g0"| 27585₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (1224, 1247, 27831, 27720, 27977, 28065) ||class="entry q0 g0"| 1224₄ ||class="entry q1 g0"| 1247₈ ||class="entry q1 g0"| 27831₁₀ ||class="entry q0 g0"| 27720₆ ||class="entry q1 g0"| 27977₈ ||class="entry q1 g0"| 28065₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (2600, 2863, 28071, 25256, 25273, 27825) ||class="entry q0 g0"| 2600₄ ||class="entry q1 g0"| 2863₈ ||class="entry q1 g0"| 28071₁₀ ||class="entry q0 g0"| 25256₆ ||class="entry q1 g0"| 25273₈ ||class="entry q1 g0"| 27825₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (3144, 3407, 27591, 25800, 25817, 27345) ||class="entry q0 g0"| 3144₄ ||class="entry q1 g0"| 3407₈ ||class="entry q1 g0"| 27591₁₀ ||class="entry q0 g0"| 25800₆ ||class="entry q1 g0"| 25817₈ ||class="entry q1 g0"| 27345₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (4320, 4343, 30879, 30816, 31073, 31113) ||class="entry q0 g0"| 4320₄ ||class="entry q1 g0"| 4343₈ ||class="entry q1 g0"| 30879₁₀ ||class="entry q0 g0"| 30816₆ ||class="entry q1 g0"| 31073₈ ||class="entry q1 g0"| 31113₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (7680, 7943, 31119, 30336, 30353, 30873) ||class="entry q0 g0"| 7680₄ ||class="entry q1 g0"| 7943₈ ||class="entry q1 g0"| 31119₁₀ ||class="entry q0 g0"| 30336₆ ||class="entry q1 g0"| 30353₈ ||class="entry q1 g0"| 30873₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (8744, 9019, 31131, 19112, 19117, 30861) ||class="entry q0 g0"| 8744₄ ||class="entry q1 g0"| 9019₈ ||class="entry q1 g0"| 31131₁₀ ||class="entry q0 g0"| 19112₆ ||class="entry q1 g0"| 19117₈ ||class="entry q1 g0"| 30861₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (12384, 12659, 27603, 22752, 22757, 27333) ||class="entry q0 g0"| 12384₄ ||class="entry q1 g0"| 12659₈ ||class="entry q1 g0"| 27603₁₀ ||class="entry q0 g0"| 22752₆ ||class="entry q1 g0"| 22757₈ ||class="entry q1 g0"| 27333₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (13824, 14099, 28083, 24192, 24197, 27813) ||class="entry q0 g0"| 13824₄ ||class="entry q1 g0"| 14099₈ ||class="entry q1 g0"| 28083₁₀ ||class="entry q0 g0"| 24192₆ ||class="entry q1 g0"| 24197₈ ||class="entry q1 g0"| 27813₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (17480, 17757, 31133, 11464, 11467, 30859) ||class="entry q0 g0"| 17480₄ ||class="entry q1 g0"| 17757₈ ||class="entry q1 g0"| 31133₁₀ ||class="entry q0 g0"| 11464₆ ||class="entry q1 g0"| 11467₈ ||class="entry q1 g0"| 30859₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (20576, 20853, 28085, 14560, 14563, 27811) ||class="entry q0 g0"| 20576₄ ||class="entry q1 g0"| 20853₈ ||class="entry q1 g0"| 28085₁₀ ||class="entry q0 g0"| 14560₆ ||class="entry q1 g0"| 14563₈ ||class="entry q1 g0"| 27811₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (22016, 22293, 27605, 16000, 16003, 27331) ||class="entry q0 g0"| 22016₄ ||class="entry q1 g0"| 22293₈ ||class="entry q1 g0"| 27605₁₀ ||class="entry q0 g0"| 16000₆ ||class="entry q1 g0"| 16003₈ ||class="entry q1 g0"| 27331₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 14, 6, 8, 12) ||class="c"| (5760, 5783, 32511, 32256, 32513, 32745) ||class="entry q0 g0"| 5760₄ ||class="entry q1 g0"| 5783₈ ||class="entry q1 g0"| 32511₁₄ ||class="entry q0 g0"| 32256₆ ||class="entry q1 g0"| 32513₈ ||class="entry q1 g0"| 32745₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 14, 6, 8, 12) ||class="c"| (6240, 6503, 32751, 28896, 28913, 32505) ||class="entry q0 g0"| 6240₄ ||class="entry q1 g0"| 6503₈ ||class="entry q1 g0"| 32751₁₄ ||class="entry q0 g0"| 28896₆ ||class="entry q1 g0"| 28913₈ ||class="entry q1 g0"| 32505₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 14, 6, 8, 12) ||class="c"| (9288, 9563, 32763, 19656, 19661, 32493) ||class="entry q0 g0"| 9288₄ ||class="entry q1 g0"| 9563₈ ||class="entry q1 g0"| 32763₁₄ ||class="entry q0 g0"| 19656₆ ||class="entry q1 g0"| 19661₈ ||class="entry q1 g0"| 32493₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 14, 6, 8, 12) ||class="c"| (16936, 17213, 32765, 10920, 10923, 32491) ||class="entry q0 g0"| 16936₄ ||class="entry q1 g0"| 17213₈ ||class="entry q1 g0"| 32765₁₄ ||class="entry q0 g0"| 10920₆ ||class="entry q1 g0"| 10923₈ ||class="entry q1 g0"| 32491₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (1678, 2409, 24593, 28174, 24831, 24839) ||class="entry q0 g0"| 1678₆ ||class="entry q1 g0"| 2409₆ ||class="entry q1 g0"| 24593₄ ||class="entry q0 g0"| 28174₈ ||class="entry q1 g0"| 24831₁₀ ||class="entry q1 g0"| 24839₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (2158, 1689, 24833, 24814, 28431, 24599) ||class="entry q0 g0"| 2158₆ ||class="entry q1 g0"| 1689₆ ||class="entry q1 g0"| 24833₄ ||class="entry q0 g0"| 24814₈ ||class="entry q1 g0"| 28431₁₀ ||class="entry q1 g0"| 24599₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (4786, 8553, 18437, 31282, 18687, 18707) ||class="entry q0 g0"| 4786₆ ||class="entry q1 g0"| 8553₆ ||class="entry q1 g0"| 18437₄ ||class="entry q0 g0"| 31282₈ ||class="entry q1 g0"| 18687₁₀ ||class="entry q1 g0"| 18707₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (5332, 16745, 10243, 31828, 10495, 10517) ||class="entry q0 g0"| 5332₆ ||class="entry q1 g0"| 16745₆ ||class="entry q1 g0"| 10243₄ ||class="entry q0 g0"| 31828₈ ||class="entry q1 g0"| 10495₁₀ ||class="entry q1 g0"| 10517₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (6946, 9753, 16517, 29602, 20367, 16787) ||class="entry q0 g0"| 6946₆ ||class="entry q1 g0"| 9753₆ ||class="entry q1 g0"| 16517₄ ||class="entry q0 g0"| 29602₈ ||class="entry q1 g0"| 20367₁₀ ||class="entry q1 g0"| 16787₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (7492, 17945, 8323, 30148, 12175, 8597) ||class="entry q0 g0"| 7492₆ ||class="entry q1 g0"| 17945₆ ||class="entry q1 g0"| 8323₄ ||class="entry q0 g0"| 30148₈ ||class="entry q1 g0"| 12175₁₀ ||class="entry q1 g0"| 8597₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (8314, 4773, 18689, 18682, 31539, 18455) ||class="entry q0 g0"| 8314₆ ||class="entry q1 g0"| 4773₆ ||class="entry q1 g0"| 18689₄ ||class="entry q0 g0"| 18682₈ ||class="entry q1 g0"| 31539₁₀ ||class="entry q1 g0"| 18455₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (9994, 6693, 16529, 20362, 29619, 16775) ||class="entry q0 g0"| 9994₆ ||class="entry q1 g0"| 6693₆ ||class="entry q1 g0"| 16529₄ ||class="entry q0 g0"| 20362₈ ||class="entry q1 g0"| 29619₁₀ ||class="entry q1 g0"| 16775₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (13648, 21029, 2179, 24016, 15283, 2453) ||class="entry q0 g0"| 13648₆ ||class="entry q1 g0"| 21029₆ ||class="entry q1 g0"| 2179₄ ||class="entry q0 g0"| 24016₈ ||class="entry q1 g0"| 15283₁₀ ||class="entry q1 g0"| 2453₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (16508, 5315, 10497, 10492, 32085, 10263) ||class="entry q0 g0"| 16508₆ ||class="entry q1 g0"| 5315₆ ||class="entry q1 g0"| 10497₄ ||class="entry q0 g0"| 10492₈ ||class="entry q1 g0"| 32085₁₀ ||class="entry q1 g0"| 10263₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (18188, 7235, 8337, 12172, 30165, 8583) ||class="entry q0 g0"| 18188₆ ||class="entry q1 g0"| 7235₆ ||class="entry q1 g0"| 8337₄ ||class="entry q0 g0"| 12172₈ ||class="entry q1 g0"| 30165₁₀ ||class="entry q1 g0"| 8583₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 4, 8, 10, 6) ||class="c"| (21296, 13379, 2181, 15280, 24021, 2451) ||class="entry q0 g0"| 21296₆ ||class="entry q1 g0"| 13379₆ ||class="entry q1 g0"| 2181₄ ||class="entry q0 g0"| 15280₈ ||class="entry q1 g0"| 24021₁₀ ||class="entry q1 g0"| 2451₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1690, 13633, 23597, 28186, 23767, 23867) ||class="entry q0 g0"| 1690₆ ||class="entry q1 g0"| 13633₆ ||class="entry q1 g0"| 23597₈ ||class="entry q0 g0"| 28186₈ ||class="entry q1 g0"| 23767₁₀ ||class="entry q1 g0"| 23867₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1692, 21281, 14923, 28188, 15031, 15197) ||class="entry q0 g0"| 1692₆ ||class="entry q1 g0"| 21281₆ ||class="entry q1 g0"| 14923₈ ||class="entry q0 g0"| 28188₈ ||class="entry q1 g0"| 15031₁₀ ||class="entry q1 g0"| 15197₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (2410, 13393, 21197, 25066, 24007, 21467) ||class="entry q0 g0"| 2410₆ ||class="entry q1 g0"| 13393₆ ||class="entry q1 g0"| 21197₈ ||class="entry q0 g0"| 25066₈ ||class="entry q1 g0"| 24007₁₀ ||class="entry q1 g0"| 21467₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (2412, 21041, 13483, 25068, 15271, 13757) ||class="entry q0 g0"| 2412₆ ||class="entry q1 g0"| 21041₆ ||class="entry q1 g0"| 13483₈ ||class="entry q0 g0"| 25068₈ ||class="entry q1 g0"| 15271₁₀ ||class="entry q1 g0"| 13757₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4774, 7489, 29753, 31270, 29911, 29999) ||class="entry q0 g0"| 4774₆ ||class="entry q1 g0"| 7489₆ ||class="entry q1 g0"| 29753₈ ||class="entry q0 g0"| 31270₈ ||class="entry q1 g0"| 29911₁₀ ||class="entry q1 g0"| 29999₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4788, 18185, 11875, 31284, 11935, 12149) ||class="entry q0 g0"| 4788₆ ||class="entry q1 g0"| 18185₆ ||class="entry q1 g0"| 11875₈ ||class="entry q0 g0"| 31284₈ ||class="entry q1 g0"| 11935₁₀ ||class="entry q1 g0"| 12149₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (5318, 6945, 29273, 31814, 29367, 29519) ||class="entry q0 g0"| 5318₆ ||class="entry q1 g0"| 6945₆ ||class="entry q1 g0"| 29273₈ ||class="entry q0 g0"| 31814₈ ||class="entry q1 g0"| 29367₁₀ ||class="entry q1 g0"| 29519₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (5330, 9993, 20069, 31826, 20127, 20339) ||class="entry q0 g0"| 5330₆ ||class="entry q1 g0"| 9993₆ ||class="entry q1 g0"| 20069₈ ||class="entry q0 g0"| 31826₈ ||class="entry q1 g0"| 20127₁₀ ||class="entry q1 g0"| 20339₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6694, 5329, 29513, 29350, 32071, 29279) ||class="entry q0 g0"| 6694₆ ||class="entry q1 g0"| 5329₆ ||class="entry q1 g0"| 29513₈ ||class="entry q0 g0"| 29350₈ ||class="entry q1 g0"| 32071₁₀ ||class="entry q1 g0"| 29279₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6948, 16505, 9955, 29604, 10735, 10229) ||class="entry q0 g0"| 6948₆ ||class="entry q1 g0"| 16505₆ ||class="entry q1 g0"| 9955₈ ||class="entry q0 g0"| 29604₈ ||class="entry q1 g0"| 10735₁₀ ||class="entry q1 g0"| 10229₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7238, 4785, 29993, 29894, 31527, 29759) ||class="entry q0 g0"| 7238₆ ||class="entry q1 g0"| 4785₆ ||class="entry q1 g0"| 29993₈ ||class="entry q0 g0"| 29894₈ ||class="entry q1 g0"| 31527₁₀ ||class="entry q1 g0"| 29759₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7490, 8313, 18149, 30146, 18927, 18419) ||class="entry q0 g0"| 7490₆ ||class="entry q1 g0"| 8313₆ ||class="entry q1 g0"| 18149₈ ||class="entry q0 g0"| 30146₈ ||class="entry q1 g0"| 18927₁₀ ||class="entry q1 g0"| 18419₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (8554, 7237, 18161, 18922, 30163, 18407) ||class="entry q0 g0"| 8554₆ ||class="entry q1 g0"| 7237₆ ||class="entry q1 g0"| 18161₈ ||class="entry q0 g0"| 18922₈ ||class="entry q1 g0"| 30163₁₀ ||class="entry q1 g0"| 18407₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (8568, 17933, 7339, 18936, 12187, 7613) ||class="entry q0 g0"| 8568₆ ||class="entry q1 g0"| 17933₆ ||class="entry q1 g0"| 7339₈ ||class="entry q0 g0"| 18936₈ ||class="entry q1 g0"| 12187₁₀ ||class="entry q1 g0"| 7613₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (9754, 5317, 20321, 20122, 32083, 20087) ||class="entry q0 g0"| 9754₆ ||class="entry q1 g0"| 5317₆ ||class="entry q1 g0"| 20321₈ ||class="entry q0 g0"| 20122₈ ||class="entry q1 g0"| 32083₁₀ ||class="entry q1 g0"| 20087₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10008, 16493, 6859, 20376, 10747, 7133) ||class="entry q0 g0"| 10008₆ ||class="entry q1 g0"| 16493₆ ||class="entry q1 g0"| 6859₈ ||class="entry q0 g0"| 20376₈ ||class="entry q1 g0"| 10747₁₀ ||class="entry q1 g0"| 7133₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13394, 1677, 23849, 23762, 28443, 23615) ||class="entry q0 g0"| 13394₆ ||class="entry q1 g0"| 1677₆ ||class="entry q1 g0"| 23849₈ ||class="entry q0 g0"| 23762₈ ||class="entry q1 g0"| 28443₁₀ ||class="entry q1 g0"| 23615₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13634, 2157, 21209, 24002, 25083, 21455) ||class="entry q0 g0"| 13634₆ ||class="entry q1 g0"| 2157₆ ||class="entry q1 g0"| 21209₈ ||class="entry q0 g0"| 24002₈ ||class="entry q1 g0"| 25083₁₀ ||class="entry q1 g0"| 21455₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (16748, 6691, 9969, 10732, 29621, 10215) ||class="entry q0 g0"| 16748₆ ||class="entry q1 g0"| 6691₆ ||class="entry q1 g0"| 9969₈ ||class="entry q0 g0"| 10732₈ ||class="entry q1 g0"| 29621₁₀ ||class="entry q1 g0"| 10215₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (16760, 9739, 6861, 10744, 20381, 7131) ||class="entry q0 g0"| 16760₆ ||class="entry q1 g0"| 9739₆ ||class="entry q1 g0"| 6861₈ ||class="entry q0 g0"| 10744₈ ||class="entry q1 g0"| 20381₁₀ ||class="entry q1 g0"| 7131₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (17948, 4771, 12129, 11932, 31541, 11895) ||class="entry q0 g0"| 17948₆ ||class="entry q1 g0"| 4771₆ ||class="entry q1 g0"| 12129₈ ||class="entry q0 g0"| 11932₈ ||class="entry q1 g0"| 31541₁₀ ||class="entry q1 g0"| 11895₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18200, 8299, 7341, 12184, 18941, 7611) ||class="entry q0 g0"| 18200₆ ||class="entry q1 g0"| 8299₆ ||class="entry q1 g0"| 7341₈ ||class="entry q0 g0"| 12184₈ ||class="entry q1 g0"| 18941₁₀ ||class="entry q1 g0"| 7611₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21044, 1675, 15177, 15028, 28445, 14943) ||class="entry q0 g0"| 21044₆ ||class="entry q1 g0"| 1675₆ ||class="entry q1 g0"| 15177₈ ||class="entry q0 g0"| 15028₈ ||class="entry q1 g0"| 28445₁₀ ||class="entry q1 g0"| 14943₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21284, 2155, 13497, 15268, 25085, 13743) ||class="entry q0 g0"| 21284₆ ||class="entry q1 g0"| 2155₆ ||class="entry q1 g0"| 13497₈ ||class="entry q0 g0"| 15268₈ ||class="entry q1 g0"| 25085₁₀ ||class="entry q1 g0"| 13743₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (1944, 25065, 2439, 28440, 2175, 2193) ||class="entry q0 g0"| 1944₆ ||class="entry q1 g0"| 25065₈ ||class="entry q1 g0"| 2439₆ ||class="entry q0 g0"| 28440₈ ||class="entry q1 g0"| 2175₈ ||class="entry q1 g0"| 2193₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (2424, 28185, 2199, 25080, 1935, 2433) ||class="entry q0 g0"| 2424₆ ||class="entry q1 g0"| 28185₈ ||class="entry q1 g0"| 2199₆ ||class="entry q0 g0"| 25080₈ ||class="entry q1 g0"| 1935₈ ||class="entry q1 g0"| 2433₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (5028, 18921, 8595, 31524, 8319, 8325) ||class="entry q0 g0"| 5028₆ ||class="entry q1 g0"| 18921₈ ||class="entry q1 g0"| 8595₆ ||class="entry q0 g0"| 31524₈ ||class="entry q1 g0"| 8319₈ ||class="entry q1 g0"| 8325₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (5570, 10729, 16789, 32066, 16511, 16515) ||class="entry q0 g0"| 5570₆ ||class="entry q1 g0"| 10729₈ ||class="entry q1 g0"| 16789₆ ||class="entry q0 g0"| 32066₈ ||class="entry q1 g0"| 16511₈ ||class="entry q1 g0"| 16515₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (6708, 20121, 10515, 29364, 9999, 10245) ||class="entry q0 g0"| 6708₆ ||class="entry q1 g0"| 20121₈ ||class="entry q1 g0"| 10515₆ ||class="entry q0 g0"| 29364₈ ||class="entry q1 g0"| 9999₈ ||class="entry q1 g0"| 10245₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (7250, 11929, 18709, 29906, 18191, 18435) ||class="entry q0 g0"| 7250₆ ||class="entry q1 g0"| 11929₈ ||class="entry q1 g0"| 18709₆ ||class="entry q0 g0"| 29906₈ ||class="entry q1 g0"| 18191₈ ||class="entry q1 g0"| 18435₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (8556, 31269, 8343, 18924, 5043, 8577) ||class="entry q0 g0"| 8556₆ ||class="entry q1 g0"| 31269₈ ||class="entry q1 g0"| 8343₆ ||class="entry q0 g0"| 18924₈ ||class="entry q1 g0"| 5043₈ ||class="entry q1 g0"| 8577₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (9756, 29349, 10503, 20124, 6963, 10257) ||class="entry q0 g0"| 9756₆ ||class="entry q1 g0"| 29349₈ ||class="entry q1 g0"| 10503₆ ||class="entry q0 g0"| 20124₈ ||class="entry q1 g0"| 6963₈ ||class="entry q1 g0"| 10257₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (13382, 15013, 24853, 23750, 21299, 24579) ||class="entry q0 g0"| 13382₆ ||class="entry q1 g0"| 15013₈ ||class="entry q1 g0"| 24853₆ ||class="entry q0 g0"| 23750₈ ||class="entry q1 g0"| 21299₈ ||class="entry q1 g0"| 24579₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (16746, 31811, 16535, 10730, 5589, 16769) ||class="entry q0 g0"| 16746₆ ||class="entry q1 g0"| 31811₈ ||class="entry q1 g0"| 16535₆ ||class="entry q0 g0"| 10730₈ ||class="entry q1 g0"| 5589₈ ||class="entry q1 g0"| 16769₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (17946, 29891, 18695, 11930, 7509, 18449) ||class="entry q0 g0"| 17946₆ ||class="entry q1 g0"| 29891₈ ||class="entry q1 g0"| 18695₆ ||class="entry q0 g0"| 11930₈ ||class="entry q1 g0"| 7509₈ ||class="entry q1 g0"| 18449₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (21030, 23747, 24851, 15014, 13653, 24581) ||class="entry q0 g0"| 21030₆ ||class="entry q1 g0"| 23747₈ ||class="entry q1 g0"| 24851₆ ||class="entry q0 g0"| 15014₈ ||class="entry q1 g0"| 13653₈ ||class="entry q1 g0"| 24581₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (940, 12407, 22811, 27436, 23009, 22541) ||class="entry q0 g0"| 940₆ ||class="entry q1 g0"| 12407₈ ||class="entry q1 g0"| 22811₈ ||class="entry q0 g0"| 27436₈ ||class="entry q1 g0"| 23009₈ ||class="entry q1 g0"| 22541₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (952, 3167, 25895, 27448, 26057, 25649) ||class="entry q0 g0"| 952₆ ||class="entry q1 g0"| 3167₈ ||class="entry q1 g0"| 25895₈ ||class="entry q0 g0"| 27448₈ ||class="entry q1 g0"| 26057₈ ||class="entry q1 g0"| 25649₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (1482, 20599, 14621, 27978, 14817, 14347) ||class="entry q0 g0"| 1482₆ ||class="entry q1 g0"| 20599₈ ||class="entry q1 g0"| 14621₈ ||class="entry q0 g0"| 27978₈ ||class="entry q1 g0"| 14817₈ ||class="entry q1 g0"| 14347₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (1496, 2623, 25415, 27992, 25513, 25169) ||class="entry q0 g0"| 1496₆ ||class="entry q1 g0"| 2623₈ ||class="entry q1 g0"| 25415₈ ||class="entry q0 g0"| 27992₈ ||class="entry q1 g0"| 25513₈ ||class="entry q1 g0"| 25169₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (2620, 14087, 20891, 25276, 24209, 20621) ||class="entry q0 g0"| 2620₆ ||class="entry q1 g0"| 14087₈ ||class="entry q1 g0"| 20891₈ ||class="entry q0 g0"| 25276₈ ||class="entry q1 g0"| 24209₈ ||class="entry q1 g0"| 20621₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (2872, 1487, 25175, 25528, 27737, 25409) ||class="entry q0 g0"| 2872₆ ||class="entry q1 g0"| 1487₈ ||class="entry q1 g0"| 25175₈ ||class="entry q0 g0"| 25528₈ ||class="entry q1 g0"| 27737₈ ||class="entry q1 g0"| 25409₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (3162, 22279, 12701, 25818, 16017, 12427) ||class="entry q0 g0"| 3162₆ ||class="entry q1 g0"| 22279₈ ||class="entry q1 g0"| 12701₈ ||class="entry q0 g0"| 25818₈ ||class="entry q1 g0"| 16017₈ ||class="entry q1 g0"| 12427₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (3416, 943, 25655, 26072, 27193, 25889) ||class="entry q0 g0"| 3416₆ ||class="entry q1 g0"| 943₈ ||class="entry q1 g0"| 25655₈ ||class="entry q0 g0"| 26072₈ ||class="entry q1 g0"| 27193₈ ||class="entry q1 g0"| 25889₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (4578, 17503, 11573, 31074, 11721, 11299) ||class="entry q0 g0"| 4578₆ ||class="entry q1 g0"| 17503₈ ||class="entry q1 g0"| 11573₈ ||class="entry q0 g0"| 31074₈ ||class="entry q1 g0"| 11721₈ ||class="entry q1 g0"| 11299₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (4580, 8767, 19283, 31076, 19369, 19013) ||class="entry q0 g0"| 4580₆ ||class="entry q1 g0"| 8767₈ ||class="entry q1 g0"| 19283₈ ||class="entry q0 g0"| 31076₈ ||class="entry q1 g0"| 19369₈ ||class="entry q1 g0"| 19013₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6018, 16959, 11093, 32514, 11177, 10819) ||class="entry q0 g0"| 6018₆ ||class="entry q1 g0"| 16959₈ ||class="entry q1 g0"| 11093₈ ||class="entry q0 g0"| 32514₈ ||class="entry q1 g0"| 11177₈ ||class="entry q1 g0"| 10819₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6020, 9311, 19763, 32516, 19913, 19493) ||class="entry q0 g0"| 6020₆ ||class="entry q1 g0"| 9311₈ ||class="entry q1 g0"| 19763₈ ||class="entry q0 g0"| 32516₈ ||class="entry q1 g0"| 19913₈ ||class="entry q1 g0"| 19493₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6032, 6263, 28943, 32528, 29153, 28697) ||class="entry q0 g0"| 6032₆ ||class="entry q1 g0"| 6263₈ ||class="entry q1 g0"| 28943₈ ||class="entry q0 g0"| 32528₈ ||class="entry q1 g0"| 29153₈ ||class="entry q1 g0"| 28697₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6258, 17199, 9653, 28914, 10937, 9379) ||class="entry q0 g0"| 6258₆ ||class="entry q1 g0"| 17199₈ ||class="entry q1 g0"| 9653₈ ||class="entry q0 g0"| 28914₈ ||class="entry q1 g0"| 10937₈ ||class="entry q1 g0"| 9379₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6260, 9551, 17363, 28916, 19673, 17093) ||class="entry q0 g0"| 6260₆ ||class="entry q1 g0"| 9551₈ ||class="entry q1 g0"| 17363₈ ||class="entry q0 g0"| 28916₈ ||class="entry q1 g0"| 19673₈ ||class="entry q1 g0"| 17093₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6512, 6023, 28703, 29168, 32273, 28937) ||class="entry q0 g0"| 6512₆ ||class="entry q1 g0"| 6023₈ ||class="entry q1 g0"| 28703₈ ||class="entry q0 g0"| 29168₈ ||class="entry q1 g0"| 32273₈ ||class="entry q1 g0"| 28937₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (7698, 17743, 9173, 30354, 11481, 8899) ||class="entry q0 g0"| 7698₆ ||class="entry q1 g0"| 17743₈ ||class="entry q1 g0"| 9173₈ ||class="entry q0 g0"| 30354₈ ||class="entry q1 g0"| 11481₈ ||class="entry q1 g0"| 8899₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (7700, 9007, 17843, 30356, 19129, 17573) ||class="entry q0 g0"| 7700₆ ||class="entry q1 g0"| 9007₈ ||class="entry q1 g0"| 17843₈ ||class="entry q0 g0"| 30356₈ ||class="entry q1 g0"| 19129₈ ||class="entry q1 g0"| 17573₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (8764, 7955, 17831, 19132, 30341, 17585) ||class="entry q0 g0"| 8764₆ ||class="entry q1 g0"| 7955₈ ||class="entry q1 g0"| 17831₈ ||class="entry q0 g0"| 19132₈ ||class="entry q1 g0"| 30341₈ ||class="entry q1 g0"| 17585₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9004, 4595, 19031, 19372, 30821, 19265) ||class="entry q0 g0"| 9004₆ ||class="entry q1 g0"| 4595₈ ||class="entry q1 g0"| 19031₈ ||class="entry q0 g0"| 19372₈ ||class="entry q1 g0"| 30821₈ ||class="entry q1 g0"| 19265₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9294, 17211, 6557, 19662, 10925, 6283) ||class="entry q0 g0"| 9294₆ ||class="entry q1 g0"| 17211₈ ||class="entry q1 g0"| 6557₈ ||class="entry q0 g0"| 19662₈ ||class="entry q1 g0"| 10925₈ ||class="entry q1 g0"| 6283₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9308, 6515, 17351, 19676, 28901, 17105) ||class="entry q0 g0"| 9308₆ ||class="entry q1 g0"| 6515₈ ||class="entry q1 g0"| 17351₈ ||class="entry q0 g0"| 19676₈ ||class="entry q1 g0"| 28901₈ ||class="entry q1 g0"| 17105₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9548, 6035, 19511, 19916, 32261, 19745) ||class="entry q0 g0"| 9548₆ ||class="entry q1 g0"| 6035₈ ||class="entry q1 g0"| 19511₈ ||class="entry q0 g0"| 19916₈ ||class="entry q1 g0"| 32261₈ ||class="entry q1 g0"| 19745₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (12390, 22291, 3509, 22758, 16005, 3235) ||class="entry q0 g0"| 12390₆ ||class="entry q1 g0"| 22291₈ ||class="entry q1 g0"| 3509₈ ||class="entry q0 g0"| 22758₈ ||class="entry q1 g0"| 16005₈ ||class="entry q1 g0"| 3235₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (12644, 955, 22559, 23012, 27181, 22793) ||class="entry q0 g0"| 12644₆ ||class="entry q1 g0"| 955₈ ||class="entry q1 g0"| 22559₈ ||class="entry q0 g0"| 23012₈ ||class="entry q1 g0"| 27181₈ ||class="entry q1 g0"| 22793₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (13830, 20851, 3029, 24198, 14565, 2755) ||class="entry q0 g0"| 13830₆ ||class="entry q1 g0"| 20851₈ ||class="entry q1 g0"| 3029₈ ||class="entry q0 g0"| 24198₈ ||class="entry q1 g0"| 14565₈ ||class="entry q1 g0"| 2755₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (13844, 2875, 20879, 24212, 25261, 20633) ||class="entry q0 g0"| 13844₆ ||class="entry q1 g0"| 2875₈ ||class="entry q1 g0"| 20879₈ ||class="entry q0 g0"| 24212₈ ||class="entry q1 g0"| 25261₈ ||class="entry q1 g0"| 20633₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (16942, 9565, 6555, 10926, 19659, 6285) ||class="entry q0 g0"| 16942₆ ||class="entry q1 g0"| 9565₈ ||class="entry q1 g0"| 6555₈ ||class="entry q0 g0"| 10926₈ ||class="entry q1 g0"| 19659₈ ||class="entry q1 g0"| 6285₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (16954, 6517, 9639, 10938, 28899, 9393) ||class="entry q0 g0"| 16954₆ ||class="entry q1 g0"| 6517₈ ||class="entry q1 g0"| 9639₈ ||class="entry q0 g0"| 10938₈ ||class="entry q1 g0"| 28899₈ ||class="entry q1 g0"| 9393₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17194, 6037, 10839, 11178, 32259, 11073) ||class="entry q0 g0"| 17194₆ ||class="entry q1 g0"| 6037₈ ||class="entry q1 g0"| 10839₈ ||class="entry q0 g0"| 11178₈ ||class="entry q1 g0"| 32259₈ ||class="entry q1 g0"| 11073₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17498, 7957, 9159, 11482, 30339, 8913) ||class="entry q0 g0"| 17498₆ ||class="entry q1 g0"| 7957₈ ||class="entry q1 g0"| 9159₈ ||class="entry q0 g0"| 11482₈ ||class="entry q1 g0"| 30339₈ ||class="entry q1 g0"| 8913₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17738, 4597, 11319, 11722, 30819, 11553) ||class="entry q0 g0"| 17738₆ ||class="entry q1 g0"| 4597₈ ||class="entry q1 g0"| 11319₈ ||class="entry q0 g0"| 11722₈ ||class="entry q1 g0"| 30819₈ ||class="entry q1 g0"| 11553₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (20582, 14101, 3027, 14566, 24195, 2757) ||class="entry q0 g0"| 20582₆ ||class="entry q1 g0"| 14101₈ ||class="entry q1 g0"| 3027₈ ||class="entry q0 g0"| 14566₈ ||class="entry q1 g0"| 24195₈ ||class="entry q1 g0"| 2757₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (20834, 1501, 14367, 14818, 27723, 14601) ||class="entry q0 g0"| 20834₆ ||class="entry q1 g0"| 1501₈ ||class="entry q1 g0"| 14367₈ ||class="entry q0 g0"| 14818₈ ||class="entry q1 g0"| 27723₈ ||class="entry q1 g0"| 14601₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (22022, 12661, 3507, 16006, 22755, 3237) ||class="entry q0 g0"| 22022₆ ||class="entry q1 g0"| 12661₈ ||class="entry q1 g0"| 3507₈ ||class="entry q0 g0"| 16006₈ ||class="entry q1 g0"| 22755₈ ||class="entry q1 g0"| 3237₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (22034, 3421, 12687, 16018, 25803, 12441) ||class="entry q0 g0"| 22034₆ ||class="entry q1 g0"| 3421₈ ||class="entry q1 g0"| 12687₈ ||class="entry q0 g0"| 16018₈ ||class="entry q1 g0"| 25803₈ ||class="entry q1 g0"| 12441₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (490, 15809, 21949, 26986, 21591, 21675) ||class="entry q0 g0"| 490₆ ||class="entry q1 g0"| 15809₈ ||class="entry q1 g0"| 21949₁₀ ||class="entry q0 g0"| 26986₈ ||class="entry q1 g0"| 21591₈ ||class="entry q1 g0"| 21675₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (492, 23457, 13275, 26988, 12855, 13005) ||class="entry q0 g0"| 492₆ ||class="entry q1 g0"| 23457₈ ||class="entry q1 g0"| 13275₁₀ ||class="entry q0 g0"| 26988₈ ||class="entry q1 g0"| 12855₈ ||class="entry q1 g0"| 13005₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (504, 26505, 4071, 27000, 3615, 3825) ||class="entry q0 g0"| 504₆ ||class="entry q1 g0"| 26505₈ ||class="entry q1 g0"| 4071₁₀ ||class="entry q0 g0"| 27000₈ ||class="entry q1 g0"| 3615₈ ||class="entry q1 g0"| 3825₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (1930, 15265, 21469, 28426, 21047, 21195) ||class="entry q0 g0"| 1930₆ ||class="entry q1 g0"| 15265₈ ||class="entry q1 g0"| 21469₁₀ ||class="entry q0 g0"| 28426₈ ||class="entry q1 g0"| 21047₈ ||class="entry q1 g0"| 21195₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (1932, 24001, 13755, 28428, 13399, 13485) ||class="entry q0 g0"| 1932₆ ||class="entry q1 g0"| 24001₈ ||class="entry q1 g0"| 13755₁₀ ||class="entry q0 g0"| 28428₈ ||class="entry q1 g0"| 13399₈ ||class="entry q1 g0"| 13485₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (2170, 15025, 23869, 24826, 21287, 23595) ||class="entry q0 g0"| 2170₆ ||class="entry q1 g0"| 15025₈ ||class="entry q1 g0"| 23869₁₀ ||class="entry q0 g0"| 24826₈ ||class="entry q1 g0"| 21287₈ ||class="entry q1 g0"| 23595₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (2172, 23761, 15195, 24828, 13639, 14925) ||class="entry q0 g0"| 2172₆ ||class="entry q1 g0"| 23761₈ ||class="entry q1 g0"| 15195₁₀ ||class="entry q0 g0"| 24828₈ ||class="entry q1 g0"| 13639₈ ||class="entry q1 g0"| 14925₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3610, 15569, 23389, 26266, 21831, 23115) ||class="entry q0 g0"| 3610₆ ||class="entry q1 g0"| 15569₈ ||class="entry q1 g0"| 23389₁₀ ||class="entry q0 g0"| 26266₈ ||class="entry q1 g0"| 21831₈ ||class="entry q1 g0"| 23115₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3612, 23217, 15675, 26268, 13095, 15405) ||class="entry q0 g0"| 3612₆ ||class="entry q1 g0"| 23217₈ ||class="entry q1 g0"| 15675₁₀ ||class="entry q0 g0"| 26268₈ ||class="entry q1 g0"| 13095₈ ||class="entry q1 g0"| 15405₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3864, 26745, 3831, 26520, 495, 4065) ||class="entry q0 g0"| 3864₆ ||class="entry q1 g0"| 26745₈ ||class="entry q1 g0"| 3831₁₀ ||class="entry q0 g0"| 26520₈ ||class="entry q1 g0"| 495₈ ||class="entry q1 g0"| 4065₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (5026, 12169, 18421, 31522, 17951, 18147) ||class="entry q0 g0"| 5026₆ ||class="entry q1 g0"| 12169₈ ||class="entry q1 g0"| 18421₁₀ ||class="entry q0 g0"| 31522₈ ||class="entry q1 g0"| 17951₈ ||class="entry q1 g0"| 18147₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (5040, 30145, 7599, 31536, 7255, 7353) ||class="entry q0 g0"| 5040₆ ||class="entry q1 g0"| 30145₈ ||class="entry q1 g0"| 7599₁₀ ||class="entry q0 g0"| 31536₈ ||class="entry q1 g0"| 7255₈ ||class="entry q1 g0"| 7353₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (5572, 20361, 10227, 32068, 9759, 9957) ||class="entry q0 g0"| 5572₆ ||class="entry q1 g0"| 20361₈ ||class="entry q1 g0"| 10227₁₀ ||class="entry q0 g0"| 32068₈ ||class="entry q1 g0"| 9759₈ ||class="entry q1 g0"| 9957₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (5584, 29601, 7119, 32080, 6711, 6873) ||class="entry q0 g0"| 5584₆ ||class="entry q1 g0"| 29601₈ ||class="entry q1 g0"| 7119₁₀ ||class="entry q0 g0"| 32080₈ ||class="entry q1 g0"| 6711₈ ||class="entry q1 g0"| 6873₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (6706, 10489, 20341, 29362, 16751, 20067) ||class="entry q0 g0"| 6706₆ ||class="entry q1 g0"| 10489₈ ||class="entry q1 g0"| 20341₁₀ ||class="entry q0 g0"| 29362₈ ||class="entry q1 g0"| 16751₈ ||class="entry q1 g0"| 20067₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (6960, 31825, 6879, 29616, 5575, 7113) ||class="entry q0 g0"| 6960₆ ||class="entry q1 g0"| 31825₈ ||class="entry q1 g0"| 6879₁₀ ||class="entry q0 g0"| 29616₈ ||class="entry q1 g0"| 5575₈ ||class="entry q1 g0"| 7113₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (7252, 18681, 12147, 29908, 8559, 11877) ||class="entry q0 g0"| 7252₆ ||class="entry q1 g0"| 18681₈ ||class="entry q1 g0"| 12147₁₀ ||class="entry q0 g0"| 29908₈ ||class="entry q1 g0"| 8559₈ ||class="entry q1 g0"| 11877₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (7504, 31281, 7359, 30160, 5031, 7593) ||class="entry q0 g0"| 7504₆ ||class="entry q1 g0"| 31281₈ ||class="entry q1 g0"| 7359₁₀ ||class="entry q0 g0"| 30160₈ ||class="entry q1 g0"| 5031₈ ||class="entry q1 g0"| 7593₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (8302, 11917, 30013, 18670, 18203, 29739) ||class="entry q0 g0"| 8302₆ ||class="entry q1 g0"| 11917₈ ||class="entry q1 g0"| 30013₁₀ ||class="entry q0 g0"| 18670₈ ||class="entry q1 g0"| 18203₈ ||class="entry q1 g0"| 29739₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (8316, 29893, 12135, 18684, 7507, 11889) ||class="entry q0 g0"| 8316₆ ||class="entry q1 g0"| 29893₈ ||class="entry q1 g0"| 12135₁₀ ||class="entry q0 g0"| 18684₈ ||class="entry q1 g0"| 7507₈ ||class="entry q1 g0"| 11889₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (9742, 10477, 29533, 20110, 16763, 29259) ||class="entry q0 g0"| 9742₆ ||class="entry q1 g0"| 10477₈ ||class="entry q1 g0"| 29533₁₀ ||class="entry q0 g0"| 20110₈ ||class="entry q1 g0"| 16763₈ ||class="entry q1 g0"| 29259₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (9996, 31813, 9975, 20364, 5587, 10209) ||class="entry q0 g0"| 9996₆ ||class="entry q1 g0"| 31813₈ ||class="entry q1 g0"| 9975₁₀ ||class="entry q0 g0"| 20364₈ ||class="entry q1 g0"| 5587₈ ||class="entry q1 g0"| 10209₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (12838, 15557, 26485, 23206, 21843, 26211) ||class="entry q0 g0"| 12838₆ ||class="entry q1 g0"| 15557₈ ||class="entry q1 g0"| 26485₁₀ ||class="entry q0 g0"| 23206₈ ||class="entry q1 g0"| 21843₈ ||class="entry q1 g0"| 26211₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (12852, 26253, 15663, 23220, 3867, 15417) ||class="entry q0 g0"| 12852₆ ||class="entry q1 g0"| 26253₈ ||class="entry q1 g0"| 15663₁₀ ||class="entry q0 g0"| 23220₈ ||class="entry q1 g0"| 3867₈ ||class="entry q1 g0"| 15417₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (13092, 26733, 13023, 23460, 507, 13257) ||class="entry q0 g0"| 13092₆ ||class="entry q1 g0"| 26733₈ ||class="entry q1 g0"| 13023₁₀ ||class="entry q0 g0"| 23460₈ ||class="entry q1 g0"| 507₈ ||class="entry q1 g0"| 13257₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (13396, 24813, 15183, 23764, 2427, 14937) ||class="entry q0 g0"| 13396₆ ||class="entry q1 g0"| 24813₈ ||class="entry q1 g0"| 15183₁₀ ||class="entry q0 g0"| 23764₈ ||class="entry q1 g0"| 2427₈ ||class="entry q1 g0"| 14937₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (13636, 28173, 13503, 24004, 1947, 13737) ||class="entry q0 g0"| 13636₆ ||class="entry q1 g0"| 28173₈ ||class="entry q1 g0"| 13503₁₀ ||class="entry q0 g0"| 24004₈ ||class="entry q1 g0"| 1947₈ ||class="entry q1 g0"| 13737₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (16494, 20107, 29531, 10478, 10013, 29261) ||class="entry q0 g0"| 16494₆ ||class="entry q1 g0"| 20107₈ ||class="entry q1 g0"| 29531₁₀ ||class="entry q0 g0"| 10478₈ ||class="entry q1 g0"| 10013₈ ||class="entry q1 g0"| 29261₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (16506, 29347, 20327, 10490, 6965, 20081) ||class="entry q0 g0"| 16506₆ ||class="entry q1 g0"| 29347₈ ||class="entry q1 g0"| 20327₁₀ ||class="entry q0 g0"| 10490₈ ||class="entry q1 g0"| 6965₈ ||class="entry q1 g0"| 20081₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (17934, 18667, 30011, 11918, 8573, 29741) ||class="entry q0 g0"| 17934₆ ||class="entry q1 g0"| 18667₈ ||class="entry q1 g0"| 30011₁₀ ||class="entry q0 g0"| 11918₈ ||class="entry q1 g0"| 8573₈ ||class="entry q1 g0"| 29741₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (18186, 31267, 18167, 12170, 5045, 18401) ||class="entry q0 g0"| 18186₆ ||class="entry q1 g0"| 31267₈ ||class="entry q1 g0"| 18167₁₀ ||class="entry q0 g0"| 12170₈ ||class="entry q1 g0"| 5045₈ ||class="entry q1 g0"| 18401₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21042, 24811, 23855, 15026, 2429, 23609) ||class="entry q0 g0"| 21042₆ ||class="entry q1 g0"| 24811₈ ||class="entry q1 g0"| 23855₁₀ ||class="entry q0 g0"| 15026₈ ||class="entry q1 g0"| 2429₈ ||class="entry q1 g0"| 23609₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21282, 28171, 21215, 15266, 1949, 21449) ||class="entry q0 g0"| 21282₆ ||class="entry q1 g0"| 28171₈ ||class="entry q1 g0"| 21215₁₀ ||class="entry q0 g0"| 15266₈ ||class="entry q1 g0"| 1949₈ ||class="entry q1 g0"| 21449₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21574, 23203, 26483, 15558, 13109, 26213) ||class="entry q0 g0"| 21574₆ ||class="entry q1 g0"| 23203₈ ||class="entry q1 g0"| 26483₁₀ ||class="entry q0 g0"| 15558₈ ||class="entry q1 g0"| 13109₈ ||class="entry q1 g0"| 26213₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21586, 26251, 23375, 15570, 3869, 23129) ||class="entry q0 g0"| 21586₆ ||class="entry q1 g0"| 26251₈ ||class="entry q1 g0"| 23375₁₀ ||class="entry q0 g0"| 15570₈ ||class="entry q1 g0"| 3869₈ ||class="entry q1 g0"| 23129₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21826, 26731, 21695, 15810, 509, 21929) ||class="entry q0 g0"| 21826₆ ||class="entry q1 g0"| 26731₈ ||class="entry q1 g0"| 21695₁₀ ||class="entry q0 g0"| 15810₈ ||class="entry q1 g0"| 509₈ ||class="entry q1 g0"| 21929₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (938, 22039, 16253, 27434, 16257, 15979) ||class="entry q0 g0"| 938₆ ||class="entry q1 g0"| 22039₈ ||class="entry q1 g0"| 16253₁₂ ||class="entry q0 g0"| 27434₈ ||class="entry q1 g0"| 16257₈ ||class="entry q1 g0"| 15979₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (1484, 13847, 24443, 27980, 24449, 24173) ||class="entry q0 g0"| 1484₆ ||class="entry q1 g0"| 13847₈ ||class="entry q1 g0"| 24443₁₂ ||class="entry q0 g0"| 27980₈ ||class="entry q1 g0"| 24449₈ ||class="entry q1 g0"| 24173₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (2618, 20839, 14333, 25274, 14577, 14059) ||class="entry q0 g0"| 2618₆ ||class="entry q1 g0"| 20839₈ ||class="entry q1 g0"| 14333₁₂ ||class="entry q0 g0"| 25274₈ ||class="entry q1 g0"| 14577₈ ||class="entry q1 g0"| 14059₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (3164, 12647, 22523, 25820, 22769, 22253) ||class="entry q0 g0"| 3164₆ ||class="entry q1 g0"| 12647₈ ||class="entry q1 g0"| 22523₁₂ ||class="entry q0 g0"| 25820₈ ||class="entry q1 g0"| 22769₈ ||class="entry q1 g0"| 22253₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (4592, 7703, 30575, 31088, 30593, 30329) ||class="entry q0 g0"| 4592₆ ||class="entry q1 g0"| 7703₈ ||class="entry q1 g0"| 30575₁₂ ||class="entry q0 g0"| 31088₈ ||class="entry q1 g0"| 30593₈ ||class="entry q1 g0"| 30329₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (7952, 4583, 30335, 30608, 30833, 30569) ||class="entry q0 g0"| 7952₆ ||class="entry q1 g0"| 4583₈ ||class="entry q1 g0"| 30335₁₂ ||class="entry q0 g0"| 30608₈ ||class="entry q1 g0"| 30833₈ ||class="entry q1 g0"| 30569₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (8750, 17755, 8189, 19118, 11469, 7915) ||class="entry q0 g0"| 8750₆ ||class="entry q1 g0"| 17755₈ ||class="entry q1 g0"| 8189₁₂ ||class="entry q0 g0"| 19118₈ ||class="entry q1 g0"| 11469₈ ||class="entry q1 g0"| 7915₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (12404, 3419, 22511, 22772, 25805, 22265) ||class="entry q0 g0"| 12404₆ ||class="entry q1 g0"| 3419₈ ||class="entry q1 g0"| 22511₁₂ ||class="entry q0 g0"| 22772₈ ||class="entry q1 g0"| 25805₈ ||class="entry q1 g0"| 22265₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (14084, 1499, 24191, 24452, 27725, 24425) ||class="entry q0 g0"| 14084₆ ||class="entry q1 g0"| 1499₈ ||class="entry q1 g0"| 24191₁₂ ||class="entry q0 g0"| 24452₈ ||class="entry q1 g0"| 27725₈ ||class="entry q1 g0"| 24425₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (17486, 9021, 8187, 11470, 19115, 7917) ||class="entry q0 g0"| 17486₆ ||class="entry q1 g0"| 9021₈ ||class="entry q1 g0"| 8187₁₂ ||class="entry q0 g0"| 11470₈ ||class="entry q1 g0"| 19115₈ ||class="entry q1 g0"| 7917₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (20594, 2877, 14319, 14578, 25259, 14073) ||class="entry q0 g0"| 20594₆ ||class="entry q1 g0"| 2877₈ ||class="entry q1 g0"| 14319₁₂ ||class="entry q0 g0"| 14578₈ ||class="entry q1 g0"| 25259₈ ||class="entry q1 g0"| 14073₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (22274, 957, 15999, 16258, 27179, 16233) ||class="entry q0 g0"| 22274₆ ||class="entry q1 g0"| 957₈ ||class="entry q1 g0"| 15999₁₂ ||class="entry q0 g0"| 16258₈ ||class="entry q1 g0"| 27179₈ ||class="entry q1 g0"| 16233₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (686, 25823, 3249, 27182, 3401, 3495) ||class="entry q0 g0"| 686₆ ||class="entry q1 g0"| 25823₁₀ ||class="entry q1 g0"| 3249₆ ||class="entry q0 g0"| 27182₈ ||class="entry q1 g0"| 3401₆ ||class="entry q1 g0"| 3495₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (698, 22775, 12429, 27194, 12641, 12699) ||class="entry q0 g0"| 698₆ ||class="entry q1 g0"| 22775₁₀ ||class="entry q1 g0"| 12429₆ ||class="entry q0 g0"| 27194₈ ||class="entry q1 g0"| 12641₆ ||class="entry q1 g0"| 12699₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1230, 25279, 2769, 27726, 2857, 3015) ||class="entry q0 g0"| 1230₆ ||class="entry q1 g0"| 25279₁₀ ||class="entry q1 g0"| 2769₆ ||class="entry q0 g0"| 27726₈ ||class="entry q1 g0"| 2857₆ ||class="entry q1 g0"| 3015₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1244, 14583, 20619, 27740, 20833, 20893) ||class="entry q0 g0"| 1244₆ ||class="entry q1 g0"| 14583₁₀ ||class="entry q1 g0"| 20619₆ ||class="entry q0 g0"| 27740₈ ||class="entry q1 g0"| 20833₆ ||class="entry q1 g0"| 20893₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (2606, 27983, 3009, 25262, 1241, 2775) ||class="entry q0 g0"| 2606₆ ||class="entry q1 g0"| 27983₁₀ ||class="entry q1 g0"| 3009₆ ||class="entry q0 g0"| 25262₈ ||class="entry q1 g0"| 1241₆ ||class="entry q1 g0"| 2775₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (2858, 24455, 14349, 25514, 13841, 14619) ||class="entry q0 g0"| 2858₆ ||class="entry q1 g0"| 24455₁₀ ||class="entry q1 g0"| 14349₆ ||class="entry q0 g0"| 25514₈ ||class="entry q1 g0"| 13841₆ ||class="entry q1 g0"| 14619₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (3150, 27439, 3489, 25806, 697, 3255) ||class="entry q0 g0"| 3150₆ ||class="entry q1 g0"| 27439₁₀ ||class="entry q1 g0"| 3489₆ ||class="entry q0 g0"| 25806₈ ||class="entry q1 g0"| 697₆ ||class="entry q1 g0"| 3255₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (3404, 16263, 22539, 26060, 22033, 22813) ||class="entry q0 g0"| 3404₆ ||class="entry q1 g0"| 16263₁₀ ||class="entry q1 g0"| 22539₆ ||class="entry q0 g0"| 26060₈ ||class="entry q1 g0"| 22033₆ ||class="entry q1 g0"| 22813₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4338, 19135, 8901, 30834, 9001, 9171) ||class="entry q0 g0"| 4338₆ ||class="entry q1 g0"| 19135₁₀ ||class="entry q1 g0"| 8901₆ ||class="entry q0 g0"| 30834₈ ||class="entry q1 g0"| 9001₆ ||class="entry q1 g0"| 9171₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4340, 11487, 17571, 30836, 17737, 17845) ||class="entry q0 g0"| 4340₆ ||class="entry q1 g0"| 11487₁₀ ||class="entry q1 g0"| 17571₆ ||class="entry q0 g0"| 30836₈ ||class="entry q1 g0"| 17737₆ ||class="entry q1 g0"| 17845₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5766, 28919, 6297, 32262, 6497, 6543) ||class="entry q0 g0"| 5766₆ ||class="entry q1 g0"| 28919₁₀ ||class="entry q1 g0"| 6297₆ ||class="entry q0 g0"| 32262₈ ||class="entry q1 g0"| 6497₆ ||class="entry q1 g0"| 6543₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5778, 19679, 9381, 32274, 9545, 9651) ||class="entry q0 g0"| 5778₆ ||class="entry q1 g0"| 19679₁₀ ||class="entry q1 g0"| 9381₆ ||class="entry q0 g0"| 32274₈ ||class="entry q1 g0"| 9545₆ ||class="entry q1 g0"| 9651₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5780, 10943, 17091, 32276, 17193, 17365) ||class="entry q0 g0"| 5780₆ ||class="entry q1 g0"| 10943₁₀ ||class="entry q1 g0"| 17091₆ ||class="entry q0 g0"| 32276₈ ||class="entry q1 g0"| 17193₆ ||class="entry q1 g0"| 17365₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6246, 32519, 6537, 28902, 5777, 6303) ||class="entry q0 g0"| 6246₆ ||class="entry q1 g0"| 32519₁₀ ||class="entry q1 g0"| 6537₆ ||class="entry q0 g0"| 28902₈ ||class="entry q1 g0"| 5777₆ ||class="entry q1 g0"| 6303₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6498, 19919, 10821, 29154, 9305, 11091) ||class="entry q0 g0"| 6498₆ ||class="entry q1 g0"| 19919₁₀ ||class="entry q1 g0"| 10821₆ ||class="entry q0 g0"| 29154₈ ||class="entry q1 g0"| 9305₆ ||class="entry q1 g0"| 11091₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6500, 11183, 19491, 29156, 16953, 19765) ||class="entry q0 g0"| 6500₆ ||class="entry q1 g0"| 11183₁₀ ||class="entry q1 g0"| 19491₆ ||class="entry q0 g0"| 29156₈ ||class="entry q1 g0"| 16953₆ ||class="entry q1 g0"| 19765₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (7938, 19375, 11301, 30594, 8761, 11571) ||class="entry q0 g0"| 7938₆ ||class="entry q1 g0"| 19375₁₀ ||class="entry q1 g0"| 11301₆ ||class="entry q0 g0"| 30594₈ ||class="entry q1 g0"| 8761₆ ||class="entry q1 g0"| 11571₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (7940, 11727, 19011, 30596, 17497, 19285) ||class="entry q0 g0"| 7940₆ ||class="entry q1 g0"| 11727₁₀ ||class="entry q1 g0"| 19011₆ ||class="entry q0 g0"| 30596₈ ||class="entry q1 g0"| 17497₆ ||class="entry q1 g0"| 19285₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (8762, 31091, 9153, 19130, 4325, 8919) ||class="entry q0 g0"| 8762₆ ||class="entry q1 g0"| 31091₁₀ ||class="entry q1 g0"| 9153₆ ||class="entry q0 g0"| 19130₈ ||class="entry q1 g0"| 4325₆ ||class="entry q1 g0"| 8919₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9002, 30611, 11313, 19370, 7685, 11559) ||class="entry q0 g0"| 9002₆ ||class="entry q1 g0"| 30611₁₀ ||class="entry q1 g0"| 11313₆ ||class="entry q0 g0"| 19370₈ ||class="entry q1 g0"| 7685₆ ||class="entry q1 g0"| 11559₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9306, 32531, 9633, 19674, 5765, 9399) ||class="entry q0 g0"| 9306₆ ||class="entry q1 g0"| 32531₁₀ ||class="entry q1 g0"| 9633₆ ||class="entry q0 g0"| 19674₈ ||class="entry q1 g0"| 5765₆ ||class="entry q1 g0"| 9399₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9546, 29171, 10833, 19914, 6245, 11079) ||class="entry q0 g0"| 9546₆ ||class="entry q1 g0"| 29171₁₀ ||class="entry q1 g0"| 10833₆ ||class="entry q0 g0"| 19914₈ ||class="entry q1 g0"| 6245₆ ||class="entry q1 g0"| 11079₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9560, 11195, 28683, 19928, 16941, 28957) ||class="entry q0 g0"| 9560₆ ||class="entry q1 g0"| 11195₁₀ ||class="entry q1 g0"| 28683₆ ||class="entry q0 g0"| 19928₈ ||class="entry q1 g0"| 16941₆ ||class="entry q1 g0"| 28957₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (12402, 27451, 12681, 22770, 685, 12447) ||class="entry q0 g0"| 12402₆ ||class="entry q1 g0"| 27451₁₀ ||class="entry q1 g0"| 12681₆ ||class="entry q0 g0"| 22770₈ ||class="entry q1 g0"| 685₆ ||class="entry q1 g0"| 12447₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (12656, 16275, 25635, 23024, 22021, 25909) ||class="entry q0 g0"| 12656₆ ||class="entry q1 g0"| 16275₁₀ ||class="entry q1 g0"| 25635₆ ||class="entry q0 g0"| 23024₈ ||class="entry q1 g0"| 22021₆ ||class="entry q1 g0"| 25909₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (14082, 25531, 14361, 24450, 2605, 14607) ||class="entry q0 g0"| 14082₆ ||class="entry q1 g0"| 25531₁₀ ||class="entry q1 g0"| 14361₆ ||class="entry q0 g0"| 24450₈ ||class="entry q1 g0"| 2605₆ ||class="entry q1 g0"| 14607₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (14096, 14835, 25155, 24464, 20581, 25429) ||class="entry q0 g0"| 14096₆ ||class="entry q1 g0"| 14835₁₀ ||class="entry q1 g0"| 25155₆ ||class="entry q0 g0"| 24464₈ ||class="entry q1 g0"| 20581₆ ||class="entry q1 g0"| 25429₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (16956, 32533, 17345, 10940, 5763, 17111) ||class="entry q0 g0"| 16956₆ ||class="entry q1 g0"| 32533₁₀ ||class="entry q1 g0"| 17345₆ ||class="entry q0 g0"| 10940₈ ||class="entry q1 g0"| 5763₆ ||class="entry q1 g0"| 17111₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17196, 29173, 19505, 11180, 6243, 19751) ||class="entry q0 g0"| 17196₆ ||class="entry q1 g0"| 29173₁₀ ||class="entry q1 g0"| 19505₆ ||class="entry q0 g0"| 11180₈ ||class="entry q1 g0"| 6243₆ ||class="entry q1 g0"| 19751₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17208, 19933, 28685, 11192, 9291, 28955) ||class="entry q0 g0"| 17208₆ ||class="entry q1 g0"| 19933₁₀ ||class="entry q1 g0"| 28685₆ ||class="entry q0 g0"| 11192₈ ||class="entry q1 g0"| 9291₆ ||class="entry q1 g0"| 28955₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17500, 31093, 17825, 11484, 4323, 17591) ||class="entry q0 g0"| 17500₆ ||class="entry q1 g0"| 31093₁₀ ||class="entry q1 g0"| 17825₆ ||class="entry q0 g0"| 11484₈ ||class="entry q1 g0"| 4323₆ ||class="entry q1 g0"| 17591₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17740, 30613, 19025, 11724, 7683, 19271) ||class="entry q0 g0"| 17740₆ ||class="entry q1 g0"| 30613₁₀ ||class="entry q1 g0"| 19025₆ ||class="entry q0 g0"| 11724₈ ||class="entry q1 g0"| 7683₆ ||class="entry q1 g0"| 19271₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (20596, 27997, 20873, 14580, 1227, 20639) ||class="entry q0 g0"| 20596₆ ||class="entry q1 g0"| 27997₁₀ ||class="entry q1 g0"| 20873₆ ||class="entry q0 g0"| 14580₈ ||class="entry q1 g0"| 1227₆ ||class="entry q1 g0"| 20639₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (20848, 24469, 25157, 14832, 13827, 25427) ||class="entry q0 g0"| 20848₆ ||class="entry q1 g0"| 24469₁₀ ||class="entry q1 g0"| 25157₆ ||class="entry q0 g0"| 14832₈ ||class="entry q1 g0"| 13827₆ ||class="entry q1 g0"| 25427₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (22276, 26077, 22553, 16260, 3147, 22799) ||class="entry q0 g0"| 22276₆ ||class="entry q1 g0"| 26077₁₀ ||class="entry q1 g0"| 22553₆ ||class="entry q0 g0"| 16260₈ ||class="entry q1 g0"| 3147₆ ||class="entry q1 g0"| 22799₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (22288, 23029, 25637, 16272, 12387, 25907) ||class="entry q0 g0"| 22288₆ ||class="entry q1 g0"| 23029₁₀ ||class="entry q1 g0"| 25637₆ ||class="entry q0 g0"| 16272₈ ||class="entry q1 g0"| 12387₆ ||class="entry q1 g0"| 25907₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (700, 16023, 22251, 27196, 22273, 22525) ||class="entry q0 g0"| 700₆ ||class="entry q1 g0"| 16023₁₀ ||class="entry q1 g0"| 22251₁₀ ||class="entry q0 g0"| 27196₈ ||class="entry q1 g0"| 22273₆ ||class="entry q1 g0"| 22525₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (1242, 24215, 14061, 27738, 14081, 14331) ||class="entry q0 g0"| 1242₆ ||class="entry q1 g0"| 24215₁₀ ||class="entry q1 g0"| 14061₁₀ ||class="entry q0 g0"| 27738₈ ||class="entry q1 g0"| 14081₆ ||class="entry q1 g0"| 14331₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (2860, 14823, 24171, 25516, 20593, 24445) ||class="entry q0 g0"| 2860₆ ||class="entry q1 g0"| 14823₁₀ ||class="entry q1 g0"| 24171₁₀ ||class="entry q0 g0"| 25516₈ ||class="entry q1 g0"| 20593₆ ||class="entry q1 g0"| 24445₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (3402, 23015, 15981, 26058, 12401, 16251) ||class="entry q0 g0"| 3402₆ ||class="entry q1 g0"| 23015₁₀ ||class="entry q1 g0"| 15981₁₀ ||class="entry q0 g0"| 26058₈ ||class="entry q1 g0"| 12401₆ ||class="entry q1 g0"| 16251₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (4326, 30359, 7929, 30822, 7937, 8175) ||class="entry q0 g0"| 4326₆ ||class="entry q1 g0"| 30359₁₀ ||class="entry q1 g0"| 7929₁₀ ||class="entry q0 g0"| 30822₈ ||class="entry q1 g0"| 7937₆ ||class="entry q1 g0"| 8175₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (7686, 31079, 8169, 30342, 4337, 7935) ||class="entry q0 g0"| 7686₆ ||class="entry q1 g0"| 31079₁₀ ||class="entry q1 g0"| 8169₁₀ ||class="entry q0 g0"| 30342₈ ||class="entry q1 g0"| 4337₆ ||class="entry q1 g0"| 7935₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (9016, 11739, 30315, 19384, 17485, 30589) ||class="entry q0 g0"| 9016₆ ||class="entry q1 g0"| 11739₁₀ ||class="entry q1 g0"| 30315₁₀ ||class="entry q0 g0"| 19384₈ ||class="entry q1 g0"| 17485₆ ||class="entry q1 g0"| 30589₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (12642, 26075, 15993, 23010, 3149, 16239) ||class="entry q0 g0"| 12642₆ ||class="entry q1 g0"| 26075₁₀ ||class="entry q1 g0"| 15993₁₀ ||class="entry q0 g0"| 23010₈ ||class="entry q1 g0"| 3149₆ ||class="entry q1 g0"| 16239₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (13842, 27995, 14313, 24210, 1229, 14079) ||class="entry q0 g0"| 13842₆ ||class="entry q1 g0"| 27995₁₀ ||class="entry q1 g0"| 14313₁₀ ||class="entry q0 g0"| 24210₈ ||class="entry q1 g0"| 1229₆ ||class="entry q1 g0"| 14079₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (17752, 19389, 30317, 11736, 8747, 30587) ||class="entry q0 g0"| 17752₆ ||class="entry q1 g0"| 19389₁₀ ||class="entry q1 g0"| 30317₁₀ ||class="entry q0 g0"| 11736₈ ||class="entry q1 g0"| 8747₆ ||class="entry q1 g0"| 30587₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (20836, 25533, 24185, 14820, 2603, 24431) ||class="entry q0 g0"| 20836₆ ||class="entry q1 g0"| 25533₁₀ ||class="entry q1 g0"| 24185₁₀ ||class="entry q0 g0"| 14820₈ ||class="entry q1 g0"| 2603₆ ||class="entry q1 g0"| 24431₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 8, 6, 12) ||class="c"| (22036, 27453, 22505, 16020, 683, 22271) ||class="entry q0 g0"| 22036₆ ||class="entry q1 g0"| 27453₁₀ ||class="entry q1 g0"| 22505₁₀ ||class="entry q0 g0"| 16020₈ ||class="entry q1 g0"| 683₆ ||class="entry q1 g0"| 22271₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (510, 489, 27009, 27006, 26751, 26775) ||class="entry q0 g0"| 510₈ ||class="entry q1 g0"| 489₆ ||class="entry q1 g0"| 27009₆ ||class="entry q0 g0"| 27006₁₀ ||class="entry q1 g0"| 26751₁₀ ||class="entry q1 g0"| 26775₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (3870, 3609, 26769, 26526, 26511, 27015) ||class="entry q0 g0"| 3870₈ ||class="entry q1 g0"| 3609₆ ||class="entry q1 g0"| 26769₆ ||class="entry q0 g0"| 26526₁₀ ||class="entry q1 g0"| 26511₁₀ ||class="entry q1 g0"| 27015₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (13110, 12837, 26757, 23478, 23475, 27027) ||class="entry q0 g0"| 13110₈ ||class="entry q1 g0"| 12837₆ ||class="entry q1 g0"| 26757₆ ||class="entry q0 g0"| 23478₁₀ ||class="entry q1 g0"| 23475₁₀ ||class="entry q1 g0"| 27027₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (21846, 21571, 26755, 15830, 15829, 27029) ||class="entry q0 g0"| 21846₈ ||class="entry q1 g0"| 21571₆ ||class="entry q1 g0"| 26755₆ ||class="entry q0 g0"| 15830₁₀ ||class="entry q1 g0"| 15829₁₀ ||class="entry q1 g0"| 27029₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (1950, 1929, 28641, 28446, 28191, 28407) ||class="entry q0 g0"| 1950₈ ||class="entry q1 g0"| 1929₆ ||class="entry q1 g0"| 28641₁₀ ||class="entry q0 g0"| 28446₁₀ ||class="entry q1 g0"| 28191₁₀ ||class="entry q1 g0"| 28407₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (2430, 2169, 28401, 25086, 25071, 28647) ||class="entry q0 g0"| 2430₈ ||class="entry q1 g0"| 2169₆ ||class="entry q1 g0"| 28401₁₀ ||class="entry q0 g0"| 25086₁₀ ||class="entry q1 g0"| 25071₁₀ ||class="entry q1 g0"| 28647₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (5046, 5025, 31689, 31542, 31287, 31455) ||class="entry q0 g0"| 5046₈ ||class="entry q1 g0"| 5025₆ ||class="entry q1 g0"| 31689₁₀ ||class="entry q0 g0"| 31542₁₀ ||class="entry q1 g0"| 31287₁₀ ||class="entry q1 g0"| 31455₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (5590, 5569, 32169, 32086, 31831, 31935) ||class="entry q0 g0"| 5590₈ ||class="entry q1 g0"| 5569₆ ||class="entry q1 g0"| 32169₁₀ ||class="entry q0 g0"| 32086₁₀ ||class="entry q1 g0"| 31831₁₀ ||class="entry q1 g0"| 31935₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (6966, 6705, 31929, 29622, 29607, 32175) ||class="entry q0 g0"| 6966₈ ||class="entry q1 g0"| 6705₆ ||class="entry q1 g0"| 31929₁₀ ||class="entry q0 g0"| 29622₁₀ ||class="entry q1 g0"| 29607₁₀ ||class="entry q1 g0"| 32175₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (7510, 7249, 31449, 30166, 30151, 31695) ||class="entry q0 g0"| 7510₈ ||class="entry q1 g0"| 7249₆ ||class="entry q1 g0"| 31449₁₀ ||class="entry q0 g0"| 30166₁₀ ||class="entry q1 g0"| 30151₁₀ ||class="entry q1 g0"| 31695₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (8574, 8301, 31437, 18942, 18939, 31707) ||class="entry q0 g0"| 8574₈ ||class="entry q1 g0"| 8301₆ ||class="entry q1 g0"| 31437₁₀ ||class="entry q0 g0"| 18942₁₀ ||class="entry q1 g0"| 18939₁₀ ||class="entry q1 g0"| 31707₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (10014, 9741, 31917, 20382, 20379, 32187) ||class="entry q0 g0"| 10014₈ ||class="entry q1 g0"| 9741₆ ||class="entry q1 g0"| 31917₁₀ ||class="entry q0 g0"| 20382₁₀ ||class="entry q1 g0"| 20379₁₀ ||class="entry q1 g0"| 32187₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (13654, 13381, 28389, 24022, 24019, 28659) ||class="entry q0 g0"| 13654₈ ||class="entry q1 g0"| 13381₆ ||class="entry q1 g0"| 28389₁₀ ||class="entry q0 g0"| 24022₁₀ ||class="entry q1 g0"| 24019₁₀ ||class="entry q1 g0"| 28659₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (16766, 16491, 31915, 10750, 10749, 32189) ||class="entry q0 g0"| 16766₈ ||class="entry q1 g0"| 16491₆ ||class="entry q1 g0"| 31915₁₀ ||class="entry q0 g0"| 10750₁₀ ||class="entry q1 g0"| 10749₁₀ ||class="entry q1 g0"| 32189₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (18206, 17931, 31435, 12190, 12189, 31709) ||class="entry q0 g0"| 18206₈ ||class="entry q1 g0"| 17931₆ ||class="entry q1 g0"| 31435₁₀ ||class="entry q0 g0"| 12190₁₀ ||class="entry q1 g0"| 12189₁₀ ||class="entry q1 g0"| 31709₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 10, 10, 10, 12) ||class="c"| (21302, 21027, 28387, 15286, 15285, 28661) ||class="entry q0 g0"| 21302₈ ||class="entry q1 g0"| 21027₆ ||class="entry q1 g0"| 28387₁₀ ||class="entry q0 g0"| 15286₁₀ ||class="entry q1 g0"| 15285₁₀ ||class="entry q1 g0"| 28661₁₂ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (958, 27199, 833, 27454, 937, 599) ||class="entry q0 g0"| 958₈ ||class="entry q1 g0"| 27199₁₀ ||class="entry q1 g0"| 833₄ ||class="entry q0 g0"| 27454₁₀ ||class="entry q1 g0"| 937₆ ||class="entry q1 g0"| 599₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (1502, 27743, 1313, 27998, 1481, 1079) ||class="entry q0 g0"| 1502₈ ||class="entry q1 g0"| 27743₁₀ ||class="entry q1 g0"| 1313₄ ||class="entry q0 g0"| 27998₁₀ ||class="entry q1 g0"| 1481₆ ||class="entry q1 g0"| 1079₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (2878, 25519, 1073, 25534, 2617, 1319) ||class="entry q0 g0"| 2878₈ ||class="entry q1 g0"| 25519₁₀ ||class="entry q1 g0"| 1073₄ ||class="entry q0 g0"| 25534₁₀ ||class="entry q1 g0"| 2617₆ ||class="entry q1 g0"| 1319₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (3422, 26063, 593, 26078, 3161, 839) ||class="entry q0 g0"| 3422₈ ||class="entry q1 g0"| 26063₁₀ ||class="entry q1 g0"| 593₄ ||class="entry q0 g0"| 26078₁₀ ||class="entry q1 g0"| 3161₆ ||class="entry q1 g0"| 839₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (4598, 30839, 4361, 31094, 4577, 4127) ||class="entry q0 g0"| 4598₈ ||class="entry q1 g0"| 30839₁₀ ||class="entry q1 g0"| 4361₄ ||class="entry q0 g0"| 31094₁₀ ||class="entry q1 g0"| 4577₆ ||class="entry q1 g0"| 4127₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (7958, 30599, 4121, 30614, 7697, 4367) ||class="entry q0 g0"| 7958₈ ||class="entry q1 g0"| 30599₁₀ ||class="entry q1 g0"| 4121₄ ||class="entry q0 g0"| 30614₁₀ ||class="entry q1 g0"| 7697₆ ||class="entry q1 g0"| 4367₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (9022, 19387, 4109, 19390, 8749, 4379) ||class="entry q0 g0"| 9022₈ ||class="entry q1 g0"| 19387₁₀ ||class="entry q1 g0"| 4109₄ ||class="entry q0 g0"| 19390₁₀ ||class="entry q1 g0"| 8749₆ ||class="entry q1 g0"| 4379₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (12662, 23027, 581, 23030, 12389, 851) ||class="entry q0 g0"| 12662₈ ||class="entry q1 g0"| 23027₁₀ ||class="entry q1 g0"| 581₄ ||class="entry q0 g0"| 23030₁₀ ||class="entry q1 g0"| 12389₆ ||class="entry q1 g0"| 851₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (14102, 24467, 1061, 24470, 13829, 1331) ||class="entry q0 g0"| 14102₈ ||class="entry q1 g0"| 24467₁₀ ||class="entry q1 g0"| 1061₄ ||class="entry q0 g0"| 24470₁₀ ||class="entry q1 g0"| 13829₆ ||class="entry q1 g0"| 1331₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (17758, 11741, 4107, 11742, 17483, 4381) ||class="entry q0 g0"| 17758₈ ||class="entry q1 g0"| 11741₁₀ ||class="entry q1 g0"| 4107₄ ||class="entry q0 g0"| 11742₁₀ ||class="entry q1 g0"| 17483₆ ||class="entry q1 g0"| 4381₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (20854, 14837, 1059, 14838, 20579, 1333) ||class="entry q0 g0"| 20854₈ ||class="entry q1 g0"| 14837₁₀ ||class="entry q1 g0"| 1059₄ ||class="entry q0 g0"| 14838₁₀ ||class="entry q1 g0"| 20579₆ ||class="entry q1 g0"| 1333₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 4, 10, 6, 6) ||class="c"| (22294, 16277, 579, 16278, 22019, 853) ||class="entry q0 g0"| 22294₈ ||class="entry q1 g0"| 16277₁₀ ||class="entry q1 g0"| 579₄ ||class="entry q0 g0"| 16278₁₀ ||class="entry q1 g0"| 22019₆ ||class="entry q1 g0"| 853₆ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6038, 32279, 5993, 32534, 6017, 5759) ||class="entry q0 g0"| 6038₈ ||class="entry q1 g0"| 32279₁₀ ||class="entry q1 g0"| 5993₈ ||class="entry q0 g0"| 32534₁₀ ||class="entry q1 g0"| 6017₆ ||class="entry q1 g0"| 5759₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6518, 29159, 5753, 29174, 6257, 5999) ||class="entry q0 g0"| 6518₈ ||class="entry q1 g0"| 29159₁₀ ||class="entry q1 g0"| 5753₈ ||class="entry q0 g0"| 29174₁₀ ||class="entry q1 g0"| 6257₆ ||class="entry q1 g0"| 5999₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (9566, 19931, 5741, 19934, 9293, 6011) ||class="entry q0 g0"| 9566₈ ||class="entry q1 g0"| 19931₁₀ ||class="entry q1 g0"| 5741₈ ||class="entry q0 g0"| 19934₁₀ ||class="entry q1 g0"| 9293₆ ||class="entry q1 g0"| 6011₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (17214, 11197, 5739, 11198, 16939, 6013) ||class="entry q0 g0"| 17214₈ ||class="entry q1 g0"| 11197₁₀ ||class="entry q1 g0"| 5739₈ ||class="entry q0 g0"| 11198₁₀ ||class="entry q1 g0"| 16939₆ ||class="entry q1 g0"| 6013₁₀ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (15572, 27005, 15423, 21588, 235, 15657) ||class="entry q0 g0"| 15572₈ ||class="entry q1 g0"| 27005₁₀ ||class="entry q1 g0"| 15423₁₀ ||class="entry q0 g1"| 21588₆ ||class="entry q1 g1"| 235₆ ||class="entry q1 g0"| 15657₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (15812, 26525, 13263, 21828, 3595, 13017) ||class="entry q0 g0"| 15812₈ ||class="entry q1 g0"| 26525₁₀ ||class="entry q1 g0"| 13263₁₀ ||class="entry q0 g1"| 21828₆ ||class="entry q1 g1"| 3595₆ ||class="entry q1 g0"| 13017₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (15824, 23477, 4083, 21840, 12835, 3813) ||class="entry q0 g0"| 15824₈ ||class="entry q1 g0"| 23477₁₀ ||class="entry q1 g0"| 4083₁₀ ||class="entry q0 g1"| 21840₆ ||class="entry q1 g1"| 12835₆ ||class="entry q1 g0"| 3813₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (23218, 27003, 23135, 12850, 237, 23369) ||class="entry q0 g0"| 23218₈ ||class="entry q1 g0"| 27003₁₀ ||class="entry q1 g0"| 23135₁₀ ||class="entry q0 g1"| 12850₆ ||class="entry q1 g1"| 237₆ ||class="entry q1 g0"| 23369₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (23458, 26523, 21935, 13090, 3597, 21689) ||class="entry q0 g0"| 23458₈ ||class="entry q1 g0"| 26523₁₀ ||class="entry q1 g0"| 21935₁₀ ||class="entry q0 g1"| 13090₆ ||class="entry q1 g1"| 3597₆ ||class="entry q1 g0"| 21689₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (23472, 15827, 4085, 13104, 21573, 3811) ||class="entry q0 g0"| 23472₈ ||class="entry q1 g0"| 15827₁₀ ||class="entry q1 g0"| 4085₁₀ ||class="entry q0 g1"| 13104₆ ||class="entry q1 g1"| 21573₆ ||class="entry q1 g0"| 3811₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26254, 26991, 26231, 3598, 249, 26465) ||class="entry q0 g0"| 26254₈ ||class="entry q1 g0"| 26991₁₀ ||class="entry q1 g0"| 26231₁₀ ||class="entry q0 g1"| 3598₆ ||class="entry q1 g1"| 249₆ ||class="entry q1 g0"| 26465₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26506, 23463, 21947, 3850, 12849, 21677) ||class="entry q0 g0"| 26506₈ ||class="entry q1 g0"| 23463₁₀ ||class="entry q1 g0"| 21947₁₀ ||class="entry q0 g1"| 3850₆ ||class="entry q1 g1"| 12849₆ ||class="entry q1 g0"| 21677₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26508, 15815, 13277, 3852, 21585, 13003) ||class="entry q0 g0"| 26508₈ ||class="entry q1 g0"| 15815₁₀ ||class="entry q1 g0"| 13277₁₀ ||class="entry q0 g1"| 3852₆ ||class="entry q1 g1"| 21585₆ ||class="entry q1 g0"| 13003₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26734, 26271, 26471, 238, 3849, 26225) ||class="entry q0 g0"| 26734₈ ||class="entry q1 g0"| 26271₁₀ ||class="entry q1 g0"| 26471₁₀ ||class="entry q0 g1"| 238₆ ||class="entry q1 g1"| 3849₆ ||class="entry q1 g0"| 26225₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26746, 23223, 23387, 250, 13089, 23117) ||class="entry q0 g0"| 26746₈ ||class="entry q1 g0"| 23223₁₀ ||class="entry q1 g0"| 23387₁₀ ||class="entry q0 g1"| 250₆ ||class="entry q1 g1"| 13089₆ ||class="entry q1 g0"| 23117₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 10, 6, 6, 8) ||class="c"| (26748, 15575, 15677, 252, 21825, 15403) ||class="entry q0 g0"| 26748₈ ||class="entry q1 g0"| 15575₁₀ ||class="entry q1 g0"| 15677₁₀ ||class="entry q0 g1"| 252₆ ||class="entry q1 g1"| 21825₆ ||class="entry q1 g0"| 15403₈ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 1, 1, 0, 0) ||class="w"| (6, 8, 2, 4, 8, 4) ||class="c"| (15552, 21845, 3, 21568, 15555, 277) ||class="entry q0 g0"| 15552₆ ||class="entry q1 g1"| 21845₈ ||class="entry q1 g1"| 3₂ ||class="entry q0 g1"| 21568₄ ||class="entry q1 g0"| 15555₈ ||class="entry q1 g0"| 277₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 1, 1, 0, 0) ||class="w"| (6, 8, 2, 4, 8, 4) ||class="c"| (23200, 13107, 5, 12832, 23205, 275) ||class="entry q0 g0"| 23200₆ ||class="entry q1 g1"| 13107₈ ||class="entry q1 g1"| 5₂ ||class="entry q0 g1"| 12832₄ ||class="entry q1 g0"| 23205₈ ||class="entry q1 g0"| 275₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 1, 1, 0, 0) ||class="w"| (6, 8, 2, 4, 8, 4) ||class="c"| (26248, 3855, 17, 3592, 26265, 263) ||class="entry q0 g0"| 26248₆ ||class="entry q1 g1"| 3855₈ ||class="entry q1 g1"| 17₂ ||class="entry q0 g1"| 3592₄ ||class="entry q1 g0"| 26265₈ ||class="entry q1 g0"| 263₄ |- |class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 1, 1, 0, 0) ||class="w"| (6, 8, 2, 4, 8, 4) ||class="c"| (26728, 255, 257, 232, 26985, 23) ||class="entry q0 g0"| 26728₆ ||class="entry q1 g1"| 255₈ ||class="entry q1 g1"| 257₂ ||class="entry q0 g1"| 232₄ ||class="entry q1 g0"| 26985₈ ||class="entry q1 g0"| 23₄ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (32808, 48724, 54678, 33662, 48386, 54976) ||class="entry q2 g1"| 32808₃ ||class="entry q2 g1"| 48724₉ ||class="entry q2 g1"| 54678₉ ||class="entry q2 g1"| 33662₉ ||class="entry q2 g1"| 48386₇ ||class="entry q2 g1"| 54976₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (33800, 54242, 47414, 34654, 53428, 47712) ||class="entry q2 g1"| 33800₃ ||class="entry q2 g1"| 54242₉ ||class="entry q2 g1"| 47414₉ ||class="entry q2 g1"| 34654₉ ||class="entry q2 g1"| 53428₇ ||class="entry q2 g1"| 47712₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (36896, 51146, 44318, 37750, 50332, 44616) ||class="entry q2 g1"| 36896₃ ||class="entry q2 g1"| 51146₉ ||class="entry q2 g1"| 44318₉ ||class="entry q2 g1"| 37750₉ ||class="entry q2 g1"| 50332₇ ||class="entry q2 g1"| 44616₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (37888, 43644, 49598, 38742, 43306, 49896) ||class="entry q2 g1"| 37888₃ ||class="entry q2 g1"| 43644₉ ||class="entry q2 g1"| 49598₉ ||class="entry q2 g1"| 38742₉ ||class="entry q2 g1"| 43306₇ ||class="entry q2 g1"| 49896₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (43048, 38464, 49578, 43902, 38166, 49916) ||class="entry q2 g1"| 43048₅ ||class="entry q2 g1"| 38464₅ ||class="entry q2 g1"| 49578₇ ||class="entry q2 g1"| 43902₁₁ ||class="entry q2 g1"| 38166₇ ||class="entry q2 g1"| 49916₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (48128, 33384, 54658, 48982, 33086, 54996) ||class="entry q2 g1"| 48128₅ ||class="entry q2 g1"| 33384₅ ||class="entry q2 g1"| 54658₇ ||class="entry q2 g1"| 48982₁₁ ||class="entry q2 g1"| 33086₇ ||class="entry q2 g1"| 54996₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (50312, 37472, 44060, 51166, 37174, 44874) ||class="entry q2 g1"| 50312₅ ||class="entry q2 g1"| 37472₅ ||class="entry q2 g1"| 44060₇ ||class="entry q2 g1"| 51166₁₁ ||class="entry q2 g1"| 37174₇ ||class="entry q2 g1"| 44874₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (53408, 34376, 47156, 54262, 34078, 47970) ||class="entry q2 g1"| 53408₅ ||class="entry q2 g1"| 34376₅ ||class="entry q2 g1"| 47156₇ ||class="entry q2 g1"| 54262₁₁ ||class="entry q2 g1"| 34078₇ ||class="entry q2 g1"| 47970₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (32828, 33404, 59818, 33642, 33066, 60156) ||class="entry q2 g1"| 32828₅ ||class="entry q2 g1"| 33404₇ ||class="entry q2 g1"| 59818₉ ||class="entry q2 g1"| 33642₇ ||class="entry q2 g1"| 33066₅ ||class="entry q2 g1"| 60156₁₁ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (34058, 34634, 60572, 34396, 33820, 61386) ||class="entry q2 g1"| 34058₅ ||class="entry q2 g1"| 34634₇ ||class="entry q2 g1"| 60572₉ ||class="entry q2 g1"| 34396₇ ||class="entry q2 g1"| 33820₅ ||class="entry q2 g1"| 61386₁₁ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37154, 37730, 63668, 37492, 36916, 64482) ||class="entry q2 g1"| 37154₅ ||class="entry q2 g1"| 37730₇ ||class="entry q2 g1"| 63668₉ ||class="entry q2 g1"| 37492₇ ||class="entry q2 g1"| 36916₅ ||class="entry q2 g1"| 64482₁₁ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37908, 38484, 64898, 38722, 38146, 65236) ||class="entry q2 g1"| 37908₅ ||class="entry q2 g1"| 38484₇ ||class="entry q2 g1"| 64898₉ ||class="entry q2 g1"| 38722₇ ||class="entry q2 g1"| 38146₅ ||class="entry q2 g1"| 65236₁₁ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32814, 55348, 46064, 33656, 56162, 45222) ||class="entry q2 g1"| 32814₅ ||class="entry q2 g1"| 55348₇ ||class="entry q2 g1"| 46064₉ ||class="entry q2 g1"| 33656₇ ||class="entry q2 g1"| 56162₉ ||class="entry q2 g1"| 45222₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32826, 58396, 36812, 33644, 59210, 35994) ||class="entry q2 g1"| 32826₅ ||class="entry q2 g1"| 58396₇ ||class="entry q2 g1"| 36812₉ ||class="entry q2 g1"| 33644₇ ||class="entry q2 g1"| 59210₉ ||class="entry q2 g1"| 35994₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33068, 35996, 58970, 33402, 36810, 58636) ||class="entry q2 g1"| 33068₅ ||class="entry q2 g1"| 35996₇ ||class="entry q2 g1"| 58970₉ ||class="entry q2 g1"| 33402₇ ||class="entry q2 g1"| 36810₉ ||class="entry q2 g1"| 58636₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33080, 45236, 55910, 33390, 46050, 55600) ||class="entry q2 g1"| 33080₅ ||class="entry q2 g1"| 45236₇ ||class="entry q2 g1"| 55910₉ ||class="entry q2 g1"| 33390₇ ||class="entry q2 g1"| 46050₉ ||class="entry q2 g1"| 55600₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33806, 46466, 57168, 34648, 46804, 56326) ||class="entry q2 g1"| 33806₅ ||class="entry q2 g1"| 46466₇ ||class="entry q2 g1"| 57168₉ ||class="entry q2 g1"| 34648₇ ||class="entry q2 g1"| 46804₉ ||class="entry q2 g1"| 56326₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33818, 35242, 58220, 34636, 35580, 57402) ||class="entry q2 g1"| 33818₅ ||class="entry q2 g1"| 35242₇ ||class="entry q2 g1"| 58220₉ ||class="entry q2 g1"| 34636₇ ||class="entry q2 g1"| 35580₉ ||class="entry q2 g1"| 57402₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34060, 57642, 35578, 34394, 57980, 35244) ||class="entry q2 g1"| 34060₅ ||class="entry q2 g1"| 57642₇ ||class="entry q2 g1"| 35578₉ ||class="entry q2 g1"| 34394₇ ||class="entry q2 g1"| 57980₉ ||class="entry q2 g1"| 35244₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34072, 56578, 46790, 34382, 56916, 46480) ||class="entry q2 g1"| 34072₅ ||class="entry q2 g1"| 56578₇ ||class="entry q2 g1"| 46790₉ ||class="entry q2 g1"| 34382₇ ||class="entry q2 g1"| 56916₉ ||class="entry q2 g1"| 46480₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (36902, 41386, 52088, 37744, 41724, 51246) ||class="entry q2 g1"| 36902₅ ||class="entry q2 g1"| 41386₇ ||class="entry q2 g1"| 52088₉ ||class="entry q2 g1"| 37744₇ ||class="entry q2 g1"| 41724₉ ||class="entry q2 g1"| 51246₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (36914, 40322, 63300, 37732, 40660, 62482) ||class="entry q2 g1"| 36914₅ ||class="entry q2 g1"| 40322₇ ||class="entry q2 g1"| 63300₉ ||class="entry q2 g1"| 37732₇ ||class="entry q2 g1"| 40660₉ ||class="entry q2 g1"| 62482₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37156, 62722, 40658, 37490, 63060, 40324) ||class="entry q2 g1"| 37156₅ ||class="entry q2 g1"| 62722₇ ||class="entry q2 g1"| 40658₉ ||class="entry q2 g1"| 37490₇ ||class="entry q2 g1"| 63060₉ ||class="entry q2 g1"| 40324₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37168, 51498, 41710, 37478, 51836, 41400) ||class="entry q2 g1"| 37168₅ ||class="entry q2 g1"| 51498₇ ||class="entry q2 g1"| 41710₉ ||class="entry q2 g1"| 37478₇ ||class="entry q2 g1"| 51836₉ ||class="entry q2 g1"| 41400₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37894, 52252, 42968, 38736, 53066, 42126) ||class="entry q2 g1"| 37894₅ ||class="entry q2 g1"| 52252₇ ||class="entry q2 g1"| 42968₉ ||class="entry q2 g1"| 38736₇ ||class="entry q2 g1"| 53066₉ ||class="entry q2 g1"| 42126₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37906, 61492, 39908, 38724, 62306, 39090) ||class="entry q2 g1"| 37906₅ ||class="entry q2 g1"| 61492₇ ||class="entry q2 g1"| 39908₉ ||class="entry q2 g1"| 38724₇ ||class="entry q2 g1"| 62306₉ ||class="entry q2 g1"| 39090₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (38148, 39092, 62066, 38482, 39906, 61732) ||class="entry q2 g1"| 38148₅ ||class="entry q2 g1"| 39092₇ ||class="entry q2 g1"| 62066₉ ||class="entry q2 g1"| 38482₇ ||class="entry q2 g1"| 39906₉ ||class="entry q2 g1"| 61732₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (38160, 42140, 52814, 38470, 42954, 52504) ||class="entry q2 g1"| 38160₅ ||class="entry q2 g1"| 42140₇ ||class="entry q2 g1"| 52814₉ ||class="entry q2 g1"| 38470₇ ||class="entry q2 g1"| 42954₉ ||class="entry q2 g1"| 52504₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (34984, 47044, 53990, 35838, 46226, 53680) ||class="entry q2 g1"| 34984₅ ||class="entry q2 g1"| 47044₉ ||class="entry q2 g1"| 53990₉ ||class="entry q2 g1"| 35838₁₁ ||class="entry q2 g1"| 46226₇ ||class="entry q2 g1"| 53680₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (35976, 55922, 48710, 36830, 55588, 48400) ||class="entry q2 g1"| 35976₅ ||class="entry q2 g1"| 55922₉ ||class="entry q2 g1"| 48710₉ ||class="entry q2 g1"| 36830₁₁ ||class="entry q2 g1"| 55588₇ ||class="entry q2 g1"| 48400₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (39072, 52826, 43630, 39926, 52492, 43320) ||class="entry q2 g1"| 39072₅ ||class="entry q2 g1"| 52826₉ ||class="entry q2 g1"| 43630₉ ||class="entry q2 g1"| 39926₁₁ ||class="entry q2 g1"| 52492₇ ||class="entry q2 g1"| 43320₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (40064, 41964, 50894, 40918, 41146, 50584) ||class="entry q2 g1"| 40064₅ ||class="entry q2 g1"| 41964₉ ||class="entry q2 g1"| 50894₉ ||class="entry q2 g1"| 40918₁₁ ||class="entry q2 g1"| 41146₇ ||class="entry q2 g1"| 50584₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (41128, 40912, 50906, 41982, 40070, 50572) ||class="entry q2 g1"| 41128₅ ||class="entry q2 g1"| 40912₉ ||class="entry q2 g1"| 50906₉ ||class="entry q2 g1"| 41982₁₁ ||class="entry q2 g1"| 40070₇ ||class="entry q2 g1"| 50572₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (42120, 62054, 43642, 42974, 61744, 43308) ||class="entry q2 g1"| 42120₅ ||class="entry q2 g1"| 62054₉ ||class="entry q2 g1"| 43642₉ ||class="entry q2 g1"| 42974₁₁ ||class="entry q2 g1"| 61744₇ ||class="entry q2 g1"| 43308₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (45216, 58958, 48722, 46070, 58648, 48388) ||class="entry q2 g1"| 45216₅ ||class="entry q2 g1"| 58958₉ ||class="entry q2 g1"| 48722₉ ||class="entry q2 g1"| 46070₁₁ ||class="entry q2 g1"| 58648₇ ||class="entry q2 g1"| 48388₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (46208, 35832, 54002, 47062, 34990, 53668) ||class="entry q2 g1"| 46208₅ ||class="entry q2 g1"| 35832₉ ||class="entry q2 g1"| 54002₉ ||class="entry q2 g1"| 47062₁₁ ||class="entry q2 g1"| 34990₇ ||class="entry q2 g1"| 53668₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (51240, 63046, 51148, 52094, 62736, 50330) ||class="entry q2 g1"| 51240₅ ||class="entry q2 g1"| 63046₉ ||class="entry q2 g1"| 51148₉ ||class="entry q2 g1"| 52094₁₁ ||class="entry q2 g1"| 62736₇ ||class="entry q2 g1"| 50330₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (52232, 39920, 43884, 53086, 39078, 43066) ||class="entry q2 g1"| 52232₅ ||class="entry q2 g1"| 39920₉ ||class="entry q2 g1"| 43884₉ ||class="entry q2 g1"| 53086₁₁ ||class="entry q2 g1"| 39078₇ ||class="entry q2 g1"| 43066₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (55328, 36824, 48964, 56182, 35982, 48146) ||class="entry q2 g1"| 55328₅ ||class="entry q2 g1"| 36824₉ ||class="entry q2 g1"| 48964₉ ||class="entry q2 g1"| 56182₁₁ ||class="entry q2 g1"| 35982₇ ||class="entry q2 g1"| 48146₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (56320, 57966, 54244, 57174, 57656, 53426) ||class="entry q2 g1"| 56320₅ ||class="entry q2 g1"| 57966₉ ||class="entry q2 g1"| 54244₉ ||class="entry q2 g1"| 57174₁₁ ||class="entry q2 g1"| 57656₇ ||class="entry q2 g1"| 53426₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (57384, 56914, 54256, 58238, 56580, 53414) ||class="entry q2 g1"| 57384₅ ||class="entry q2 g1"| 56914₉ ||class="entry q2 g1"| 54256₉ ||class="entry q2 g1"| 58238₁₁ ||class="entry q2 g1"| 56580₇ ||class="entry q2 g1"| 53414₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (58376, 46052, 48976, 59230, 45234, 48134) ||class="entry q2 g1"| 58376₅ ||class="entry q2 g1"| 46052₉ ||class="entry q2 g1"| 48976₉ ||class="entry q2 g1"| 59230₁₁ ||class="entry q2 g1"| 45234₇ ||class="entry q2 g1"| 48134₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (61472, 42956, 43896, 62326, 42138, 43054) ||class="entry q2 g1"| 61472₅ ||class="entry q2 g1"| 42956₉ ||class="entry q2 g1"| 43896₉ ||class="entry q2 g1"| 62326₁₁ ||class="entry q2 g1"| 42138₇ ||class="entry q2 g1"| 43054₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (62464, 51834, 51160, 63318, 51500, 50318) ||class="entry q2 g1"| 62464₅ ||class="entry q2 g1"| 51834₉ ||class="entry q2 g1"| 51160₉ ||class="entry q2 g1"| 63318₁₁ ||class="entry q2 g1"| 51500₇ ||class="entry q2 g1"| 50318₇ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (44040, 64502, 44298, 44894, 63648, 44636) ||class="entry q2 g1"| 44040₅ ||class="entry q2 g1"| 64502₁₃ ||class="entry q2 g1"| 44298₇ ||class="entry q2 g1"| 44894₁₁ ||class="entry q2 g1"| 63648₇ ||class="entry q2 g1"| 44636₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (47136, 61406, 47394, 47990, 60552, 47732) ||class="entry q2 g1"| 47136₅ ||class="entry q2 g1"| 61406₁₃ ||class="entry q2 g1"| 47394₇ ||class="entry q2 g1"| 47990₁₁ ||class="entry q2 g1"| 60552₇ ||class="entry q2 g1"| 47732₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (49320, 65494, 49340, 50174, 64640, 50154) ||class="entry q2 g1"| 49320₅ ||class="entry q2 g1"| 65494₁₃ ||class="entry q2 g1"| 49340₇ ||class="entry q2 g1"| 50174₁₁ ||class="entry q2 g1"| 64640₇ ||class="entry q2 g1"| 50154₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (54400, 60414, 54420, 55254, 59560, 55234) ||class="entry q2 g1"| 54400₅ ||class="entry q2 g1"| 60414₁₃ ||class="entry q2 g1"| 54420₇ ||class="entry q2 g1"| 55254₁₁ ||class="entry q2 g1"| 59560₇ ||class="entry q2 g1"| 55234₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (35560, 56338, 47142, 35262, 57156, 47984) ||class="entry q2 g1"| 35560₇ ||class="entry q2 g1"| 56338₇ ||class="entry q2 g1"| 47142₇ ||class="entry q2 g1"| 35262₉ ||class="entry q2 g1"| 57156₉ ||class="entry q2 g1"| 47984₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36552, 45476, 54406, 36254, 45810, 55248) ||class="entry q2 g1"| 36552₇ ||class="entry q2 g1"| 45476₇ ||class="entry q2 g1"| 54406₇ ||class="entry q2 g1"| 36254₉ ||class="entry q2 g1"| 45810₉ ||class="entry q2 g1"| 55248₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (39648, 42380, 49326, 39350, 42714, 50168) ||class="entry q2 g1"| 39648₇ ||class="entry q2 g1"| 42380₇ ||class="entry q2 g1"| 49326₇ ||class="entry q2 g1"| 39350₉ ||class="entry q2 g1"| 42714₉ ||class="entry q2 g1"| 50168₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (40640, 51258, 44046, 40342, 52076, 44888) ||class="entry q2 g1"| 40640₇ ||class="entry q2 g1"| 51258₇ ||class="entry q2 g1"| 44046₇ ||class="entry q2 g1"| 40342₉ ||class="entry q2 g1"| 52076₉ ||class="entry q2 g1"| 44888₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (41704, 62470, 44058, 41406, 63312, 44876) ||class="entry q2 g1"| 41704₇ ||class="entry q2 g1"| 62470₇ ||class="entry q2 g1"| 44058₇ ||class="entry q2 g1"| 41406₉ ||class="entry q2 g1"| 63312₉ ||class="entry q2 g1"| 44876₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (42696, 39344, 49338, 42398, 39654, 50156) ||class="entry q2 g1"| 42696₇ ||class="entry q2 g1"| 39344₇ ||class="entry q2 g1"| 49338₇ ||class="entry q2 g1"| 42398₉ ||class="entry q2 g1"| 39654₉ ||class="entry q2 g1"| 50156₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45792, 36248, 54418, 45494, 36558, 55236) ||class="entry q2 g1"| 45792₇ ||class="entry q2 g1"| 36248₇ ||class="entry q2 g1"| 54418₇ ||class="entry q2 g1"| 45494₉ ||class="entry q2 g1"| 36558₉ ||class="entry q2 g1"| 55236₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (46784, 57390, 47154, 46486, 58232, 47972) ||class="entry q2 g1"| 46784₇ ||class="entry q2 g1"| 57390₇ ||class="entry q2 g1"| 47154₇ ||class="entry q2 g1"| 46486₉ ||class="entry q2 g1"| 58232₉ ||class="entry q2 g1"| 47972₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (51816, 40336, 44300, 51518, 40646, 44634) ||class="entry q2 g1"| 51816₇ ||class="entry q2 g1"| 40336₇ ||class="entry q2 g1"| 44300₇ ||class="entry q2 g1"| 51518₉ ||class="entry q2 g1"| 40646₉ ||class="entry q2 g1"| 44634₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (52808, 61478, 49580, 52510, 62320, 49914) ||class="entry q2 g1"| 52808₇ ||class="entry q2 g1"| 61478₇ ||class="entry q2 g1"| 49580₇ ||class="entry q2 g1"| 52510₉ ||class="entry q2 g1"| 62320₉ ||class="entry q2 g1"| 49914₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (55904, 58382, 54660, 55606, 59224, 54994) ||class="entry q2 g1"| 55904₇ ||class="entry q2 g1"| 58382₇ ||class="entry q2 g1"| 54660₇ ||class="entry q2 g1"| 55606₉ ||class="entry q2 g1"| 59224₉ ||class="entry q2 g1"| 54994₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (56896, 35256, 47396, 56598, 35566, 47730) ||class="entry q2 g1"| 56896₇ ||class="entry q2 g1"| 35256₇ ||class="entry q2 g1"| 47396₇ ||class="entry q2 g1"| 56598₉ ||class="entry q2 g1"| 35566₉ ||class="entry q2 g1"| 47730₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (57960, 46468, 47408, 57662, 46802, 47718) ||class="entry q2 g1"| 57960₇ ||class="entry q2 g1"| 46468₇ ||class="entry q2 g1"| 47408₇ ||class="entry q2 g1"| 57662₉ ||class="entry q2 g1"| 46802₉ ||class="entry q2 g1"| 47718₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (58952, 55346, 54672, 58654, 56164, 54982) ||class="entry q2 g1"| 58952₇ ||class="entry q2 g1"| 55346₇ ||class="entry q2 g1"| 54672₇ ||class="entry q2 g1"| 58654₉ ||class="entry q2 g1"| 56164₉ ||class="entry q2 g1"| 54982₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (62048, 52250, 49592, 61750, 53068, 49902) ||class="entry q2 g1"| 62048₇ ||class="entry q2 g1"| 52250₇ ||class="entry q2 g1"| 49592₇ ||class="entry q2 g1"| 61750₉ ||class="entry q2 g1"| 53068₉ ||class="entry q2 g1"| 49902₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (63040, 41388, 44312, 62742, 41722, 44622) ||class="entry q2 g1"| 63040₇ ||class="entry q2 g1"| 41388₇ ||class="entry q2 g1"| 44312₇ ||class="entry q2 g1"| 62742₉ ||class="entry q2 g1"| 41722₉ ||class="entry q2 g1"| 44622₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (43068, 43624, 64918, 43882, 43326, 65216) ||class="entry q2 g1"| 43068₇ ||class="entry q2 g1"| 43624₇ ||class="entry q2 g1"| 64918₁₁ ||class="entry q2 g1"| 43882₉ ||class="entry q2 g1"| 43326₉ ||class="entry q2 g1"| 65216₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (48148, 48704, 59838, 48962, 48406, 60136) ||class="entry q2 g1"| 48148₇ ||class="entry q2 g1"| 48704₇ ||class="entry q2 g1"| 59838₁₁ ||class="entry q2 g1"| 48962₉ ||class="entry q2 g1"| 48406₉ ||class="entry q2 g1"| 60136₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (50570, 50888, 63926, 50908, 50590, 64224) ||class="entry q2 g1"| 50570₇ ||class="entry q2 g1"| 50888₇ ||class="entry q2 g1"| 63926₁₁ ||class="entry q2 g1"| 50908₉ ||class="entry q2 g1"| 50590₉ ||class="entry q2 g1"| 64224₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (53666, 53984, 60830, 54004, 53686, 61128) ||class="entry q2 g1"| 53666₇ ||class="entry q2 g1"| 53984₇ ||class="entry q2 g1"| 60830₁₁ ||class="entry q2 g1"| 54004₉ ||class="entry q2 g1"| 53686₉ ||class="entry q2 g1"| 61128₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35002, 60812, 35004, 35820, 61146, 35818) ||class="entry q2 g1"| 35002₇ ||class="entry q2 g1"| 60812₉ ||class="entry q2 g1"| 35004₇ ||class="entry q2 g1"| 35820₉ ||class="entry q2 g1"| 61146₁₁ ||class="entry q2 g1"| 35818₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (36236, 59578, 36234, 36570, 60396, 36572) ||class="entry q2 g1"| 36236₇ ||class="entry q2 g1"| 59578₉ ||class="entry q2 g1"| 36234₇ ||class="entry q2 g1"| 36570₉ ||class="entry q2 g1"| 60396₁₁ ||class="entry q2 g1"| 36572₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39332, 64658, 39330, 39666, 65476, 39668) ||class="entry q2 g1"| 39332₇ ||class="entry q2 g1"| 64658₉ ||class="entry q2 g1"| 39330₇ ||class="entry q2 g1"| 39666₉ ||class="entry q2 g1"| 65476₁₁ ||class="entry q2 g1"| 39668₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (40082, 63908, 40084, 40900, 64242, 40898) ||class="entry q2 g1"| 40082₇ ||class="entry q2 g1"| 63908₉ ||class="entry q2 g1"| 40084₇ ||class="entry q2 g1"| 40900₉ ||class="entry q2 g1"| 64242₁₁ ||class="entry q2 g1"| 40898₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41134, 63920, 41148, 41976, 64230, 41962) ||class="entry q2 g1"| 41134₇ ||class="entry q2 g1"| 63920₉ ||class="entry q2 g1"| 41148₇ ||class="entry q2 g1"| 41976₉ ||class="entry q2 g1"| 64230₁₁ ||class="entry q2 g1"| 41962₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42392, 64646, 42378, 42702, 65488, 42716) ||class="entry q2 g1"| 42392₇ ||class="entry q2 g1"| 64646₉ ||class="entry q2 g1"| 42378₇ ||class="entry q2 g1"| 42702₉ ||class="entry q2 g1"| 65488₁₁ ||class="entry q2 g1"| 42716₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (45488, 59566, 45474, 45798, 60408, 45812) ||class="entry q2 g1"| 45488₇ ||class="entry q2 g1"| 59566₉ ||class="entry q2 g1"| 45474₇ ||class="entry q2 g1"| 45798₉ ||class="entry q2 g1"| 60408₁₁ ||class="entry q2 g1"| 45812₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (46214, 60824, 46228, 47056, 61134, 47042) ||class="entry q2 g1"| 46214₇ ||class="entry q2 g1"| 60824₉ ||class="entry q2 g1"| 46228₇ ||class="entry q2 g1"| 47056₉ ||class="entry q2 g1"| 61134₁₁ ||class="entry q2 g1"| 47042₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (51512, 63654, 51260, 51822, 64496, 52074) ||class="entry q2 g1"| 51512₇ ||class="entry q2 g1"| 63654₉ ||class="entry q2 g1"| 51260₇ ||class="entry q2 g1"| 51822₉ ||class="entry q2 g1"| 64496₁₁ ||class="entry q2 g1"| 52074₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (52238, 64912, 52490, 53080, 65222, 52828) ||class="entry q2 g1"| 52238₇ ||class="entry q2 g1"| 64912₉ ||class="entry q2 g1"| 52490₇ ||class="entry q2 g1"| 53080₉ ||class="entry q2 g1"| 65222₁₁ ||class="entry q2 g1"| 52828₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (55334, 59832, 55586, 56176, 60142, 55924) ||class="entry q2 g1"| 55334₇ ||class="entry q2 g1"| 59832₉ ||class="entry q2 g1"| 55586₇ ||class="entry q2 g1"| 56176₉ ||class="entry q2 g1"| 60142₁₁ ||class="entry q2 g1"| 55924₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (56592, 60558, 56340, 56902, 61400, 57154) ||class="entry q2 g1"| 56592₇ ||class="entry q2 g1"| 60558₉ ||class="entry q2 g1"| 56340₇ ||class="entry q2 g1"| 56902₉ ||class="entry q2 g1"| 61400₁₁ ||class="entry q2 g1"| 57154₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (57644, 60570, 57404, 57978, 61388, 58218) ||class="entry q2 g1"| 57644₇ ||class="entry q2 g1"| 60570₉ ||class="entry q2 g1"| 57404₇ ||class="entry q2 g1"| 57978₉ ||class="entry q2 g1"| 61388₁₁ ||class="entry q2 g1"| 58218₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (58394, 59820, 58634, 59212, 60154, 58972) ||class="entry q2 g1"| 58394₇ ||class="entry q2 g1"| 59820₉ ||class="entry q2 g1"| 58634₇ ||class="entry q2 g1"| 59212₉ ||class="entry q2 g1"| 60154₁₁ ||class="entry q2 g1"| 58972₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (61490, 64900, 61730, 62308, 65234, 62068) ||class="entry q2 g1"| 61490₇ ||class="entry q2 g1"| 64900₉ ||class="entry q2 g1"| 61730₇ ||class="entry q2 g1"| 62308₉ ||class="entry q2 g1"| 65234₁₁ ||class="entry q2 g1"| 62068₉ |- |class="f"| 854 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (62724, 63666, 62484, 63058, 64484, 63298) ||class="entry q2 g1"| 62724₇ ||class="entry q2 g1"| 63666₉ ||class="entry q2 g1"| 62484₇ ||class="entry q2 g1"| 63058₉ ||class="entry q2 g1"| 64484₁₁ ||class="entry q2 g1"| 63298₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (32960, 55101, 54657, 60182, 48637, 55233) ||class="entry q2 g1"| 32960₃ ||class="entry q3 g1"| 55101₁₁ ||class="entry q3 g1"| 54657₇ ||class="entry q2 g1"| 60182₉ ||class="entry q3 g1"| 48637₁₃ ||class="entry q3 g1"| 55233₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (35328, 46459, 47153, 57814, 57275, 47729) ||class="entry q2 g1"| 35328₃ ||class="entry q3 g1"| 46459₁₁ ||class="entry q3 g1"| 47153₇ ||class="entry q2 g1"| 57814₉ ||class="entry q3 g1"| 57275₁₃ ||class="entry q3 g1"| 47729₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (41472, 40303, 44045, 51670, 63407, 44621) ||class="entry q2 g1"| 41472₃ ||class="entry q3 g1"| 40303₁₁ ||class="entry q3 g1"| 44045₇ ||class="entry q2 g1"| 51670₉ ||class="entry q3 g1"| 63407₁₃ ||class="entry q3 g1"| 44621₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (49216, 38591, 49323, 43926, 64639, 49899) ||class="entry q2 g1"| 49216₃ ||class="entry q3 g1"| 38591₁₁ ||class="entry q3 g1"| 49323₇ ||class="entry q2 g1"| 43926₉ ||class="entry q3 g1"| 64639₁₃ ||class="entry q3 g1"| 49899₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (33408, 48363, 48961, 59734, 54827, 48385) ||class="entry q2 g1"| 33408₃ ||class="entry q3 g1"| 48363₁₁ ||class="entry q3 g1"| 48961₉ ||class="entry q2 g1"| 59734₉ ||class="entry q3 g1"| 54827₉ ||class="entry q3 g1"| 48385₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (34880, 57005, 54001, 58262, 46189, 53425) ||class="entry q2 g1"| 34880₃ ||class="entry q3 g1"| 57005₁₁ ||class="entry q3 g1"| 54001₉ ||class="entry q2 g1"| 58262₉ ||class="entry q3 g1"| 46189₉ ||class="entry q3 g1"| 53425₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (41024, 63161, 50893, 52118, 40057, 50317) ||class="entry q2 g1"| 41024₃ ||class="entry q3 g1"| 63161₁₁ ||class="entry q3 g1"| 50893₉ ||class="entry q2 g1"| 52118₉ ||class="entry q3 g1"| 40057₉ ||class="entry q3 g1"| 50317₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (49664, 64873, 43627, 43478, 38825, 43051) ||class="entry q2 g1"| 49664₃ ||class="entry q3 g1"| 64873₁₁ ||class="entry q3 g1"| 43627₉ ||class="entry q2 g1"| 43478₉ ||class="entry q3 g1"| 38825₉ ||class="entry q3 g1"| 43051₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (33428, 32963, 33661, 59714, 59907, 33085) ||class="entry q2 g1"| 33428₅ ||class="entry q3 g1"| 32963₅ ||class="entry q3 g1"| 33661₉ ||class="entry q2 g1"| 59714₇ ||class="entry q3 g1"| 59907₇ ||class="entry q3 g1"| 33085₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (35138, 35333, 34651, 58004, 57541, 34075) ||class="entry q2 g1"| 35138₅ ||class="entry q3 g1"| 35333₅ ||class="entry q3 g1"| 34651₉ ||class="entry q2 g1"| 58004₇ ||class="entry q3 g1"| 57541₇ ||class="entry q3 g1"| 34075₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (41282, 41489, 37735, 51860, 51409, 37159) ||class="entry q2 g1"| 41282₅ ||class="entry q3 g1"| 41489₅ ||class="entry q3 g1"| 37735₉ ||class="entry q2 g1"| 51860₇ ||class="entry q3 g1"| 51409₇ ||class="entry q3 g1"| 37159₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (49684, 49473, 38487, 43458, 43905, 37911) ||class="entry q2 g1"| 49684₅ ||class="entry q3 g1"| 49473₅ ||class="entry q3 g1"| 38487₉ ||class="entry q2 g1"| 43458₇ ||class="entry q3 g1"| 43905₇ ||class="entry q3 g1"| 37911₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (33218, 33685, 32811, 59924, 59733, 33387) ||class="entry q2 g1"| 33218₅ ||class="entry q3 g1"| 33685₇ ||class="entry q3 g1"| 32811₅ ||class="entry q2 g1"| 59924₇ ||class="entry q3 g1"| 59733₉ ||class="entry q3 g1"| 33387₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (35348, 35155, 33805, 57794, 58259, 34381) ||class="entry q2 g1"| 35348₅ ||class="entry q3 g1"| 35155₇ ||class="entry q3 g1"| 33805₅ ||class="entry q2 g1"| 57794₇ ||class="entry q3 g1"| 58259₉ ||class="entry q3 g1"| 34381₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (41492, 41287, 36913, 51650, 52103, 37489) ||class="entry q2 g1"| 41492₅ ||class="entry q3 g1"| 41287₇ ||class="entry q3 g1"| 36913₅ ||class="entry q2 g1"| 51650₇ ||class="entry q3 g1"| 52103₉ ||class="entry q3 g1"| 37489₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (49474, 49687, 38145, 43668, 43223, 38721) ||class="entry q2 g1"| 49474₅ ||class="entry q3 g1"| 49687₇ ||class="entry q3 g1"| 38145₅ ||class="entry q2 g1"| 43668₇ ||class="entry q3 g1"| 43223₉ ||class="entry q3 g1"| 38721₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (33668, 36387, 35981, 59474, 58595, 36557) ||class="entry q2 g1"| 33668₅ ||class="entry q3 g1"| 36387₇ ||class="entry q3 g1"| 35981₇ ||class="entry q2 g1"| 59474₇ ||class="entry q3 g1"| 58595₉ ||class="entry q3 g1"| 36557₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (33680, 45579, 45233, 59462, 55499, 45809) ||class="entry q2 g1"| 33680₅ ||class="entry q3 g1"| 45579₇ ||class="entry q3 g1"| 45233₇ ||class="entry q2 g1"| 59462₇ ||class="entry q3 g1"| 55499₉ ||class="entry q3 g1"| 45809₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34898, 34021, 34987, 58244, 60965, 35563) ||class="entry q2 g1"| 34898₅ ||class="entry q3 g1"| 34021₇ ||class="entry q3 g1"| 34987₇ ||class="entry q2 g1"| 58244₇ ||class="entry q3 g1"| 60965₉ ||class="entry q3 g1"| 35563₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (35152, 53325, 56577, 57990, 47757, 57153) ||class="entry q2 g1"| 35152₅ ||class="entry q3 g1"| 53325₇ ||class="entry q3 g1"| 56577₇ ||class="entry q2 g1"| 57990₇ ||class="entry q3 g1"| 47757₉ ||class="entry q3 g1"| 57153₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41030, 37081, 41131, 52112, 64025, 41707) ||class="entry q2 g1"| 41030₅ ||class="entry q3 g1"| 37081₇ ||class="entry q3 g1"| 41131₇ ||class="entry q2 g1"| 52112₇ ||class="entry q3 g1"| 64025₉ ||class="entry q3 g1"| 41707₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41284, 50289, 62721, 51858, 44721, 63297) ||class="entry q2 g1"| 41284₅ ||class="entry q3 g1"| 50289₇ ||class="entry q3 g1"| 62721₇ ||class="entry q2 g1"| 51858₇ ||class="entry q3 g1"| 44721₉ ||class="entry q3 g1"| 63297₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49670, 39689, 52237, 43472, 61897, 52813) ||class="entry q2 g1"| 49670₅ ||class="entry q3 g1"| 39689₇ ||class="entry q3 g1"| 52237₇ ||class="entry q2 g1"| 43472₇ ||class="entry q3 g1"| 61897₉ ||class="entry q3 g1"| 52813₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49682, 42785, 61489, 43460, 52705, 62065) ||class="entry q2 g1"| 49682₅ ||class="entry q3 g1"| 42785₇ ||class="entry q3 g1"| 61489₇ ||class="entry q2 g1"| 43460₇ ||class="entry q3 g1"| 52705₉ ||class="entry q3 g1"| 62065₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (33666, 59459, 60139, 59476, 33411, 59563) ||class="entry q2 g1"| 33666₅ ||class="entry q3 g1"| 59459₇ ||class="entry q3 g1"| 60139₁₁ ||class="entry q2 g1"| 59476₇ ||class="entry q3 g1"| 33411₅ ||class="entry q3 g1"| 59563₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (34900, 57989, 61133, 58242, 34885, 60557) ||class="entry q2 g1"| 34900₅ ||class="entry q3 g1"| 57989₇ ||class="entry q3 g1"| 61133₁₁ ||class="entry q2 g1"| 58242₇ ||class="entry q3 g1"| 34885₅ ||class="entry q3 g1"| 60557₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (41044, 51857, 64241, 52098, 41041, 63665) ||class="entry q2 g1"| 41044₅ ||class="entry q3 g1"| 51857₇ ||class="entry q3 g1"| 64241₁₁ ||class="entry q2 g1"| 52098₇ ||class="entry q3 g1"| 41041₅ ||class="entry q3 g1"| 63665₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (49922, 43457, 65473, 43220, 49921, 64897) ||class="entry q2 g1"| 49922₅ ||class="entry q3 g1"| 43457₇ ||class="entry q3 g1"| 65473₁₁ ||class="entry q2 g1"| 43220₇ ||class="entry q3 g1"| 49921₅ ||class="entry q3 g1"| 64897₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (34016, 47755, 47393, 61238, 53323, 47969) ||class="entry q2 g1"| 34016₅ ||class="entry q3 g1"| 47755₉ ||class="entry q3 g1"| 47393₇ ||class="entry q2 g1"| 61238₁₁ ||class="entry q3 g1"| 53323₇ ||class="entry q3 g1"| 47969₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (36384, 55501, 54417, 58870, 45581, 54993) ||class="entry q2 g1"| 36384₅ ||class="entry q3 g1"| 55501₉ ||class="entry q3 g1"| 54417₇ ||class="entry q2 g1"| 58870₁₁ ||class="entry q3 g1"| 45581₇ ||class="entry q3 g1"| 54993₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (37064, 44707, 44297, 64286, 50275, 44873) ||class="entry q2 g1"| 37064₅ ||class="entry q3 g1"| 44707₉ ||class="entry q3 g1"| 44297₇ ||class="entry q2 g1"| 64286₁₁ ||class="entry q3 g1"| 50275₇ ||class="entry q3 g1"| 44873₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (39432, 52453, 49337, 61918, 42533, 49913) ||class="entry q2 g1"| 39432₅ ||class="entry q3 g1"| 52453₉ ||class="entry q3 g1"| 49337₇ ||class="entry q2 g1"| 61918₁₁ ||class="entry q3 g1"| 42533₇ ||class="entry q3 g1"| 49913₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (42528, 61657, 49325, 52726, 39449, 49901) ||class="entry q2 g1"| 42528₅ ||class="entry q3 g1"| 61657₉ ||class="entry q3 g1"| 49325₇ ||class="entry q2 g1"| 52726₁₁ ||class="entry q3 g1"| 39449₇ ||class="entry q3 g1"| 49901₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (45576, 58609, 54405, 55774, 36401, 54981) ||class="entry q2 g1"| 45576₅ ||class="entry q3 g1"| 58609₉ ||class="entry q3 g1"| 54405₇ ||class="entry q2 g1"| 55774₁₁ ||class="entry q3 g1"| 36401₇ ||class="entry q3 g1"| 54981₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (50272, 64265, 44043, 44982, 37321, 44619) ||class="entry q2 g1"| 50272₅ ||class="entry q3 g1"| 64265₉ ||class="entry q3 g1"| 44043₇ ||class="entry q2 g1"| 44982₁₁ ||class="entry q3 g1"| 37321₇ ||class="entry q3 g1"| 44619₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (53320, 61217, 47139, 48030, 34273, 47715) ||class="entry q2 g1"| 53320₅ ||class="entry q3 g1"| 61217₉ ||class="entry q3 g1"| 47139₇ ||class="entry q2 g1"| 48030₁₁ ||class="entry q3 g1"| 34273₇ ||class="entry q3 g1"| 47715₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33414, 55947, 55591, 59728, 45131, 56167) ||class="entry q2 g1"| 33414₅ ||class="entry q3 g1"| 55947₉ ||class="entry q3 g1"| 55591₉ ||class="entry q2 g1"| 59728₇ ||class="entry q3 g1"| 45131₇ ||class="entry q3 g1"| 56167₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33426, 59043, 58651, 59716, 35939, 59227) ||class="entry q2 g1"| 33426₅ ||class="entry q3 g1"| 59043₉ ||class="entry q3 g1"| 58651₉ ||class="entry q2 g1"| 59716₇ ||class="entry q3 g1"| 35939₇ ||class="entry q3 g1"| 59227₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (34886, 47309, 46231, 58256, 53773, 46807) ||class="entry q2 g1"| 34886₅ ||class="entry q3 g1"| 47309₉ ||class="entry q3 g1"| 46231₉ ||class="entry q2 g1"| 58256₇ ||class="entry q3 g1"| 53773₇ ||class="entry q3 g1"| 46807₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (35140, 60517, 57661, 58002, 34469, 58237) ||class="entry q2 g1"| 35140₅ ||class="entry q3 g1"| 60517₉ ||class="entry q3 g1"| 57661₉ ||class="entry q2 g1"| 58002₇ ||class="entry q3 g1"| 34469₇ ||class="entry q3 g1"| 58237₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (41042, 44273, 40087, 52100, 50737, 40663) ||class="entry q2 g1"| 41042₅ ||class="entry q3 g1"| 44273₉ ||class="entry q3 g1"| 40087₉ ||class="entry q2 g1"| 52100₇ ||class="entry q3 g1"| 50737₇ ||class="entry q3 g1"| 40663₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (41296, 63577, 51517, 51846, 37529, 52093) ||class="entry q2 g1"| 41296₅ ||class="entry q3 g1"| 63577₉ ||class="entry q3 g1"| 51517₉ ||class="entry q2 g1"| 51846₇ ||class="entry q3 g1"| 37529₇ ||class="entry q3 g1"| 52093₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49924, 53153, 39335, 43218, 42337, 39911) ||class="entry q2 g1"| 49924₅ ||class="entry q3 g1"| 53153₉ ||class="entry q3 g1"| 39335₉ ||class="entry q2 g1"| 43218₇ ||class="entry q3 g1"| 42337₇ ||class="entry q3 g1"| 39911₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49936, 62345, 42395, 43206, 39241, 42971) ||class="entry q2 g1"| 49936₅ ||class="entry q3 g1"| 62345₉ ||class="entry q3 g1"| 42395₉ ||class="entry q2 g1"| 43206₇ ||class="entry q3 g1"| 39241₇ ||class="entry q3 g1"| 42971₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (34464, 53597, 54241, 60790, 48029, 53665) ||class="entry q2 g1"| 34464₅ ||class="entry q3 g1"| 53597₉ ||class="entry q3 g1"| 54241₉ ||class="entry q2 g1"| 60790₁₁ ||class="entry q3 g1"| 48029₁₁ ||class="entry q3 g1"| 53665₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (35936, 45851, 48721, 59318, 55771, 48145) ||class="entry q2 g1"| 35936₅ ||class="entry q3 g1"| 45851₉ ||class="entry q3 g1"| 48721₉ ||class="entry q2 g1"| 59318₁₁ ||class="entry q3 g1"| 55771₁₁ ||class="entry q3 g1"| 48145₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (37512, 50549, 51145, 63838, 44981, 50569) ||class="entry q2 g1"| 37512₅ ||class="entry q3 g1"| 50549₉ ||class="entry q3 g1"| 51145₉ ||class="entry q2 g1"| 63838₁₁ ||class="entry q3 g1"| 44981₁₁ ||class="entry q3 g1"| 50569₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38984, 42803, 43641, 62366, 52723, 43065) ||class="entry q2 g1"| 38984₅ ||class="entry q3 g1"| 42803₉ ||class="entry q3 g1"| 43641₉ ||class="entry q2 g1"| 62366₁₁ ||class="entry q3 g1"| 52723₁₁ ||class="entry q3 g1"| 43065₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (42080, 39695, 43629, 53174, 61903, 43053) ||class="entry q2 g1"| 42080₅ ||class="entry q3 g1"| 39695₉ ||class="entry q3 g1"| 43629₉ ||class="entry q2 g1"| 53174₁₁ ||class="entry q3 g1"| 61903₁₁ ||class="entry q3 g1"| 43053₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (45128, 36647, 48709, 56222, 58855, 48133) ||class="entry q2 g1"| 45128₅ ||class="entry q3 g1"| 36647₉ ||class="entry q3 g1"| 48709₉ ||class="entry q2 g1"| 56222₁₁ ||class="entry q3 g1"| 58855₁₁ ||class="entry q3 g1"| 48133₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (50720, 37087, 50891, 44534, 64031, 50315) ||class="entry q2 g1"| 50720₅ ||class="entry q3 g1"| 37087₉ ||class="entry q3 g1"| 50891₉ ||class="entry q2 g1"| 44534₁₁ ||class="entry q3 g1"| 64031₁₁ ||class="entry q3 g1"| 50315₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (53768, 34039, 53987, 47582, 60983, 53411) ||class="entry q2 g1"| 53768₅ ||class="entry q3 g1"| 34039₉ ||class="entry q3 g1"| 53987₉ ||class="entry q2 g1"| 47582₁₁ ||class="entry q3 g1"| 60983₁₁ ||class="entry q3 g1"| 53411₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (32980, 60181, 59837, 60162, 33237, 60413) ||class="entry q2 g1"| 32980₅ ||class="entry q3 g1"| 60181₉ ||class="entry q3 g1"| 59837₁₁ ||class="entry q2 g1"| 60162₇ ||class="entry q3 g1"| 33237₇ ||class="entry q3 g1"| 60413₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (35586, 57811, 60827, 57556, 35603, 61403) ||class="entry q2 g1"| 35586₅ ||class="entry q3 g1"| 57811₉ ||class="entry q3 g1"| 60827₁₁ ||class="entry q2 g1"| 57556₇ ||class="entry q3 g1"| 35603₇ ||class="entry q3 g1"| 61403₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (41730, 51655, 63911, 51412, 41735, 64487) ||class="entry q2 g1"| 41730₅ ||class="entry q3 g1"| 51655₉ ||class="entry q3 g1"| 63911₁₁ ||class="entry q2 g1"| 51412₇ ||class="entry q3 g1"| 41735₇ ||class="entry q3 g1"| 64487₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (49236, 43671, 64663, 43906, 49239, 65239) ||class="entry q2 g1"| 49236₅ ||class="entry q3 g1"| 43671₉ ||class="entry q3 g1"| 64663₁₁ ||class="entry q2 g1"| 43906₇ ||class="entry q3 g1"| 49239₇ ||class="entry q3 g1"| 65239₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32966, 45405, 46055, 60176, 56221, 45479) ||class="entry q2 g1"| 32966₅ ||class="entry q3 g1"| 45405₉ ||class="entry q3 g1"| 46055₁₁ ||class="entry q2 g1"| 60176₇ ||class="entry q3 g1"| 56221₁₁ ||class="entry q3 g1"| 45479₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32978, 36213, 36827, 60164, 59317, 36251) ||class="entry q2 g1"| 32978₅ ||class="entry q3 g1"| 36213₉ ||class="entry q3 g1"| 36827₁₁ ||class="entry q2 g1"| 60164₇ ||class="entry q3 g1"| 59317₁₁ ||class="entry q3 g1"| 36251₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35334, 54043, 56919, 57808, 47579, 56343) ||class="entry q2 g1"| 35334₅ ||class="entry q3 g1"| 54043₉ ||class="entry q3 g1"| 56919₁₁ ||class="entry q2 g1"| 57808₇ ||class="entry q3 g1"| 47579₁₁ ||class="entry q3 g1"| 56343₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35588, 34739, 35837, 57554, 60787, 35261) ||class="entry q2 g1"| 35588₅ ||class="entry q3 g1"| 34739₉ ||class="entry q3 g1"| 35837₁₁ ||class="entry q2 g1"| 57554₇ ||class="entry q3 g1"| 60787₁₁ ||class="entry q3 g1"| 35261₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41490, 50983, 63063, 51652, 44519, 62487) ||class="entry q2 g1"| 41490₅ ||class="entry q3 g1"| 50983₉ ||class="entry q3 g1"| 63063₁₁ ||class="entry q2 g1"| 51652₇ ||class="entry q3 g1"| 44519₁₁ ||class="entry q3 g1"| 62487₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41744, 37775, 41981, 51398, 63823, 41405) ||class="entry q2 g1"| 41744₅ ||class="entry q3 g1"| 37775₉ ||class="entry q3 g1"| 41981₁₁ ||class="entry q2 g1"| 51398₇ ||class="entry q3 g1"| 63823₁₁ ||class="entry q3 g1"| 41405₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (49476, 42103, 62311, 43666, 52919, 61735) ||class="entry q2 g1"| 49476₅ ||class="entry q3 g1"| 42103₉ ||class="entry q3 g1"| 62311₁₁ ||class="entry q2 g1"| 43666₇ ||class="entry q3 g1"| 52919₁₁ ||class="entry q3 g1"| 61735₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (49488, 39007, 53083, 43654, 62111, 52507) ||class="entry q2 g1"| 49488₅ ||class="entry q3 g1"| 39007₉ ||class="entry q3 g1"| 53083₁₁ ||class="entry q2 g1"| 43654₇ ||class="entry q3 g1"| 62111₁₁ ||class="entry q3 g1"| 52507₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33220, 58869, 58957, 59922, 36661, 58381) ||class="entry q2 g1"| 33220₅ ||class="entry q3 g1"| 58869₁₁ ||class="entry q3 g1"| 58957₉ ||class="entry q2 g1"| 59922₇ ||class="entry q3 g1"| 36661₉ ||class="entry q3 g1"| 58381₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33232, 55773, 55921, 59910, 45853, 55345) ||class="entry q2 g1"| 33232₅ ||class="entry q3 g1"| 55773₁₁ ||class="entry q3 g1"| 55921₉ ||class="entry q2 g1"| 59910₇ ||class="entry q3 g1"| 45853₉ ||class="entry q3 g1"| 55345₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (35346, 61235, 57963, 57796, 34291, 57387) ||class="entry q2 g1"| 35346₅ ||class="entry q3 g1"| 61235₁₁ ||class="entry q3 g1"| 57963₉ ||class="entry q2 g1"| 57796₇ ||class="entry q3 g1"| 34291₉ ||class="entry q3 g1"| 57387₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (35600, 48027, 47041, 57542, 53595, 46465) ||class="entry q2 g1"| 35600₅ ||class="entry q3 g1"| 48027₁₁ ||class="entry q3 g1"| 47041₉ ||class="entry q2 g1"| 57542₇ ||class="entry q3 g1"| 53595₉ ||class="entry q3 g1"| 46465₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (41478, 64271, 51819, 51664, 37327, 51243) ||class="entry q2 g1"| 41478₅ ||class="entry q3 g1"| 64271₁₁ ||class="entry q3 g1"| 51819₉ ||class="entry q2 g1"| 51664₇ ||class="entry q3 g1"| 37327₉ ||class="entry q3 g1"| 51243₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (41732, 44967, 40897, 51410, 50535, 40321) ||class="entry q2 g1"| 41732₅ ||class="entry q3 g1"| 44967₁₁ ||class="entry q3 g1"| 40897₉ ||class="entry q2 g1"| 51410₇ ||class="entry q3 g1"| 50535₉ ||class="entry q3 g1"| 40321₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (49222, 61663, 42701, 43920, 39455, 42125) ||class="entry q2 g1"| 49222₅ ||class="entry q3 g1"| 61663₁₁ ||class="entry q3 g1"| 42701₉ ||class="entry q2 g1"| 43920₇ ||class="entry q3 g1"| 39455₉ ||class="entry q3 g1"| 42125₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (49234, 52471, 39665, 43908, 42551, 39089) ||class="entry q2 g1"| 49234₅ ||class="entry q3 g1"| 52471₁₁ ||class="entry q3 g1"| 39665₉ ||class="entry q2 g1"| 43908₇ ||class="entry q3 g1"| 42551₉ ||class="entry q3 g1"| 39089₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (43200, 65321, 49597, 49942, 38377, 50173) ||class="entry q2 g1"| 43200₅ ||class="entry q3 g1"| 65321₁₁ ||class="entry q3 g1"| 49597₉ ||class="entry q2 g1"| 49942₇ ||class="entry q3 g1"| 38377₉ ||class="entry q3 g1"| 50173₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (51840, 62713, 44315, 41302, 40505, 44891) ||class="entry q2 g1"| 51840₅ ||class="entry q3 g1"| 62713₁₁ ||class="entry q3 g1"| 44315₉ ||class="entry q2 g1"| 41302₇ ||class="entry q3 g1"| 40505₉ ||class="entry q3 g1"| 44891₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (57984, 56557, 47399, 35158, 46637, 47975) ||class="entry q2 g1"| 57984₅ ||class="entry q3 g1"| 56557₁₁ ||class="entry q3 g1"| 47399₉ ||class="entry q2 g1"| 35158₇ ||class="entry q3 g1"| 46637₉ ||class="entry q3 g1"| 47975₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (59456, 48811, 54423, 33686, 54379, 54999) ||class="entry q2 g1"| 59456₅ ||class="entry q3 g1"| 48811₁₁ ||class="entry q3 g1"| 54423₉ ||class="entry q2 g1"| 33686₇ ||class="entry q3 g1"| 54379₉ ||class="entry q3 g1"| 54999₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (43648, 38143, 43901, 49494, 65087, 43325) ||class="entry q2 g1"| 43648₅ ||class="entry q3 g1"| 38143₁₁ ||class="entry q3 g1"| 43901₁₁ ||class="entry q2 g1"| 49494₇ ||class="entry q3 g1"| 65087₁₃ ||class="entry q3 g1"| 43325₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (51392, 40751, 51163, 41750, 62959, 50587) ||class="entry q2 g1"| 51392₅ ||class="entry q3 g1"| 40751₁₁ ||class="entry q3 g1"| 51163₁₁ ||class="entry q2 g1"| 41750₇ ||class="entry q3 g1"| 62959₁₃ ||class="entry q3 g1"| 50587₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (57536, 46907, 54247, 35606, 56827, 53671) ||class="entry q2 g1"| 57536₅ ||class="entry q3 g1"| 46907₁₁ ||class="entry q3 g1"| 54247₁₁ ||class="entry q2 g1"| 35606₇ ||class="entry q3 g1"| 56827₁₃ ||class="entry q3 g1"| 53671₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (59904, 54653, 48727, 33238, 49085, 48151) ||class="entry q2 g1"| 59904₅ ||class="entry q3 g1"| 54653₁₁ ||class="entry q3 g1"| 48727₁₁ ||class="entry q2 g1"| 33238₇ ||class="entry q3 g1"| 49085₁₃ ||class="entry q3 g1"| 48151₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (34276, 34883, 35565, 60978, 57987, 34989) ||class="entry q2 g1"| 34276₇ ||class="entry q3 g1"| 34883₅ ||class="entry q3 g1"| 35565₉ ||class="entry q2 g1"| 60978₉ ||class="entry q3 g1"| 57987₇ ||class="entry q3 g1"| 34989₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (36402, 33413, 36555, 58852, 59461, 35979) ||class="entry q2 g1"| 36402₇ ||class="entry q3 g1"| 33413₅ ||class="entry q3 g1"| 36555₉ ||class="entry q2 g1"| 58852₉ ||class="entry q3 g1"| 59461₇ ||class="entry q3 g1"| 35979₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (37336, 41027, 41721, 64014, 51843, 41145) ||class="entry q2 g1"| 37336₇ ||class="entry q3 g1"| 41027₅ ||class="entry q3 g1"| 41721₉ ||class="entry q2 g1"| 64014₉ ||class="entry q3 g1"| 51843₇ ||class="entry q3 g1"| 41145₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (39704, 49669, 53065, 61646, 43205, 52489) ||class="entry q2 g1"| 39704₇ ||class="entry q3 g1"| 49669₅ ||class="entry q3 g1"| 53065₉ ||class="entry q2 g1"| 61646₉ ||class="entry q3 g1"| 43205₇ ||class="entry q3 g1"| 52489₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (42788, 49681, 62305, 52466, 43217, 61729) ||class="entry q2 g1"| 42788₇ ||class="entry q3 g1"| 49681₅ ||class="entry q3 g1"| 62305₉ ||class="entry q2 g1"| 52466₉ ||class="entry q3 g1"| 43217₇ ||class="entry q3 g1"| 61729₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (45582, 33425, 45795, 55768, 59473, 45219) ||class="entry q2 g1"| 45582₇ ||class="entry q3 g1"| 33425₅ ||class="entry q3 g1"| 45795₉ ||class="entry q2 g1"| 55768₉ ||class="entry q3 g1"| 59473₇ ||class="entry q3 g1"| 45219₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (50290, 41281, 63057, 44964, 52097, 62481) ||class="entry q2 g1"| 50290₇ ||class="entry q3 g1"| 41281₅ ||class="entry q3 g1"| 63057₉ ||class="entry q2 g1"| 44964₉ ||class="entry q3 g1"| 52097₇ ||class="entry q3 g1"| 62481₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (53326, 35137, 56901, 48024, 58241, 56325) ||class="entry q2 g1"| 53326₇ ||class="entry q3 g1"| 35137₅ ||class="entry q3 g1"| 56901₉ ||class="entry q2 g1"| 48024₉ ||class="entry q3 g1"| 58241₇ ||class="entry q3 g1"| 56325₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (34036, 34467, 34077, 61218, 60515, 34653) ||class="entry q2 g1"| 34036₇ ||class="entry q3 g1"| 34467₇ ||class="entry q3 g1"| 34077₇ ||class="entry q2 g1"| 61218₉ ||class="entry q3 g1"| 60515₉ ||class="entry q3 g1"| 34653₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36642, 35941, 33083, 58612, 59045, 33659) ||class="entry q2 g1"| 36642₇ ||class="entry q3 g1"| 35941₇ ||class="entry q3 g1"| 33083₇ ||class="entry q2 g1"| 58612₉ ||class="entry q3 g1"| 59045₉ ||class="entry q3 g1"| 33659₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (37084, 37515, 37173, 64266, 63563, 37749) ||class="entry q2 g1"| 37084₇ ||class="entry q3 g1"| 37515₇ ||class="entry q3 g1"| 37173₇ ||class="entry q2 g1"| 64266₉ ||class="entry q3 g1"| 63563₉ ||class="entry q3 g1"| 37749₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (39690, 38989, 38163, 61660, 62093, 38739) ||class="entry q2 g1"| 39690₇ ||class="entry q3 g1"| 38989₇ ||class="entry q3 g1"| 38163₇ ||class="entry q2 g1"| 61660₉ ||class="entry q3 g1"| 62093₉ ||class="entry q3 g1"| 38739₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (42786, 42097, 38151, 52468, 52913, 38727) ||class="entry q2 g1"| 42786₇ ||class="entry q3 g1"| 42097₇ ||class="entry q3 g1"| 38151₇ ||class="entry q2 g1"| 52468₉ ||class="entry q3 g1"| 52913₉ ||class="entry q3 g1"| 38727₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45834, 45145, 33071, 55516, 55961, 33647) ||class="entry q2 g1"| 45834₇ ||class="entry q3 g1"| 45145₇ ||class="entry q3 g1"| 33071₇ ||class="entry q2 g1"| 55516₉ ||class="entry q3 g1"| 55961₉ ||class="entry q3 g1"| 33647₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (50292, 50977, 36919, 44962, 44513, 37495) ||class="entry q2 g1"| 50292₇ ||class="entry q3 g1"| 50977₇ ||class="entry q3 g1"| 36919₇ ||class="entry q2 g1"| 44962₉ ||class="entry q3 g1"| 44513₉ ||class="entry q3 g1"| 37495₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (53340, 54025, 33823, 48010, 47561, 34399) ||class="entry q2 g1"| 53340₇ ||class="entry q3 g1"| 54025₇ ||class="entry q3 g1"| 33823₇ ||class="entry q2 g1"| 48010₉ ||class="entry q3 g1"| 47561₉ ||class="entry q3 g1"| 34399₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (38120, 49941, 49577, 65342, 43477, 50153) ||class="entry q2 g1"| 38120₇ ||class="entry q3 g1"| 49941₇ ||class="entry q3 g1"| 49577₇ ||class="entry q2 g1"| 65342₁₃ ||class="entry q3 g1"| 43477₉ ||class="entry q3 g1"| 50153₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (40488, 41299, 44057, 62974, 52115, 44633) ||class="entry q2 g1"| 40488₇ ||class="entry q3 g1"| 41299₇ ||class="entry q3 g1"| 44057₇ ||class="entry q2 g1"| 62974₁₃ ||class="entry q3 g1"| 52115₉ ||class="entry q3 g1"| 44633₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (46632, 35143, 47141, 56830, 58247, 47717) ||class="entry q2 g1"| 46632₇ ||class="entry q3 g1"| 35143₇ ||class="entry q3 g1"| 47141₇ ||class="entry q2 g1"| 56830₁₃ ||class="entry q3 g1"| 58247₉ ||class="entry q3 g1"| 47717₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (54376, 33431, 54403, 49086, 59479, 54979) ||class="entry q2 g1"| 54376₇ ||class="entry q3 g1"| 33431₇ ||class="entry q3 g1"| 54403₇ ||class="entry q2 g1"| 49086₁₃ ||class="entry q3 g1"| 59479₉ ||class="entry q3 g1"| 54979₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (34482, 35605, 35259, 60772, 57813, 35835) ||class="entry q2 g1"| 34482₇ ||class="entry q3 g1"| 35605₇ ||class="entry q3 g1"| 35259₉ ||class="entry q2 g1"| 60772₉ ||class="entry q3 g1"| 57813₉ ||class="entry q3 g1"| 35835₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (36196, 33235, 36253, 59058, 60179, 36829) ||class="entry q2 g1"| 36196₇ ||class="entry q3 g1"| 33235₇ ||class="entry q3 g1"| 36253₉ ||class="entry q2 g1"| 59058₉ ||class="entry q3 g1"| 60179₉ ||class="entry q3 g1"| 36829₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (37518, 41749, 41391, 63832, 51669, 41967) ||class="entry q2 g1"| 37518₇ ||class="entry q3 g1"| 41749₇ ||class="entry q3 g1"| 41391₉ ||class="entry q2 g1"| 63832₉ ||class="entry q3 g1"| 51669₉ ||class="entry q3 g1"| 41967₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38990, 49491, 52255, 62360, 43923, 52831) ||class="entry q2 g1"| 38990₇ ||class="entry q3 g1"| 49491₇ ||class="entry q3 g1"| 52255₉ ||class="entry q2 g1"| 62360₉ ||class="entry q3 g1"| 43923₉ ||class="entry q3 g1"| 52831₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (42098, 49479, 61495, 53156, 43911, 62071) ||class="entry q2 g1"| 42098₇ ||class="entry q3 g1"| 49479₇ ||class="entry q3 g1"| 61495₉ ||class="entry q2 g1"| 53156₉ ||class="entry q3 g1"| 43911₉ ||class="entry q3 g1"| 62071₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (45400, 33223, 45493, 55950, 60167, 46069) ||class="entry q2 g1"| 45400₇ ||class="entry q3 g1"| 33223₇ ||class="entry q3 g1"| 45493₉ ||class="entry q2 g1"| 55950₉ ||class="entry q3 g1"| 60167₉ ||class="entry q3 g1"| 46069₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (50980, 41495, 62727, 44274, 51415, 63303) ||class="entry q2 g1"| 50980₇ ||class="entry q3 g1"| 41495₇ ||class="entry q3 g1"| 62727₉ ||class="entry q2 g1"| 44274₉ ||class="entry q3 g1"| 51415₉ ||class="entry q3 g1"| 63303₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (54040, 35351, 56595, 47310, 57559, 57171) ||class="entry q2 g1"| 54040₇ ||class="entry q3 g1"| 35351₇ ||class="entry q3 g1"| 56595₉ ||class="entry q2 g1"| 47310₉ ||class="entry q3 g1"| 57559₉ ||class="entry q3 g1"| 57171₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (38568, 43203, 43881, 64894, 49667, 43305) ||class="entry q2 g1"| 38568₇ ||class="entry q3 g1"| 43203₇ ||class="entry q3 g1"| 43881₉ ||class="entry q2 g1"| 64894₁₃ ||class="entry q3 g1"| 49667₅ ||class="entry q3 g1"| 43305₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (40040, 51845, 50905, 63422, 41029, 50329) ||class="entry q2 g1"| 40040₇ ||class="entry q3 g1"| 51845₇ ||class="entry q3 g1"| 50905₉ ||class="entry q2 g1"| 63422₁₃ ||class="entry q3 g1"| 41029₅ ||class="entry q3 g1"| 50329₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (46184, 58001, 53989, 57278, 34897, 53413) ||class="entry q2 g1"| 46184₇ ||class="entry q3 g1"| 58001₇ ||class="entry q3 g1"| 53989₉ ||class="entry q2 g1"| 57278₁₃ ||class="entry q3 g1"| 34897₅ ||class="entry q3 g1"| 53413₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (54824, 59713, 48707, 48638, 33665, 48131) ||class="entry q2 g1"| 54824₇ ||class="entry q3 g1"| 59713₇ ||class="entry q3 g1"| 48707₉ ||class="entry q2 g1"| 48638₁₃ ||class="entry q3 g1"| 33665₅ ||class="entry q3 g1"| 48131₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (34034, 57539, 58235, 61220, 35331, 57659) ||class="entry q2 g1"| 34034₇ ||class="entry q3 g1"| 57539₇ ||class="entry q3 g1"| 58235₁₁ ||class="entry q2 g1"| 61220₉ ||class="entry q3 g1"| 35331₅ ||class="entry q3 g1"| 57659₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (36644, 59909, 59229, 58610, 32965, 58653) ||class="entry q2 g1"| 36644₇ ||class="entry q3 g1"| 59909₇ ||class="entry q3 g1"| 59229₁₁ ||class="entry q2 g1"| 58610₉ ||class="entry q3 g1"| 32965₅ ||class="entry q3 g1"| 58653₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (37070, 51395, 52079, 64280, 41475, 51503) ||class="entry q2 g1"| 37070₇ ||class="entry q3 g1"| 51395₇ ||class="entry q3 g1"| 52079₁₁ ||class="entry q2 g1"| 64280₉ ||class="entry q3 g1"| 41475₅ ||class="entry q3 g1"| 51503₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (39438, 43653, 42719, 61912, 49221, 42143) ||class="entry q2 g1"| 39438₇ ||class="entry q3 g1"| 43653₇ ||class="entry q3 g1"| 42719₁₁ ||class="entry q2 g1"| 61912₉ ||class="entry q3 g1"| 49221₅ ||class="entry q3 g1"| 42143₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (42546, 43665, 39671, 52708, 49233, 39095) ||class="entry q2 g1"| 42546₇ ||class="entry q3 g1"| 43665₇ ||class="entry q3 g1"| 39671₁₁ ||class="entry q2 g1"| 52708₉ ||class="entry q3 g1"| 49233₅ ||class="entry q3 g1"| 39095₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (45848, 59921, 56181, 55502, 32977, 55605) ||class="entry q2 g1"| 45848₇ ||class="entry q3 g1"| 59921₇ ||class="entry q3 g1"| 56181₁₁ ||class="entry q2 g1"| 55502₉ ||class="entry q3 g1"| 32977₅ ||class="entry q3 g1"| 55605₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (50532, 51649, 40903, 44722, 41729, 40327) ||class="entry q2 g1"| 50532₇ ||class="entry q3 g1"| 51649₇ ||class="entry q3 g1"| 40903₁₁ ||class="entry q2 g1"| 44722₉ ||class="entry q3 g1"| 41729₅ ||class="entry q3 g1"| 40327₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (53592, 57793, 47059, 47758, 35585, 46483) ||class="entry q2 g1"| 53592₇ ||class="entry q3 g1"| 57793₇ ||class="entry q3 g1"| 47059₁₁ ||class="entry q2 g1"| 47758₉ ||class="entry q3 g1"| 35585₅ ||class="entry q3 g1"| 46483₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (34724, 58261, 57389, 60530, 35157, 57965) ||class="entry q2 g1"| 34724₇ ||class="entry q3 g1"| 58261₉ ||class="entry q3 g1"| 57389₇ ||class="entry q2 g1"| 60530₉ ||class="entry q3 g1"| 35157₇ ||class="entry q3 g1"| 57965₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (35954, 59731, 58379, 59300, 33683, 58955) ||class="entry q2 g1"| 35954₇ ||class="entry q3 g1"| 59731₉ ||class="entry q3 g1"| 58379₇ ||class="entry q2 g1"| 59300₉ ||class="entry q3 g1"| 33683₇ ||class="entry q3 g1"| 58955₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (37784, 52117, 51257, 63566, 41301, 51833) ||class="entry q2 g1"| 37784₇ ||class="entry q3 g1"| 52117₉ ||class="entry q3 g1"| 51257₇ ||class="entry q2 g1"| 63566₉ ||class="entry q3 g1"| 41301₇ ||class="entry q3 g1"| 51833₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39256, 43475, 42377, 62094, 49939, 42953) ||class="entry q2 g1"| 39256₇ ||class="entry q3 g1"| 43475₉ ||class="entry q3 g1"| 42377₇ ||class="entry q2 g1"| 62094₉ ||class="entry q3 g1"| 49939₇ ||class="entry q3 g1"| 42953₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42340, 43463, 39329, 52914, 49927, 39905) ||class="entry q2 g1"| 42340₇ ||class="entry q3 g1"| 43463₉ ||class="entry q3 g1"| 39329₇ ||class="entry q2 g1"| 52914₉ ||class="entry q3 g1"| 49927₇ ||class="entry q3 g1"| 39905₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (45134, 59719, 55331, 56216, 33671, 55907) ||class="entry q2 g1"| 45134₇ ||class="entry q3 g1"| 59719₉ ||class="entry q3 g1"| 55331₇ ||class="entry q2 g1"| 56216₉ ||class="entry q3 g1"| 33671₇ ||class="entry q3 g1"| 55907₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (50738, 51863, 40081, 44516, 41047, 40657) ||class="entry q2 g1"| 50738₇ ||class="entry q3 g1"| 51863₉ ||class="entry q3 g1"| 40081₇ ||class="entry q2 g1"| 44516₉ ||class="entry q3 g1"| 41047₇ ||class="entry q3 g1"| 40657₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (53774, 58007, 46213, 47576, 34903, 46789) ||class="entry q2 g1"| 53774₇ ||class="entry q3 g1"| 58007₉ ||class="entry q3 g1"| 46213₇ ||class="entry q2 g1"| 47576₉ ||class="entry q3 g1"| 34903₇ ||class="entry q3 g1"| 46789₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (34722, 34293, 34379, 60532, 61237, 33803) ||class="entry q2 g1"| 34722₇ ||class="entry q3 g1"| 34293₉ ||class="entry q3 g1"| 34379₇ ||class="entry q2 g1"| 60532₉ ||class="entry q3 g1"| 61237₁₁ ||class="entry q3 g1"| 33803₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (35956, 36659, 33389, 59298, 58867, 32813) ||class="entry q2 g1"| 35956₇ ||class="entry q3 g1"| 36659₉ ||class="entry q3 g1"| 33389₇ ||class="entry q2 g1"| 59298₉ ||class="entry q3 g1"| 58867₁₁ ||class="entry q3 g1"| 32813₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (37770, 37341, 37475, 63580, 64285, 36899) ||class="entry q2 g1"| 37770₇ ||class="entry q3 g1"| 37341₉ ||class="entry q3 g1"| 37475₇ ||class="entry q2 g1"| 63580₉ ||class="entry q3 g1"| 64285₁₁ ||class="entry q3 g1"| 36899₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (39004, 39707, 38469, 62346, 61915, 37893) ||class="entry q2 g1"| 39004₇ ||class="entry q3 g1"| 39707₉ ||class="entry q3 g1"| 38469₇ ||class="entry q2 g1"| 62346₉ ||class="entry q3 g1"| 61915₁₁ ||class="entry q3 g1"| 37893₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (42100, 42791, 38481, 53154, 52711, 37905) ||class="entry q2 g1"| 42100₇ ||class="entry q3 g1"| 42791₉ ||class="entry q3 g1"| 38481₇ ||class="entry q2 g1"| 53154₉ ||class="entry q3 g1"| 52711₁₁ ||class="entry q3 g1"| 37905₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (45148, 45839, 33401, 56202, 55759, 32825) ||class="entry q2 g1"| 45148₇ ||class="entry q3 g1"| 45839₉ ||class="entry q3 g1"| 33401₇ ||class="entry q2 g1"| 56202₉ ||class="entry q3 g1"| 55759₁₁ ||class="entry q3 g1"| 32825₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (50978, 50295, 37729, 44276, 44727, 37153) ||class="entry q2 g1"| 50978₇ ||class="entry q3 g1"| 50295₉ ||class="entry q3 g1"| 37729₇ ||class="entry q2 g1"| 44276₉ ||class="entry q3 g1"| 44727₁₁ ||class="entry q3 g1"| 37153₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (54026, 53343, 34633, 47324, 47775, 34057) ||class="entry q2 g1"| 54026₇ ||class="entry q3 g1"| 53343₉ ||class="entry q3 g1"| 34633₇ ||class="entry q2 g1"| 47324₉ ||class="entry q3 g1"| 47775₁₁ ||class="entry q3 g1"| 34057₅ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (34274, 60963, 60555, 60980, 34019, 61131) ||class="entry q2 g1"| 34274₇ ||class="entry q3 g1"| 60963₉ ||class="entry q3 g1"| 60555₉ ||class="entry q2 g1"| 60980₉ ||class="entry q3 g1"| 34019₇ ||class="entry q3 g1"| 61131₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (36404, 58597, 59565, 58850, 36389, 60141) ||class="entry q2 g1"| 36404₇ ||class="entry q3 g1"| 58597₉ ||class="entry q3 g1"| 59565₉ ||class="entry q2 g1"| 58850₉ ||class="entry q3 g1"| 36389₇ ||class="entry q3 g1"| 60141₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (37322, 64011, 63651, 64028, 37067, 64227) ||class="entry q2 g1"| 37322₇ ||class="entry q3 g1"| 64011₉ ||class="entry q3 g1"| 63651₉ ||class="entry q2 g1"| 64028₉ ||class="entry q3 g1"| 37067₇ ||class="entry q3 g1"| 64227₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (39452, 61645, 64645, 61898, 39437, 65221) ||class="entry q2 g1"| 39452₇ ||class="entry q3 g1"| 61645₉ ||class="entry q3 g1"| 64645₉ ||class="entry q2 g1"| 61898₉ ||class="entry q3 g1"| 39437₇ ||class="entry q3 g1"| 65221₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (42548, 52465, 64657, 52706, 42545, 65233) ||class="entry q2 g1"| 42548₇ ||class="entry q3 g1"| 52465₉ ||class="entry q3 g1"| 64657₉ ||class="entry q2 g1"| 52706₉ ||class="entry q3 g1"| 42545₇ ||class="entry q3 g1"| 65233₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (45596, 55513, 59577, 55754, 45593, 60153) ||class="entry q2 g1"| 45596₇ ||class="entry q3 g1"| 55513₉ ||class="entry q3 g1"| 59577₉ ||class="entry q2 g1"| 55754₉ ||class="entry q3 g1"| 45593₇ ||class="entry q3 g1"| 60153₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (50530, 44961, 63905, 44724, 50529, 64481) ||class="entry q2 g1"| 50530₇ ||class="entry q3 g1"| 44961₉ ||class="entry q3 g1"| 63905₉ ||class="entry q2 g1"| 44724₉ ||class="entry q3 g1"| 50529₇ ||class="entry q3 g1"| 64481₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (53578, 48009, 60809, 47772, 53577, 61385) ||class="entry q2 g1"| 53578₇ ||class="entry q3 g1"| 48009₉ ||class="entry q3 g1"| 60809₉ ||class="entry q2 g1"| 47772₉ ||class="entry q3 g1"| 53577₇ ||class="entry q3 g1"| 61385₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (34288, 46187, 46801, 60966, 57003, 46225) ||class="entry q2 g1"| 34288₇ ||class="entry q3 g1"| 46187₉ ||class="entry q3 g1"| 46801₉ ||class="entry q2 g1"| 60966₉ ||class="entry q3 g1"| 57003₁₁ ||class="entry q3 g1"| 46225₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (36656, 54829, 56161, 58598, 48365, 55585) ||class="entry q2 g1"| 36656₇ ||class="entry q3 g1"| 54829₉ ||class="entry q3 g1"| 56161₉ ||class="entry q2 g1"| 58598₉ ||class="entry q3 g1"| 48365₁₁ ||class="entry q3 g1"| 55585₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (37324, 40043, 40645, 64026, 63147, 40069) ||class="entry q2 g1"| 37324₇ ||class="entry q3 g1"| 40043₉ ||class="entry q3 g1"| 40645₉ ||class="entry q2 g1"| 64026₉ ||class="entry q3 g1"| 63147₁₁ ||class="entry q3 g1"| 40069₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (39450, 38573, 39651, 61900, 64621, 39075) ||class="entry q2 g1"| 39450₇ ||class="entry q3 g1"| 38573₉ ||class="entry q3 g1"| 39651₉ ||class="entry q2 g1"| 61900₉ ||class="entry q3 g1"| 64621₁₁ ||class="entry q3 g1"| 39075₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (42534, 38585, 42699, 52720, 64633, 42123) ||class="entry q2 g1"| 42534₇ ||class="entry q3 g1"| 38585₉ ||class="entry q3 g1"| 42699₉ ||class="entry q2 g1"| 52720₉ ||class="entry q3 g1"| 64633₁₁ ||class="entry q3 g1"| 42123₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (45836, 54841, 59209, 55514, 48377, 58633) ||class="entry q2 g1"| 45836₇ ||class="entry q3 g1"| 54841₉ ||class="entry q3 g1"| 59209₉ ||class="entry q2 g1"| 55514₉ ||class="entry q3 g1"| 48377₁₁ ||class="entry q3 g1"| 58633₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (50278, 40297, 51821, 44976, 63401, 51245) ||class="entry q2 g1"| 50278₇ ||class="entry q3 g1"| 40297₉ ||class="entry q3 g1"| 51821₉ ||class="entry q2 g1"| 44976₉ ||class="entry q3 g1"| 63401₁₁ ||class="entry q3 g1"| 51245₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (53338, 46441, 57977, 48012, 57257, 57401) ||class="entry q2 g1"| 53338₇ ||class="entry q3 g1"| 46441₉ ||class="entry q3 g1"| 57977₉ ||class="entry q2 g1"| 48012₉ ||class="entry q3 g1"| 57257₁₁ ||class="entry q3 g1"| 57401₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (44256, 37535, 44317, 50998, 63583, 44893) ||class="entry q2 g1"| 44256₇ ||class="entry q3 g1"| 37535₉ ||class="entry q3 g1"| 44317₉ ||class="entry q2 g1"| 50998₉ ||class="entry q3 g1"| 63583₁₁ ||class="entry q3 g1"| 44893₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (47304, 34487, 47413, 54046, 60535, 47989) ||class="entry q2 g1"| 47304₇ ||class="entry q3 g1"| 34487₉ ||class="entry q3 g1"| 47413₉ ||class="entry q2 g1"| 54046₉ ||class="entry q3 g1"| 60535₁₁ ||class="entry q3 g1"| 47989₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (52896, 39247, 49595, 42358, 62351, 50171) ||class="entry q2 g1"| 52896₇ ||class="entry q3 g1"| 39247₉ ||class="entry q3 g1"| 49595₉ ||class="entry q2 g1"| 42358₉ ||class="entry q3 g1"| 62351₁₁ ||class="entry q3 g1"| 50171₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (55944, 36199, 54675, 45406, 59303, 55251) ||class="entry q2 g1"| 55944₇ ||class="entry q3 g1"| 36199₉ ||class="entry q3 g1"| 54675₉ ||class="entry q2 g1"| 45406₉ ||class="entry q3 g1"| 59303₁₁ ||class="entry q3 g1"| 55251₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (59040, 45403, 54663, 36214, 56219, 55239) ||class="entry q2 g1"| 59040₇ ||class="entry q3 g1"| 45403₉ ||class="entry q3 g1"| 54663₉ ||class="entry q2 g1"| 36214₉ ||class="entry q3 g1"| 56219₁₁ ||class="entry q3 g1"| 55239₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (60512, 54045, 47159, 34742, 47581, 47735) ||class="entry q2 g1"| 60512₇ ||class="entry q3 g1"| 54045₉ ||class="entry q3 g1"| 47159₉ ||class="entry q2 g1"| 34742₉ ||class="entry q3 g1"| 47581₁₁ ||class="entry q3 g1"| 47735₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (62088, 42355, 49583, 39262, 53171, 50159) ||class="entry q2 g1"| 62088₇ ||class="entry q3 g1"| 42355₉ ||class="entry q3 g1"| 49583₉ ||class="entry q2 g1"| 39262₉ ||class="entry q3 g1"| 53171₁₁ ||class="entry q3 g1"| 50159₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (63560, 50997, 44063, 37790, 44533, 44639) ||class="entry q2 g1"| 63560₇ ||class="entry q3 g1"| 50997₉ ||class="entry q3 g1"| 44063₉ ||class="entry q2 g1"| 37790₉ ||class="entry q3 g1"| 44533₁₁ ||class="entry q3 g1"| 44639₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (44704, 63817, 51165, 50550, 37769, 50589) ||class="entry q2 g1"| 44704₇ ||class="entry q3 g1"| 63817₉ ||class="entry q3 g1"| 51165₁₁ ||class="entry q2 g1"| 50550₉ ||class="entry q3 g1"| 37769₇ ||class="entry q3 g1"| 50589₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (47752, 60769, 54261, 53598, 34721, 53685) ||class="entry q2 g1"| 47752₇ ||class="entry q3 g1"| 60769₉ ||class="entry q3 g1"| 54261₁₁ ||class="entry q2 g1"| 53598₉ ||class="entry q3 g1"| 34721₇ ||class="entry q3 g1"| 53685₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (52448, 62105, 43899, 42806, 39001, 43323) ||class="entry q2 g1"| 52448₇ ||class="entry q3 g1"| 62105₉ ||class="entry q3 g1"| 43899₁₁ ||class="entry q2 g1"| 42806₉ ||class="entry q3 g1"| 39001₇ ||class="entry q3 g1"| 43323₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (55496, 59057, 48979, 45854, 35953, 48403) ||class="entry q2 g1"| 55496₇ ||class="entry q3 g1"| 59057₉ ||class="entry q3 g1"| 48979₁₁ ||class="entry q2 g1"| 45854₉ ||class="entry q3 g1"| 35953₇ ||class="entry q3 g1"| 48403₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (58592, 55949, 48967, 36662, 45133, 48391) ||class="entry q2 g1"| 58592₇ ||class="entry q3 g1"| 55949₉ ||class="entry q3 g1"| 48967₁₁ ||class="entry q2 g1"| 36662₉ ||class="entry q3 g1"| 45133₇ ||class="entry q3 g1"| 48391₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (60960, 47307, 54007, 34294, 53771, 53431) ||class="entry q2 g1"| 60960₇ ||class="entry q3 g1"| 47307₉ ||class="entry q3 g1"| 54007₁₁ ||class="entry q2 g1"| 34294₉ ||class="entry q3 g1"| 53771₇ ||class="entry q3 g1"| 53431₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (61640, 52901, 43887, 39710, 42085, 43311) ||class="entry q2 g1"| 61640₇ ||class="entry q3 g1"| 52901₉ ||class="entry q3 g1"| 43887₁₁ ||class="entry q2 g1"| 39710₉ ||class="entry q3 g1"| 42085₇ ||class="entry q3 g1"| 43311₉ |- |class="f"| 854 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q3 g1"| 49083₁₃ ||class="entry q3 g1"| 55927₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (37530, 40765, 40339, 63820, 62973, 40915) ||class="entry q2 g1"| 37530₇ ||class="entry q3 g1"| 40765₁₁ ||class="entry q3 g1"| 40339₉ ||class="entry q2 g1"| 63820₉ ||class="entry q3 g1"| 62973₁₃ ||class="entry q3 g1"| 40915₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (39244, 38395, 39349, 62106, 65339, 39925) ||class="entry q2 g1"| 39244₇ ||class="entry q3 g1"| 38395₁₁ ||class="entry q3 g1"| 39349₉ ||class="entry q2 g1"| 62106₉ ||class="entry q3 g1"| 65339₁₃ ||class="entry q3 g1"| 39925₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (42352, 38383, 42397, 52902, 65327, 42973) ||class="entry q2 g1"| 42352₇ ||class="entry q3 g1"| 38383₁₁ 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56575, 58223) ||class="entry q2 g1"| 54028₇ ||class="entry q3 g1"| 46655₁₁ ||class="entry q3 g1"| 57647₉ ||class="entry q2 g1"| 47322₉ ||class="entry q3 g1"| 56575₁₃ ||class="entry q3 g1"| 58223₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (34022, 56555, 57159, 61232, 46635, 56583) ||class="entry q2 g1"| 34022₇ ||class="entry q3 g1"| 56555₁₁ ||class="entry q3 g1"| 57159₁₁ ||class="entry q2 g1"| 61232₉ ||class="entry q3 g1"| 46635₉ ||class="entry q3 g1"| 56583₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (36390, 48813, 45815, 58864, 54381, 45239) ||class="entry q2 g1"| 36390₇ ||class="entry q3 g1"| 48813₁₁ ||class="entry q3 g1"| 45815₁₁ ||class="entry q2 g1"| 58864₉ ||class="entry q3 g1"| 54381₉ ||class="entry q3 g1"| 45239₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (37082, 62699, 63315, 64268, 40491, 62739) ||class="entry q2 g1"| 37082₇ ||class="entry q3 g1"| 62699₁₁ ||class="entry q3 g1"| 63315₁₁ ||class="entry q2 g1"| 64268₉ ||class="entry q3 g1"| 40491₉ ||class="entry q3 g1"| 62739₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (39692, 65069, 62325, 61658, 38125, 61749) ||class="entry q2 g1"| 39692₇ ||class="entry q3 g1"| 65069₁₁ ||class="entry q3 g1"| 62325₁₁ ||class="entry q2 g1"| 61658₉ ||class="entry q3 g1"| 38125₉ ||class="entry q3 g1"| 61749₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (42800, 65081, 53085, 52454, 38137, 52509) ||class="entry q2 g1"| 42800₇ ||class="entry q3 g1"| 65081₁₁ ||class="entry q3 g1"| 53085₁₁ ||class="entry q2 g1"| 52454₉ ||class="entry q3 g1"| 38137₉ ||class="entry q3 g1"| 52509₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (45594, 48825, 36575, 55756, 54393, 35999) ||class="entry q2 g1"| 45594₇ ||class="entry q3 g1"| 48825₁₁ ||class="entry q3 g1"| 36575₁₁ ||class="entry q2 g1"| 55756₉ ||class="entry q3 g1"| 54393₉ ||class="entry q3 g1"| 35999₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (50544, 62953, 41979, 44710, 40745, 41403) ||class="entry q2 g1"| 50544₇ ||class="entry q3 g1"| 62953₁₁ ||class="entry q3 g1"| 41979₁₁ ||class="entry q2 g1"| 44710₉ ||class="entry q3 g1"| 40745₉ ||class="entry q3 g1"| 41403₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (53580, 56809, 35823, 47770, 46889, 35247) ||class="entry q2 g1"| 53580₇ ||class="entry q3 g1"| 56809₁₁ ||class="entry q3 g1"| 35823₁₁ ||class="entry q2 g1"| 47770₉ ||class="entry q3 g1"| 46889₉ ||class="entry q3 g1"| 35247₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (34484, 60789, 61405, 60770, 34741, 60829) ||class="entry q2 g1"| 34484₇ ||class="entry q3 g1"| 60789₁₁ ||class="entry q3 g1"| 61405₁₃ ||class="entry q2 g1"| 60770₉ ||class="entry q3 g1"| 34741₉ ||class="entry q3 g1"| 60829₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (36194, 59315, 60411, 59060, 36211, 59835) ||class="entry q2 g1"| 36194₇ ||class="entry q3 g1"| 59315₁₁ ||class="entry q3 g1"| 60411₁₃ ||class="entry q2 g1"| 59060₉ ||class="entry q3 g1"| 36211₉ ||class="entry q3 g1"| 59835₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (37532, 63837, 64501, 63818, 37789, 63925) ||class="entry q2 g1"| 37532₇ ||class="entry q3 g1"| 63837₁₁ ||class="entry q3 g1"| 64501₁₃ ||class="entry q2 g1"| 63818₉ ||class="entry q3 g1"| 37789₉ ||class="entry q3 g1"| 63925₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39242, 62363, 65491, 62108, 39259, 64915) ||class="entry q2 g1"| 39242₇ ||class="entry q3 g1"| 62363₁₁ ||class="entry q3 g1"| 65491₁₃ ||class="entry q2 g1"| 62108₉ ||class="entry q3 g1"| 39259₉ ||class="entry q3 g1"| 64915₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (42338, 53159, 65479, 52916, 42343, 64903) ||class="entry q2 g1"| 42338₇ ||class="entry q3 g1"| 53159₁₁ ||class="entry q3 g1"| 65479₁₃ ||class="entry q2 g1"| 52916₉ ||class="entry q3 g1"| 42343₉ ||class="entry q3 g1"| 64903₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (45386, 56207, 60399, 55964, 45391, 59823) ||class="entry q2 g1"| 45386₇ ||class="entry q3 g1"| 56207₁₁ ||class="entry q3 g1"| 60399₁₃ ||class="entry q2 g1"| 55964₉ ||class="entry q3 g1"| 45391₉ ||class="entry q3 g1"| 59823₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50740, 44279, 64247, 44514, 50743, 63671) ||class="entry q2 g1"| 50740₇ ||class="entry q3 g1"| 44279₁₁ ||class="entry q3 g1"| 64247₁₃ ||class="entry q2 g1"| 44514₉ ||class="entry q3 g1"| 50743₉ ||class="entry q3 g1"| 63671₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (53788, 47327, 61151, 47562, 53791, 60575) ||class="entry q2 g1"| 53788₇ ||class="entry q3 g1"| 47327₁₁ ||class="entry q3 g1"| 61151₁₃ ||class="entry q2 g1"| 47562₉ ||class="entry q3 g1"| 53791₉ ||class="entry q3 g1"| 60575₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (34736, 57277, 56337, 60518, 46461, 56913) ||class="entry q2 g1"| 34736₇ ||class="entry q3 g1"| 57277₁₃ ||class="entry q3 g1"| 56337₇ ||class="entry q2 g1"| 60518₉ ||class="entry q3 g1"| 46461₁₁ ||class="entry q3 g1"| 56913₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (36208, 48635, 45473, 59046, 55099, 46049) ||class="entry q2 g1"| 36208₇ ||class="entry q3 g1"| 48635₁₃ ||class="entry q3 g1"| 45473₇ ||class="entry q2 g1"| 59046₉ ||class="entry q3 g1"| 55099₁₁ ||class="entry q3 g1"| 46049₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (37772, 63421, 62469, 63578, 40317, 63045) ||class="entry q2 g1"| 37772₇ ||class="entry q3 g1"| 63421₁₃ ||class="entry q3 g1"| 62469₇ ||class="entry q2 g1"| 63578₉ ||class="entry q3 g1"| 40317₁₁ ||class="entry q3 g1"| 63045₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (39002, 64891, 61475, 62348, 38843, 62051) ||class="entry q2 g1"| 39002₇ ||class="entry q3 g1"| 64891₁₃ ||class="entry q3 g1"| 61475₇ ||class="entry q2 g1"| 62348₉ ||class="entry q3 g1"| 38843₁₁ ||class="entry q3 g1"| 62051₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (42086, 64879, 52235, 53168, 38831, 52811) ||class="entry q2 g1"| 42086₇ ||class="entry q3 g1"| 64879₁₃ ||class="entry q3 g1"| 52235₇ ||class="entry q2 g1"| 53168₉ ||class="entry q3 g1"| 38831₁₁ ||class="entry q3 g1"| 52811₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (45388, 48623, 36233, 55962, 55087, 36809) ||class="entry q2 g1"| 45388₇ ||class="entry q3 g1"| 48623₁₃ ||class="entry q3 g1"| 36233₇ ||class="entry q2 g1"| 55962₉ ||class="entry q3 g1"| 55087₁₁ ||class="entry q3 g1"| 36809₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (50726, 63167, 41133, 44528, 40063, 41709) ||class="entry q2 g1"| 50726₇ ||class="entry q3 g1"| 63167₁₃ ||class="entry q3 g1"| 41133₇ ||class="entry q2 g1"| 44528₉ ||class="entry q3 g1"| 40063₁₁ ||class="entry q3 g1"| 41709₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (53786, 57023, 35001, 47564, 46207, 35577) ||class="entry q2 g1"| 53786₇ ||class="entry q3 g1"| 57023₁₃ ||class="entry q3 g1"| 35001₇ ||class="entry q2 g1"| 47564₉ ||class="entry q3 g1"| 46207₁₁ ||class="entry q3 g1"| 35577₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38828, 39435, 39077, 64634, 61643, 39653) ||class="entry q2 g1"| 38828₉ ||class="entry q3 g1"| 39435₇ ||class="entry q3 g1"| 39077₇ ||class="entry q2 g1"| 64634₁₁ ||class="entry q3 g1"| 61643₉ ||class="entry q3 g1"| 39653₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38840, 42531, 42137, 64622, 52451, 42713) ||class="entry q2 g1"| 38840₉ ||class="entry q3 g1"| 42531₇ ||class="entry q3 g1"| 42137₇ ||class="entry q2 g1"| 64622₁₁ ||class="entry q3 g1"| 52451₉ ||class="entry q3 g1"| 42713₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40058, 37069, 40067, 63404, 64013, 40643) ||class="entry q2 g1"| 40058₉ ||class="entry q3 g1"| 37069₇ ||class="entry q3 g1"| 40067₇ ||class="entry q2 g1"| 63404₁₁ ||class="entry q3 g1"| 64013₉ ||class="entry q3 g1"| 40643₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40312, 50277, 51497, 63150, 44709, 52073) ||class="entry q2 g1"| 40312₉ ||class="entry q3 g1"| 50277₇ ||class="entry q3 g1"| 51497₇ ||class="entry q2 g1"| 63150₁₁ ||class="entry q3 g1"| 44709₉ ||class="entry q3 g1"| 52073₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46190, 34033, 46211, 57272, 60977, 46787) ||class="entry q2 g1"| 46190₉ ||class="entry q3 g1"| 34033₇ ||class="entry q3 g1"| 46211₇ ||class="entry q2 g1"| 57272₁₁ ||class="entry q3 g1"| 60977₉ ||class="entry q3 g1"| 46787₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46444, 53337, 57641, 57018, 47769, 58217) ||class="entry q2 g1"| 46444₉ ||class="entry q3 g1"| 53337₇ ||class="entry q3 g1"| 57641₇ ||class="entry q2 g1"| 57018₁₁ ||class="entry q3 g1"| 47769₉ ||class="entry q3 g1"| 58217₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (54830, 36641, 55333, 48632, 58849, 55909) ||class="entry q2 g1"| 54830₉ ||class="entry q3 g1"| 36641₇ ||class="entry q3 g1"| 55333₇ ||class="entry q2 g1"| 48632₁₁ ||class="entry q3 g1"| 58849₉ ||class="entry q3 g1"| 55909₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (54842, 45833, 58393, 48620, 55753, 58969) ||class="entry q2 g1"| 54842₉ ||class="entry q3 g1"| 45833₇ ||class="entry q3 g1"| 58393₇ ||class="entry q2 g1"| 48620₁₁ ||class="entry q3 g1"| 55753₉ ||class="entry q3 g1"| 58969₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (48360, 60161, 54677, 55102, 33217, 55253) ||class="entry q2 g1"| 48360₉ ||class="entry q3 g1"| 60161₇ ||class="entry q3 g1"| 54677₉ ||class="entry q2 g1"| 55102₁₁ ||class="entry q3 g1"| 33217₅ ||class="entry q3 g1"| 55253₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (57000, 57553, 47411, 46462, 35345, 47987) ||class="entry q2 g1"| 57000₉ ||class="entry q3 g1"| 57553₇ ||class="entry q3 g1"| 47411₉ ||class="entry q2 g1"| 46462₁₁ ||class="entry q3 g1"| 35345₅ ||class="entry q3 g1"| 47987₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (63144, 51397, 44303, 40318, 41477, 44879) ||class="entry q2 g1"| 63144₉ ||class="entry q3 g1"| 51397₇ ||class="entry q3 g1"| 44303₉ ||class="entry q2 g1"| 40318₁₁ ||class="entry q3 g1"| 41477₅ ||class="entry q3 g1"| 44879₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (64616, 43651, 49343, 38846, 49219, 49919) ||class="entry q2 g1"| 64616₉ ||class="entry q3 g1"| 43651₇ ||class="entry q3 g1"| 49343₉ ||class="entry q2 g1"| 38846₁₁ ||class="entry q3 g1"| 49219₅ ||class="entry q3 g1"| 49919₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (48808, 32983, 48981, 54654, 59927, 48405) ||class="entry q2 g1"| 48808₉ ||class="entry q3 g1"| 32983₇ ||class="entry q3 g1"| 48981₁₁ ||class="entry q2 g1"| 54654₁₁ ||class="entry q3 g1"| 59927₉ ||class="entry q3 g1"| 48405₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (56552, 35591, 54259, 46910, 57799, 53683) ||class="entry q2 g1"| 56552₉ ||class="entry q3 g1"| 35591₇ ||class="entry q3 g1"| 54259₁₁ ||class="entry q2 g1"| 46910₁₁ ||class="entry q3 g1"| 57799₉ ||class="entry q3 g1"| 53683₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (62696, 41747, 51151, 40766, 51667, 50575) ||class="entry q2 g1"| 62696₉ ||class="entry q3 g1"| 41747₇ ||class="entry q3 g1"| 51151₁₁ ||class="entry q2 g1"| 40766₁₁ ||class="entry q3 g1"| 51667₉ ||class="entry q3 g1"| 50575₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (65064, 49493, 43647, 38398, 43925, 43071) ||class="entry q2 g1"| 65064₉ ||class="entry q3 g1"| 49493₇ ||class="entry q3 g1"| 43647₁₁ ||class="entry q2 g1"| 38398₁₁ ||class="entry q3 g1"| 43925₉ ||class="entry q3 g1"| 43071₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38574, 52899, 52495, 64888, 42083, 53071) ||class="entry q2 g1"| 38574₉ ||class="entry q3 g1"| 52899₉ ||class="entry q3 g1"| 52495₉ ||class="entry q2 g1"| 64888₁₁ ||class="entry q3 g1"| 42083₇ ||class="entry q3 g1"| 53071₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38586, 62091, 61747, 64876, 38987, 62323) ||class="entry q2 g1"| 38586₉ ||class="entry q3 g1"| 62091₉ ||class="entry q3 g1"| 61747₉ ||class="entry q2 g1"| 64876₁₁ ||class="entry q3 g1"| 38987₇ ||class="entry q3 g1"| 62323₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40046, 44261, 41151, 63416, 50725, 41727) ||class="entry q2 g1"| 40046₉ ||class="entry q3 g1"| 44261₉ ||class="entry q3 g1"| 41151₉ ||class="entry q2 g1"| 63416₁₁ ||class="entry q3 g1"| 50725₇ ||class="entry q3 g1"| 41727₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40300, 63565, 62741, 63162, 37517, 63317) ||class="entry q2 g1"| 40300₉ ||class="entry q3 g1"| 63565₉ ||class="entry q3 g1"| 62741₉ ||class="entry q2 g1"| 63162₁₁ ||class="entry q3 g1"| 37517₇ ||class="entry q3 g1"| 63317₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46202, 47321, 35007, 57260, 53785, 35583) ||class="entry q2 g1"| 46202₉ ||class="entry q3 g1"| 47321₉ ||class="entry q3 g1"| 35007₉ ||class="entry q2 g1"| 57260₁₁ ||class="entry q3 g1"| 53785₇ ||class="entry q3 g1"| 35583₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46456, 60529, 56597, 57006, 34481, 57173) ||class="entry q2 g1"| 46456₉ ||class="entry q3 g1"| 60529₉ ||class="entry q3 g1"| 56597₉ ||class="entry q2 g1"| 57006₁₁ ||class="entry q3 g1"| 34481₇ ||class="entry q3 g1"| 57173₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (55084, 56201, 36239, 48378, 45385, 36815) ||class="entry q2 g1"| 55084₉ ||class="entry q3 g1"| 56201₉ ||class="entry q3 g1"| 36239₉ ||class="entry q2 g1"| 48378₁₁ ||class="entry q3 g1"| 45385₇ ||class="entry q3 g1"| 36815₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (55096, 59297, 45491, 48366, 36193, 46067) ||class="entry q2 g1"| 55096₉ ||class="entry q3 g1"| 59297₉ ||class="entry q3 g1"| 45491₉ ||class="entry q2 g1"| 48366₁₁ ||class="entry q3 g1"| 36193₇ ||class="entry q3 g1"| 46067₁₁ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (38588, 38123, 38741, 64874, 65067, 38165) ||class="entry q2 g1"| 38588₉ ||class="entry q3 g1"| 38123₉ ||class="entry q3 g1"| 38741₉ ||class="entry q2 g1"| 64874₁₁ ||class="entry q3 g1"| 65067₁₁ ||class="entry q3 g1"| 38165₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (40298, 40493, 37747, 63164, 62701, 37171) ||class="entry q2 g1"| 40298₉ ||class="entry q3 g1"| 40493₉ ||class="entry q3 g1"| 37747₉ ||class="entry q2 g1"| 63164₁₁ ||class="entry q3 g1"| 62701₁₁ ||class="entry q3 g1"| 37171₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (46442, 46649, 34639, 57020, 56569, 34063) ||class="entry q2 g1"| 46442₉ ||class="entry q3 g1"| 46649₉ ||class="entry q3 g1"| 34639₉ ||class="entry q2 g1"| 57020₁₁ ||class="entry q3 g1"| 56569₁₁ ||class="entry q3 g1"| 34063₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (54844, 54633, 33407, 48618, 49065, 32831) ||class="entry q2 g1"| 54844₉ ||class="entry q3 g1"| 54633₉ ||class="entry q3 g1"| 33407₉ ||class="entry q2 g1"| 48618₁₁ ||class="entry q3 g1"| 49065₁₁ ||class="entry q3 g1"| 32831₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (38126, 42357, 42959, 65336, 53173, 42383) ||class="entry q2 g1"| 38126₉ ||class="entry q3 g1"| 42357₉ ||class="entry q3 g1"| 42959₁₁ ||class="entry q2 g1"| 65336₁₁ ||class="entry q3 g1"| 53173₁₁ ||class="entry q3 g1"| 42383₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (38138, 39261, 39923, 65324, 62365, 39347) ||class="entry q2 g1"| 38138₉ ||class="entry q3 g1"| 39261₉ ||class="entry q3 g1"| 39923₁₁ ||class="entry q2 g1"| 65324₁₁ ||class="entry q3 g1"| 62365₁₁ ||class="entry q3 g1"| 39347₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40494, 50995, 51839, 62968, 44531, 51263) ||class="entry q2 g1"| 40494₉ ||class="entry q3 g1"| 50995₉ ||class="entry q3 g1"| 51839₁₁ ||class="entry q2 g1"| 62968₁₁ ||class="entry q3 g1"| 44531₁₁ ||class="entry q3 g1"| 51263₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40748, 37787, 40917, 62714, 63835, 40341) ||class="entry q2 g1"| 40748₉ ||class="entry q3 g1"| 37787₉ ||class="entry q3 g1"| 40917₁₁ ||class="entry q2 g1"| 62714₁₁ ||class="entry q3 g1"| 63835₁₁ ||class="entry q3 g1"| 40341₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46650, 54031, 57983, 56812, 47567, 57407) ||class="entry q2 g1"| 46650₉ ||class="entry q3 g1"| 54031₉ ||class="entry q3 g1"| 57983₁₁ ||class="entry q2 g1"| 56812₁₁ ||class="entry q3 g1"| 47567₁₁ ||class="entry q3 g1"| 57407₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46904, 34727, 47061, 56558, 60775, 46485) ||class="entry q2 g1"| 46904₉ ||class="entry q3 g1"| 34727₉ ||class="entry q3 g1"| 47061₁₁ ||class="entry q2 g1"| 56558₁₁ ||class="entry q3 g1"| 60775₁₁ ||class="entry q3 g1"| 46485₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54636, 45151, 59215, 48826, 55967, 58639) ||class="entry q2 g1"| 54636₉ ||class="entry q3 g1"| 45151₉ ||class="entry q3 g1"| 59215₁₁ ||class="entry q2 g1"| 48826₁₁ ||class="entry q3 g1"| 55967₁₁ ||class="entry q3 g1"| 58639₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54648, 35959, 56179, 48814, 59063, 55603) ||class="entry q2 g1"| 54648₉ ||class="entry q3 g1"| 35959₉ ||class="entry q3 g1"| 56179₁₁ ||class="entry q2 g1"| 48814₁₁ ||class="entry q3 g1"| 59063₁₁ ||class="entry q3 g1"| 55603₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (38378, 38845, 37891, 65084, 64893, 38467) ||class="entry q2 g1"| 38378₉ ||class="entry q3 g1"| 38845₁₁ ||class="entry q3 g1"| 37891₅ ||class="entry q2 g1"| 65084₁₁ ||class="entry q3 g1"| 64893₁₃ ||class="entry q3 g1"| 38467₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (40508, 40315, 36901, 62954, 63419, 37477) ||class="entry q2 g1"| 40508₉ ||class="entry q3 g1"| 40315₁₁ ||class="entry q3 g1"| 36901₅ ||class="entry q2 g1"| 62954₁₁ ||class="entry q3 g1"| 63419₁₃ ||class="entry q3 g1"| 37477₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (46652, 46447, 33817, 56810, 57263, 34393) ||class="entry q2 g1"| 46652₉ ||class="entry q3 g1"| 46447₁₁ ||class="entry q3 g1"| 33817₅ ||class="entry q2 g1"| 56810₁₁ ||class="entry q3 g1"| 57263₁₃ ||class="entry q3 g1"| 34393₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54634, 54847, 33065, 48828, 48383, 33641) ||class="entry q2 g1"| 54634₉ ||class="entry q3 g1"| 54847₁₁ ||class="entry q3 g1"| 33065₅ ||class="entry q2 g1"| 48828₁₁ ||class="entry q3 g1"| 48383₁₃ ||class="entry q3 g1"| 33641₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (38380, 61917, 62053, 65082, 39709, 61477) ||class="entry q2 g1"| 38380₉ ||class="entry q3 g1"| 61917₁₁ ||class="entry q3 g1"| 62053₉ ||class="entry q2 g1"| 65082₁₁ ||class="entry q3 g1"| 39709₉ ||class="entry q3 g1"| 61477₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (38392, 52725, 52825, 65070, 42805, 52249) ||class="entry q2 g1"| 38392₉ ||class="entry q3 g1"| 52725₁₁ ||class="entry q3 g1"| 52825₉ ||class="entry q2 g1"| 65070₁₁ ||class="entry q3 g1"| 42805₉ ||class="entry q3 g1"| 52249₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40506, 64283, 63043, 62956, 37339, 62467) ||class="entry q2 g1"| 40506₉ ||class="entry q3 g1"| 64283₁₁ ||class="entry q3 g1"| 63043₉ ||class="entry q2 g1"| 62956₁₁ ||class="entry q3 g1"| 37339₉ ||class="entry q3 g1"| 62467₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40760, 44979, 41961, 62702, 50547, 41385) ||class="entry q2 g1"| 40760₉ ||class="entry q3 g1"| 44979₁₁ ||class="entry q3 g1"| 41961₉ ||class="entry q2 g1"| 62702₁₁ ||class="entry q3 g1"| 50547₉ ||class="entry q3 g1"| 41385₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46638, 61223, 56899, 56824, 34279, 56323) ||class="entry q2 g1"| 46638₉ ||class="entry q3 g1"| 61223₁₁ ||class="entry q3 g1"| 56899₉ ||class="entry q2 g1"| 56824₁₁ ||class="entry q3 g1"| 34279₉ ||class="entry q3 g1"| 56323₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46892, 48015, 35817, 56570, 53583, 35241) ||class="entry q2 g1"| 46892₉ ||class="entry q3 g1"| 48015₁₁ ||class="entry q3 g1"| 35817₉ ||class="entry q2 g1"| 56570₁₁ ||class="entry q3 g1"| 53583₉ ||class="entry q3 g1"| 35241₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54382, 58615, 45797, 49080, 36407, 45221) ||class="entry q2 g1"| 54382₉ ||class="entry q3 g1"| 58615₁₁ ||class="entry q3 g1"| 45797₉ ||class="entry q2 g1"| 49080₁₁ ||class="entry q3 g1"| 36407₉ ||class="entry q3 g1"| 45221₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54394, 55519, 36569, 49068, 45599, 35993) ||class="entry q2 g1"| 54394₉ ||class="entry q3 g1"| 55519₁₁ ||class="entry q3 g1"| 36569₉ ||class="entry q2 g1"| 49068₁₁ ||class="entry q3 g1"| 45599₉ ||class="entry q3 g1"| 35993₇ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38826, 64619, 65219, 64636, 38571, 64643) ||class="entry q2 g1"| 38826₉ ||class="entry q3 g1"| 64619₁₁ ||class="entry q3 g1"| 65219₁₁ ||class="entry q2 g1"| 64636₁₁ ||class="entry q3 g1"| 38571₉ ||class="entry q3 g1"| 64643₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (40060, 63149, 64229, 63402, 40045, 63653) ||class="entry q2 g1"| 40060₉ ||class="entry q3 g1"| 63149₁₁ ||class="entry q3 g1"| 64229₁₁ ||class="entry q2 g1"| 63402₁₁ ||class="entry q3 g1"| 40045₉ ||class="entry q3 g1"| 63653₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (46204, 57017, 61145, 57258, 46201, 60569) ||class="entry q2 g1"| 46204₉ ||class="entry q3 g1"| 57017₁₁ ||class="entry q3 g1"| 61145₁₁ ||class="entry q2 g1"| 57258₁₁ ||class="entry q3 g1"| 46201₉ ||class="entry q3 g1"| 60569₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (55082, 48617, 60393, 48380, 55081, 59817) ||class="entry q2 g1"| 55082₉ ||class="entry q3 g1"| 48617₁₁ ||class="entry q3 g1"| 60393₁₁ ||class="entry q2 g1"| 48380₁₁ ||class="entry q3 g1"| 55081₉ ||class="entry q3 g1"| 59817₉ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (38140, 65341, 64917, 65322, 38397, 65493) ||class="entry q2 g1"| 38140₉ ||class="entry q3 g1"| 65341₁₃ ||class="entry q3 g1"| 64917₁₁ ||class="entry q2 g1"| 65322₁₁ ||class="entry q3 g1"| 38397₁₁ ||class="entry q3 g1"| 65493₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (40746, 62971, 63923, 62716, 40763, 64499) ||class="entry q2 g1"| 40746₉ ||class="entry q3 g1"| 62971₁₃ ||class="entry q3 g1"| 63923₁₁ ||class="entry q2 g1"| 62716₁₁ ||class="entry q3 g1"| 40763₁₁ ||class="entry q3 g1"| 64499₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (46890, 56815, 60815, 56572, 46895, 61391) ||class="entry q2 g1"| 46890₉ ||class="entry q3 g1"| 56815₁₃ ||class="entry q3 g1"| 60815₁₁ ||class="entry q2 g1"| 56572₁₁ ||class="entry q3 g1"| 46895₁₁ ||class="entry q3 g1"| 61391₁₃ |- |class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (54396, 48831, 59583, 49066, 54399, 60159) ||class="entry q2 g1"| 54396₉ ||class="entry q3 g1"| 48831₁₃ ||class="entry q3 g1"| 59583₁₁ ||class="entry q2 g1"| 49066₁₁ ||class="entry q3 g1"| 54399₁₁ ||class="entry q3 g1"| 60159₁₃ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (32840, 56882, 45974, 34174, 56068, 46752) ||class="entry q2 g1"| 32840₃ ||class="entry q2 g1"| 56882₉ ||class="entry q2 g1"| 45974₉ ||class="entry q2 g1"| 34174₉ ||class="entry q2 g1"| 56068₇ ||class="entry q2 g1"| 46752₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (33288, 46564, 55638, 34622, 45266, 56416) ||class="entry q2 g1"| 33288₃ ||class="entry q2 g1"| 46564₉ ||class="entry q2 g1"| 55638₉ ||class="entry q2 g1"| 34622₉ ||class="entry q2 g1"| 45266₇ ||class="entry q2 g1"| 56416₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (36928, 42924, 51998, 38262, 41626, 52776) ||class="entry q2 g1"| 36928₃ ||class="entry q2 g1"| 42924₉ ||class="entry q2 g1"| 51998₉ ||class="entry q2 g1"| 38262₉ ||class="entry q2 g1"| 41626₇ ||class="entry q2 g1"| 52776₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (37376, 52346, 41438, 38710, 51532, 42216) ||class="entry q2 g1"| 37376₃ ||class="entry q2 g1"| 52346₉ ||class="entry q2 g1"| 41438₉ ||class="entry q2 g1"| 38710₉ ||class="entry q2 g1"| 51532₇ ||class="entry q2 g1"| 42216₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (41608, 37984, 51738, 42942, 37206, 53036) ||class="entry q2 g1"| 41608₅ ||class="entry q2 g1"| 37984₅ ||class="entry q2 g1"| 51738₇ ||class="entry q2 g1"| 42942₁₁ ||class="entry q2 g1"| 37206₇ ||class="entry q2 g1"| 53036₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (45248, 34344, 55378, 46582, 33566, 56676) ||class="entry q2 g1"| 45248₅ ||class="entry q2 g1"| 34344₅ ||class="entry q2 g1"| 55378₇ ||class="entry q2 g1"| 46582₁₁ ||class="entry q2 g1"| 33566₇ ||class="entry q2 g1"| 56676₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (51272, 38432, 41420, 52606, 37654, 42234) ||class="entry q2 g1"| 51272₅ ||class="entry q2 g1"| 38432₅ ||class="entry q2 g1"| 41420₇ ||class="entry q2 g1"| 52606₁₁ ||class="entry q2 g1"| 37654₇ ||class="entry q2 g1"| 42234₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (55808, 33896, 45956, 57142, 33118, 46770) ||class="entry q2 g1"| 55808₅ ||class="entry q2 g1"| 33896₅ ||class="entry q2 g1"| 45956₇ ||class="entry q2 g1"| 57142₁₁ ||class="entry q2 g1"| 33118₇ ||class="entry q2 g1"| 46770₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (32858, 33914, 59852, 34156, 33100, 60666) ||class="entry q2 g1"| 32858₅ ||class="entry q2 g1"| 33914₇ ||class="entry q2 g1"| 59852₉ ||class="entry q2 g1"| 34156₇ ||class="entry q2 g1"| 33100₅ ||class="entry q2 g1"| 60666₁₁ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (33548, 34604, 60058, 34362, 33306, 61356) ||class="entry q2 g1"| 33548₅ ||class="entry q2 g1"| 34604₇ ||class="entry q2 g1"| 60058₉ ||class="entry q2 g1"| 34362₇ ||class="entry q2 g1"| 33306₅ ||class="entry q2 g1"| 61356₁₁ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37188, 38244, 63698, 38002, 36946, 64996) ||class="entry q2 g1"| 37188₅ ||class="entry q2 g1"| 38244₇ ||class="entry q2 g1"| 63698₉ ||class="entry q2 g1"| 38002₇ ||class="entry q2 g1"| 36946₅ ||class="entry q2 g1"| 64996₁₁ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37394, 38450, 64388, 38692, 37636, 65202) ||class="entry q2 g1"| 37394₅ ||class="entry q2 g1"| 38450₇ ||class="entry q2 g1"| 64388₉ ||class="entry q2 g1"| 38692₇ ||class="entry q2 g1"| 37636₅ ||class="entry q2 g1"| 65202₁₁ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32846, 47186, 54768, 34168, 48484, 53446) ||class="entry q2 g1"| 32846₅ ||class="entry q2 g1"| 47186₇ ||class="entry q2 g1"| 54768₉ ||class="entry q2 g1"| 34168₇ ||class="entry q2 g1"| 48484₉ ||class="entry q2 g1"| 53446₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32860, 57882, 36778, 34154, 59180, 35484) ||class="entry q2 g1"| 32860₅ ||class="entry q2 g1"| 57882₇ ||class="entry q2 g1"| 36778₉ ||class="entry q2 g1"| 34154₇ ||class="entry q2 g1"| 59180₉ ||class="entry q2 g1"| 35484₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33098, 35482, 58940, 33916, 36780, 58122) ||class="entry q2 g1"| 33098₅ ||class="entry q2 g1"| 35482₇ ||class="entry q2 g1"| 58940₉ ||class="entry q2 g1"| 33916₇ ||class="entry q2 g1"| 36780₉ ||class="entry q2 g1"| 58122₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33112, 53458, 48230, 33902, 54756, 47440) ||class="entry q2 g1"| 33112₅ ||class="entry q2 g1"| 53458₇ ||class="entry q2 g1"| 48230₉ ||class="entry q2 g1"| 33902₇ ||class="entry q2 g1"| 54756₉ ||class="entry q2 g1"| 47440₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33294, 54148, 48944, 34616, 54962, 47622) ||class="entry q2 g1"| 33294₅ ||class="entry q2 g1"| 54148₇ ||class="entry q2 g1"| 48944₉ ||class="entry q2 g1"| 34616₇ ||class="entry q2 g1"| 54962₉ ||class="entry q2 g1"| 47622₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33308, 35276, 58730, 34602, 36090, 57436) ||class="entry q2 g1"| 33308₅ ||class="entry q2 g1"| 35276₇ ||class="entry q2 g1"| 58730₉ ||class="entry q2 g1"| 34602₇ ||class="entry q2 g1"| 36090₉ ||class="entry q2 g1"| 57436₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33546, 57676, 36092, 34364, 58490, 35274) ||class="entry q2 g1"| 33546₅ ||class="entry q2 g1"| 57676₇ ||class="entry q2 g1"| 36092₉ ||class="entry q2 g1"| 34364₇ ||class="entry q2 g1"| 58490₉ ||class="entry q2 g1"| 35274₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33560, 47876, 54950, 34350, 48690, 54160) ||class="entry q2 g1"| 33560₅ ||class="entry q2 g1"| 47876₇ ||class="entry q2 g1"| 54950₉ ||class="entry q2 g1"| 34350₇ ||class="entry q2 g1"| 48690₉ ||class="entry q2 g1"| 54160₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (36934, 49612, 44408, 38256, 50426, 43086) ||class="entry q2 g1"| 36934₅ ||class="entry q2 g1"| 49612₇ ||class="entry q2 g1"| 44408₉ ||class="entry q2 g1"| 38256₇ ||class="entry q2 g1"| 50426₉ ||class="entry q2 g1"| 43086₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (36948, 39812, 63266, 38242, 40626, 61972) ||class="entry q2 g1"| 36948₅ ||class="entry q2 g1"| 39812₇ ||class="entry q2 g1"| 63266₉ ||class="entry q2 g1"| 38242₇ ||class="entry q2 g1"| 40626₉ ||class="entry q2 g1"| 61972₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37186, 62212, 40628, 38004, 63026, 39810) ||class="entry q2 g1"| 37186₅ ||class="entry q2 g1"| 62212₇ ||class="entry q2 g1"| 40628₉ ||class="entry q2 g1"| 38004₇ ||class="entry q2 g1"| 63026₉ ||class="entry q2 g1"| 39810₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37200, 43340, 50414, 37990, 44154, 49624) ||class="entry q2 g1"| 37200₅ ||class="entry q2 g1"| 43340₇ ||class="entry q2 g1"| 50414₉ ||class="entry q2 g1"| 37990₇ ||class="entry q2 g1"| 44154₉ ||class="entry q2 g1"| 49624₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37382, 43546, 51128, 38704, 44844, 49806) ||class="entry q2 g1"| 37382₅ ||class="entry q2 g1"| 43546₇ ||class="entry q2 g1"| 51128₉ ||class="entry q2 g1"| 38704₇ ||class="entry q2 g1"| 44844₉ ||class="entry q2 g1"| 49806₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37396, 61522, 40418, 38690, 62820, 39124) ||class="entry q2 g1"| 37396₅ ||class="entry q2 g1"| 61522₇ ||class="entry q2 g1"| 40418₉ ||class="entry q2 g1"| 38690₇ ||class="entry q2 g1"| 62820₉ ||class="entry q2 g1"| 39124₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37634, 39122, 62580, 38452, 40420, 61762) ||class="entry q2 g1"| 37634₅ ||class="entry q2 g1"| 39122₇ ||class="entry q2 g1"| 62580₉ ||class="entry q2 g1"| 38452₇ ||class="entry q2 g1"| 40420₉ ||class="entry q2 g1"| 61762₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37648, 49818, 44590, 38438, 51116, 43800) ||class="entry q2 g1"| 37648₅ ||class="entry q2 g1"| 49818₇ ||class="entry q2 g1"| 44590₉ ||class="entry q2 g1"| 38438₇ ||class="entry q2 g1"| 51116₉ ||class="entry q2 g1"| 43800₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (35016, 55202, 46310, 36350, 53908, 45520) ||class="entry q2 g1"| 35016₅ ||class="entry q2 g1"| 55202₉ ||class="entry q2 g1"| 46310₉ ||class="entry q2 g1"| 36350₁₁ ||class="entry q2 g1"| 53908₇ ||class="entry q2 g1"| 45520₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (35464, 48244, 56870, 36798, 47426, 56080) ||class="entry q2 g1"| 35464₅ ||class="entry q2 g1"| 48244₉ ||class="entry q2 g1"| 56870₉ ||class="entry q2 g1"| 36798₁₁ ||class="entry q2 g1"| 47426₇ ||class="entry q2 g1"| 56080₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (39104, 44604, 52334, 40438, 43786, 51544) ||class="entry q2 g1"| 39104₅ ||class="entry q2 g1"| 44604₉ ||class="entry q2 g1"| 52334₉ ||class="entry q2 g1"| 40438₁₁ ||class="entry q2 g1"| 43786₇ ||class="entry q2 g1"| 51544₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (39552, 50666, 42670, 40886, 49372, 41880) ||class="entry q2 g1"| 39552₅ ||class="entry q2 g1"| 50666₉ ||class="entry q2 g1"| 42670₉ ||class="entry q2 g1"| 40886₁₁ ||class="entry q2 g1"| 49372₇ ||class="entry q2 g1"| 41880₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43080, 63014, 42922, 44414, 62224, 41628) ||class="entry q2 g1"| 43080₅ ||class="entry q2 g1"| 63014₉ ||class="entry q2 g1"| 42922₉ ||class="entry q2 g1"| 44414₁₁ ||class="entry q2 g1"| 62224₇ ||class="entry q2 g1"| 41628₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43528, 40432, 52586, 44862, 39110, 51292) ||class="entry q2 g1"| 43528₅ ||class="entry q2 g1"| 40432₉ ||class="entry q2 g1"| 52586₉ ||class="entry q2 g1"| 44862₁₁ ||class="entry q2 g1"| 39110₇ ||class="entry q2 g1"| 51292₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (47168, 36792, 57122, 48502, 35470, 55828) ||class="entry q2 g1"| 47168₅ ||class="entry q2 g1"| 36792₉ ||class="entry q2 g1"| 57122₉ ||class="entry q2 g1"| 48502₁₁ ||class="entry q2 g1"| 35470₇ ||class="entry q2 g1"| 55828₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (47616, 58478, 46562, 48950, 57688, 45268) ||class="entry q2 g1"| 47616₅ ||class="entry q2 g1"| 58478₉ ||class="entry q2 g1"| 46562₉ ||class="entry q2 g1"| 48950₁₁ ||class="entry q2 g1"| 57688₇ ||class="entry q2 g1"| 45268₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (49352, 40880, 42684, 50686, 39558, 41866) ||class="entry q2 g1"| 49352₅ ||class="entry q2 g1"| 40880₉ ||class="entry q2 g1"| 42684₉ ||class="entry q2 g1"| 50686₁₁ ||class="entry q2 g1"| 39558₇ ||class="entry q2 g1"| 41866₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (49800, 62566, 52348, 51134, 61776, 51530) ||class="entry q2 g1"| 49800₅ ||class="entry q2 g1"| 62566₉ ||class="entry q2 g1"| 52348₉ ||class="entry q2 g1"| 51134₁₁ ||class="entry q2 g1"| 61776₇ ||class="entry q2 g1"| 51530₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (53440, 58926, 56884, 54774, 58136, 56066) ||class="entry q2 g1"| 53440₅ ||class="entry q2 g1"| 58926₉ ||class="entry q2 g1"| 56884₉ ||class="entry q2 g1"| 54774₁₁ ||class="entry q2 g1"| 58136₇ ||class="entry q2 g1"| 56066₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (53888, 36344, 46324, 55222, 35022, 45506) ||class="entry q2 g1"| 53888₅ ||class="entry q2 g1"| 36344₉ ||class="entry q2 g1"| 46324₉ ||class="entry q2 g1"| 55222₁₁ ||class="entry q2 g1"| 35022₇ ||class="entry q2 g1"| 45506₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (57416, 48692, 46576, 58750, 47874, 45254) ||class="entry q2 g1"| 57416₅ ||class="entry q2 g1"| 48692₉ ||class="entry q2 g1"| 46576₉ ||class="entry q2 g1"| 58750₁₁ ||class="entry q2 g1"| 47874₇ ||class="entry q2 g1"| 45254₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (57864, 54754, 57136, 59198, 53460, 55814) ||class="entry q2 g1"| 57864₅ ||class="entry q2 g1"| 54754₉ ||class="entry q2 g1"| 57136₉ ||class="entry q2 g1"| 59198₁₁ ||class="entry q2 g1"| 53460₇ ||class="entry q2 g1"| 55814₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (61504, 51114, 52600, 62838, 49820, 51278) ||class="entry q2 g1"| 61504₅ ||class="entry q2 g1"| 51114₉ ||class="entry q2 g1"| 52600₉ ||class="entry q2 g1"| 62838₁₁ ||class="entry q2 g1"| 49820₇ ||class="entry q2 g1"| 51278₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (61952, 44156, 42936, 63286, 43338, 41614) ||class="entry q2 g1"| 61952₅ ||class="entry q2 g1"| 44156₉ ||class="entry q2 g1"| 42936₉ ||class="entry q2 g1"| 63286₁₁ ||class="entry q2 g1"| 43338₇ ||class="entry q2 g1"| 41614₇ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (41160, 65462, 41178, 42494, 64128, 42476) ||class="entry q2 g1"| 41160₅ ||class="entry q2 g1"| 65462₁₃ ||class="entry q2 g1"| 41178₇ ||class="entry q2 g1"| 42494₁₁ ||class="entry q2 g1"| 64128₇ ||class="entry q2 g1"| 42476₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (45696, 60926, 45714, 47030, 59592, 47012) ||class="entry q2 g1"| 45696₅ ||class="entry q2 g1"| 60926₁₃ ||class="entry q2 g1"| 45714₇ ||class="entry q2 g1"| 47030₁₁ ||class="entry q2 g1"| 59592₇ ||class="entry q2 g1"| 47012₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (51720, 65014, 51980, 53054, 63680, 52794) ||class="entry q2 g1"| 51720₅ ||class="entry q2 g1"| 65014₁₃ ||class="entry q2 g1"| 51980₇ ||class="entry q2 g1"| 53054₁₁ ||class="entry q2 g1"| 63680₇ ||class="entry q2 g1"| 52794₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (55360, 61374, 55620, 56694, 60040, 56434) ||class="entry q2 g1"| 55360₅ ||class="entry q2 g1"| 61374₁₃ ||class="entry q2 g1"| 55620₇ ||class="entry q2 g1"| 56694₁₁ ||class="entry q2 g1"| 60040₇ ||class="entry q2 g1"| 56434₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36072, 47636, 55366, 35294, 48930, 56688) ||class="entry q2 g1"| 36072₇ ||class="entry q2 g1"| 47636₇ ||class="entry q2 g1"| 55366₇ ||class="entry q2 g1"| 35294₉ ||class="entry q2 g1"| 48930₉ ||class="entry q2 g1"| 56688₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36520, 53698, 45702, 35742, 54516, 47024) ||class="entry q2 g1"| 36520₇ ||class="entry q2 g1"| 53698₇ ||class="entry q2 g1"| 45702₇ ||class="entry q2 g1"| 35742₉ ||class="entry q2 g1"| 54516₉ ||class="entry q2 g1"| 47024₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (40160, 50058, 41166, 39382, 50876, 42488) ||class="entry q2 g1"| 40160₇ ||class="entry q2 g1"| 50058₇ ||class="entry q2 g1"| 41166₇ ||class="entry q2 g1"| 39382₉ ||class="entry q2 g1"| 50876₉ ||class="entry q2 g1"| 42488₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (40608, 43100, 51726, 39830, 44394, 53048) ||class="entry q2 g1"| 40608₇ ||class="entry q2 g1"| 43100₇ ||class="entry q2 g1"| 51726₇ ||class="entry q2 g1"| 39830₉ ||class="entry q2 g1"| 44394₉ ||class="entry q2 g1"| 53048₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (44136, 39824, 51978, 43358, 40614, 52796) ||class="entry q2 g1"| 44136₇ ||class="entry q2 g1"| 39824₇ ||class="entry q2 g1"| 51978₇ ||class="entry q2 g1"| 43358₉ ||class="entry q2 g1"| 40614₉ ||class="entry q2 g1"| 52796₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (44584, 61510, 41418, 43806, 62832, 42236) ||class="entry q2 g1"| 44584₇ ||class="entry q2 g1"| 61510₇ ||class="entry q2 g1"| 41418₇ ||class="entry q2 g1"| 43806₉ ||class="entry q2 g1"| 62832₉ ||class="entry q2 g1"| 42236₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (48224, 57870, 45954, 47446, 59192, 46772) ||class="entry q2 g1"| 48224₇ ||class="entry q2 g1"| 57870₇ ||class="entry q2 g1"| 45954₇ ||class="entry q2 g1"| 47446₉ ||class="entry q2 g1"| 59192₉ ||class="entry q2 g1"| 46772₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (48672, 35288, 55618, 47894, 36078, 56436) ||class="entry q2 g1"| 48672₇ ||class="entry q2 g1"| 35288₇ ||class="entry q2 g1"| 55618₇ ||class="entry q2 g1"| 47894₉ ||class="entry q2 g1"| 36078₉ ||class="entry q2 g1"| 56436₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (50408, 61958, 51740, 49630, 63280, 53034) ||class="entry q2 g1"| 50408₇ ||class="entry q2 g1"| 61958₇ ||class="entry q2 g1"| 51740₇ ||class="entry q2 g1"| 49630₉ ||class="entry q2 g1"| 63280₉ ||class="entry q2 g1"| 53034₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (50856, 39376, 41180, 50078, 40166, 42474) ||class="entry q2 g1"| 50856₇ ||class="entry q2 g1"| 39376₇ ||class="entry q2 g1"| 41180₇ ||class="entry q2 g1"| 50078₉ ||class="entry q2 g1"| 40166₉ ||class="entry q2 g1"| 42474₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54496, 35736, 45716, 53718, 36526, 47010) ||class="entry q2 g1"| 54496₇ ||class="entry q2 g1"| 35736₇ ||class="entry q2 g1"| 45716₇ ||class="entry q2 g1"| 53718₉ ||class="entry q2 g1"| 36526₉ ||class="entry q2 g1"| 47010₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54944, 57422, 55380, 54166, 58744, 56674) ||class="entry q2 g1"| 54944₇ ||class="entry q2 g1"| 57422₇ ||class="entry q2 g1"| 55380₇ ||class="entry q2 g1"| 54166₉ ||class="entry q2 g1"| 58744₉ ||class="entry q2 g1"| 56674₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (58472, 54146, 55632, 57694, 54964, 56422) ||class="entry q2 g1"| 58472₇ ||class="entry q2 g1"| 54146₇ ||class="entry q2 g1"| 55632₇ ||class="entry q2 g1"| 57694₉ ||class="entry q2 g1"| 54964₉ ||class="entry q2 g1"| 56422₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (58920, 47188, 45968, 58142, 48482, 46758) ||class="entry q2 g1"| 58920₇ ||class="entry q2 g1"| 47188₇ ||class="entry q2 g1"| 45968₇ ||class="entry q2 g1"| 58142₉ ||class="entry q2 g1"| 48482₉ ||class="entry q2 g1"| 46758₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (62560, 43548, 41432, 61782, 44842, 42222) ||class="entry q2 g1"| 62560₇ ||class="entry q2 g1"| 43548₇ ||class="entry q2 g1"| 41432₇ ||class="entry q2 g1"| 61782₉ ||class="entry q2 g1"| 44842₉ ||class="entry q2 g1"| 42222₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (63008, 49610, 51992, 62230, 50428, 52782) ||class="entry q2 g1"| 63008₇ ||class="entry q2 g1"| 49610₇ ||class="entry q2 g1"| 51992₇ ||class="entry q2 g1"| 62230₉ ||class="entry q2 g1"| 50428₉ ||class="entry q2 g1"| 52782₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (41868, 42664, 63958, 42682, 41886, 64736) ||class="entry q2 g1"| 41868₇ ||class="entry q2 g1"| 42664₇ ||class="entry q2 g1"| 63958₁₁ ||class="entry q2 g1"| 42682₉ ||class="entry q2 g1"| 41886₉ ||class="entry q2 g1"| 64736₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (45508, 46304, 60318, 46322, 45526, 61096) ||class="entry q2 g1"| 45508₇ ||class="entry q2 g1"| 46304₇ ||class="entry q2 g1"| 60318₁₁ ||class="entry q2 g1"| 46322₉ ||class="entry q2 g1"| 45526₉ ||class="entry q2 g1"| 61096₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (51290, 52328, 64406, 52588, 51550, 65184) ||class="entry q2 g1"| 51290₇ ||class="entry q2 g1"| 52328₇ ||class="entry q2 g1"| 64406₁₁ ||class="entry q2 g1"| 52588₉ ||class="entry q2 g1"| 51550₉ ||class="entry q2 g1"| 65184₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (55826, 56864, 59870, 57124, 56086, 60648) ||class="entry q2 g1"| 55826₇ ||class="entry q2 g1"| 56864₇ ||class="entry q2 g1"| 59870₁₁ ||class="entry q2 g1"| 57124₉ ||class="entry q2 g1"| 56086₉ ||class="entry q2 g1"| 60648₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35036, 60298, 35034, 36330, 61116, 36332) ||class="entry q2 g1"| 35036₇ ||class="entry q2 g1"| 60298₉ ||class="entry q2 g1"| 35034₇ ||class="entry q2 g1"| 36330₉ ||class="entry q2 g1"| 61116₁₁ ||class="entry q2 g1"| 36332₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35722, 59612, 35724, 36540, 60906, 36538) ||class="entry q2 g1"| 35722₇ ||class="entry q2 g1"| 59612₉ ||class="entry q2 g1"| 35724₇ ||class="entry q2 g1"| 36540₉ ||class="entry q2 g1"| 60906₁₁ ||class="entry q2 g1"| 36538₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39362, 64148, 39364, 40180, 65442, 40178) ||class="entry q2 g1"| 39362₇ ||class="entry q2 g1"| 64148₉ ||class="entry q2 g1"| 39364₇ ||class="entry q2 g1"| 40180₉ ||class="entry q2 g1"| 65442₁₁ ||class="entry q2 g1"| 40178₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39572, 63938, 39570, 40866, 64756, 40868) ||class="entry q2 g1"| 39572₇ ||class="entry q2 g1"| 63938₉ ||class="entry q2 g1"| 39570₇ ||class="entry q2 g1"| 40866₉ ||class="entry q2 g1"| 64756₁₁ ||class="entry q2 g1"| 40868₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43352, 63686, 43098, 44142, 65008, 44396) ||class="entry q2 g1"| 43352₇ ||class="entry q2 g1"| 63686₉ ||class="entry q2 g1"| 43098₇ ||class="entry q2 g1"| 44142₉ ||class="entry q2 g1"| 65008₁₁ ||class="entry q2 g1"| 44396₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43534, 64400, 43788, 44856, 65190, 44602) ||class="entry q2 g1"| 43534₇ ||class="entry q2 g1"| 64400₉ ||class="entry q2 g1"| 43788₇ ||class="entry q2 g1"| 44856₉ ||class="entry q2 g1"| 65190₁₁ ||class="entry q2 g1"| 44602₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (47174, 59864, 47428, 48496, 60654, 48242) ||class="entry q2 g1"| 47174₇ ||class="entry q2 g1"| 59864₉ ||class="entry q2 g1"| 47428₇ ||class="entry q2 g1"| 48496₉ ||class="entry q2 g1"| 60654₁₁ ||class="entry q2 g1"| 48242₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (47888, 60046, 47634, 48678, 61368, 48932) ||class="entry q2 g1"| 47888₇ ||class="entry q2 g1"| 60046₉ ||class="entry q2 g1"| 47634₇ ||class="entry q2 g1"| 48678₉ ||class="entry q2 g1"| 61368₁₁ ||class="entry q2 g1"| 48932₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (49358, 63952, 49370, 50680, 64742, 50668) ||class="entry q2 g1"| 49358₇ ||class="entry q2 g1"| 63952₉ ||class="entry q2 g1"| 49370₇ ||class="entry q2 g1"| 50680₉ ||class="entry q2 g1"| 64742₁₁ ||class="entry q2 g1"| 50668₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50072, 64134, 50060, 50862, 65456, 50874) ||class="entry q2 g1"| 50072₇ ||class="entry q2 g1"| 64134₉ ||class="entry q2 g1"| 50060₇ ||class="entry q2 g1"| 50862₉ ||class="entry q2 g1"| 65456₁₁ ||class="entry q2 g1"| 50874₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (53712, 59598, 53700, 54502, 60920, 54514) ||class="entry q2 g1"| 53712₇ ||class="entry q2 g1"| 59598₉ ||class="entry q2 g1"| 53700₇ ||class="entry q2 g1"| 54502₉ ||class="entry q2 g1"| 60920₁₁ ||class="entry q2 g1"| 54514₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (53894, 60312, 53906, 55216, 61102, 55204) ||class="entry q2 g1"| 53894₇ ||class="entry q2 g1"| 60312₉ ||class="entry q2 g1"| 53906₇ ||class="entry q2 g1"| 55216₉ ||class="entry q2 g1"| 61102₁₁ ||class="entry q2 g1"| 55204₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (57674, 60060, 57434, 58492, 61354, 58732) ||class="entry q2 g1"| 57674₇ ||class="entry q2 g1"| 60060₉ ||class="entry q2 g1"| 57434₇ ||class="entry q2 g1"| 58492₉ ||class="entry q2 g1"| 61354₁₁ ||class="entry q2 g1"| 58732₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (57884, 59850, 58124, 59178, 60668, 58938) ||class="entry q2 g1"| 57884₇ ||class="entry q2 g1"| 59850₉ ||class="entry q2 g1"| 58124₇ ||class="entry q2 g1"| 59178₉ ||class="entry q2 g1"| 60668₁₁ ||class="entry q2 g1"| 58938₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (61524, 64386, 61764, 62818, 65204, 62578) ||class="entry q2 g1"| 61524₇ ||class="entry q2 g1"| 64386₉ ||class="entry q2 g1"| 61764₇ ||class="entry q2 g1"| 62818₉ ||class="entry q2 g1"| 65204₁₁ ||class="entry q2 g1"| 62578₉ |- |class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (62210, 63700, 61970, 63028, 64994, 63268) ||class="entry q2 g1"| 62210₇ ||class="entry q2 g1"| 63700₉ ||class="entry q2 g1"| 61970₇ ||class="entry q2 g1"| 63028₉ ||class="entry q2 g1"| 64994₁₁ ||class="entry q2 g1"| 63268₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (32928, 46939, 45953, 60694, 56315, 47009) ||class="entry q2 g1"| 32928₃ ||class="entry q3 g1"| 46939₁₁ ||class="entry q3 g1"| 45953₇ ||class="entry q2 g1"| 60694₉ ||class="entry q3 g1"| 56315₁₃ ||class="entry q3 g1"| 47009₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (35840, 54141, 55377, 57782, 49117, 56433) ||class="entry q2 g1"| 35840₃ ||class="entry q3 g1"| 54141₁₁ ||class="entry q3 g1"| 55377₇ ||class="entry q2 g1"| 57782₉ ||class="entry q3 g1"| 49117₁₃ ||class="entry q3 g1"| 56433₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (40992, 38623, 41165, 52630, 64127, 42221) ||class="entry q2 g1"| 40992₃ ||class="entry q3 g1"| 38623₁₁ ||class="entry q3 g1"| 41165₇ ||class="entry q2 g1"| 52630₉ ||class="entry q3 g1"| 64127₁₃ ||class="entry q3 g1"| 42221₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (50176, 39791, 51723, 43446, 63439, 52779) ||class="entry q2 g1"| 50176₃ ||class="entry q3 g1"| 39791₁₁ ||class="entry q3 g1"| 51723₇ ||class="entry q2 g1"| 43446₉ ||class="entry q3 g1"| 63439₁₃ ||class="entry q3 g1"| 52779₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (33920, 56045, 57121, 59702, 46669, 56065) ||class="entry q2 g1"| 33920₃ ||class="entry q3 g1"| 56045₁₁ ||class="entry q3 g1"| 57121₉ ||class="entry q2 g1"| 59702₉ ||class="entry q3 g1"| 46669₉ ||class="entry q3 g1"| 56065₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (34848, 48843, 46321, 58774, 53867, 45265) ||class="entry q2 g1"| 34848₃ ||class="entry q3 g1"| 48843₁₁ ||class="entry q3 g1"| 46321₉ ||class="entry q2 g1"| 58774₉ ||class="entry q3 g1"| 53867₉ ||class="entry q3 g1"| 45265₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (41984, 64361, 52333, 51638, 38857, 51277) ||class="entry q2 g1"| 41984₃ ||class="entry q3 g1"| 64361₁₁ ||class="entry q3 g1"| 52333₉ ||class="entry q2 g1"| 51638₉ ||class="entry q3 g1"| 38857₉ ||class="entry q3 g1"| 51277₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (49184, 63193, 42667, 44438, 39545, 41611) ||class="entry q2 g1"| 49184₃ ||class="entry q3 g1"| 63193₁₁ ||class="entry q3 g1"| 42667₉ ||class="entry q2 g1"| 44438₉ ||class="entry q3 g1"| 39545₉ ||class="entry q3 g1"| 41611₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (33938, 32933, 34171, 59684, 60421, 33115) ||class="entry q2 g1"| 33938₅ ||class="entry q3 g1"| 32933₅ ||class="entry q3 g1"| 34171₉ ||class="entry q2 g1"| 59684₇ ||class="entry q3 g1"| 60421₇ ||class="entry q3 g1"| 33115₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (35108, 35843, 34621, 58514, 57507, 33565) ||class="entry q2 g1"| 35108₅ ||class="entry q3 g1"| 35843₅ ||class="entry q3 g1"| 34621₉ ||class="entry q2 g1"| 58514₇ ||class="entry q3 g1"| 57507₇ ||class="entry q3 g1"| 33565₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (42002, 41249, 38455, 51620, 52609, 37399) ||class="entry q2 g1"| 42002₅ ||class="entry q3 g1"| 41249₅ ||class="entry q3 g1"| 38455₉ ||class="entry q2 g1"| 51620₇ ||class="entry q3 g1"| 52609₇ ||class="entry q3 g1"| 37399₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (49444, 50193, 38247, 44178, 43185, 37191) ||class="entry q2 g1"| 49444₅ ||class="entry q3 g1"| 50193₅ ||class="entry q3 g1"| 38247₉ ||class="entry q2 g1"| 44178₇ ||class="entry q3 g1"| 43185₇ ||class="entry q3 g1"| 37191₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (33188, 34195, 32845, 60434, 59699, 33901) ||class="entry q2 g1"| 33188₅ ||class="entry q3 g1"| 34195₇ ||class="entry q3 g1"| 32845₅ ||class="entry q2 g1"| 60434₇ ||class="entry q3 g1"| 59699₉ ||class="entry q3 g1"| 33901₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (35858, 35125, 33291, 57764, 58773, 34347) ||class="entry q2 g1"| 35858₅ ||class="entry q3 g1"| 35125₇ ||class="entry q3 g1"| 33291₅ ||class="entry q2 g1"| 57764₇ ||class="entry q3 g1"| 58773₉ ||class="entry q3 g1"| 34347₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (41252, 42007, 37633, 52370, 51383, 38689) ||class="entry q2 g1"| 41252₅ ||class="entry q3 g1"| 42007₇ ||class="entry q3 g1"| 37633₅ ||class="entry q2 g1"| 52370₇ ||class="entry q3 g1"| 51383₉ ||class="entry q3 g1"| 38689₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (50194, 49447, 36945, 43428, 44423, 38001) ||class="entry q2 g1"| 50194₅ ||class="entry q3 g1"| 49447₇ ||class="entry q3 g1"| 36945₅ ||class="entry q2 g1"| 43428₇ ||class="entry q3 g1"| 44423₉ ||class="entry q3 g1"| 38001₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34178, 36421, 35467, 59444, 58085, 36523) ||class="entry q2 g1"| 34178₅ ||class="entry q3 g1"| 36421₇ ||class="entry q3 g1"| 35467₇ ||class="entry q2 g1"| 59444₇ ||class="entry q3 g1"| 58085₉ ||class="entry q3 g1"| 36523₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34192, 54285, 53457, 59430, 47277, 54513) ||class="entry q2 g1"| 34192₅ ||class="entry q3 g1"| 54285₇ ||class="entry q3 g1"| 53457₇ ||class="entry q2 g1"| 59430₇ ||class="entry q3 g1"| 47277₉ ||class="entry q3 g1"| 54513₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34868, 33507, 35021, 58754, 60995, 36077) ||class="entry q2 g1"| 34868₅ ||class="entry q3 g1"| 33507₇ ||class="entry q3 g1"| 35021₇ ||class="entry q2 g1"| 58754₇ ||class="entry q3 g1"| 60995₉ ||class="entry q3 g1"| 36077₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (35120, 45099, 47873, 58502, 56459, 48929) ||class="entry q2 g1"| 35120₅ ||class="entry q3 g1"| 45099₇ ||class="entry q3 g1"| 47873₇ ||class="entry q2 g1"| 58502₇ ||class="entry q3 g1"| 56459₉ ||class="entry q3 g1"| 48929₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41990, 40201, 43531, 51632, 61865, 44587) ||class="entry q2 g1"| 41990₅ ||class="entry q3 g1"| 40201₇ ||class="entry q3 g1"| 43531₇ ||class="entry q2 g1"| 51632₇ ||class="entry q3 g1"| 61865₉ ||class="entry q3 g1"| 44587₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (42004, 51009, 61521, 51618, 44001, 62577) ||class="entry q2 g1"| 42004₅ ||class="entry q3 g1"| 51009₇ ||class="entry q3 g1"| 61521₇ ||class="entry q2 g1"| 51618₇ ||class="entry q3 g1"| 44001₉ ||class="entry q3 g1"| 62577₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49190, 37049, 49357, 44432, 64537, 50413) ||class="entry q2 g1"| 49190₅ ||class="entry q3 g1"| 37049₇ ||class="entry q3 g1"| 49357₇ ||class="entry q2 g1"| 44432₇ ||class="entry q3 g1"| 64537₉ ||class="entry q3 g1"| 50413₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49442, 41585, 62209, 44180, 52945, 63265) ||class="entry q2 g1"| 49442₅ ||class="entry q3 g1"| 41585₇ ||class="entry q3 g1"| 62209₇ ||class="entry q2 g1"| 44180₇ ||class="entry q3 g1"| 52945₉ ||class="entry q3 g1"| 63265₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (34180, 59429, 60653, 59442, 33925, 59597) ||class="entry q2 g1"| 34180₅ ||class="entry q3 g1"| 59429₇ ||class="entry q3 g1"| 60653₁₁ ||class="entry q2 g1"| 59442₇ ||class="entry q3 g1"| 33925₅ ||class="entry q3 g1"| 59597₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (34866, 58499, 61099, 58756, 34851, 60043) ||class="entry q2 g1"| 34866₅ ||class="entry q3 g1"| 58499₇ ||class="entry q3 g1"| 61099₁₁ ||class="entry q2 g1"| 58756₇ ||class="entry q3 g1"| 34851₅ ||class="entry q3 g1"| 60043₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (42244, 51617, 65441, 51378, 42241, 64385) ||class="entry q2 g1"| 42244₅ ||class="entry q3 g1"| 51617₇ ||class="entry q3 g1"| 65441₁₁ ||class="entry q2 g1"| 51378₇ ||class="entry q3 g1"| 42241₅ ||class="entry q3 g1"| 64385₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (49202, 44177, 64753, 44420, 49201, 63697) ||class="entry q2 g1"| 49202₅ ||class="entry q3 g1"| 44177₇ ||class="entry q3 g1"| 64753₁₁ ||class="entry q2 g1"| 44420₇ ||class="entry q3 g1"| 49201₅ ||class="entry q3 g1"| 63697₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33504, 56461, 55617, 61270, 45101, 56673) ||class="entry q2 g1"| 33504₅ ||class="entry q3 g1"| 56461₉ ||class="entry q3 g1"| 55617₇ ||class="entry q2 g1"| 61270₁₁ ||class="entry q3 g1"| 45101₇ ||class="entry q3 g1"| 56673₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (36416, 47275, 45713, 58358, 54283, 46769) ||class="entry q2 g1"| 36416₅ ||class="entry q3 g1"| 47275₉ ||class="entry q3 g1"| 45713₇ ||class="entry q2 g1"| 58358₁₁ ||class="entry q3 g1"| 54283₇ ||class="entry q3 g1"| 46769₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (37032, 52933, 51977, 64798, 41573, 53033) ||class="entry q2 g1"| 37032₅ ||class="entry q3 g1"| 52933₉ ||class="entry q3 g1"| 51977₇ ||class="entry q2 g1"| 64798₁₁ ||class="entry q3 g1"| 41573₇ ||class="entry q3 g1"| 53033₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (39944, 43747, 41177, 61886, 50755, 42233) ||class="entry q2 g1"| 39944₅ ||class="entry q3 g1"| 43747₉ ||class="entry q3 g1"| 41177₇ ||class="entry q2 g1"| 61886₁₁ ||class="entry q3 g1"| 50755₇ ||class="entry q3 g1"| 42233₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (41568, 64777, 51725, 53206, 37289, 52781) ||class="entry q2 g1"| 41568₅ ||class="entry q3 g1"| 64777₉ ||class="entry q3 g1"| 51725₇ ||class="entry q2 g1"| 53206₁₁ ||class="entry q3 g1"| 37289₇ ||class="entry q3 g1"| 52781₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (45096, 61249, 55365, 56734, 33761, 56421) ||class="entry q2 g1"| 45096₅ ||class="entry q3 g1"| 61249₉ ||class="entry q3 g1"| 55365₇ ||class="entry q2 g1"| 56734₁₁ ||class="entry q3 g1"| 33761₇ ||class="entry q3 g1"| 56421₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (50752, 61625, 41163, 44022, 39961, 42219) ||class="entry q2 g1"| 50752₅ ||class="entry q3 g1"| 61625₉ ||class="entry q3 g1"| 41163₇ ||class="entry q2 g1"| 44022₁₁ ||class="entry q3 g1"| 39961₇ ||class="entry q3 g1"| 42219₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (54280, 58097, 45699, 47550, 36433, 46755) ||class="entry q2 g1"| 54280₅ ||class="entry q3 g1"| 58097₉ ||class="entry q3 g1"| 45699₇ ||class="entry q2 g1"| 47550₁₁ ||class="entry q3 g1"| 36433₇ ||class="entry q3 g1"| 46755₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33926, 48269, 47431, 59696, 53293, 48487) ||class="entry q2 g1"| 33926₅ ||class="entry q3 g1"| 48269₉ ||class="entry q3 g1"| 47431₉ ||class="entry q2 g1"| 59696₇ ||class="entry q3 g1"| 53293₇ ||class="entry q3 g1"| 48487₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33940, 59077, 58141, 59682, 35429, 59197) ||class="entry q2 g1"| 33940₅ ||class="entry q3 g1"| 59077₉ ||class="entry q3 g1"| 58141₉ ||class="entry q2 g1"| 59682₇ ||class="entry q3 g1"| 35429₇ ||class="entry q3 g1"| 59197₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (34854, 55467, 53911, 58768, 46091, 54967) ||class="entry q2 g1"| 34854₅ ||class="entry q3 g1"| 55467₉ ||class="entry q3 g1"| 53911₉ ||class="entry q2 g1"| 58768₇ ||class="entry q3 g1"| 46091₇ ||class="entry q3 g1"| 54967₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (35106, 60003, 57691, 58516, 34499, 58747) ||class="entry q2 g1"| 35106₅ ||class="entry q3 g1"| 60003₉ ||class="entry q3 g1"| 57691₉ ||class="entry q2 g1"| 58516₇ ||class="entry q3 g1"| 34499₇ ||class="entry q3 g1"| 58747₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (42242, 44993, 39367, 51380, 50017, 40423) ||class="entry q2 g1"| 42242₅ ||class="entry q3 g1"| 44993₉ ||class="entry q3 g1"| 39367₉ ||class="entry q2 g1"| 51380₇ ||class="entry q3 g1"| 50017₇ ||class="entry q3 g1"| 40423₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (42256, 62857, 50077, 51366, 39209, 51133) ||class="entry q2 g1"| 42256₅ ||class="entry q3 g1"| 62857₉ ||class="entry q3 g1"| 50077₉ ||class="entry q2 g1"| 51366₇ ||class="entry q3 g1"| 39209₇ ||class="entry q3 g1"| 51133₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49204, 51953, 39575, 44418, 42577, 40631) ||class="entry q2 g1"| 49204₅ ||class="entry q3 g1"| 51953₉ ||class="entry q3 g1"| 39575₉ ||class="entry q2 g1"| 44418₇ ||class="entry q3 g1"| 42577₇ ||class="entry q3 g1"| 40631₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49456, 63545, 43355, 44166, 38041, 44411) ||class="entry q2 g1"| 49456₅ ||class="entry q3 g1"| 63545₉ ||class="entry q3 g1"| 43355₉ ||class="entry q2 g1"| 44166₇ ||class="entry q3 g1"| 38041₇ ||class="entry q3 g1"| 44411₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (34496, 45371, 46561, 60278, 56731, 45505) ||class="entry q2 g1"| 34496₅ ||class="entry q3 g1"| 45371₉ ||class="entry q3 g1"| 46561₉ ||class="entry q2 g1"| 60278₁₁ ||class="entry q3 g1"| 56731₁₁ ||class="entry q3 g1"| 45505₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (35424, 54557, 56881, 59350, 47549, 55825) ||class="entry q2 g1"| 35424₅ ||class="entry q3 g1"| 54557₉ ||class="entry q3 g1"| 56881₉ ||class="entry q2 g1"| 59350₁₁ ||class="entry q3 g1"| 47549₁₁ ||class="entry q3 g1"| 55825₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38024, 41843, 42921, 63806, 53203, 41865) ||class="entry q2 g1"| 38024₅ ||class="entry q3 g1"| 41843₉ ||class="entry q3 g1"| 42921₉ ||class="entry q2 g1"| 63806₁₁ ||class="entry q3 g1"| 53203₁₁ ||class="entry q3 g1"| 41865₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38952, 51029, 52345, 62878, 44021, 51289) ||class="entry q2 g1"| 38952₅ ||class="entry q3 g1"| 51029₉ ||class="entry q3 g1"| 52345₉ ||class="entry q2 g1"| 62878₁₁ ||class="entry q3 g1"| 44021₁₁ ||class="entry q3 g1"| 51289₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (42560, 37055, 42669, 52214, 64543, 41613) ||class="entry q2 g1"| 42560₅ ||class="entry q3 g1"| 37055₉ ||class="entry q3 g1"| 42669₉ ||class="entry q2 g1"| 52214₁₁ ||class="entry q3 g1"| 64543₁₁ ||class="entry q3 g1"| 41613₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (46088, 33527, 46309, 55742, 61015, 45253) ||class="entry q2 g1"| 46088₅ ||class="entry q3 g1"| 33527₉ ||class="entry q3 g1"| 46309₉ ||class="entry q2 g1"| 55742₁₁ ||class="entry q3 g1"| 61015₁₁ ||class="entry q3 g1"| 45253₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (49760, 40207, 52331, 45014, 61871, 51275) ||class="entry q2 g1"| 49760₅ ||class="entry q3 g1"| 40207₉ ||class="entry q3 g1"| 52331₉ ||class="entry q2 g1"| 45014₁₁ ||class="entry q3 g1"| 61871₁₁ ||class="entry q3 g1"| 51275₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (53288, 36679, 56867, 48542, 58343, 55811) ||class="entry q2 g1"| 53288₅ ||class="entry q3 g1"| 36679₉ ||class="entry q3 g1"| 56867₉ ||class="entry q2 g1"| 48542₁₁ ||class="entry q3 g1"| 58343₁₁ ||class="entry q3 g1"| 55811₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (32946, 60691, 59867, 60676, 33203, 60923) ||class="entry q2 g1"| 32946₅ ||class="entry q3 g1"| 60691₉ ||class="entry q3 g1"| 59867₁₁ ||class="entry q2 g1"| 60676₇ ||class="entry q3 g1"| 33203₇ ||class="entry q3 g1"| 60923₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (36100, 57781, 60317, 57522, 36117, 61373) ||class="entry q2 g1"| 36100₅ ||class="entry q3 g1"| 57781₉ ||class="entry q3 g1"| 60317₁₁ ||class="entry q2 g1"| 57522₇ ||class="entry q3 g1"| 36117₇ ||class="entry q3 g1"| 61373₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (41010, 52375, 64151, 52612, 41015, 65207) ||class="entry q2 g1"| 41010₅ ||class="entry q3 g1"| 52375₉ ||class="entry q3 g1"| 64151₁₁ ||class="entry q2 g1"| 52612₇ ||class="entry q3 g1"| 41015₇ ||class="entry q3 g1"| 65207₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (50436, 43431, 63943, 43186, 50439, 64999) ||class="entry q2 g1"| 50436₅ ||class="entry q3 g1"| 43431₉ ||class="entry q3 g1"| 63943₁₁ ||class="entry q2 g1"| 43186₇ ||class="entry q3 g1"| 50439₇ ||class="entry q3 g1"| 64999₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32934, 53563, 54759, 60688, 48539, 53703) ||class="entry q2 g1"| 32934₅ ||class="entry q3 g1"| 53563₉ ||class="entry q3 g1"| 54759₁₁ ||class="entry q2 g1"| 60688₇ ||class="entry q3 g1"| 48539₁₁ ||class="entry q3 g1"| 53703₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32948, 35699, 36797, 60674, 59347, 35741) ||class="entry q2 g1"| 32948₅ ||class="entry q3 g1"| 35699₉ ||class="entry q3 g1"| 36797₁₁ ||class="entry q2 g1"| 60674₇ ||class="entry q3 g1"| 59347₁₁ ||class="entry q3 g1"| 35741₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35846, 46365, 48695, 57776, 55741, 47639) ||class="entry q2 g1"| 35846₅ ||class="entry q3 g1"| 46365₉ ||class="entry q3 g1"| 48695₁₁ ||class="entry q2 g1"| 57776₇ ||class="entry q3 g1"| 55741₁₁ ||class="entry q3 g1"| 47639₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (36098, 34773, 36347, 57524, 60277, 35291) ||class="entry q2 g1"| 36098₅ ||class="entry q3 g1"| 34773₉ ||class="entry q3 g1"| 36347₁₁ ||class="entry q2 g1"| 57524₇ ||class="entry q3 g1"| 60277₁₁ ||class="entry q3 g1"| 35291₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41250, 49783, 62823, 52372, 44759, 61767) ||class="entry q2 g1"| 41250₅ ||class="entry q3 g1"| 49783₉ ||class="entry q3 g1"| 62823₁₁ ||class="entry q2 g1"| 52372₇ ||class="entry q3 g1"| 44759₁₁ ||class="entry q3 g1"| 61767₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41264, 38975, 44861, 52358, 62623, 43805) ||class="entry q2 g1"| 41264₅ ||class="entry q3 g1"| 38975₉ ||class="entry q3 g1"| 44861₁₁ ||class="entry q2 g1"| 52358₇ ||class="entry q3 g1"| 62623₁₁ ||class="entry q3 g1"| 43805₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (50196, 42823, 63031, 43426, 52199, 61975) ||class="entry q2 g1"| 50196₅ ||class="entry q3 g1"| 42823₉ ||class="entry q3 g1"| 63031₁₁ ||class="entry q2 g1"| 43426₇ ||class="entry q3 g1"| 52199₁₁ ||class="entry q3 g1"| 61975₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (50448, 38287, 50683, 43174, 63791, 49627) ||class="entry q2 g1"| 50448₅ ||class="entry q3 g1"| 38287₉ ||class="entry q3 g1"| 50683₁₁ ||class="entry q2 g1"| 43174₇ ||class="entry q3 g1"| 63791₁₁ ||class="entry q3 g1"| 49627₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33186, 58355, 58923, 60436, 36691, 57867) ||class="entry q2 g1"| 33186₅ ||class="entry q3 g1"| 58355₁₁ ||class="entry q3 g1"| 58923₉ ||class="entry q2 g1"| 60436₇ ||class="entry q3 g1"| 36691₉ ||class="entry q3 g1"| 57867₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33200, 47547, 48241, 60422, 54555, 47185) ||class="entry q2 g1"| 33200₅ ||class="entry q3 g1"| 47547₁₁ ||class="entry q3 g1"| 48241₉ ||class="entry q2 g1"| 60422₇ ||class="entry q3 g1"| 54555₉ ||class="entry q3 g1"| 47185₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (35860, 61269, 58477, 57762, 33781, 57421) ||class="entry q2 g1"| 35860₅ ||class="entry q3 g1"| 61269₁₁ ||class="entry q3 g1"| 58477₉ ||class="entry q2 g1"| 57762₇ ||class="entry q3 g1"| 33781₉ ||class="entry q3 g1"| 57421₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (36112, 56733, 55201, 57510, 45373, 54145) ||class="entry q2 g1"| 36112₅ ||class="entry q3 g1"| 56733₁₁ ||class="entry q3 g1"| 55201₉ ||class="entry q2 g1"| 57510₇ ||class="entry q3 g1"| 45373₉ ||class="entry q3 g1"| 54145₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (40998, 61631, 50859, 52624, 39967, 49803) ||class="entry q2 g1"| 40998₅ ||class="entry q3 g1"| 61631₁₁ ||class="entry q3 g1"| 50859₉ ||class="entry q2 g1"| 52624₇ ||class="entry q3 g1"| 39967₉ ||class="entry q3 g1"| 49803₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (41012, 43767, 40177, 52610, 50775, 39121) ||class="entry q2 g1"| 41012₅ ||class="entry q3 g1"| 43767₁₁ ||class="entry q3 g1"| 40177₉ ||class="entry q2 g1"| 52610₇ ||class="entry q3 g1"| 50775₉ ||class="entry q3 g1"| 39121₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (50182, 64783, 44141, 43440, 37295, 43085) ||class="entry q2 g1"| 50182₅ ||class="entry q3 g1"| 64783₁₁ ||class="entry q3 g1"| 44141₉ ||class="entry q2 g1"| 43440₇ ||class="entry q3 g1"| 37295₉ ||class="entry q3 g1"| 43085₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (50434, 53191, 40865, 43188, 41831, 39809) ||class="entry q2 g1"| 50434₅ ||class="entry q3 g1"| 53191₁₁ ||class="entry q3 g1"| 40865₉ ||class="entry q2 g1"| 43188₇ ||class="entry q3 g1"| 41831₉ ||class="entry q3 g1"| 39809₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (44160, 62201, 51997, 49462, 40537, 53053) ||class="entry q2 g1"| 44160₅ ||class="entry q3 g1"| 62201₁₁ ||class="entry q3 g1"| 51997₉ ||class="entry q2 g1"| 49462₇ ||class="entry q3 g1"| 40537₉ ||class="entry q3 g1"| 53053₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (51360, 65353, 41435, 42262, 37865, 42491) ||class="entry q2 g1"| 51360₅ ||class="entry q3 g1"| 65353₁₁ ||class="entry q3 g1"| 41435₉ ||class="entry q2 g1"| 42262₇ ||class="entry q3 g1"| 37865₉ ||class="entry q3 g1"| 42491₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (58496, 47851, 55623, 35126, 54859, 56679) ||class="entry q2 g1"| 58496₅ ||class="entry q3 g1"| 47851₁₁ ||class="entry q3 g1"| 55623₉ ||class="entry q2 g1"| 35126₇ ||class="entry q3 g1"| 54859₉ ||class="entry q3 g1"| 56679₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (59424, 57037, 45719, 34198, 45677, 46775) ||class="entry q2 g1"| 59424₅ ||class="entry q3 g1"| 57037₁₁ ||class="entry q3 g1"| 45719₉ ||class="entry q2 g1"| 34198₇ ||class="entry q3 g1"| 45677₉ ||class="entry q3 g1"| 46775₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (43168, 40783, 42941, 50454, 62447, 41885) ||class="entry q2 g1"| 43168₅ ||class="entry q3 g1"| 40783₁₁ ||class="entry q3 g1"| 42941₁₁ ||class="entry q2 g1"| 50454₇ ||class="entry q3 g1"| 62447₁₃ ||class="entry q3 g1"| 41885₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (52352, 37631, 52603, 41270, 65119, 51547) ||class="entry q2 g1"| 52352₅ ||class="entry q3 g1"| 37631₁₁ ||class="entry q3 g1"| 52603₁₁ ||class="entry q2 g1"| 41270₇ ||class="entry q3 g1"| 65119₁₃ ||class="entry q3 g1"| 51547₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (57504, 55133, 46567, 36118, 48125, 45511) ||class="entry q2 g1"| 57504₅ ||class="entry q3 g1"| 55133₁₁ ||class="entry q3 g1"| 46567₁₁ ||class="entry q2 g1"| 36118₇ ||class="entry q3 g1"| 48125₁₃ ||class="entry q3 g1"| 45511₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (60416, 45947, 56887, 33206, 57307, 55831) ||class="entry q2 g1"| 60416₅ ||class="entry q3 g1"| 45947₁₁ ||class="entry q3 g1"| 56887₁₁ ||class="entry q2 g1"| 33206₇ ||class="entry q3 g1"| 57307₁₃ ||class="entry q3 g1"| 55831₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (33762, 34853, 36075, 61012, 58501, 35019) ||class="entry q2 g1"| 33762₇ ||class="entry q3 g1"| 34853₅ ||class="entry q3 g1"| 36075₉ ||class="entry q2 g1"| 61012₉ ||class="entry q3 g1"| 58501₇ ||class="entry q3 g1"| 35019₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (36436, 33923, 36525, 58338, 59427, 35469) ||class="entry q2 g1"| 36436₇ ||class="entry q3 g1"| 33923₅ ||class="entry q3 g1"| 36525₉ ||class="entry q2 g1"| 58338₉ ||class="entry q3 g1"| 59427₇ ||class="entry q3 g1"| 35469₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (37304, 49189, 50425, 64526, 44165, 49369) ||class="entry q2 g1"| 37304₇ ||class="entry q3 g1"| 49189₅ ||class="entry q3 g1"| 50425₉ ||class="entry q2 g1"| 64526₉ ||class="entry q3 g1"| 44165₇ ||class="entry q3 g1"| 49369₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (40216, 41987, 44841, 61614, 51363, 43785) ||class="entry q2 g1"| 40216₇ ||class="entry q3 g1"| 41987₅ ||class="entry q3 g1"| 44841₉ ||class="entry q2 g1"| 61614₉ ||class="entry q3 g1"| 51363₇ ||class="entry q3 g1"| 43785₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (41588, 49441, 63025, 53186, 44417, 61969) ||class="entry q2 g1"| 41588₇ ||class="entry q3 g1"| 49441₅ ||class="entry q3 g1"| 63025₉ ||class="entry q2 g1"| 53186₉ ||class="entry q3 g1"| 44417₇ ||class="entry q3 g1"| 61969₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (45102, 35105, 48675, 56728, 58753, 47619) ||class="entry q2 g1"| 45102₇ ||class="entry q3 g1"| 35105₅ ||class="entry q3 g1"| 48675₉ ||class="entry q2 g1"| 56728₉ ||class="entry q3 g1"| 58753₇ ||class="entry q3 g1"| 47619₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (51010, 42001, 62817, 43764, 51377, 61761) ||class="entry q2 g1"| 51010₇ ||class="entry q3 g1"| 42001₅ ||class="entry q3 g1"| 62817₉ ||class="entry q2 g1"| 43764₉ ||class="entry q3 g1"| 51377₇ ||class="entry q3 g1"| 61761₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (54286, 33937, 54501, 47544, 59441, 53445) ||class="entry q2 g1"| 54286₇ ||class="entry q3 g1"| 33937₅ ||class="entry q3 g1"| 54501₉ ||class="entry q2 g1"| 47544₉ ||class="entry q3 g1"| 59441₇ ||class="entry q3 g1"| 53445₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33522, 34501, 33563, 61252, 60005, 34619) ||class="entry q2 g1"| 33522₇ ||class="entry q3 g1"| 34501₇ ||class="entry q3 g1"| 33563₇ ||class="entry q2 g1"| 61252₉ ||class="entry q3 g1"| 60005₉ ||class="entry q3 g1"| 34619₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36676, 35427, 33117, 58098, 59075, 34173) ||class="entry q2 g1"| 36676₇ ||class="entry q3 g1"| 35427₇ ||class="entry q3 g1"| 33117₇ ||class="entry q2 g1"| 58098₉ ||class="entry q3 g1"| 59075₉ ||class="entry q3 g1"| 34173₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (37050, 38029, 37203, 64780, 63533, 38259) ||class="entry q2 g1"| 37050₇ ||class="entry q3 g1"| 38029₇ ||class="entry q3 g1"| 37203₇ ||class="entry q2 g1"| 64780₉ ||class="entry q3 g1"| 63533₉ ||class="entry q3 g1"| 38259₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (40204, 38955, 37653, 61626, 62603, 38709) ||class="entry q2 g1"| 40204₇ ||class="entry q3 g1"| 38955₇ ||class="entry q3 g1"| 37653₇ ||class="entry q2 g1"| 61626₉ ||class="entry q3 g1"| 62603₉ ||class="entry q3 g1"| 38709₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (41586, 42817, 36951, 53188, 52193, 38007) ||class="entry q2 g1"| 41586₇ ||class="entry q3 g1"| 42817₇ ||class="entry q3 g1"| 36951₇ ||class="entry q2 g1"| 53188₉ ||class="entry q3 g1"| 52193₉ ||class="entry q3 g1"| 38007₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45114, 46345, 33311, 56716, 55721, 34367) ||class="entry q2 g1"| 45114₇ ||class="entry q3 g1"| 46345₇ ||class="entry q3 g1"| 33311₇ ||class="entry q2 g1"| 56716₉ ||class="entry q3 g1"| 55721₉ ||class="entry q3 g1"| 34367₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (51012, 49777, 37639, 43762, 44753, 38695) ||class="entry q2 g1"| 51012₇ ||class="entry q3 g1"| 49777₇ ||class="entry q3 g1"| 37639₇ ||class="entry q2 g1"| 43762₉ ||class="entry q3 g1"| 44753₉ ||class="entry q3 g1"| 38695₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54540, 53305, 33103, 47290, 48281, 34159) ||class="entry q2 g1"| 54540₇ ||class="entry q3 g1"| 53305₇ ||class="entry q3 g1"| 33103₇ ||class="entry q2 g1"| 47290₉ ||class="entry q3 g1"| 48281₉ ||class="entry q3 g1"| 34159₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (37608, 42259, 41417, 65374, 51635, 42473) ||class="entry q2 g1"| 37608₇ ||class="entry q3 g1"| 42259₇ ||class="entry q3 g1"| 41417₇ ||class="entry q2 g1"| 65374₁₃ ||class="entry q3 g1"| 51635₉ ||class="entry q3 g1"| 42473₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (40520, 49461, 51737, 62462, 44437, 52793) ||class="entry q2 g1"| 40520₇ ||class="entry q3 g1"| 49461₇ ||class="entry q3 g1"| 51737₇ ||class="entry q2 g1"| 62462₁₃ ||class="entry q3 g1"| 44437₉ ||class="entry q3 g1"| 52793₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (45672, 33943, 45701, 57310, 59447, 46757) ||class="entry q2 g1"| 45672₇ ||class="entry q3 g1"| 33943₇ ||class="entry q3 g1"| 45701₇ ||class="entry q2 g1"| 57310₁₃ ||class="entry q3 g1"| 59447₉ ||class="entry q3 g1"| 46757₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (54856, 35111, 55363, 48126, 58759, 56419) ||class="entry q2 g1"| 54856₇ ||class="entry q3 g1"| 35111₇ ||class="entry q3 g1"| 55363₇ ||class="entry q2 g1"| 48126₁₃ ||class="entry q3 g1"| 58759₉ ||class="entry q3 g1"| 56419₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (34516, 36115, 35293, 60258, 57779, 36349) ||class="entry q2 g1"| 34516₇ ||class="entry q3 g1"| 36115₇ ||class="entry q3 g1"| 35293₉ ||class="entry q2 g1"| 60258₉ ||class="entry q3 g1"| 57779₉ ||class="entry q3 g1"| 36349₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (35682, 33205, 35739, 59092, 60693, 36795) ||class="entry q2 g1"| 35682₇ ||class="entry q3 g1"| 33205₇ ||class="entry q3 g1"| 35739₉ ||class="entry q2 g1"| 59092₉ ||class="entry q3 g1"| 60693₉ ||class="entry q3 g1"| 36795₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38030, 50451, 49615, 63800, 43443, 50671) ||class="entry q2 g1"| 38030₇ ||class="entry q3 g1"| 50451₇ ||class="entry q3 g1"| 49615₉ ||class="entry q2 g1"| 63800₉ ||class="entry q3 g1"| 43443₉ ||class="entry q3 g1"| 50671₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38958, 41269, 43551, 62872, 52629, 44607) ||class="entry q2 g1"| 38958₇ ||class="entry q3 g1"| 41269₇ ||class="entry q3 g1"| 43551₉ ||class="entry q2 g1"| 62872₉ ||class="entry q3 g1"| 52629₉ ||class="entry q3 g1"| 44607₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (42818, 50199, 62215, 51956, 43191, 63271) ||class="entry q2 g1"| 42818₇ ||class="entry q3 g1"| 50199₇ ||class="entry q3 g1"| 62215₉ ||class="entry q2 g1"| 51956₉ ||class="entry q3 g1"| 43191₉ ||class="entry q3 g1"| 63271₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (46360, 35863, 47893, 55470, 57527, 48949) ||class="entry q2 g1"| 46360₇ ||class="entry q3 g1"| 35863₇ ||class="entry q3 g1"| 47893₉ ||class="entry q2 g1"| 55470₉ ||class="entry q3 g1"| 57527₉ ||class="entry q3 g1"| 48949₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (49780, 41255, 61527, 44994, 52615, 62583) ||class="entry q2 g1"| 49780₇ ||class="entry q3 g1"| 41255₇ ||class="entry q3 g1"| 61527₉ ||class="entry q2 g1"| 44994₉ ||class="entry q3 g1"| 52615₉ ||class="entry q3 g1"| 62583₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (53560, 33191, 53715, 48270, 60679, 54771) ||class="entry q2 g1"| 53560₇ ||class="entry q3 g1"| 33191₇ ||class="entry q3 g1"| 53715₉ ||class="entry q2 g1"| 48270₉ ||class="entry q3 g1"| 60679₉ ||class="entry q3 g1"| 54771₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (38600, 51365, 52585, 64382, 41989, 51529) ||class="entry q2 g1"| 38600₇ ||class="entry q3 g1"| 51365₇ ||class="entry q3 g1"| 52585₉ ||class="entry q2 g1"| 64382₁₃ ||class="entry q3 g1"| 41989₅ ||class="entry q3 g1"| 51529₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (39528, 44163, 42681, 63454, 49187, 41625) ||class="entry q2 g1"| 39528₇ ||class="entry q3 g1"| 44163₇ ||class="entry q3 g1"| 42681₉ ||class="entry q2 g1"| 63454₁₃ ||class="entry q3 g1"| 49187₅ ||class="entry q3 g1"| 41625₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (46664, 59681, 56869, 56318, 34177, 55813) ||class="entry q2 g1"| 46664₇ ||class="entry q3 g1"| 59681₇ ||class="entry q3 g1"| 56869₉ ||class="entry q2 g1"| 56318₁₃ ||class="entry q3 g1"| 34177₅ ||class="entry q3 g1"| 55813₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (53864, 58513, 46307, 49118, 34865, 45251) ||class="entry q2 g1"| 53864₇ ||class="entry q3 g1"| 58513₇ ||class="entry q3 g1"| 46307₉ ||class="entry q2 g1"| 49118₁₃ ||class="entry q3 g1"| 34865₅ ||class="entry q3 g1"| 45251₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (33524, 57509, 58749, 61250, 35845, 57693) ||class="entry q2 g1"| 33524₇ ||class="entry q3 g1"| 57509₇ ||class="entry q3 g1"| 58749₁₁ ||class="entry q2 g1"| 61250₉ ||class="entry q3 g1"| 35845₅ ||class="entry q3 g1"| 57693₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (36674, 60419, 59195, 58100, 32931, 58139) ||class="entry q2 g1"| 36674₇ ||class="entry q3 g1"| 60419₇ ||class="entry q3 g1"| 59195₁₁ ||class="entry q2 g1"| 58100₉ ||class="entry q3 g1"| 32931₅ ||class="entry q3 g1"| 58139₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (37038, 43173, 44399, 64792, 50181, 43343) ||class="entry q2 g1"| 37038₇ ||class="entry q3 g1"| 43173₇ ||class="entry q3 g1"| 44399₁₁ ||class="entry q2 g1"| 64792₉ ||class="entry q3 g1"| 50181₅ ||class="entry q3 g1"| 43343₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (39950, 52355, 50879, 61880, 40995, 49823) ||class="entry q2 g1"| 39950₇ ||class="entry q3 g1"| 52355₇ ||class="entry q3 g1"| 50879₁₁ ||class="entry q2 g1"| 61880₉ ||class="entry q3 g1"| 40995₅ ||class="entry q3 g1"| 49823₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (41826, 43425, 40871, 52948, 50433, 39815) ||class="entry q2 g1"| 41826₇ ||class="entry q3 g1"| 43425₇ ||class="entry q3 g1"| 40871₁₁ ||class="entry q2 g1"| 52948₉ ||class="entry q3 g1"| 50433₅ ||class="entry q3 g1"| 39815₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (45368, 57761, 55221, 56462, 36097, 54165) ||class="entry q2 g1"| 45368₇ ||class="entry q3 g1"| 57761₇ ||class="entry q3 g1"| 55221₁₁ ||class="entry q2 g1"| 56462₉ ||class="entry q3 g1"| 36097₅ ||class="entry q3 g1"| 54165₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (50772, 52369, 40183, 44002, 41009, 39127) ||class="entry q2 g1"| 50772₇ ||class="entry q3 g1"| 52369₇ ||class="entry q3 g1"| 40183₁₁ ||class="entry q2 g1"| 44002₉ ||class="entry q3 g1"| 41009₅ ||class="entry q3 g1"| 39127₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (54552, 60433, 48499, 47278, 32945, 47443) ||class="entry q2 g1"| 54552₇ ||class="entry q3 g1"| 60433₇ ||class="entry q3 g1"| 48499₁₁ ||class="entry q2 g1"| 47278₉ ||class="entry q3 g1"| 32945₅ ||class="entry q3 g1"| 47443₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (34754, 58771, 57419, 60020, 35123, 58475) ||class="entry q2 g1"| 34754₇ ||class="entry q3 g1"| 58771₉ ||class="entry q3 g1"| 57419₇ ||class="entry q2 g1"| 60020₉ ||class="entry q3 g1"| 35123₇ ||class="entry q3 g1"| 58475₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (35444, 59701, 57869, 59330, 34197, 58925) ||class="entry q2 g1"| 35444₇ ||class="entry q3 g1"| 59701₉ ||class="entry q3 g1"| 57869₇ ||class="entry q2 g1"| 59330₉ ||class="entry q3 g1"| 34197₇ ||class="entry q3 g1"| 58925₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (38296, 44435, 43097, 63534, 49459, 44153) ||class="entry q2 g1"| 38296₇ ||class="entry q3 g1"| 44435₉ ||class="entry q3 g1"| 43097₇ ||class="entry q2 g1"| 63534₉ ||class="entry q3 g1"| 49459₇ ||class="entry q3 g1"| 44153₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39224, 51637, 50057, 62606, 42261, 51113) ||class="entry q2 g1"| 39224₇ ||class="entry q3 g1"| 51637₉ ||class="entry q3 g1"| 50057₇ ||class="entry q2 g1"| 62606₉ ||class="entry q3 g1"| 42261₇ ||class="entry q3 g1"| 51113₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42580, 44183, 39569, 52194, 49207, 40625) ||class="entry q2 g1"| 42580₇ ||class="entry q3 g1"| 44183₉ ||class="entry q3 g1"| 39569₇ ||class="entry q2 g1"| 52194₉ ||class="entry q3 g1"| 49207₇ ||class="entry q3 g1"| 40625₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (46094, 58519, 53891, 55736, 34871, 54947) ||class="entry q2 g1"| 46094₇ ||class="entry q3 g1"| 58519₉ ||class="entry q3 g1"| 53891₇ ||class="entry q2 g1"| 55736₉ ||class="entry q3 g1"| 34871₇ ||class="entry q3 g1"| 54947₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (50018, 51623, 39361, 44756, 42247, 40417) ||class="entry q2 g1"| 50018₇ ||class="entry q3 g1"| 51623₉ ||class="entry q3 g1"| 39361₇ ||class="entry q2 g1"| 44756₉ ||class="entry q3 g1"| 42247₇ ||class="entry q3 g1"| 40417₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (53294, 59687, 47173, 48536, 34183, 48229) ||class="entry q2 g1"| 53294₇ ||class="entry q3 g1"| 59687₉ ||class="entry q3 g1"| 47173₇ ||class="entry q2 g1"| 48536₉ ||class="entry q3 g1"| 34183₇ ||class="entry q3 g1"| 48229₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (34756, 33779, 34349, 60018, 61267, 33293) ||class="entry q2 g1"| 34756₇ ||class="entry q3 g1"| 33779₉ ||class="entry q3 g1"| 34349₇ ||class="entry q2 g1"| 60018₉ ||class="entry q3 g1"| 61267₁₁ ||class="entry q3 g1"| 33293₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (35442, 36693, 33899, 59332, 58357, 32843) ||class="entry q2 g1"| 35442₇ ||class="entry q3 g1"| 36693₉ ||class="entry q3 g1"| 33899₇ ||class="entry q2 g1"| 59332₉ ||class="entry q3 g1"| 58357₁₁ ||class="entry q3 g1"| 32843₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38284, 37307, 37989, 63546, 64795, 36933) ||class="entry q2 g1"| 38284₇ ||class="entry q3 g1"| 37307₉ ||class="entry q3 g1"| 37989₇ ||class="entry q2 g1"| 63546₉ ||class="entry q3 g1"| 64795₁₁ ||class="entry q3 g1"| 36933₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38970, 40221, 38435, 62860, 61885, 37379) ||class="entry q2 g1"| 38970₇ ||class="entry q3 g1"| 40221₉ ||class="entry q3 g1"| 38435₇ ||class="entry q2 g1"| 62860₉ ||class="entry q3 g1"| 61885₁₁ ||class="entry q3 g1"| 37379₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (42820, 41591, 38241, 51954, 52951, 37185) ||class="entry q2 g1"| 42820₇ ||class="entry q3 g1"| 41591₉ ||class="entry q3 g1"| 38241₇ ||class="entry q2 g1"| 51954₉ ||class="entry q3 g1"| 52951₁₁ ||class="entry q3 g1"| 37185₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (46348, 45119, 34601, 55482, 56479, 33545) ||class="entry q2 g1"| 46348₇ ||class="entry q3 g1"| 45119₉ ||class="entry q3 g1"| 34601₇ ||class="entry q2 g1"| 55482₉ ||class="entry q3 g1"| 56479₁₁ ||class="entry q3 g1"| 33545₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (49778, 51015, 38449, 44996, 44007, 37393) ||class="entry q2 g1"| 49778₇ ||class="entry q3 g1"| 51015₉ ||class="entry q3 g1"| 38449₇ ||class="entry q2 g1"| 44996₉ ||class="entry q3 g1"| 44007₁₁ ||class="entry q3 g1"| 37393₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (53306, 54543, 33913, 48524, 47535, 32857) ||class="entry q2 g1"| 53306₇ ||class="entry q3 g1"| 54543₉ ||class="entry q3 g1"| 33913₇ ||class="entry q2 g1"| 48524₉ ||class="entry q3 g1"| 47535₁₁ ||class="entry q3 g1"| 32857₅ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (33764, 60997, 60045, 61010, 33509, 61101) ||class="entry q2 g1"| 33764₇ ||class="entry q3 g1"| 60997₉ ||class="entry q3 g1"| 60045₉ ||class="entry q2 g1"| 61010₉ ||class="entry q3 g1"| 33509₇ ||class="entry q3 g1"| 61101₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (36434, 58083, 59595, 58340, 36419, 60651) ||class="entry q2 g1"| 36434₇ ||class="entry q3 g1"| 58083₉ ||class="entry q3 g1"| 59595₉ ||class="entry q2 g1"| 58340₉ ||class="entry q3 g1"| 36419₇ ||class="entry q3 g1"| 60651₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (37292, 64525, 63685, 64538, 37037, 64741) ||class="entry q2 g1"| 37292₇ ||class="entry q3 g1"| 64525₉ ||class="entry q3 g1"| 63685₉ ||class="entry q2 g1"| 64538₉ ||class="entry q3 g1"| 37037₇ ||class="entry q3 g1"| 64741₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (39962, 61611, 64131, 61868, 39947, 65187) ||class="entry q2 g1"| 39962₇ ||class="entry q3 g1"| 61611₉ ||class="entry q3 g1"| 64131₉ ||class="entry q2 g1"| 61868₉ ||class="entry q3 g1"| 39947₇ ||class="entry q3 g1"| 65187₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (41828, 53185, 63937, 52946, 41825, 64993) ||class="entry q2 g1"| 41828₇ ||class="entry q3 g1"| 53185₉ ||class="entry q3 g1"| 63937₉ ||class="entry q2 g1"| 52946₉ ||class="entry q3 g1"| 41825₇ ||class="entry q3 g1"| 64993₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (45356, 56713, 60297, 56474, 45353, 61353) ||class="entry q2 g1"| 45356₇ ||class="entry q3 g1"| 56713₉ ||class="entry q3 g1"| 60297₉ ||class="entry q2 g1"| 56474₉ ||class="entry q3 g1"| 45353₇ ||class="entry q3 g1"| 61353₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (50770, 43761, 64145, 44004, 50769, 65201) ||class="entry q2 g1"| 50770₇ ||class="entry q3 g1"| 43761₉ ||class="entry q3 g1"| 64145₉ ||class="entry q2 g1"| 44004₉ ||class="entry q3 g1"| 50769₇ ||class="entry q3 g1"| 65201₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (54298, 47289, 59609, 47532, 54297, 60665) ||class="entry q2 g1"| 54298₇ ||class="entry q3 g1"| 47289₉ ||class="entry q3 g1"| 59609₉ ||class="entry q2 g1"| 47532₉ ||class="entry q3 g1"| 54297₇ ||class="entry q3 g1"| 60665₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (33776, 53869, 54961, 60998, 48845, 53905) ||class="entry q2 g1"| 33776₇ ||class="entry q3 g1"| 53869₉ ||class="entry q3 g1"| 54961₉ ||class="entry q2 g1"| 60998₉ ||class="entry q3 g1"| 48845₁₁ ||class="entry q3 g1"| 53905₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (36688, 46667, 48481, 58086, 56043, 47425) ||class="entry q2 g1"| 36688₇ ||class="entry q3 g1"| 46667₉ ||class="entry q3 g1"| 48481₉ ||class="entry q2 g1"| 58086₉ ||class="entry q3 g1"| 56043₁₁ ||class="entry q3 g1"| 47425₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (37290, 39533, 40611, 64540, 63181, 39555) ||class="entry q2 g1"| 37290₇ ||class="entry q3 g1"| 39533₉ ||class="entry q3 g1"| 40611₉ ||class="entry q2 g1"| 64540₉ ||class="entry q3 g1"| 63181₁₁ ||class="entry q3 g1"| 39555₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (39964, 38603, 40165, 61866, 64107, 39109) ||class="entry q2 g1"| 39964₇ ||class="entry q3 g1"| 38603₉ ||class="entry q3 g1"| 40165₉ ||class="entry q2 g1"| 61866₉ ||class="entry q3 g1"| 64107₁₁ ||class="entry q3 g1"| 39109₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (41574, 39785, 44139, 53200, 63433, 43083) ||class="entry q2 g1"| 41574₇ ||class="entry q3 g1"| 39785₉ ||class="entry q3 g1"| 44139₉ ||class="entry q2 g1"| 53200₉ ||class="entry q3 g1"| 63433₁₁ ||class="entry q3 g1"| 43083₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (45116, 54121, 58489, 56714, 49097, 57433) ||class="entry q2 g1"| 45116₇ ||class="entry q3 g1"| 54121₉ ||class="entry q3 g1"| 58489₉ ||class="entry q2 g1"| 56714₉ ||class="entry q3 g1"| 49097₁₁ ||class="entry q3 g1"| 57433₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (50758, 38617, 50861, 44016, 64121, 49805) ||class="entry q2 g1"| 50758₇ ||class="entry q3 g1"| 38617₉ ||class="entry q3 g1"| 50861₉ ||class="entry q2 g1"| 44016₉ ||class="entry q3 g1"| 64121₁₁ ||class="entry q3 g1"| 49805₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (54538, 46681, 59177, 47292, 56057, 58121) ||class="entry q2 g1"| 54538₇ ||class="entry q3 g1"| 46681₉ ||class="entry q3 g1"| 59177₉ ||class="entry q2 g1"| 47292₉ ||class="entry q3 g1"| 56057₁₁ ||class="entry q3 g1"| 58121₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (44736, 39215, 41437, 50038, 62863, 42493) ||class="entry q2 g1"| 44736₇ ||class="entry q3 g1"| 39215₉ ||class="entry q3 g1"| 41437₉ ||class="entry q2 g1"| 50038₉ ||class="entry q3 g1"| 62863₁₁ ||class="entry q3 g1"| 42493₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (48264, 35687, 45973, 53566, 59335, 47029) ||class="entry q2 g1"| 48264₇ ||class="entry q3 g1"| 35687₉ ||class="entry q3 g1"| 45973₉ ||class="entry q2 g1"| 53566₉ ||class="entry q3 g1"| 59335₁₁ ||class="entry q3 g1"| 47029₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (51936, 38047, 51995, 42838, 63551, 53051) ||class="entry q2 g1"| 51936₇ ||class="entry q3 g1"| 38047₉ ||class="entry q3 g1"| 51995₉ ||class="entry q2 g1"| 42838₉ ||class="entry q3 g1"| 63551₁₁ ||class="entry q3 g1"| 53051₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (55464, 34519, 55635, 46366, 60023, 56691) ||class="entry q2 g1"| 55464₇ ||class="entry q3 g1"| 34519₉ ||class="entry q3 g1"| 55635₉ ||class="entry q2 g1"| 46366₉ ||class="entry q3 g1"| 60023₁₁ ||class="entry q3 g1"| 56691₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (59072, 53565, 45959, 35702, 48541, 47015) ||class="entry q2 g1"| 59072₇ ||class="entry q3 g1"| 53565₉ ||class="entry q3 g1"| 45959₉ ||class="entry q2 g1"| 35702₉ ||class="entry q3 g1"| 48541₁₁ ||class="entry q3 g1"| 47015₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (60000, 46363, 55383, 34774, 55739, 56439) ||class="entry q2 g1"| 60000₇ ||class="entry q3 g1"| 46363₉ ||class="entry q3 g1"| 55383₉ ||class="entry q2 g1"| 34774₉ ||class="entry q3 g1"| 55739₁₁ ||class="entry q3 g1"| 56439₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (62600, 50037, 41423, 39230, 45013, 42479) ||class="entry q2 g1"| 62600₇ ||class="entry q3 g1"| 50037₉ ||class="entry q3 g1"| 41423₉ ||class="entry q2 g1"| 39230₉ ||class="entry q3 g1"| 45013₁₁ ||class="entry q3 g1"| 42479₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (63528, 42835, 51743, 38302, 52211, 52799) ||class="entry q2 g1"| 63528₇ ||class="entry q3 g1"| 42835₉ ||class="entry q3 g1"| 51743₉ ||class="entry q2 g1"| 38302₉ ||class="entry q3 g1"| 52211₁₁ ||class="entry q3 g1"| 52799₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (43744, 62617, 52605, 51030, 38969, 51549) ||class="entry q2 g1"| 43744₇ ||class="entry q3 g1"| 62617₉ ||class="entry q3 g1"| 52605₁₁ ||class="entry q2 g1"| 51030₉ ||class="entry q3 g1"| 38969₇ ||class="entry q3 g1"| 51549₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (47272, 59089, 57141, 54558, 35441, 56085) ||class="entry q2 g1"| 47272₇ ||class="entry q3 g1"| 59089₉ ||class="entry q3 g1"| 57141₁₁ ||class="entry q2 g1"| 54558₉ ||class="entry q3 g1"| 35441₇ ||class="entry q3 g1"| 56085₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (52928, 63785, 42939, 41846, 38281, 41883) ||class="entry q2 g1"| 52928₇ ||class="entry q3 g1"| 63785₉ ||class="entry q3 g1"| 42939₁₁ ||class="entry q2 g1"| 41846₉ ||class="entry q3 g1"| 38281₇ ||class="entry q3 g1"| 41883₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (56456, 60257, 46579, 45374, 34753, 45523) ||class="entry q2 g1"| 56456₇ ||class="entry q3 g1"| 60257₉ ||class="entry q3 g1"| 46579₁₁ ||class="entry q2 g1"| 45374₉ ||class="entry q3 g1"| 34753₇ ||class="entry q3 g1"| 45523₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (58080, 48267, 57127, 36694, 53291, 56071) ||class="entry q2 g1"| 58080₇ ||class="entry q3 g1"| 48267₉ ||class="entry q3 g1"| 57127₁₁ ||class="entry q2 g1"| 36694₉ ||class="entry q3 g1"| 53291₇ ||class="entry q3 g1"| 56071₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (60992, 55469, 46327, 33782, 46093, 45271) ||class="entry q2 g1"| 60992₇ ||class="entry q3 g1"| 55469₉ ||class="entry q3 g1"| 46327₁₁ ||class="entry q2 g1"| 33782₉ ||class="entry q3 g1"| 46093₇ ||class="entry q3 g1"| 45271₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (61608, 44739, 52591, 40222, 49763, 51535) ||class="entry q2 g1"| 61608₇ ||class="entry q3 g1"| 44739₉ ||class="entry q3 g1"| 52591₁₁ ||class="entry q2 g1"| 40222₉ ||class="entry q3 g1"| 49763₇ ||class="entry q3 g1"| 51535₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64520, 51941, 42687, 37310, 42565, 41631) ||class="entry q2 g1"| 64520₇ ||class="entry q3 g1"| 51941₉ ||class="entry q3 g1"| 42687₁₁ ||class="entry q2 g1"| 37310₉ ||class="entry q3 g1"| 42565₇ ||class="entry q3 g1"| 41631₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (34502, 55131, 54151, 60272, 48123, 55207) ||class="entry q2 g1"| 34502₇ ||class="entry q3 g1"| 55131₁₁ ||class="entry q3 g1"| 54151₉ ||class="entry q2 g1"| 60272₉ ||class="entry q3 g1"| 48123₁₃ ||class="entry q3 g1"| 55207₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (35430, 45949, 47191, 59344, 57309, 48247) ||class="entry q2 g1"| 35430₇ ||class="entry q3 g1"| 45949₁₁ ||class="entry q3 g1"| 47191₉ ||class="entry q2 g1"| 59344₉ ||class="entry q3 g1"| 57309₁₃ ||class="entry q3 g1"| 48247₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (38044, 40795, 39829, 63786, 62459, 40885) ||class="entry q2 g1"| 38044₇ ||class="entry q3 g1"| 40795₁₁ ||class="entry q3 g1"| 39829₉ ||class="entry q2 g1"| 63786₉ ||class="entry q3 g1"| 62459₁₃ ||class="entry q3 g1"| 40885₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (39210, 37885, 39379, 62620, 65373, 40435) ||class="entry q2 g1"| 39210₇ ||class="entry q3 g1"| 37885₁₁ ||class="entry q3 g1"| 39379₉ ||class="entry q2 g1"| 62620₉ ||class="entry q3 g1"| 65373₁₃ ||class="entry q3 g1"| 40435₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (42832, 40543, 43357, 51942, 62207, 44413) ||class="entry q2 g1"| 42832₇ ||class="entry q3 g1"| 40543₁₁ ||class="entry q3 g1"| 43357₉ ||class="entry q2 g1"| 51942₉ ||class="entry q3 g1"| 62207₁₃ ||class="entry q3 g1"| 44413₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (46346, 54879, 57679, 55484, 47871, 58735) ||class="entry q2 g1"| 46346₇ ||class="entry q3 g1"| 54879₁₁ ||class="entry q3 g1"| 57679₉ ||class="entry q2 g1"| 55484₉ ||class="entry q3 g1"| 47871₁₃ ||class="entry q3 g1"| 58735₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (50032, 37871, 50075, 44742, 65359, 51131) ||class="entry q2 g1"| 50032₇ ||class="entry q3 g1"| 37871₁₁ ||class="entry q3 g1"| 50075₉ ||class="entry q2 g1"| 44742₉ ||class="entry q3 g1"| 65359₁₃ ||class="entry q3 g1"| 51131₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (53308, 45935, 57887, 48522, 57295, 58943) ||class="entry q2 g1"| 53308₇ ||class="entry q3 g1"| 45935₁₁ ||class="entry q3 g1"| 57887₉ ||class="entry q2 g1"| 48522₉ ||class="entry q3 g1"| 57295₁₃ ||class="entry q3 g1"| 58943₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33510, 47853, 48935, 61264, 54861, 47879) ||class="entry q2 g1"| 33510₇ ||class="entry q3 g1"| 47853₁₁ ||class="entry q3 g1"| 48935₁₁ ||class="entry q2 g1"| 61264₉ ||class="entry q3 g1"| 54861₉ ||class="entry q3 g1"| 47879₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (36422, 57035, 54519, 58352, 45675, 53463) ||class="entry q2 g1"| 36422₇ ||class="entry q3 g1"| 57035₁₁ ||class="entry q3 g1"| 54519₁₁ ||class="entry q2 g1"| 58352₉ ||class="entry q3 g1"| 45675₉ ||class="entry q3 g1"| 53463₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (37052, 62189, 63285, 64778, 40525, 62229) ||class="entry q2 g1"| 37052₇ ||class="entry q3 g1"| 62189₁₁ ||class="entry q3 g1"| 63285₁₁ ||class="entry q2 g1"| 64778₉ ||class="entry q3 g1"| 40525₉ ||class="entry q3 g1"| 62229₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (40202, 65099, 62835, 61628, 37611, 61779) ||class="entry q2 g1"| 40202₇ ||class="entry q3 g1"| 65099₁₁ ||class="entry q3 g1"| 62835₁₁ ||class="entry q2 g1"| 61628₉ ||class="entry q3 g1"| 37611₉ ||class="entry q3 g1"| 61779₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (41840, 62441, 50685, 52934, 40777, 49629) ||class="entry q2 g1"| 41840₇ ||class="entry q3 g1"| 62441₁₁ ||class="entry q3 g1"| 50685₁₁ ||class="entry q2 g1"| 52934₉ ||class="entry q3 g1"| 40777₉ ||class="entry q3 g1"| 49629₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (45354, 48105, 36335, 56476, 55113, 35279) ||class="entry q2 g1"| 45354₇ ||class="entry q3 g1"| 48105₁₁ ||class="entry q3 g1"| 36335₁₁ ||class="entry q2 g1"| 56476₉ ||class="entry q3 g1"| 55113₉ ||class="entry q3 g1"| 35279₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (51024, 65113, 44859, 43750, 37625, 43803) ||class="entry q2 g1"| 51024₇ ||class="entry q3 g1"| 65113₁₁ ||class="entry q3 g1"| 44859₁₁ ||class="entry q2 g1"| 43750₉ ||class="entry q3 g1"| 37625₉ ||class="entry q3 g1"| 43803₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54300, 57049, 36543, 47530, 45689, 35487) ||class="entry q2 g1"| 54300₇ ||class="entry q3 g1"| 57049₁₁ ||class="entry q3 g1"| 36543₁₁ ||class="entry q2 g1"| 47530₉ ||class="entry q3 g1"| 45689₉ ||class="entry q3 g1"| 35487₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (34514, 60275, 61371, 60260, 34771, 60315) ||class="entry q2 g1"| 34514₇ ||class="entry q3 g1"| 60275₁₁ ||class="entry q3 g1"| 61371₁₃ ||class="entry q2 g1"| 60260₉ ||class="entry q3 g1"| 34771₉ ||class="entry q3 g1"| 60315₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (35684, 59349, 60925, 59090, 35701, 59869) ||class="entry q2 g1"| 35684₇ ||class="entry q3 g1"| 59349₁₁ ||class="entry q3 g1"| 60925₁₃ ||class="entry q2 g1"| 59090₉ ||class="entry q3 g1"| 35701₉ ||class="entry q3 g1"| 59869₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (38042, 63803, 65011, 63788, 38299, 63955) ||class="entry q2 g1"| 38042₇ ||class="entry q3 g1"| 63803₁₁ ||class="entry q3 g1"| 65011₁₃ ||class="entry q2 g1"| 63788₉ ||class="entry q3 g1"| 38299₉ ||class="entry q3 g1"| 63955₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39212, 62877, 65461, 62618, 39229, 64405) ||class="entry q2 g1"| 39212₇ ||class="entry q3 g1"| 62877₁₁ ||class="entry q3 g1"| 65461₁₃ ||class="entry q2 g1"| 62618₉ ||class="entry q3 g1"| 39229₉ ||class="entry q3 g1"| 64405₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (42578, 51959, 64759, 52196, 42583, 63703) ||class="entry q2 g1"| 42578₇ ||class="entry q3 g1"| 51959₁₁ ||class="entry q3 g1"| 64759₁₃ ||class="entry q2 g1"| 52196₉ ||class="entry q3 g1"| 42583₉ ||class="entry q3 g1"| 63703₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (46106, 55487, 61119, 55724, 46111, 60063) ||class="entry q2 g1"| 46106₇ ||class="entry q3 g1"| 55487₁₁ ||class="entry q3 g1"| 61119₁₃ ||class="entry q2 g1"| 55724₉ ||class="entry q3 g1"| 46111₉ ||class="entry q3 g1"| 60063₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50020, 44999, 65447, 44754, 50023, 64391) ||class="entry q2 g1"| 50020₇ ||class="entry q3 g1"| 44999₁₁ ||class="entry q3 g1"| 65447₁₃ ||class="entry q2 g1"| 44754₉ ||class="entry q3 g1"| 50023₉ ||class="entry q3 g1"| 64391₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (53548, 48527, 60911, 48282, 53551, 59855) ||class="entry q2 g1"| 53548₇ ||class="entry q3 g1"| 48527₁₁ ||class="entry q3 g1"| 60911₁₃ ||class="entry q2 g1"| 48282₉ ||class="entry q3 g1"| 53551₉ ||class="entry q3 g1"| 59855₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (34768, 49115, 47633, 60006, 54139, 48689) ||class="entry q2 g1"| 34768₇ ||class="entry q3 g1"| 49115₁₃ ||class="entry q3 g1"| 47633₇ ||class="entry q2 g1"| 60006₉ ||class="entry q3 g1"| 54139₁₁ ||class="entry q3 g1"| 48689₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (35696, 56317, 53697, 59078, 46941, 54753) ||class="entry q2 g1"| 35696₇ ||class="entry q3 g1"| 56317₁₃ ||class="entry q3 g1"| 53697₇ ||class="entry q2 g1"| 59078₉ ||class="entry q3 g1"| 46941₁₁ ||class="entry q3 g1"| 54753₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38282, 63451, 61955, 63548, 39803, 63011) ||class="entry q2 g1"| 38282₇ ||class="entry q3 g1"| 63451₁₃ ||class="entry q3 g1"| 61955₇ ||class="entry q2 g1"| 63548₉ ||class="entry q3 g1"| 39803₁₁ ||class="entry q3 g1"| 63011₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38972, 64381, 61509, 62858, 38877, 62565) ||class="entry q2 g1"| 38972₇ ||class="entry q3 g1"| 64381₁₃ ||class="entry q3 g1"| 61509₇ ||class="entry q2 g1"| 62858₉ ||class="entry q3 g1"| 38877₁₁ ||class="entry q3 g1"| 62565₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (42566, 63199, 49355, 52208, 39551, 50411) ||class="entry q2 g1"| 42566₇ ||class="entry q3 g1"| 63199₁₃ ||class="entry q3 g1"| 49355₇ ||class="entry q2 g1"| 52208₉ ||class="entry q3 g1"| 39551₁₁ ||class="entry q3 g1"| 50411₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (46108, 48863, 35033, 55722, 53887, 36089) ||class="entry q2 g1"| 46108₇ ||class="entry q3 g1"| 48863₁₃ ||class="entry q3 g1"| 35033₇ ||class="entry q2 g1"| 55722₉ ||class="entry q3 g1"| 53887₁₁ ||class="entry q3 g1"| 36089₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (49766, 64367, 43533, 45008, 38863, 44589) ||class="entry q2 g1"| 49766₇ ||class="entry q3 g1"| 64367₁₃ ||class="entry q3 g1"| 43533₇ ||class="entry q2 g1"| 45008₉ ||class="entry q3 g1"| 38863₁₁ ||class="entry q3 g1"| 44589₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (53546, 56303, 35721, 48284, 46927, 36777) ||class="entry q2 g1"| 53546₇ ||class="entry q3 g1"| 56303₁₃ ||class="entry q3 g1"| 35721₇ ||class="entry q2 g1"| 48284₉ ||class="entry q3 g1"| 46927₁₁ ||class="entry q3 g1"| 36777₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38858, 39949, 39107, 64124, 61613, 40163) ||class="entry q2 g1"| 38858₉ ||class="entry q3 g1"| 39949₇ ||class="entry q3 g1"| 39107₇ ||class="entry q2 g1"| 64124₁₁ ||class="entry q3 g1"| 61613₉ ||class="entry q3 g1"| 40163₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38872, 50757, 49817, 64110, 43749, 50873) ||class="entry q2 g1"| 38872₉ ||class="entry q3 g1"| 50757₇ ||class="entry q3 g1"| 49817₇ ||class="entry q2 g1"| 64110₁₁ ||class="entry q3 g1"| 43749₉ ||class="entry q3 g1"| 50873₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (39548, 37035, 39557, 63434, 64523, 40613) ||class="entry q2 g1"| 39548₉ ||class="entry q3 g1"| 37035₇ ||class="entry q3 g1"| 39557₇ ||class="entry q2 g1"| 63434₁₁ ||class="entry q3 g1"| 64523₉ ||class="entry q3 g1"| 40613₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (39800, 41571, 43337, 63182, 52931, 44393) ||class="entry q2 g1"| 39800₉ ||class="entry q3 g1"| 41571₇ ||class="entry q3 g1"| 43337₇ ||class="entry q2 g1"| 63182₁₁ ||class="entry q3 g1"| 52931₉ ||class="entry q3 g1"| 44393₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46670, 36673, 47171, 56312, 58337, 48227) ||class="entry q2 g1"| 46670₉ ||class="entry q3 g1"| 36673₇ ||class="entry q3 g1"| 47171₇ ||class="entry q2 g1"| 56312₁₁ ||class="entry q3 g1"| 58337₉ ||class="entry q3 g1"| 48227₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46684, 54537, 57881, 56298, 47529, 58937) ||class="entry q2 g1"| 46684₉ ||class="entry q3 g1"| 54537₇ ||class="entry q3 g1"| 57881₇ ||class="entry q2 g1"| 56298₁₁ ||class="entry q3 g1"| 47529₉ ||class="entry q3 g1"| 58937₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (53870, 33521, 53893, 49112, 61009, 54949) ||class="entry q2 g1"| 53870₉ ||class="entry q3 g1"| 33521₇ ||class="entry q3 g1"| 53893₇ ||class="entry q2 g1"| 49112₁₁ ||class="entry q3 g1"| 61009₉ ||class="entry q3 g1"| 54949₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (54122, 45113, 57673, 48860, 56473, 58729) ||class="entry q2 g1"| 54122₉ ||class="entry q3 g1"| 45113₇ ||class="entry q3 g1"| 57673₇ ||class="entry q2 g1"| 48860₁₁ ||class="entry q3 g1"| 56473₉ ||class="entry q3 g1"| 58729₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (48840, 57521, 55637, 54142, 35857, 56693) ||class="entry q2 g1"| 48840₉ ||class="entry q3 g1"| 57521₇ ||class="entry q3 g1"| 55637₉ ||class="entry q2 g1"| 54142₁₁ ||class="entry q3 g1"| 35857₅ ||class="entry q3 g1"| 56693₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (56040, 60673, 45971, 46942, 33185, 47027) ||class="entry q2 g1"| 56040₉ ||class="entry q3 g1"| 60673₇ ||class="entry q3 g1"| 45971₉ ||class="entry q2 g1"| 46942₁₁ ||class="entry q3 g1"| 33185₅ ||class="entry q3 g1"| 47027₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (63176, 43171, 51983, 39806, 50179, 53039) ||class="entry q2 g1"| 63176₉ ||class="entry q3 g1"| 43171₇ ||class="entry q3 g1"| 51983₉ ||class="entry q2 g1"| 39806₁₁ ||class="entry q3 g1"| 50179₅ ||class="entry q3 g1"| 53039₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (64104, 52357, 41183, 38878, 40997, 42239) ||class="entry q2 g1"| 64104₉ ||class="entry q3 g1"| 52357₇ ||class="entry q3 g1"| 41183₉ ||class="entry q2 g1"| 38878₁₁ ||class="entry q3 g1"| 40997₅ ||class="entry q3 g1"| 42239₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (47848, 36103, 46581, 55134, 57767, 45525) ||class="entry q2 g1"| 47848₉ ||class="entry q3 g1"| 36103₇ ||class="entry q3 g1"| 46581₁₁ ||class="entry q2 g1"| 55134₁₁ ||class="entry q3 g1"| 57767₉ ||class="entry q3 g1"| 45525₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (57032, 32951, 57139, 45950, 60439, 56083) ||class="entry q2 g1"| 57032₉ ||class="entry q3 g1"| 32951₇ ||class="entry q3 g1"| 57139₁₁ ||class="entry q2 g1"| 45950₁₁ ||class="entry q3 g1"| 60439₉ ||class="entry q3 g1"| 56083₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (62184, 50453, 42927, 40798, 43445, 41871) ||class="entry q2 g1"| 62184₉ ||class="entry q3 g1"| 50453₇ ||class="entry q3 g1"| 42927₁₁ ||class="entry q2 g1"| 40798₁₁ ||class="entry q3 g1"| 43445₉ ||class="entry q3 g1"| 41871₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (65096, 41267, 52351, 37886, 52627, 51295) ||class="entry q2 g1"| 65096₉ ||class="entry q3 g1"| 41267₇ ||class="entry q3 g1"| 52351₁₁ ||class="entry q2 g1"| 37886₁₁ ||class="entry q3 g1"| 52627₉ ||class="entry q3 g1"| 51295₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38606, 44741, 43791, 64376, 49765, 44847) ||class="entry q2 g1"| 38606₉ ||class="entry q3 g1"| 44741₉ ||class="entry q3 g1"| 43791₉ ||class="entry q2 g1"| 64376₁₁ ||class="entry q3 g1"| 49765₇ ||class="entry q3 g1"| 44847₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38620, 62605, 61781, 64362, 38957, 62837) ||class="entry q2 g1"| 38620₉ ||class="entry q3 g1"| 62605₉ ||class="entry q3 g1"| 61781₉ ||class="entry q2 g1"| 64362₁₁ ||class="entry q3 g1"| 38957₇ ||class="entry q3 g1"| 62837₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (39534, 51939, 49375, 63448, 42563, 50431) ||class="entry q2 g1"| 39534₉ ||class="entry q3 g1"| 51939₉ ||class="entry q3 g1"| 49375₉ ||class="entry q2 g1"| 63448₁₁ ||class="entry q3 g1"| 42563₇ ||class="entry q3 g1"| 50431₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (39786, 63531, 62227, 63196, 38027, 63283) ||class="entry q2 g1"| 39786₉ ||class="entry q3 g1"| 63531₉ ||class="entry q3 g1"| 62227₉ ||class="entry q2 g1"| 63196₁₁ ||class="entry q3 g1"| 38027₇ ||class="entry q3 g1"| 63283₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46922, 48521, 35727, 56060, 53545, 36783) ||class="entry q2 g1"| 46922₉ ||class="entry q3 g1"| 48521₉ ||class="entry q3 g1"| 35727₉ ||class="entry q2 g1"| 56060₁₁ ||class="entry q3 g1"| 53545₇ ||class="entry q3 g1"| 36783₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46936, 59329, 53717, 56046, 35681, 54773) ||class="entry q2 g1"| 46936₉ ||class="entry q3 g1"| 59329₉ ||class="entry q3 g1"| 53717₉ ||class="entry q2 g1"| 56046₁₁ ||class="entry q3 g1"| 35681₇ ||class="entry q3 g1"| 54773₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (53884, 55481, 35039, 49098, 46105, 36095) ||class="entry q2 g1"| 53884₉ ||class="entry q3 g1"| 55481₉ ||class="entry q3 g1"| 35039₉ ||class="entry q2 g1"| 49098₁₁ ||class="entry q3 g1"| 46105₇ ||class="entry q3 g1"| 36095₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (54136, 60017, 47891, 48846, 34513, 48947) ||class="entry q2 g1"| 54136₉ ||class="entry q3 g1"| 60017₉ ||class="entry q3 g1"| 47891₉ ||class="entry q2 g1"| 48846₁₁ ||class="entry q3 g1"| 34513₇ ||class="entry q3 g1"| 48947₁₁ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (38618, 37613, 38707, 64364, 65101, 37651) ||class="entry q2 g1"| 38618₉ ||class="entry q3 g1"| 37613₉ ||class="entry q3 g1"| 38707₉ ||class="entry q2 g1"| 64364₁₁ ||class="entry q3 g1"| 65101₁₁ ||class="entry q3 g1"| 37651₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (39788, 40523, 38261, 63194, 62187, 37205) ||class="entry q2 g1"| 39788₉ ||class="entry q3 g1"| 40523₉ ||class="entry q3 g1"| 38261₉ ||class="entry q2 g1"| 63194₁₁ ||class="entry q3 g1"| 62187₁₁ ||class="entry q3 g1"| 37205₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (46682, 45929, 33919, 56300, 57289, 32863) ||class="entry q2 g1"| 46682₉ ||class="entry q3 g1"| 45929₉ ||class="entry q3 g1"| 33919₉ ||class="entry q2 g1"| 56300₁₁ ||class="entry q3 g1"| 57289₁₁ ||class="entry q3 g1"| 32863₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (54124, 54873, 34607, 48858, 47865, 33551) ||class="entry q2 g1"| 54124₉ ||class="entry q3 g1"| 54873₉ ||class="entry q3 g1"| 34607₉ ||class="entry q2 g1"| 48858₁₁ ||class="entry q3 g1"| 47865₁₁ ||class="entry q3 g1"| 33551₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (37614, 50035, 51119, 65368, 45011, 50063) ||class="entry q2 g1"| 37614₉ ||class="entry q3 g1"| 50035₉ ||class="entry q3 g1"| 51119₁₁ ||class="entry q2 g1"| 65368₁₁ ||class="entry q3 g1"| 45011₁₁ ||class="entry q3 g1"| 50063₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (37628, 39227, 40437, 65354, 62875, 39381) ||class="entry q2 g1"| 37628₉ ||class="entry q3 g1"| 39227₉ ||class="entry q3 g1"| 40437₁₁ ||class="entry q2 g1"| 65354₁₁ ||class="entry q3 g1"| 62875₁₁ ||class="entry q3 g1"| 39381₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40526, 42837, 44159, 62456, 52213, 43103) ||class="entry q2 g1"| 40526₉ ||class="entry q3 g1"| 42837₉ ||class="entry q3 g1"| 44159₁₁ ||class="entry q2 g1"| 62456₁₁ ||class="entry q3 g1"| 52213₁₁ ||class="entry q3 g1"| 43103₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40778, 38301, 40883, 62204, 63805, 39827) ||class="entry q2 g1"| 40778₉ ||class="entry q3 g1"| 38301₉ ||class="entry q3 g1"| 40883₁₁ ||class="entry q2 g1"| 62204₁₁ ||class="entry q3 g1"| 63805₁₁ ||class="entry q3 g1"| 39827₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (45930, 53311, 59183, 57052, 48287, 58127) ||class="entry q2 g1"| 45930₉ ||class="entry q3 g1"| 53311₉ ||class="entry q3 g1"| 59183₁₁ ||class="entry q2 g1"| 57052₁₁ ||class="entry q3 g1"| 48287₁₁ ||class="entry q3 g1"| 58127₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (45944, 35447, 48501, 57038, 59095, 47445) ||class="entry q2 g1"| 45944₉ ||class="entry q3 g1"| 35447₉ ||class="entry q3 g1"| 48501₁₁ ||class="entry q2 g1"| 57038₁₁ ||class="entry q3 g1"| 59095₁₁ ||class="entry q3 g1"| 47445₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54876, 46351, 58495, 48106, 55727, 57439) ||class="entry q2 g1"| 54876₉ ||class="entry q3 g1"| 46351₉ ||class="entry q3 g1"| 58495₁₁ ||class="entry q2 g1"| 48106₁₁ ||class="entry q3 g1"| 55727₁₁ ||class="entry q3 g1"| 57439₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (55128, 34759, 55219, 47854, 60263, 54163) ||class="entry q2 g1"| 55128₉ ||class="entry q3 g1"| 34759₉ ||class="entry q3 g1"| 55219₁₁ ||class="entry q2 g1"| 47854₁₁ ||class="entry q3 g1"| 60263₁₁ ||class="entry q3 g1"| 54163₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (37868, 38875, 37381, 65114, 64379, 38437) ||class="entry q2 g1"| 37868₉ ||class="entry q3 g1"| 38875₁₁ ||class="entry q3 g1"| 37381₅ ||class="entry q2 g1"| 65114₁₁ ||class="entry q3 g1"| 64379₁₃ ||class="entry q3 g1"| 38437₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (40538, 39805, 36931, 62444, 63453, 37987) ||class="entry q2 g1"| 40538₉ ||class="entry q3 g1"| 39805₁₁ ||class="entry q3 g1"| 36931₅ ||class="entry q2 g1"| 62444₁₁ ||class="entry q3 g1"| 63453₁₃ ||class="entry q3 g1"| 37987₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (45932, 46687, 33097, 57050, 56063, 34153) ||class="entry q2 g1"| 45932₉ ||class="entry q3 g1"| 46687₁₁ ||class="entry q3 g1"| 33097₅ ||class="entry q2 g1"| 57050₁₁ ||class="entry q3 g1"| 56063₁₃ ||class="entry q3 g1"| 34153₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54874, 54127, 33305, 48108, 49103, 34361) ||class="entry q2 g1"| 54874₉ ||class="entry q3 g1"| 54127₁₁ ||class="entry q3 g1"| 33305₅ ||class="entry q2 g1"| 48108₁₁ ||class="entry q3 g1"| 49103₁₃ ||class="entry q3 g1"| 34361₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (37866, 61883, 62563, 65116, 40219, 61507) ||class="entry q2 g1"| 37866₉ ||class="entry q3 g1"| 61883₁₁ ||class="entry q3 g1"| 62563₉ ||class="entry q2 g1"| 65116₁₁ ||class="entry q3 g1"| 40219₉ ||class="entry q3 g1"| 61507₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (37880, 44019, 44601, 65102, 51027, 43545) ||class="entry q2 g1"| 37880₉ ||class="entry q3 g1"| 44019₁₁ ||class="entry q3 g1"| 44601₉ ||class="entry q2 g1"| 65102₁₁ ||class="entry q3 g1"| 51027₉ ||class="entry q3 g1"| 43545₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40540, 64797, 63013, 62442, 37309, 61957) ||class="entry q2 g1"| 40540₉ ||class="entry q3 g1"| 64797₁₁ ||class="entry q3 g1"| 63013₉ ||class="entry q2 g1"| 62442₁₁ ||class="entry q3 g1"| 37309₉ ||class="entry q3 g1"| 61957₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40792, 53205, 50665, 62190, 41845, 49609) ||class="entry q2 g1"| 40792₉ ||class="entry q3 g1"| 53205₁₁ ||class="entry q3 g1"| 50665₉ ||class="entry q2 g1"| 62190₁₁ ||class="entry q3 g1"| 41845₉ ||class="entry q3 g1"| 49609₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (45678, 58103, 54499, 57304, 36439, 53443) ||class="entry q2 g1"| 45678₉ ||class="entry q3 g1"| 58103₁₁ ||class="entry q3 g1"| 54499₉ ||class="entry q2 g1"| 57304₁₁ ||class="entry q3 g1"| 36439₉ ||class="entry q3 g1"| 53443₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (45692, 47295, 36537, 57290, 54303, 35481) ||class="entry q2 g1"| 45692₉ ||class="entry q3 g1"| 47295₁₁ ||class="entry q3 g1"| 36537₉ ||class="entry q2 g1"| 57290₁₁ ||class="entry q3 g1"| 54303₉ ||class="entry q3 g1"| 35481₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54862, 61255, 48677, 48120, 33767, 47621) ||class="entry q2 g1"| 54862₉ ||class="entry q3 g1"| 61255₁₁ ||class="entry q3 g1"| 48677₉ ||class="entry q2 g1"| 48120₁₁ ||class="entry q3 g1"| 33767₉ ||class="entry q3 g1"| 47621₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (55114, 56719, 36329, 47868, 45359, 35273) ||class="entry q2 g1"| 55114₉ ||class="entry q3 g1"| 56719₁₁ ||class="entry q3 g1"| 36329₉ ||class="entry q2 g1"| 47868₁₁ ||class="entry q3 g1"| 45359₉ ||class="entry q3 g1"| 35273₇ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38860, 64109, 65189, 64122, 38605, 64133) ||class="entry q2 g1"| 38860₉ ||class="entry q3 g1"| 64109₁₁ ||class="entry q3 g1"| 65189₁₁ ||class="entry q2 g1"| 64122₁₁ ||class="entry q3 g1"| 38605₉ ||class="entry q3 g1"| 64133₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (39546, 63179, 64739, 63436, 39531, 63683) ||class="entry q2 g1"| 39546₉ ||class="entry q3 g1"| 63179₁₁ ||class="entry q3 g1"| 64739₁₁ ||class="entry q2 g1"| 63436₁₁ ||class="entry q3 g1"| 39531₉ ||class="entry q3 g1"| 63683₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (46924, 56297, 60905, 56058, 46921, 59849) ||class="entry q2 g1"| 46924₉ ||class="entry q3 g1"| 56297₁₁ ||class="entry q3 g1"| 60905₁₁ ||class="entry q2 g1"| 56058₁₁ ||class="entry q3 g1"| 46921₉ ||class="entry q3 g1"| 59849₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (53882, 48857, 61113, 49100, 53881, 60057) ||class="entry q2 g1"| 53882₉ ||class="entry q3 g1"| 48857₁₁ ||class="entry q3 g1"| 61113₁₁ ||class="entry q2 g1"| 49100₁₁ ||class="entry q3 g1"| 53881₉ ||class="entry q3 g1"| 60057₉ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (37626, 65371, 64403, 65356, 37883, 65459) ||class="entry q2 g1"| 37626₉ ||class="entry q3 g1"| 65371₁₃ ||class="entry q3 g1"| 64403₁₁ ||class="entry q2 g1"| 65356₁₁ ||class="entry q3 g1"| 37883₁₁ ||class="entry q3 g1"| 65459₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (40780, 62461, 63957, 62202, 40797, 65013) ||class="entry q2 g1"| 40780₉ ||class="entry q3 g1"| 62461₁₃ ||class="entry q3 g1"| 63957₁₁ ||class="entry q2 g1"| 62202₁₁ ||class="entry q3 g1"| 40797₁₁ ||class="entry q3 g1"| 65013₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (45690, 57055, 59615, 57292, 45695, 60671) ||class="entry q2 g1"| 45690₉ ||class="entry q3 g1"| 57055₁₃ ||class="entry q3 g1"| 59615₁₁ ||class="entry q2 g1"| 57292₁₁ ||class="entry q3 g1"| 45695₁₁ ||class="entry q3 g1"| 60671₁₃ |- |class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (55116, 48111, 60303, 47866, 55119, 61359) ||class="entry q2 g1"| 55116₉ ||class="entry q3 g1"| 48111₁₃ ||class="entry q3 g1"| 60303₁₁ ||class="entry q2 g1"| 47866₁₁ ||class="entry q3 g1"| 55119₁₁ ||class="entry q3 g1"| 61359₁₃ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (96, 24678, 26112, 1536, 26118, 24672) ||class="entry q0 g0"| 96₂ ||class="entry q0 g0"| 24678₆ ||class="entry q0 g0"| 26112₄ ||class="entry q0 g0"| 1536₂ ||class="entry q0 g0"| 26118₆ ||class="entry q0 g0"| 24672₄ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (544, 2992, 3264, 1088, 3536, 2720) ||class="entry q0 g0"| 544₂ ||class="entry q0 g0"| 2992₆ ||class="entry q0 g0"| 3264₄ ||class="entry q0 g0"| 1088₂ ||class="entry q0 g0"| 3536₆ ||class="entry q0 g0"| 2720₄ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (8416, 16866, 30028, 9856, 18306, 29484) ||class="entry q0 g0"| 8416₄ ||class="entry q0 g0"| 16866₆ ||class="entry q0 g0"| 30028₈ ||class="entry q0 g0"| 9856₄ ||class="entry q0 g0"| 18306₆ ||class="entry q0 g0"| 29484₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (8864, 10804, 8076, 9408, 11348, 6636) ||class="entry q0 g0"| 8864₄ ||class="entry q0 g0"| 10804₆ ||class="entry q0 g0"| 8076₈ ||class="entry q0 g0"| 9408₄ ||class="entry q0 g0"| 11348₆ ||class="entry q0 g0"| 6636₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (10336, 18546, 29244, 11776, 19986, 29788) ||class="entry q0 g0"| 10336₄ ||class="entry q0 g0"| 18546₆ ||class="entry q0 g0"| 29244₈ ||class="entry q0 g0"| 11776₄ ||class="entry q0 g0"| 19986₆ ||class="entry q0 g0"| 29788₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (10784, 9124, 6396, 11328, 9668, 7836) ||class="entry q0 g0"| 10784₄ ||class="entry q0 g0"| 9124₆ ||class="entry q0 g0"| 6396₈ ||class="entry q0 g0"| 11328₄ ||class="entry q0 g0"| 9668₆ ||class="entry q0 g0"| 7836₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (16608, 8676, 29482, 18048, 10116, 30026) ||class="entry q0 g0"| 16608₄ ||class="entry q0 g0"| 8676₆ ||class="entry q0 g0"| 29482₈ ||class="entry q0 g0"| 18048₄ ||class="entry q0 g0"| 10116₆ ||class="entry q0 g0"| 30026₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (17056, 18994, 6634, 17600, 19538, 8074) ||class="entry q0 g0"| 17056₄ ||class="entry q0 g0"| 18994₆ ||class="entry q0 g0"| 6634₈ ||class="entry q0 g0"| 17600₄ ||class="entry q0 g0"| 19538₆ ||class="entry q0 g0"| 8074₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (18528, 10356, 29786, 19968, 11796, 29242) ||class="entry q0 g0"| 18528₄ ||class="entry q0 g0"| 10356₆ ||class="entry q0 g0"| 29786₈ ||class="entry q0 g0"| 19968₄ ||class="entry q0 g0"| 11796₆ ||class="entry q0 g0"| 29242₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (18976, 17314, 7834, 19520, 17858, 6394) ||class="entry q0 g0"| 18976₄ ||class="entry q0 g0"| 17314₆ ||class="entry q0 g0"| 7834₈ ||class="entry q0 g0"| 19520₄ ||class="entry q0 g0"| 17858₆ ||class="entry q0 g0"| 6394₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (4200, 6648, 7816, 5640, 8088, 6376) ||class="entry q0 g0"| 4200₄ ||class="entry q0 g0"| 6648₈ ||class="entry q0 g0"| 7816₆ ||class="entry q0 g0"| 5640₄ ||class="entry q0 g0"| 8088₈ ||class="entry q0 g0"| 6376₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (4648, 29230, 29768, 5192, 29774, 29224) ||class="entry q0 g0"| 4648₄ ||class="entry q0 g0"| 29230₈ ||class="entry q0 g0"| 29768₆ ||class="entry q0 g0"| 5192₄ ||class="entry q0 g0"| 29774₈ ||class="entry q0 g0"| 29224₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (114, 14894, 15450, 1554, 15438, 14906) ||class="entry q0 g0"| 114₄ ||class="entry q0 g0"| 14894₈ ||class="entry q0 g0"| 15450₈ ||class="entry q0 g0"| 1554₄ ||class="entry q0 g0"| 15438₈ ||class="entry q0 g0"| 14906₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (116, 23630, 23100, 1556, 23086, 23644) ||class="entry q0 g0"| 116₄ ||class="entry q0 g0"| 23630₈ ||class="entry q0 g0"| 23100₈ ||class="entry q0 g0"| 1556₄ ||class="entry q0 g0"| 23086₈ ||class="entry q0 g0"| 23644₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (354, 13518, 13226, 1794, 12974, 13770) ||class="entry q0 g0"| 354₄ ||class="entry q0 g0"| 13518₈ ||class="entry q0 g0"| 13226₈ ||class="entry q0 g0"| 1794₄ ||class="entry q0 g0"| 12974₈ ||class="entry q0 g0"| 13770₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (356, 21166, 21964, 1796, 21710, 21420) ||class="entry q0 g0"| 356₄ ||class="entry q0 g0"| 21166₈ ||class="entry q0 g0"| 21964₈ ||class="entry q0 g0"| 1796₄ ||class="entry q0 g0"| 21710₈ ||class="entry q0 g0"| 21420₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (368, 28294, 27120, 1808, 26854, 28560) ||class="entry q0 g0"| 368₄ ||class="entry q0 g0"| 28294₈ ||class="entry q0 g0"| 27120₈ ||class="entry q0 g0"| 1808₄ ||class="entry q0 g0"| 26854₈ ||class="entry q0 g0"| 28560₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (550, 28112, 27302, 1094, 27568, 27846) ||class="entry q0 g0"| 550₄ ||class="entry q0 g0"| 28112₈ ||class="entry q0 g0"| 27302₈ ||class="entry q0 g0"| 1094₄ ||class="entry q0 g0"| 27568₈ ||class="entry q0 g0"| 27846₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (562, 20984, 22170, 1106, 22424, 20730) ||class="entry q0 g0"| 562₄ ||class="entry q0 g0"| 20984₈ ||class="entry q0 g0"| 22170₈ ||class="entry q0 g0"| 1106₄ ||class="entry q0 g0"| 22424₈ ||class="entry q0 g0"| 20730₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (564, 14232, 12540, 1108, 12792, 13980) ||class="entry q0 g0"| 564₄ ||class="entry q0 g0"| 14232₈ ||class="entry q0 g0"| 12540₈ ||class="entry q0 g0"| 1108₄ ||class="entry q0 g0"| 12792₈ ||class="entry q0 g0"| 13980₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (802, 24344, 22890, 1346, 22904, 24330) ||class="entry q0 g0"| 802₄ ||class="entry q0 g0"| 24344₈ ||class="entry q0 g0"| 22890₈ ||class="entry q0 g0"| 1346₄ ||class="entry q0 g0"| 22904₈ ||class="entry q0 g0"| 24330₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (804, 14712, 16140, 1348, 16152, 14700) ||class="entry q0 g0"| 804₄ ||class="entry q0 g0"| 14712₈ ||class="entry q0 g0"| 16140₈ ||class="entry q0 g0"| 1348₄ ||class="entry q0 g0"| 16152₈ ||class="entry q0 g0"| 14700₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (2272, 27126, 24944, 3712, 28566, 26384) ||class="entry q0 g0"| 2272₄ ||class="entry q0 g0"| 27126₁₀ ||class="entry q0 g0"| 24944₆ ||class="entry q0 g0"| 3712₄ ||class="entry q0 g0"| 28566₁₀ ||class="entry q0 g0"| 26384₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (25120, 27574, 2726, 25664, 28118, 3270) ||class="entry q0 g0"| 25120₄ ||class="entry q0 g0"| 27574₁₀ ||class="entry q0 g0"| 2726₆ ||class="entry q0 g0"| 25664₄ ||class="entry q0 g0"| 28118₁₀ ||class="entry q0 g0"| 3270₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4218, 17328, 17618, 5658, 17872, 17074) ||class="entry q0 g0"| 4218₆ ||class="entry q0 g0"| 17328₆ ||class="entry q0 g0"| 17618₆ ||class="entry q0 g0"| 5658₆ ||class="entry q0 g0"| 17872₆ ||class="entry q0 g0"| 17074₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4220, 9680, 8884, 5660, 9136, 9428) ||class="entry q0 g0"| 4220₆ ||class="entry q0 g0"| 9680₆ ||class="entry q0 g0"| 8884₆ ||class="entry q0 g0"| 5660₆ ||class="entry q0 g0"| 9136₆ ||class="entry q0 g0"| 9428₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4458, 19792, 19234, 5898, 19248, 19778) ||class="entry q0 g0"| 4458₆ ||class="entry q0 g0"| 19792₆ ||class="entry q0 g0"| 19234₆ ||class="entry q0 g0"| 5898₆ ||class="entry q0 g0"| 19248₆ ||class="entry q0 g0"| 19778₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4460, 11056, 11588, 5900, 11600, 11044) ||class="entry q0 g0"| 4460₆ ||class="entry q0 g0"| 11056₆ ||class="entry q0 g0"| 11588₆ ||class="entry q0 g0"| 5900₆ ||class="entry q0 g0"| 11600₆ ||class="entry q0 g0"| 11044₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4666, 10342, 11794, 5210, 11782, 10354) ||class="entry q0 g0"| 4666₆ ||class="entry q0 g0"| 10342₆ ||class="entry q0 g0"| 11794₆ ||class="entry q0 g0"| 5210₆ ||class="entry q0 g0"| 11782₆ ||class="entry q0 g0"| 10354₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4668, 19974, 18548, 5212, 18534, 19988) ||class="entry q0 g0"| 4668₆ ||class="entry q0 g0"| 19974₆ ||class="entry q0 g0"| 18548₆ ||class="entry q0 g0"| 5212₆ ||class="entry q0 g0"| 18534₆ ||class="entry q0 g0"| 19988₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4906, 9862, 8674, 5450, 8422, 10114) ||class="entry q0 g0"| 4906₆ ||class="entry q0 g0"| 9862₆ ||class="entry q0 g0"| 8674₆ ||class="entry q0 g0"| 5450₆ ||class="entry q0 g0"| 8422₆ ||class="entry q0 g0"| 10114₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (4908, 16614, 18308, 5452, 18054, 16868) ||class="entry q0 g0"| 4908₆ ||class="entry q0 g0"| 16614₆ ||class="entry q0 g0"| 18308₆ ||class="entry q0 g0"| 5452₆ ||class="entry q0 g0"| 18054₆ ||class="entry q0 g0"| 16868₆ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (374, 2278, 3990, 1814, 3718, 2550) ||class="entry q0 g0"| 374₆ ||class="entry q0 g0"| 2278₆ ||class="entry q0 g0"| 3990₈ ||class="entry q0 g0"| 1814₆ ||class="entry q0 g0"| 3718₆ ||class="entry q0 g0"| 2550₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (822, 25392, 25942, 1366, 25936, 25398) ||class="entry q0 g0"| 822₆ ||class="entry q0 g0"| 25392₆ ||class="entry q0 g0"| 25942₈ ||class="entry q0 g0"| 1366₆ ||class="entry q0 g0"| 25936₆ ||class="entry q0 g0"| 25398₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (2738, 22632, 20970, 3282, 24072, 22410) ||class="entry q0 g0"| 2738₆ ||class="entry q0 g0"| 22632₆ ||class="entry q0 g0"| 20970₈ ||class="entry q0 g0"| 3282₆ ||class="entry q0 g0"| 24072₆ ||class="entry q0 g0"| 22410₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (2740, 15880, 14220, 3284, 14440, 12780) ||class="entry q0 g0"| 2740₆ ||class="entry q0 g0"| 15880₆ ||class="entry q0 g0"| 14220₈ ||class="entry q0 g0"| 3284₆ ||class="entry q0 g0"| 14440₆ ||class="entry q0 g0"| 12780₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (2978, 22152, 24090, 3522, 20712, 22650) ||class="entry q0 g0"| 2978₆ ||class="entry q0 g0"| 22152₆ ||class="entry q0 g0"| 24090₈ ||class="entry q0 g0"| 3522₆ ||class="entry q0 g0"| 20712₆ ||class="entry q0 g0"| 22650₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (2980, 12520, 14460, 3524, 13960, 15900) ||class="entry q0 g0"| 2980₆ ||class="entry q0 g0"| 12520₆ ||class="entry q0 g0"| 14460₈ ||class="entry q0 g0"| 3524₆ ||class="entry q0 g0"| 13960₆ ||class="entry q0 g0"| 15900₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (24690, 23080, 14908, 26130, 23624, 15452) ||class="entry q0 g0"| 24690₆ ||class="entry q0 g0"| 23080₆ ||class="entry q0 g0"| 14908₈ ||class="entry q0 g0"| 26130₆ ||class="entry q0 g0"| 23624₆ ||class="entry q0 g0"| 15452₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (24692, 15432, 23642, 26132, 14888, 23098) ||class="entry q0 g0"| 24692₆ ||class="entry q0 g0"| 15432₆ ||class="entry q0 g0"| 23642₈ ||class="entry q0 g0"| 26132₆ ||class="entry q0 g0"| 14888₆ ||class="entry q0 g0"| 23098₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (24930, 21704, 13772, 26370, 21160, 13228) ||class="entry q0 g0"| 24930₆ ||class="entry q0 g0"| 21704₆ ||class="entry q0 g0"| 13772₈ ||class="entry q0 g0"| 26370₆ ||class="entry q0 g0"| 21160₆ ||class="entry q0 g0"| 13228₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (24932, 12968, 21418, 26372, 13512, 21962) ||class="entry q0 g0"| 24932₆ ||class="entry q0 g0"| 12968₆ ||class="entry q0 g0"| 21418₈ ||class="entry q0 g0"| 26372₆ ||class="entry q0 g0"| 13512₆ ||class="entry q0 g0"| 21962₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (26848, 2544, 26390, 28288, 3984, 24950) ||class="entry q0 g0"| 26848₆ ||class="entry q0 g0"| 2544₆ ||class="entry q0 g0"| 26390₈ ||class="entry q0 g0"| 28288₆ ||class="entry q0 g0"| 3984₆ ||class="entry q0 g0"| 24950₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (27296, 25126, 3542, 27840, 25670, 2998) ||class="entry q0 g0"| 27296₆ ||class="entry q0 g0"| 25126₆ ||class="entry q0 g0"| 3542₈ ||class="entry q0 g0"| 27840₆ ||class="entry q0 g0"| 25670₆ ||class="entry q0 g0"| 2998₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (8688, 20226, 31420, 10128, 18786, 31964) ||class="entry q0 g0"| 8688₆ ||class="entry q0 g0"| 20226₆ ||class="entry q0 g0"| 31420₁₀ ||class="entry q0 g0"| 10128₆ ||class="entry q0 g0"| 18786₆ ||class="entry q0 g0"| 31964₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (8870, 19540, 31210, 9414, 18996, 32650) ||class="entry q0 g0"| 8870₆ ||class="entry q0 g0"| 19540₆ ||class="entry q0 g0"| 31210₁₀ ||class="entry q0 g0"| 9414₆ ||class="entry q0 g0"| 18996₆ ||class="entry q0 g0"| 32650₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (10608, 18066, 32204, 12048, 16626, 31660) ||class="entry q0 g0"| 10608₆ ||class="entry q0 g0"| 18066₆ ||class="entry q0 g0"| 32204₁₀ ||class="entry q0 g0"| 12048₆ ||class="entry q0 g0"| 16626₆ ||class="entry q0 g0"| 31660₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (10790, 17860, 32410, 11334, 17316, 30970) ||class="entry q0 g0"| 10790₆ ||class="entry q0 g0"| 17860₆ ||class="entry q0 g0"| 32410₁₀ ||class="entry q0 g0"| 11334₆ ||class="entry q0 g0"| 17316₆ ||class="entry q0 g0"| 30970₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (16880, 12036, 31962, 18320, 10596, 31418) ||class="entry q0 g0"| 16880₆ ||class="entry q0 g0"| 12036₆ ||class="entry q0 g0"| 31962₁₀ ||class="entry q0 g0"| 18320₆ ||class="entry q0 g0"| 10596₆ ||class="entry q0 g0"| 31418₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (17062, 11346, 32652, 17606, 10802, 31212) ||class="entry q0 g0"| 17062₆ ||class="entry q0 g0"| 11346₆ ||class="entry q0 g0"| 32652₁₀ ||class="entry q0 g0"| 17606₆ ||class="entry q0 g0"| 10802₆ ||class="entry q0 g0"| 31212₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (18800, 9876, 31658, 20240, 8436, 32202) ||class="entry q0 g0"| 18800₆ ||class="entry q0 g0"| 9876₆ ||class="entry q0 g0"| 31658₁₀ ||class="entry q0 g0"| 20240₆ ||class="entry q0 g0"| 8436₆ ||class="entry q0 g0"| 32202₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (18982, 9666, 30972, 19526, 9122, 32412) ||class="entry q0 g0"| 18982₆ ||class="entry q0 g0"| 9666₆ ||class="entry q0 g0"| 30972₁₀ ||class="entry q0 g0"| 19526₆ ||class="entry q0 g0"| 9122₆ ||class="entry q0 g0"| 32412₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (8434, 7082, 12054, 9874, 7626, 10614) ||class="entry q0 g0"| 8434₆ ||class="entry q0 g0"| 7082₈ ||class="entry q0 g0"| 12054₈ ||class="entry q0 g0"| 9874₆ ||class="entry q0 g0"| 7626₈ ||class="entry q0 g0"| 10614₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (8882, 28796, 17878, 9426, 30236, 17334) ||class="entry q0 g0"| 8882₆ ||class="entry q0 g0"| 28796₈ ||class="entry q0 g0"| 17878₈ ||class="entry q0 g0"| 9426₆ ||class="entry q0 g0"| 30236₈ ||class="entry q0 g0"| 17334₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (10594, 7386, 10134, 12034, 6842, 8694) ||class="entry q0 g0"| 10594₆ ||class="entry q0 g0"| 7386₈ ||class="entry q0 g0"| 10134₈ ||class="entry q0 g0"| 12034₆ ||class="entry q0 g0"| 6842₈ ||class="entry q0 g0"| 8694₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (11042, 30476, 19798, 11586, 29036, 19254) ||class="entry q0 g0"| 11042₆ ||class="entry q0 g0"| 30476₈ ||class="entry q0 g0"| 19798₈ ||class="entry q0 g0"| 11586₆ ||class="entry q0 g0"| 29036₈ ||class="entry q0 g0"| 19254₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (16628, 7628, 20246, 18068, 7084, 18806) ||class="entry q0 g0"| 16628₆ ||class="entry q0 g0"| 7628₈ ||class="entry q0 g0"| 20246₈ ||class="entry q0 g0"| 18068₆ ||class="entry q0 g0"| 7084₈ ||class="entry q0 g0"| 18806₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (17076, 30234, 9686, 17620, 28794, 9142) ||class="entry q0 g0"| 17076₆ ||class="entry q0 g0"| 30234₈ ||class="entry q0 g0"| 9686₈ ||class="entry q0 g0"| 17620₆ ||class="entry q0 g0"| 28794₈ ||class="entry q0 g0"| 9142₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (18788, 6844, 18326, 20228, 7388, 16886) ||class="entry q0 g0"| 18788₆ ||class="entry q0 g0"| 6844₈ ||class="entry q0 g0"| 18326₈ ||class="entry q0 g0"| 20228₆ ||class="entry q0 g0"| 7388₈ ||class="entry q0 g0"| 16886₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (19236, 29034, 11606, 19780, 30474, 11062) ||class="entry q0 g0"| 19236₆ ||class="entry q0 g0"| 29034₈ ||class="entry q0 g0"| 11606₈ ||class="entry q0 g0"| 19780₆ ||class="entry q0 g0"| 30474₈ ||class="entry q0 g0"| 11062₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (2290, 13246, 15146, 3730, 13790, 15690) ||class="entry q0 g0"| 2290₆ ||class="entry q0 g0"| 13246₁₀ ||class="entry q0 g0"| 15146₈ ||class="entry q0 g0"| 3730₆ ||class="entry q0 g0"| 13790₁₀ ||class="entry q0 g0"| 15690₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (2292, 21982, 23884, 3732, 21438, 23340) ||class="entry q0 g0"| 2292₆ ||class="entry q0 g0"| 21982₁₀ ||class="entry q0 g0"| 23884₈ ||class="entry q0 g0"| 3732₆ ||class="entry q0 g0"| 21438₁₀ ||class="entry q0 g0"| 23340₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (2530, 15710, 13530, 3970, 15166, 12986) ||class="entry q0 g0"| 2530₆ ||class="entry q0 g0"| 15710₁₀ ||class="entry q0 g0"| 13530₈ ||class="entry q0 g0"| 3970₆ ||class="entry q0 g0"| 15166₁₀ ||class="entry q0 g0"| 12986₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (2532, 23358, 21180, 3972, 23902, 21724) ||class="entry q0 g0"| 2532₆ ||class="entry q0 g0"| 23358₁₀ ||class="entry q0 g0"| 21180₈ ||class="entry q0 g0"| 3972₆ ||class="entry q0 g0"| 23902₁₀ ||class="entry q0 g0"| 21724₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (25138, 12798, 20732, 25682, 14238, 22172) ||class="entry q0 g0"| 25138₆ ||class="entry q0 g0"| 12798₁₀ ||class="entry q0 g0"| 20732₈ ||class="entry q0 g0"| 25682₆ ||class="entry q0 g0"| 14238₁₀ ||class="entry q0 g0"| 22172₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (25140, 22430, 13978, 25684, 20990, 12538) ||class="entry q0 g0"| 25140₆ ||class="entry q0 g0"| 22430₁₀ ||class="entry q0 g0"| 13978₈ ||class="entry q0 g0"| 25684₆ ||class="entry q0 g0"| 20990₁₀ ||class="entry q0 g0"| 12538₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (25378, 16158, 24332, 25922, 14718, 22892) ||class="entry q0 g0"| 25378₆ ||class="entry q0 g0"| 16158₁₀ ||class="entry q0 g0"| 24332₈ ||class="entry q0 g0"| 25922₆ ||class="entry q0 g0"| 14718₁₀ ||class="entry q0 g0"| 22892₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (25380, 22910, 14698, 25924, 24350, 16138) ||class="entry q0 g0"| 25380₆ ||class="entry q0 g0"| 22910₁₀ ||class="entry q0 g0"| 14698₈ ||class="entry q0 g0"| 25924₆ ||class="entry q0 g0"| 24350₁₀ ||class="entry q0 g0"| 16138₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (4206, 32664, 30958, 5646, 31224, 32398) ||class="entry q0 g0"| 4206₆ ||class="entry q0 g0"| 32664₁₀ ||class="entry q0 g0"| 30958₁₀ ||class="entry q0 g0"| 5646₆ ||class="entry q0 g0"| 31224₁₀ ||class="entry q0 g0"| 32398₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (4920, 31950, 31672, 5464, 31406, 32216) ||class="entry q0 g0"| 4920₆ ||class="entry q0 g0"| 31950₁₀ ||class="entry q0 g0"| 31672₁₀ ||class="entry q0 g0"| 5464₆ ||class="entry q0 g0"| 31406₁₀ ||class="entry q0 g0"| 32216₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (6824, 31678, 29496, 7368, 32222, 30040) ||class="entry q0 g0"| 6824₆ ||class="entry q0 g0"| 31678₁₂ ||class="entry q0 g0"| 29496₈ ||class="entry q0 g0"| 7368₆ ||class="entry q0 g0"| 32222₁₂ ||class="entry q0 g0"| 30040₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (28776, 31230, 6382, 30216, 32670, 7822) ||class="entry q0 g0"| 28776₆ ||class="entry q0 g0"| 31230₁₂ ||class="entry q0 g0"| 6382₈ ||class="entry q0 g0"| 30216₆ ||class="entry q0 g0"| 32670₁₂ ||class="entry q0 g0"| 7822₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (12526, 24092, 27554, 13966, 22652, 28098) ||class="entry q0 g0"| 12526₈ ||class="entry q0 g0"| 24092₈ ||class="entry q0 g0"| 27554₈ ||class="entry q0 g0"| 13966₈ ||class="entry q0 g0"| 22652₈ ||class="entry q0 g0"| 28098₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (12778, 27860, 22638, 14218, 27316, 24078) ||class="entry q0 g0"| 12778₈ ||class="entry q0 g0"| 27860₈ ||class="entry q0 g0"| 22638₈ ||class="entry q0 g0"| 14218₈ ||class="entry q0 g0"| 27316₈ ||class="entry q0 g0"| 24078₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (12988, 28546, 23352, 13532, 27106, 23896) ||class="entry q0 g0"| 12988₈ ||class="entry q0 g0"| 28546₈ ||class="entry q0 g0"| 23352₈ ||class="entry q0 g0"| 13532₈ ||class="entry q0 g0"| 27106₈ ||class="entry q0 g0"| 23896₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (13240, 23882, 26868, 13784, 23338, 28308) ||class="entry q0 g0"| 13240₈ ||class="entry q0 g0"| 23882₈ ||class="entry q0 g0"| 26868₈ ||class="entry q0 g0"| 13784₈ ||class="entry q0 g0"| 23338₈ ||class="entry q0 g0"| 28308₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (14446, 22412, 27858, 15886, 20972, 27314) ||class="entry q0 g0"| 14446₈ ||class="entry q0 g0"| 22412₈ ||class="entry q0 g0"| 27858₈ ||class="entry q0 g0"| 15886₈ ||class="entry q0 g0"| 20972₈ ||class="entry q0 g0"| 27314₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (14458, 27556, 20718, 15898, 28100, 22158) ||class="entry q0 g0"| 14458₈ ||class="entry q0 g0"| 27556₈ ||class="entry q0 g0"| 20718₈ ||class="entry q0 g0"| 15898₈ ||class="entry q0 g0"| 28100₈ ||class="entry q0 g0"| 22158₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (15148, 26866, 21432, 15692, 28306, 21976) ||class="entry q0 g0"| 15148₈ ||class="entry q0 g0"| 26866₈ ||class="entry q0 g0"| 21432₈ ||class="entry q0 g0"| 15692₈ ||class="entry q0 g0"| 28306₈ ||class="entry q0 g0"| 21976₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (15160, 21722, 28548, 15704, 21178, 27108) ||class="entry q0 g0"| 15160₈ ||class="entry q0 g0"| 21722₈ ||class="entry q0 g0"| 28548₈ ||class="entry q0 g0"| 15704₈ ||class="entry q0 g0"| 21178₈ ||class="entry q0 g0"| 27108₈ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (4478, 29048, 30494, 5918, 30488, 29054) ||class="entry q0 g0"| 4478₈ ||class="entry q0 g0"| 29048₈ ||class="entry q0 g0"| 30494₁₀ ||class="entry q0 g0"| 5918₈ ||class="entry q0 g0"| 30488₈ ||class="entry q0 g0"| 29054₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (4926, 6830, 7646, 5470, 7374, 7102) ||class="entry q0 g0"| 4926₈ ||class="entry q0 g0"| 6830₈ ||class="entry q0 g0"| 7646₁₀ ||class="entry q0 g0"| 5470₈ ||class="entry q0 g0"| 7374₈ ||class="entry q0 g0"| 7102₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (30952, 28782, 8094, 32392, 30222, 6654) ||class="entry q0 g0"| 30952₈ ||class="entry q0 g0"| 28782₈ ||class="entry q0 g0"| 8094₁₀ ||class="entry q0 g0"| 32392₈ ||class="entry q0 g0"| 30222₈ ||class="entry q0 g0"| 6654₁₀ |- |class="f"| 1632 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (31400, 7096, 30046, 31944, 7640, 29502) ||class="entry q0 g0"| 31400₈ ||class="entry q0 g0"| 7096₈ ||class="entry q0 g0"| 30046₁₀ ||class="entry q0 g0"| 31944₈ ||class="entry q0 g0"| 7640₈ ||class="entry q0 g0"| 29502₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (4224, 28817, 7839, 32352, 8039, 6633) ||class="entry q0 g0"| 4224₂ ||class="entry q1 g0"| 28817₆ ||class="entry q1 g0"| 7839₁₀ ||class="entry q0 g0"| 32352₈ ||class="entry q1 g0"| 8039₁₀ ||class="entry q1 g0"| 6633₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (6144, 30977, 6639, 30432, 5879, 7833) ||class="entry q0 g0"| 6144₂ ||class="entry q1 g0"| 30977₆ ||class="entry q1 g0"| 6639₁₀ ||class="entry q0 g0"| 30432₈ ||class="entry q1 g0"| 5879₁₀ ||class="entry q1 g0"| 7833₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (8200, 10379, 30043, 20200, 18301, 29229) ||class="entry q0 g0"| 8200₂ ||class="entry q1 g0"| 10379₆ ||class="entry q1 g0"| 30043₁₀ ||class="entry q0 g0"| 20200₈ ||class="entry q1 g0"| 18301₁₀ ||class="entry q1 g0"| 29229₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (16392, 18573, 29501, 12008, 10107, 29771) ||class="entry q0 g0"| 16392₂ ||class="entry q1 g0"| 18573₆ ||class="entry q1 g0"| 29501₁₀ ||class="entry q0 g0"| 12008₈ ||class="entry q1 g0"| 10107₁₀ ||class="entry q1 g0"| 29771₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (14464, 22661, 2723, 22112, 14195, 3541) ||class="entry q0 g0"| 14464₄ ||class="entry q1 g0"| 22661₆ ||class="entry q1 g0"| 2723₆ ||class="entry q0 g0"| 22112₆ ||class="entry q1 g0"| 14195₁₀ ||class="entry q1 g0"| 3541₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (22656, 14467, 3269, 13920, 22389, 2995) ||class="entry q0 g0"| 22656₄ ||class="entry q1 g0"| 14467₆ ||class="entry q1 g0"| 3269₆ ||class="entry q0 g0"| 13920₆ ||class="entry q1 g0"| 22389₁₀ ||class="entry q1 g0"| 2995₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (24712, 26889, 24689, 3688, 1791, 26375) ||class="entry q0 g0"| 24712₄ ||class="entry q1 g0"| 26889₆ ||class="entry q1 g0"| 24689₆ ||class="entry q0 g0"| 3688₆ ||class="entry q1 g0"| 1791₁₀ ||class="entry q1 g0"| 26375₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (26632, 24729, 26369, 1768, 3951, 24695) ||class="entry q0 g0"| 26632₄ ||class="entry q1 g0"| 24729₆ ||class="entry q1 g0"| 26369₆ ||class="entry q0 g0"| 1768₆ ||class="entry q1 g0"| 3951₁₀ ||class="entry q1 g0"| 24695₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (10376, 8475, 29227, 18024, 20205, 30045) ||class="entry q0 g0"| 10376₄ ||class="entry q1 g0"| 8475₆ ||class="entry q1 g0"| 29227₈ ||class="entry q0 g0"| 18024₆ ||class="entry q1 g0"| 20205₁₀ ||class="entry q1 g0"| 30045₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (18568, 16669, 29773, 9832, 12011, 29499) ||class="entry q0 g0"| 18568₄ ||class="entry q1 g0"| 16669₆ ||class="entry q1 g0"| 29773₈ ||class="entry q0 g0"| 9832₆ ||class="entry q1 g0"| 12011₁₀ ||class="entry q1 g0"| 29499₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (28800, 4247, 6393, 7776, 32609, 8079) ||class="entry q0 g0"| 28800₄ ||class="entry q1 g0"| 4247₆ ||class="entry q1 g0"| 6393₈ ||class="entry q0 g0"| 7776₆ ||class="entry q1 g0"| 32609₁₀ ||class="entry q1 g0"| 8079₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (30720, 6407, 8073, 5856, 30449, 6399) ||class="entry q0 g0"| 30720₄ ||class="entry q1 g0"| 6407₆ ||class="entry q1 g0"| 8073₈ ||class="entry q0 g0"| 5856₆ ||class="entry q1 g0"| 30449₁₀ ||class="entry q1 g0"| 6399₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (4482, 9273, 19253, 32610, 19407, 19523) ||class="entry q0 g0"| 4482₄ ||class="entry q1 g0"| 9273₆ ||class="entry q1 g0"| 19253₈ ||class="entry q0 g0"| 32610₁₀ ||class="entry q1 g0"| 19407₁₀ ||class="entry q1 g0"| 19523₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (4484, 16985, 11603, 32612, 11695, 10789) ||class="entry q0 g0"| 4484₄ ||class="entry q1 g0"| 16985₆ ||class="entry q1 g0"| 11603₈ ||class="entry q0 g0"| 32612₁₀ ||class="entry q1 g0"| 11695₁₀ ||class="entry q1 g0"| 10789₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (6162, 9033, 17333, 30450, 19647, 17603) ||class="entry q0 g0"| 6162₄ ||class="entry q1 g0"| 9033₆ ||class="entry q1 g0"| 17333₈ ||class="entry q0 g0"| 30450₁₀ ||class="entry q1 g0"| 19647₁₀ ||class="entry q1 g0"| 17603₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (6164, 17705, 9683, 30452, 10975, 8869) ||class="entry q0 g0"| 6164₄ ||class="entry q1 g0"| 17705₆ ||class="entry q1 g0"| 9683₈ ||class="entry q0 g0"| 30452₁₀ ||class="entry q1 g0"| 10975₁₀ ||class="entry q1 g0"| 8869₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (8220, 5283, 18791, 20220, 31573, 19985) ||class="entry q0 g0"| 8220₄ ||class="entry q1 g0"| 5283₆ ||class="entry q1 g0"| 18791₈ ||class="entry q0 g0"| 20220₁₀ ||class="entry q1 g0"| 31573₁₀ ||class="entry q1 g0"| 19985₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (8460, 6723, 18071, 20460, 30133, 16865) ||class="entry q0 g0"| 8460₄ ||class="entry q1 g0"| 6723₆ ||class="entry q1 g0"| 18071₈ ||class="entry q0 g0"| 20460₁₀ ||class="entry q1 g0"| 30133₁₀ ||class="entry q1 g0"| 16865₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16410, 4805, 10599, 12026, 32051, 11793) ||class="entry q0 g0"| 16410₄ ||class="entry q1 g0"| 4805₆ ||class="entry q1 g0"| 10599₈ ||class="entry q0 g0"| 12026₁₀ ||class="entry q1 g0"| 32051₁₀ ||class="entry q1 g0"| 11793₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16650, 7205, 9879, 12266, 29651, 8673) ||class="entry q0 g0"| 16650₄ ||class="entry q1 g0"| 7205₆ ||class="entry q1 g0"| 9879₈ ||class="entry q0 g0"| 12266₁₀ ||class="entry q1 g0"| 29651₁₀ ||class="entry q1 g0"| 8673₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (4242, 10969, 17605, 32370, 17711, 17331) ||class="entry q0 g0"| 4242₄ ||class="entry q1 g0"| 10969₈ ||class="entry q1 g0"| 17605₆ ||class="entry q0 g0"| 32370₁₀ ||class="entry q1 g0"| 17711₈ ||class="entry q1 g0"| 17331₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (4244, 19641, 8867, 32372, 9039, 9685) ||class="entry q0 g0"| 4244₄ ||class="entry q1 g0"| 19641₈ ||class="entry q1 g0"| 8867₆ ||class="entry q0 g0"| 32372₁₀ ||class="entry q1 g0"| 9039₈ ||class="entry q1 g0"| 9685₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (6402, 11689, 19525, 30690, 16991, 19251) ||class="entry q0 g0"| 6402₄ ||class="entry q1 g0"| 11689₈ ||class="entry q1 g0"| 19525₆ ||class="entry q0 g0"| 30690₁₀ ||class="entry q1 g0"| 16991₈ ||class="entry q1 g0"| 19251₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (6404, 19401, 10787, 30692, 9279, 11605) ||class="entry q0 g0"| 6404₄ ||class="entry q1 g0"| 19401₈ ||class="entry q1 g0"| 10787₆ ||class="entry q0 g0"| 30692₁₀ ||class="entry q1 g0"| 9279₈ ||class="entry q1 g0"| 11605₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (8218, 29379, 12033, 20218, 7477, 10359) ||class="entry q0 g0"| 8218₄ ||class="entry q1 g0"| 29379₈ ||class="entry q1 g0"| 12033₆ ||class="entry q0 g0"| 20218₁₀ ||class="entry q1 g0"| 7477₈ ||class="entry q1 g0"| 10359₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (8458, 31779, 8433, 20458, 5077, 10119) ||class="entry q0 g0"| 8458₄ ||class="entry q1 g0"| 31779₈ ||class="entry q1 g0"| 8433₆ ||class="entry q0 g0"| 20458₁₀ ||class="entry q1 g0"| 5077₈ ||class="entry q1 g0"| 10119₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (16412, 29861, 20225, 12028, 6995, 18551) ||class="entry q0 g0"| 16412₄ ||class="entry q1 g0"| 29861₈ ||class="entry q1 g0"| 20225₆ ||class="entry q0 g0"| 12028₁₀ ||class="entry q1 g0"| 6995₈ ||class="entry q1 g0"| 18551₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (16652, 31301, 16625, 12268, 5555, 18311) ||class="entry q0 g0"| 16652₄ ||class="entry q1 g0"| 31301₈ ||class="entry q1 g0"| 16625₆ ||class="entry q0 g0"| 12268₁₀ ||class="entry q1 g0"| 5555₈ ||class="entry q1 g0"| 18311₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (712, 25305, 3287, 27688, 3375, 2977) ||class="entry q0 g0"| 712₄ ||class="entry q1 g0"| 25305₈ ||class="entry q1 g0"| 3287₈ ||class="entry q0 g0"| 27688₆ ||class="entry q1 g0"| 3375₈ ||class="entry q1 g0"| 2977₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (1192, 25785, 2743, 27208, 2895, 3521) ||class="entry q0 g0"| 1192₄ ||class="entry q1 g0"| 25785₈ ||class="entry q1 g0"| 2743₈ ||class="entry q0 g0"| 27208₆ ||class="entry q1 g0"| 2895₈ ||class="entry q1 g0"| 3521₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (2632, 27465, 2983, 25768, 1215, 3281) ||class="entry q0 g0"| 2632₄ ||class="entry q1 g0"| 27465₈ ||class="entry q1 g0"| 2983₈ ||class="entry q0 g0"| 25768₆ ||class="entry q1 g0"| 1215₈ ||class="entry q1 g0"| 3281₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (3112, 27945, 3527, 25288, 735, 2737) ||class="entry q0 g0"| 3112₄ ||class="entry q1 g0"| 27945₈ ||class="entry q1 g0"| 3527₈ ||class="entry q0 g0"| 25288₆ ||class="entry q1 g0"| 735₈ ||class="entry q1 g0"| 2737₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (12864, 15043, 26387, 23712, 21813, 24677) ||class="entry q0 g0"| 12864₄ ||class="entry q1 g0"| 15043₈ ||class="entry q1 g0"| 26387₈ ||class="entry q0 g0"| 23712₆ ||class="entry q1 g0"| 21813₈ ||class="entry q1 g0"| 24677₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (13344, 15523, 24947, 23232, 21333, 26117) ||class="entry q0 g0"| 13344₄ ||class="entry q1 g0"| 15523₈ ||class="entry q1 g0"| 24947₈ ||class="entry q0 g0"| 23232₆ ||class="entry q1 g0"| 21333₈ ||class="entry q1 g0"| 26117₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (21056, 23237, 24949, 15520, 13619, 26115) ||class="entry q0 g0"| 21056₄ ||class="entry q1 g0"| 23237₈ ||class="entry q1 g0"| 24949₈ ||class="entry q0 g0"| 15520₆ ||class="entry q1 g0"| 13619₈ ||class="entry q1 g0"| 26115₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (21536, 23717, 26389, 15040, 13139, 24675) ||class="entry q0 g0"| 21536₄ ||class="entry q1 g0"| 23717₈ ||class="entry q1 g0"| 26389₈ ||class="entry q0 g0"| 15040₆ ||class="entry q1 g0"| 13139₈ ||class="entry q1 g0"| 24675₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (4800, 6983, 29791, 31776, 29873, 29481) ||class="entry q0 g0"| 4800₄ ||class="entry q1 g0"| 6983₈ ||class="entry q1 g0"| 29791₁₀ ||class="entry q0 g0"| 31776₆ ||class="entry q1 g0"| 29873₈ ||class="entry q1 g0"| 29481₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (5280, 7463, 29247, 31296, 29393, 30025) ||class="entry q0 g0"| 5280₄ ||class="entry q1 g0"| 7463₈ ||class="entry q1 g0"| 29247₁₀ ||class="entry q0 g0"| 31296₆ ||class="entry q1 g0"| 29393₈ ||class="entry q1 g0"| 30025₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (6720, 4823, 29487, 29856, 32033, 29785) ||class="entry q0 g0"| 6720₄ ||class="entry q1 g0"| 4823₈ ||class="entry q1 g0"| 29487₁₀ ||class="entry q0 g0"| 29856₆ ||class="entry q1 g0"| 32033₈ ||class="entry q1 g0"| 29785₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (7200, 5303, 30031, 29376, 31553, 29241) ||class="entry q0 g0"| 7200₄ ||class="entry q1 g0"| 5303₈ ||class="entry q1 g0"| 30031₁₀ ||class="entry q0 g0"| 29376₆ ||class="entry q1 g0"| 31553₈ ||class="entry q1 g0"| 29241₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (8776, 17245, 8091, 19624, 11435, 6381) ||class="entry q0 g0"| 8776₄ ||class="entry q1 g0"| 17245₈ ||class="entry q1 g0"| 8091₁₀ ||class="entry q0 g0"| 19624₆ ||class="entry q1 g0"| 11435₈ ||class="entry q1 g0"| 6381₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (9256, 17725, 6651, 19144, 10955, 7821) ||class="entry q0 g0"| 9256₄ ||class="entry q1 g0"| 17725₈ ||class="entry q1 g0"| 6651₁₀ ||class="entry q0 g0"| 19144₆ ||class="entry q1 g0"| 10955₈ ||class="entry q1 g0"| 7821₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (16968, 9051, 6653, 11432, 19629, 7819) ||class="entry q0 g0"| 16968₄ ||class="entry q1 g0"| 9051₈ ||class="entry q1 g0"| 6653₁₀ ||class="entry q0 g0"| 11432₆ ||class="entry q1 g0"| 19629₈ ||class="entry q1 g0"| 7819₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (17448, 9531, 8093, 10952, 19149, 6379) ||class="entry q0 g0"| 17448₄ ||class="entry q1 g0"| 9531₈ ||class="entry q1 g0"| 8093₁₀ ||class="entry q0 g0"| 10952₆ ||class="entry q1 g0"| 19149₈ ||class="entry q1 g0"| 6379₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (4230, 5873, 30969, 32358, 30983, 32655) ||class="entry q0 g0"| 4230₄ ||class="entry q1 g0"| 5873₈ ||class="entry q1 g0"| 30969₁₀ ||class="entry q0 g0"| 32358₁₀ ||class="entry q1 g0"| 30983₈ ||class="entry q1 g0"| 32655₁₂ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (6150, 8033, 32649, 30438, 28823, 30975) ||class="entry q0 g0"| 6150₄ ||class="entry q1 g0"| 8033₈ ||class="entry q1 g0"| 32649₁₀ ||class="entry q0 g0"| 30438₁₀ ||class="entry q1 g0"| 28823₈ ||class="entry q1 g0"| 30975₁₂ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (8472, 9835, 31403, 20472, 18845, 32221) ||class="entry q0 g0"| 8472₄ ||class="entry q1 g0"| 9835₈ ||class="entry q1 g0"| 31403₁₀ ||class="entry q0 g0"| 20472₁₀ ||class="entry q1 g0"| 18845₈ ||class="entry q1 g0"| 32221₁₂ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (16664, 18029, 31949, 12280, 10651, 31675) ||class="entry q0 g0"| 16664₄ ||class="entry q1 g0"| 18029₈ ||class="entry q1 g0"| 31949₁₀ ||class="entry q0 g0"| 12280₁₀ ||class="entry q1 g0"| 10651₈ ||class="entry q1 g0"| 31675₁₂ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (4496, 32369, 4463, 32624, 4487, 5657) ||class="entry q0 g0"| 4496₄ ||class="entry q1 g0"| 32369₁₀ ||class="entry q1 g0"| 4463₈ ||class="entry q0 g0"| 32624₁₀ ||class="entry q1 g0"| 4487₆ ||class="entry q1 g0"| 5657₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (6416, 30689, 5663, 30704, 6167, 4457) ||class="entry q0 g0"| 6416₄ ||class="entry q1 g0"| 30689₁₀ ||class="entry q1 g0"| 5663₈ ||class="entry q0 g0"| 30704₁₀ ||class="entry q1 g0"| 6167₆ ||class="entry q1 g0"| 4457₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (8206, 20203, 4925, 20206, 8477, 5195) ||class="entry q0 g0"| 8206₄ ||class="entry q1 g0"| 20203₁₀ ||class="entry q1 g0"| 4925₈ ||class="entry q0 g0"| 20206₁₀ ||class="entry q1 g0"| 8477₆ ||class="entry q1 g0"| 5195₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (16398, 12013, 5467, 12014, 16667, 4653) ||class="entry q0 g0"| 16398₄ ||class="entry q1 g0"| 12013₁₀ ||class="entry q1 g0"| 5467₈ ||class="entry q0 g0"| 12014₁₀ ||class="entry q1 g0"| 16667₆ ||class="entry q1 g0"| 4653₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (394, 23975, 13245, 28522, 12881, 13515) ||class="entry q0 g0"| 394₄ ||class="entry q1 g0"| 23975₁₀ ||class="entry q1 g0"| 13245₁₀ ||class="entry q0 g0"| 28522₁₀ ||class="entry q1 g0"| 12881₆ ||class="entry q1 g0"| 13515₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (396, 15303, 21979, 28524, 21553, 21165) ||class="entry q0 g0"| 396₄ ||class="entry q1 g0"| 15303₁₀ ||class="entry q1 g0"| 21979₁₀ ||class="entry q0 g0"| 28524₁₀ ||class="entry q1 g0"| 21553₆ ||class="entry q1 g0"| 21165₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (408, 2031, 27111, 28536, 26649, 28305) ||class="entry q0 g0"| 408₄ ||class="entry q1 g0"| 2031₁₀ ||class="entry q1 g0"| 27111₁₀ ||class="entry q0 g0"| 28536₁₀ ||class="entry q1 g0"| 26649₆ ||class="entry q1 g0"| 28305₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2074, 23255, 15165, 26362, 13601, 15435) ||class="entry q0 g0"| 2074₄ ||class="entry q1 g0"| 23255₁₀ ||class="entry q1 g0"| 15165₁₀ ||class="entry q0 g0"| 26362₁₀ ||class="entry q1 g0"| 13601₆ ||class="entry q1 g0"| 15435₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2076, 15543, 23899, 26364, 21313, 23085) ||class="entry q0 g0"| 2076₄ ||class="entry q1 g0"| 15543₁₀ ||class="entry q1 g0"| 23899₁₀ ||class="entry q0 g0"| 26364₁₀ ||class="entry q1 g0"| 21313₆ ||class="entry q1 g0"| 23085₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2328, 3711, 28311, 26616, 24969, 27105) ||class="entry q0 g0"| 2328₄ ||class="entry q1 g0"| 3711₁₀ ||class="entry q1 g0"| 28311₁₀ ||class="entry q0 g0"| 26616₁₀ ||class="entry q1 g0"| 24969₆ ||class="entry q1 g0"| 27105₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (12294, 14197, 27573, 24294, 22659, 27843) ||class="entry q0 g0"| 12294₄ ||class="entry q1 g0"| 14197₁₀ ||class="entry q1 g0"| 27573₁₀ ||class="entry q0 g0"| 24294₁₀ ||class="entry q1 g0"| 22659₆ ||class="entry q1 g0"| 27843₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (12308, 27965, 12783, 24308, 715, 13977) ||class="entry q0 g0"| 12308₄ ||class="entry q1 g0"| 27965₁₀ ||class="entry q1 g0"| 12783₁₀ ||class="entry q0 g0"| 24308₁₀ ||class="entry q1 g0"| 715₆ ||class="entry q1 g0"| 13977₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (12548, 25565, 15903, 24548, 3115, 14697) ||class="entry q0 g0"| 12548₄ ||class="entry q1 g0"| 25565₁₀ ||class="entry q1 g0"| 15903₁₀ ||class="entry q0 g0"| 24548₁₀ ||class="entry q1 g0"| 3115₆ ||class="entry q1 g0"| 14697₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (20486, 22387, 28115, 16102, 14469, 27301) ||class="entry q0 g0"| 20486₄ ||class="entry q1 g0"| 22387₁₀ ||class="entry q1 g0"| 28115₁₀ ||class="entry q0 g0"| 16102₁₀ ||class="entry q1 g0"| 14469₆ ||class="entry q1 g0"| 27301₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (20498, 27483, 20975, 16114, 1197, 22169) ||class="entry q0 g0"| 20498₄ ||class="entry q1 g0"| 27483₁₀ ||class="entry q1 g0"| 20975₁₀ ||class="entry q0 g0"| 16114₁₀ ||class="entry q1 g0"| 1197₆ ||class="entry q1 g0"| 22169₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (20738, 26043, 24095, 16354, 2637, 22889) ||class="entry q0 g0"| 20738₄ ||class="entry q1 g0"| 26043₁₀ ||class="entry q1 g0"| 24095₁₀ ||class="entry q0 g0"| 16354₁₀ ||class="entry q1 g0"| 2637₆ ||class="entry q1 g0"| 22889₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (4502, 6161, 30473, 32630, 30695, 28799) ||class="entry q0 g0"| 4502₆ ||class="entry q1 g0"| 6161₄ ||class="entry q1 g0"| 30473₈ ||class="entry q0 g0"| 32630₁₂ ||class="entry q1 g0"| 30695₁₂ ||class="entry q1 g0"| 28799₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (6422, 4481, 28793, 30710, 32375, 30479) ||class="entry q0 g0"| 6422₆ ||class="entry q1 g0"| 4481₄ ||class="entry q1 g0"| 28793₈ ||class="entry q0 g0"| 30710₁₂ ||class="entry q1 g0"| 32375₁₂ ||class="entry q1 g0"| 30479₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (8478, 16395, 7373, 20478, 12285, 7099) ||class="entry q0 g0"| 8478₆ ||class="entry q1 g0"| 16395₄ ||class="entry q1 g0"| 7373₈ ||class="entry q0 g0"| 20478₁₂ ||class="entry q1 g0"| 12285₁₂ ||class="entry q1 g0"| 7099₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (16670, 8205, 6827, 12286, 20475, 7645) ||class="entry q0 g0"| 16670₆ ||class="entry q1 g0"| 8205₄ ||class="entry q1 g0"| 6827₈ ||class="entry q0 g0"| 12286₁₂ ||class="entry q1 g0"| 20475₁₂ ||class="entry q1 g0"| 7645₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (4818, 16655, 11781, 31794, 12025, 10611) ||class="entry q0 g0"| 4818₆ ||class="entry q1 g0"| 16655₆ ||class="entry q1 g0"| 11781₆ ||class="entry q0 g0"| 31794₈ ||class="entry q1 g0"| 12025₁₀ ||class="entry q1 g0"| 10611₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (5300, 8463, 19971, 31316, 20217, 18805) ||class="entry q0 g0"| 5300₆ ||class="entry q1 g0"| 8463₆ ||class="entry q1 g0"| 19971₆ ||class="entry q0 g0"| 31316₈ ||class="entry q1 g0"| 20217₁₀ ||class="entry q1 g0"| 18805₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (6980, 8223, 16611, 30116, 20457, 18325) ||class="entry q0 g0"| 6980₆ ||class="entry q1 g0"| 8223₆ ||class="entry q1 g0"| 16611₆ ||class="entry q0 g0"| 30116₈ ||class="entry q1 g0"| 20457₁₀ ||class="entry q1 g0"| 18325₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (7458, 16415, 8421, 29634, 12265, 10131) ||class="entry q0 g0"| 7458₆ ||class="entry q1 g0"| 16415₆ ||class="entry q1 g0"| 8421₆ ||class="entry q0 g0"| 29634₈ ||class="entry q1 g0"| 12265₁₀ ||class="entry q1 g0"| 10131₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (8794, 6421, 17857, 19642, 30435, 17079) ||class="entry q0 g0"| 8794₆ ||class="entry q1 g0"| 6421₆ ||class="entry q1 g0"| 17857₆ ||class="entry q0 g0"| 19642₈ ||class="entry q1 g0"| 30435₁₀ ||class="entry q1 g0"| 17079₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (9514, 4501, 19537, 19402, 32355, 19239) ||class="entry q0 g0"| 9514₆ ||class="entry q1 g0"| 4501₆ ||class="entry q1 g0"| 19537₆ ||class="entry q0 g0"| 19402₈ ||class="entry q1 g0"| 32355₁₀ ||class="entry q1 g0"| 19239₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (17228, 4499, 10801, 11692, 32357, 11591) ||class="entry q0 g0"| 17228₆ ||class="entry q1 g0"| 4499₆ ||class="entry q1 g0"| 10801₆ ||class="entry q0 g0"| 11692₈ ||class="entry q1 g0"| 32357₁₀ ||class="entry q1 g0"| 11591₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (17468, 6419, 9121, 10972, 30437, 9431) ||class="entry q0 g0"| 17468₆ ||class="entry q1 g0"| 6419₆ ||class="entry q1 g0"| 9121₆ ||class="entry q0 g0"| 10972₈ ||class="entry q1 g0"| 30437₁₀ ||class="entry q1 g0"| 9431₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (718, 1209, 27313, 27694, 27471, 28103) ||class="entry q0 g0"| 718₆ ||class="entry q1 g0"| 1209₆ ||class="entry q1 g0"| 27313₈ ||class="entry q0 g0"| 27694₈ ||class="entry q1 g0"| 27471₁₀ ||class="entry q1 g0"| 28103₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (730, 14481, 22157, 27706, 22375, 20987) ||class="entry q0 g0"| 730₆ ||class="entry q1 g0"| 14481₆ ||class="entry q1 g0"| 22157₈ ||class="entry q0 g0"| 27706₈ ||class="entry q1 g0"| 22375₁₀ ||class="entry q1 g0"| 20987₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1198, 729, 27857, 27214, 27951, 27559) ||class="entry q0 g0"| 1198₆ ||class="entry q1 g0"| 729₆ ||class="entry q1 g0"| 27857₈ ||class="entry q0 g0"| 27214₈ ||class="entry q1 g0"| 27951₁₀ ||class="entry q1 g0"| 27559₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1212, 22673, 13963, 27228, 14183, 12797) ||class="entry q0 g0"| 1212₆ ||class="entry q1 g0"| 22673₆ ||class="entry q1 g0"| 13963₈ ||class="entry q0 g0"| 27228₈ ||class="entry q1 g0"| 14183₁₀ ||class="entry q1 g0"| 12797₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (2638, 3369, 28097, 25774, 25311, 27319) ||class="entry q0 g0"| 2638₆ ||class="entry q1 g0"| 3369₆ ||class="entry q1 g0"| 28097₈ ||class="entry q0 g0"| 25774₈ ||class="entry q1 g0"| 25311₁₀ ||class="entry q1 g0"| 27319₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (2892, 22913, 14443, 26028, 13943, 16157) ||class="entry q0 g0"| 2892₆ ||class="entry q1 g0"| 22913₆ ||class="entry q1 g0"| 14443₈ ||class="entry q0 g0"| 26028₈ ||class="entry q1 g0"| 13943₁₀ ||class="entry q1 g0"| 16157₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3118, 2889, 27553, 25294, 25791, 27863) ||class="entry q0 g0"| 3118₆ ||class="entry q1 g0"| 2889₆ ||class="entry q1 g0"| 27553₈ ||class="entry q0 g0"| 25294₈ ||class="entry q1 g0"| 25791₁₀ ||class="entry q1 g0"| 27863₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3370, 14721, 22637, 25546, 22135, 24347) ||class="entry q0 g0"| 3370₆ ||class="entry q1 g0"| 14721₆ ||class="entry q1 g0"| 22637₈ ||class="entry q0 g0"| 25546₈ ||class="entry q1 g0"| 22135₁₀ ||class="entry q1 g0"| 24347₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12882, 24715, 15689, 23730, 3965, 14911) ||class="entry q0 g0"| 12882₆ ||class="entry q1 g0"| 24715₆ ||class="entry q1 g0"| 15689₈ ||class="entry q0 g0"| 23730₈ ||class="entry q1 g0"| 3965₁₀ ||class="entry q1 g0"| 14911₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13136, 13347, 26851, 23984, 23509, 28565) ||class="entry q0 g0"| 13136₆ ||class="entry q1 g0"| 13347₆ ||class="entry q1 g0"| 26851₈ ||class="entry q0 g0"| 23984₈ ||class="entry q1 g0"| 23509₁₀ ||class="entry q1 g0"| 28565₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13602, 26635, 13529, 23490, 2045, 13231) ||class="entry q0 g0"| 13602₆ ||class="entry q1 g0"| 26635₆ ||class="entry q1 g0"| 13529₈ ||class="entry q0 g0"| 23490₈ ||class="entry q1 g0"| 2045₁₀ ||class="entry q1 g0"| 13231₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13616, 12867, 28291, 23504, 23989, 27125) ||class="entry q0 g0"| 13616₆ ||class="entry q1 g0"| 12867₆ ||class="entry q1 g0"| 28291₈ ||class="entry q0 g0"| 23504₈ ||class="entry q1 g0"| 23989₁₀ ||class="entry q1 g0"| 27125₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (14482, 717, 20729, 22130, 27963, 22415) ||class="entry q0 g0"| 14482₆ ||class="entry q1 g0"| 717₆ ||class="entry q1 g0"| 20729₈ ||class="entry q0 g0"| 22130₈ ||class="entry q1 g0"| 27963₁₀ ||class="entry q1 g0"| 22415₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (14722, 3117, 24329, 22370, 25563, 22655) ||class="entry q0 g0"| 14722₆ ||class="entry q1 g0"| 3117₆ ||class="entry q1 g0"| 24329₈ ||class="entry q0 g0"| 22370₈ ||class="entry q1 g0"| 25563₁₀ ||class="entry q1 g0"| 22655₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21316, 26637, 21177, 15780, 2043, 21967) ||class="entry q0 g0"| 21316₆ ||class="entry q1 g0"| 26637₆ ||class="entry q1 g0"| 21177₈ ||class="entry q0 g0"| 15780₈ ||class="entry q1 g0"| 2043₁₀ ||class="entry q1 g0"| 21967₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21328, 21541, 28293, 15792, 15315, 27123) ||class="entry q0 g0"| 21328₆ ||class="entry q1 g0"| 21541₆ ||class="entry q1 g0"| 28293₈ ||class="entry q0 g0"| 15792₈ ||class="entry q1 g0"| 15315₁₀ ||class="entry q1 g0"| 27123₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21556, 24717, 23337, 15060, 3963, 23647) ||class="entry q0 g0"| 21556₆ ||class="entry q1 g0"| 24717₆ ||class="entry q1 g0"| 23337₈ ||class="entry q0 g0"| 15060₈ ||class="entry q1 g0"| 3963₁₀ ||class="entry q1 g0"| 23647₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21808, 21061, 26853, 15312, 15795, 28563) ||class="entry q0 g0"| 21808₆ ||class="entry q1 g0"| 21061₆ ||class="entry q1 g0"| 26853₈ ||class="entry q0 g0"| 15312₈ ||class="entry q1 g0"| 15795₁₀ ||class="entry q1 g0"| 28563₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (22676, 1195, 12537, 13940, 27485, 14223) ||class="entry q0 g0"| 22676₆ ||class="entry q1 g0"| 1195₆ ||class="entry q1 g0"| 12537₈ ||class="entry q0 g0"| 13940₈ ||class="entry q1 g0"| 27485₁₀ ||class="entry q1 g0"| 14223₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (22916, 2635, 16137, 14180, 26045, 14463) ||class="entry q0 g0"| 22916₆ ||class="entry q1 g0"| 2635₆ ||class="entry q1 g0"| 16137₈ ||class="entry q0 g0"| 14180₈ ||class="entry q1 g0"| 26045₁₀ ||class="entry q1 g0"| 14463₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (24730, 13121, 14891, 3706, 23735, 15709) ||class="entry q0 g0"| 24730₆ ||class="entry q1 g0"| 13121₆ ||class="entry q1 g0"| 14891₈ ||class="entry q0 g0"| 3706₈ ||class="entry q1 g0"| 23735₁₀ ||class="entry q1 g0"| 15709₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (24732, 21793, 23629, 3708, 15063, 23355) ||class="entry q0 g0"| 24732₆ ||class="entry q1 g0"| 21793₆ ||class="entry q1 g0"| 23629₈ ||class="entry q0 g0"| 3708₈ ||class="entry q1 g0"| 15063₁₀ ||class="entry q1 g0"| 23355₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26890, 13361, 12971, 2026, 23495, 13789) ||class="entry q0 g0"| 26890₆ ||class="entry q1 g0"| 13361₆ ||class="entry q1 g0"| 12971₈ ||class="entry q0 g0"| 2026₈ ||class="entry q1 g0"| 23495₁₀ ||class="entry q1 g0"| 13789₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26892, 21073, 21709, 2028, 15783, 21435) ||class="entry q0 g0"| 26892₆ ||class="entry q1 g0"| 21073₆ ||class="entry q1 g0"| 21709₈ ||class="entry q0 g0"| 2028₈ ||class="entry q1 g0"| 15783₁₀ ||class="entry q1 g0"| 21435₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (984, 27705, 807, 27960, 975, 1105) ||class="entry q0 g0"| 984₆ ||class="entry q1 g0"| 27705₈ ||class="entry q1 g0"| 807₆ ||class="entry q0 g0"| 27960₈ ||class="entry q1 g0"| 975₈ ||class="entry q1 g0"| 1105₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (1464, 27225, 1351, 27480, 1455, 561) ||class="entry q0 g0"| 1464₆ ||class="entry q1 g0"| 27225₈ ||class="entry q1 g0"| 1351₆ ||class="entry q0 g0"| 27480₈ ||class="entry q1 g0"| 1455₈ ||class="entry q1 g0"| 561₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (2904, 26025, 1111, 26040, 2655, 801) ||class="entry q0 g0"| 2904₆ ||class="entry q1 g0"| 26025₈ ||class="entry q1 g0"| 1111₆ ||class="entry q0 g0"| 26040₈ ||class="entry q1 g0"| 2655₈ ||class="entry q1 g0"| 801₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (3384, 25545, 567, 25560, 3135, 1345) ||class="entry q0 g0"| 3384₆ ||class="entry q1 g0"| 25545₈ ||class="entry q1 g0"| 567₆ ||class="entry q0 g0"| 25560₈ ||class="entry q1 g0"| 3135₈ ||class="entry q1 g0"| 1345₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (12870, 23715, 373, 23718, 13141, 1539) ||class="entry q0 g0"| 12870₆ ||class="entry q1 g0"| 23715₈ ||class="entry q1 g0"| 373₆ ||class="entry q0 g0"| 23718₈ ||class="entry q1 g0"| 13141₈ ||class="entry q1 g0"| 1539₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (13350, 23235, 1813, 23238, 13621, 99) ||class="entry q0 g0"| 13350₆ ||class="entry q1 g0"| 23235₈ ||class="entry q1 g0"| 1813₆ ||class="entry q0 g0"| 23238₈ ||class="entry q1 g0"| 13621₈ ||class="entry q1 g0"| 99₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (14736, 22117, 1363, 22384, 14739, 549) ||class="entry q0 g0"| 14736₆ ||class="entry q1 g0"| 22117₈ ||class="entry q1 g0"| 1363₆ ||class="entry q0 g0"| 22384₈ ||class="entry q1 g0"| 14739₈ ||class="entry q1 g0"| 549₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (21062, 15525, 1811, 15526, 21331, 101) ||class="entry q0 g0"| 21062₆ ||class="entry q1 g0"| 15525₈ ||class="entry q1 g0"| 1811₆ ||class="entry q0 g0"| 15526₈ ||class="entry q1 g0"| 21331₈ ||class="entry q1 g0"| 101₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (21542, 15045, 371, 15046, 21811, 1541) ||class="entry q0 g0"| 21542₆ ||class="entry q1 g0"| 15045₈ ||class="entry q1 g0"| 371₆ ||class="entry q0 g0"| 15046₈ ||class="entry q1 g0"| 21811₈ ||class="entry q1 g0"| 1541₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (22928, 13923, 821, 14192, 22933, 1091) ||class="entry q0 g0"| 22928₆ ||class="entry q1 g0"| 13923₈ ||class="entry q1 g0"| 821₆ ||class="entry q0 g0"| 14192₈ ||class="entry q1 g0"| 22933₈ ||class="entry q1 g0"| 1091₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (24718, 3945, 1559, 3694, 24735, 353) ||class="entry q0 g0"| 24718₆ ||class="entry q1 g0"| 3945₈ ||class="entry q1 g0"| 1559₆ ||class="entry q0 g0"| 3694₈ ||class="entry q1 g0"| 24735₈ ||class="entry q1 g0"| 353₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (26638, 1785, 359, 1774, 26895, 1553) ||class="entry q0 g0"| 26638₆ ||class="entry q1 g0"| 1785₈ ||class="entry q1 g0"| 359₆ ||class="entry q0 g0"| 1774₈ ||class="entry q1 g0"| 26895₈ ||class="entry q1 g0"| 1553₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (414, 24975, 3969, 28542, 3705, 2295) ||class="entry q0 g0"| 414₆ ||class="entry q1 g0"| 24975₈ ||class="entry q1 g0"| 3969₆ ||class="entry q0 g0"| 28542₁₂ ||class="entry q1 g0"| 3705₈ ||class="entry q1 g0"| 2295₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (2334, 26655, 2289, 26622, 2025, 3975) ||class="entry q0 g0"| 2334₆ ||class="entry q1 g0"| 26655₈ ||class="entry q1 g0"| 2289₆ ||class="entry q0 g0"| 26622₁₂ ||class="entry q1 g0"| 2025₈ ||class="entry q1 g0"| 3975₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (12566, 14741, 25669, 24566, 22115, 25395) ||class="entry q0 g0"| 12566₆ ||class="entry q1 g0"| 14741₈ ||class="entry q1 g0"| 25669₆ ||class="entry q0 g0"| 24566₁₂ ||class="entry q1 g0"| 22115₈ ||class="entry q1 g0"| 25395₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (20758, 22931, 25123, 16374, 13925, 25941) ||class="entry q0 g0"| 20758₆ ||class="entry q1 g0"| 22931₈ ||class="entry q1 g0"| 25123₆ ||class="entry q0 g0"| 16374₁₂ ||class="entry q1 g0"| 13925₈ ||class="entry q1 g0"| 25941₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (5060, 10639, 18323, 32036, 18041, 16613) ||class="entry q0 g0"| 5060₆ ||class="entry q1 g0"| 10639₈ ||class="entry q1 g0"| 18323₈ ||class="entry q0 g0"| 32036₈ ||class="entry q1 g0"| 18041₈ ||class="entry q1 g0"| 16613₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (5538, 18831, 10133, 31554, 9849, 8419) ||class="entry q0 g0"| 5538₆ ||class="entry q1 g0"| 18831₈ ||class="entry q1 g0"| 10133₈ ||class="entry q0 g0"| 31554₈ ||class="entry q1 g0"| 9849₈ ||class="entry q1 g0"| 8419₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6738, 18591, 10613, 29874, 10089, 11779) ||class="entry q0 g0"| 6738₆ ||class="entry q1 g0"| 18591₈ ||class="entry q1 g0"| 10613₈ ||class="entry q0 g0"| 29874₈ ||class="entry q1 g0"| 10089₈ ||class="entry q1 g0"| 11779₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (7220, 10399, 18803, 29396, 18281, 19973) ||class="entry q0 g0"| 7220₆ ||class="entry q1 g0"| 10399₈ ||class="entry q1 g0"| 18803₈ ||class="entry q0 g0"| 29396₈ ||class="entry q1 g0"| 18281₈ ||class="entry q1 g0"| 19973₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9036, 29077, 11351, 19884, 7779, 11041) ||class="entry q0 g0"| 9036₆ ||class="entry q1 g0"| 29077₈ ||class="entry q1 g0"| 11351₈ ||class="entry q0 g0"| 19884₈ ||class="entry q1 g0"| 7779₈ ||class="entry q1 g0"| 11041₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9276, 30997, 9671, 19164, 5859, 8881) ||class="entry q0 g0"| 9276₆ ||class="entry q1 g0"| 30997₈ ||class="entry q1 g0"| 9671₈ ||class="entry q0 g0"| 19164₈ ||class="entry q1 g0"| 5859₈ ||class="entry q1 g0"| 8881₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (10396, 7475, 19991, 18044, 29381, 18785) ||class="entry q0 g0"| 10396₆ ||class="entry q1 g0"| 7475₈ ||class="entry q1 g0"| 19991₈ ||class="entry q0 g0"| 18044₈ ||class="entry q1 g0"| 29381₈ ||class="entry q1 g0"| 18785₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (10636, 5075, 16871, 18284, 31781, 18065) ||class="entry q0 g0"| 10636₆ ||class="entry q1 g0"| 5075₈ ||class="entry q1 g0"| 16871₈ ||class="entry q0 g0"| 18284₈ ||class="entry q1 g0"| 31781₈ ||class="entry q1 g0"| 18065₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (16986, 30995, 17319, 11450, 5861, 17617) ||class="entry q0 g0"| 16986₆ ||class="entry q1 g0"| 30995₈ ||class="entry q1 g0"| 17319₈ ||class="entry q0 g0"| 11450₈ ||class="entry q1 g0"| 5861₈ ||class="entry q1 g0"| 17617₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17706, 29075, 18999, 11210, 7781, 19777) ||class="entry q0 g0"| 17706₆ ||class="entry q1 g0"| 29075₈ ||class="entry q1 g0"| 18999₈ ||class="entry q0 g0"| 11210₈ ||class="entry q1 g0"| 7781₈ ||class="entry q1 g0"| 19777₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18586, 6997, 11799, 9850, 29859, 10593) ||class="entry q0 g0"| 18586₆ ||class="entry q1 g0"| 6997₈ ||class="entry q1 g0"| 11799₈ ||class="entry q0 g0"| 9850₈ ||class="entry q1 g0"| 29859₈ ||class="entry q1 g0"| 10593₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18826, 5557, 8679, 10090, 31299, 9873) ||class="entry q0 g0"| 18826₆ ||class="entry q1 g0"| 5557₈ ||class="entry q1 g0"| 8679₈ ||class="entry q0 g0"| 10090₈ ||class="entry q1 g0"| 31299₈ ||class="entry q1 g0"| 9873₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (29058, 17471, 19795, 8034, 11209, 18981) ||class="entry q0 g0"| 29058₆ ||class="entry q1 g0"| 17471₈ ||class="entry q1 g0"| 19795₈ ||class="entry q0 g0"| 8034₈ ||class="entry q1 g0"| 11209₈ ||class="entry q1 g0"| 18981₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (29060, 8799, 11061, 8036, 19881, 11331) ||class="entry q0 g0"| 29060₆ ||class="entry q1 g0"| 8799₈ ||class="entry q1 g0"| 11061₈ ||class="entry q0 g0"| 8036₈ ||class="entry q1 g0"| 19881₈ ||class="entry q1 g0"| 11331₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (30738, 17231, 17875, 5874, 11449, 17061) ||class="entry q0 g0"| 30738₆ ||class="entry q1 g0"| 17231₈ ||class="entry q1 g0"| 17875₈ ||class="entry q0 g0"| 5874₈ ||class="entry q1 g0"| 11449₈ ||class="entry q1 g0"| 17061₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (30740, 9519, 9141, 5876, 19161, 9411) ||class="entry q0 g0"| 30740₆ ||class="entry q1 g0"| 9519₈ ||class="entry q1 g0"| 9141₈ ||class="entry q0 g0"| 5876₈ ||class="entry q1 g0"| 19161₈ ||class="entry q1 g0"| 9411₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (970, 13937, 22909, 27946, 22919, 24075) ||class="entry q0 g0"| 970₆ ||class="entry q1 g0"| 13937₈ ||class="entry q1 g0"| 22909₁₀ ||class="entry q0 g0"| 27946₈ ||class="entry q1 g0"| 22919₈ ||class="entry q1 g0"| 24075₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (1452, 22129, 14715, 27468, 14727, 15885) ||class="entry q0 g0"| 1452₆ ||class="entry q1 g0"| 22129₈ ||class="entry q1 g0"| 14715₁₀ ||class="entry q0 g0"| 27468₈ ||class="entry q1 g0"| 14727₈ ||class="entry q1 g0"| 15885₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (2652, 22369, 14235, 25788, 14487, 12525) ||class="entry q0 g0"| 2652₆ ||class="entry q1 g0"| 22369₈ ||class="entry q1 g0"| 14235₁₀ ||class="entry q0 g0"| 25788₈ ||class="entry q1 g0"| 14487₈ ||class="entry q1 g0"| 12525₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3130, 14177, 22429, 25306, 22679, 20715) ||class="entry q0 g0"| 3130₆ ||class="entry q1 g0"| 14177₈ ||class="entry q1 g0"| 22429₁₀ ||class="entry q0 g0"| 25306₈ ||class="entry q1 g0"| 22679₈ ||class="entry q1 g0"| 20715₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (12884, 1771, 23343, 23732, 26909, 23641) ||class="entry q0 g0"| 12884₆ ||class="entry q1 g0"| 1771₈ ||class="entry q1 g0"| 23343₁₀ ||class="entry q0 g0"| 23732₈ ||class="entry q1 g0"| 26909₈ ||class="entry q1 g0"| 23641₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (13604, 3691, 21183, 23492, 24989, 21961) ||class="entry q0 g0"| 13604₆ ||class="entry q1 g0"| 3691₈ ||class="entry q1 g0"| 21183₁₀ ||class="entry q0 g0"| 23492₈ ||class="entry q1 g0"| 24989₈ ||class="entry q1 g0"| 21961₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (14484, 25773, 13983, 22132, 2907, 12777) ||class="entry q0 g0"| 14484₆ ||class="entry q1 g0"| 25773₈ ||class="entry q1 g0"| 13983₁₀ ||class="entry q0 g0"| 22132₈ ||class="entry q1 g0"| 2907₈ ||class="entry q1 g0"| 12777₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (14724, 27213, 14703, 22372, 1467, 15897) ||class="entry q0 g0"| 14724₆ ||class="entry q1 g0"| 27213₈ ||class="entry q1 g0"| 14703₁₀ ||class="entry q0 g0"| 22372₈ ||class="entry q1 g0"| 1467₈ ||class="entry q1 g0"| 15897₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21314, 3693, 13535, 15778, 24987, 13225) ||class="entry q0 g0"| 21314₆ ||class="entry q1 g0"| 3693₈ ||class="entry q1 g0"| 13535₁₀ ||class="entry q0 g0"| 15778₈ ||class="entry q1 g0"| 24987₈ ||class="entry q1 g0"| 13225₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21554, 1773, 15695, 15058, 26907, 14905) ||class="entry q0 g0"| 21554₆ ||class="entry q1 g0"| 1773₈ ||class="entry q1 g0"| 15695₁₀ ||class="entry q0 g0"| 15058₈ ||class="entry q1 g0"| 26907₈ ||class="entry q1 g0"| 14905₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (22674, 25291, 22175, 13938, 3389, 20969) ||class="entry q0 g0"| 22674₆ ||class="entry q1 g0"| 25291₈ ||class="entry q1 g0"| 22175₁₀ ||class="entry q0 g0"| 13938₈ ||class="entry q1 g0"| 3389₈ ||class="entry q1 g0"| 20969₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (22914, 27691, 22895, 14178, 989, 24089) ||class="entry q0 g0"| 22914₆ ||class="entry q1 g0"| 27691₈ ||class="entry q1 g0"| 22895₁₀ ||class="entry q0 g0"| 14178₈ ||class="entry q1 g0"| 989₈ ||class="entry q1 g0"| 24089₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (24970, 15777, 13787, 3946, 21079, 12973) ||class="entry q0 g0"| 24970₆ ||class="entry q1 g0"| 15777₈ ||class="entry q1 g0"| 13787₁₀ ||class="entry q0 g0"| 3946₈ ||class="entry q1 g0"| 21079₈ ||class="entry q1 g0"| 12973₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (24972, 23489, 21437, 3948, 13367, 21707) ||class="entry q0 g0"| 24972₆ ||class="entry q1 g0"| 23489₈ ||class="entry q1 g0"| 21437₁₀ ||class="entry q0 g0"| 3948₈ ||class="entry q1 g0"| 13367₈ ||class="entry q1 g0"| 21707₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26650, 15057, 15707, 1786, 21799, 14893) ||class="entry q0 g0"| 26650₆ ||class="entry q1 g0"| 15057₈ ||class="entry q1 g0"| 15707₁₀ ||class="entry q0 g0"| 1786₈ ||class="entry q1 g0"| 21799₈ ||class="entry q1 g0"| 14893₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26652, 23729, 23357, 1788, 13127, 23627) ||class="entry q0 g0"| 26652₆ ||class="entry q1 g0"| 23729₈ ||class="entry q1 g0"| 23357₁₀ ||class="entry q0 g0"| 1788₈ ||class="entry q1 g0"| 13127₈ ||class="entry q1 g0"| 23627₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (5072, 5543, 31663, 32048, 31313, 31961) ||class="entry q0 g0"| 5072₆ ||class="entry q1 g0"| 5543₈ ||class="entry q1 g0"| 31663₁₂ ||class="entry q0 g0"| 32048₈ ||class="entry q1 g0"| 31313₈ ||class="entry q1 g0"| 31961₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (5552, 5063, 32207, 31568, 31793, 31417) ||class="entry q0 g0"| 5552₆ ||class="entry q1 g0"| 5063₈ ||class="entry q1 g0"| 32207₁₂ ||class="entry q0 g0"| 31568₈ ||class="entry q1 g0"| 31793₈ ||class="entry q1 g0"| 31417₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (6992, 7223, 31967, 30128, 29633, 31657) ||class="entry q0 g0"| 6992₆ ||class="entry q1 g0"| 7223₈ ||class="entry q1 g0"| 31967₁₂ ||class="entry q0 g0"| 30128₈ ||class="entry q1 g0"| 29633₈ ||class="entry q1 g0"| 31657₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (7472, 6743, 31423, 29648, 30113, 32201) ||class="entry q0 g0"| 7472₆ ||class="entry q1 g0"| 6743₈ ||class="entry q1 g0"| 31423₁₂ ||class="entry q0 g0"| 29648₈ ||class="entry q1 g0"| 30113₈ ||class="entry q1 g0"| 32201₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (8782, 9533, 31229, 19630, 19147, 32395) ||class="entry q0 g0"| 8782₆ ||class="entry q1 g0"| 9533₈ ||class="entry q1 g0"| 31229₁₂ ||class="entry q0 g0"| 19630₈ ||class="entry q1 g0"| 19147₈ ||class="entry q1 g0"| 32395₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (9262, 9053, 32669, 19150, 19627, 30955) ||class="entry q0 g0"| 9262₆ ||class="entry q1 g0"| 9053₈ ||class="entry q1 g0"| 32669₁₂ ||class="entry q0 g0"| 19150₈ ||class="entry q1 g0"| 19627₈ ||class="entry q1 g0"| 30955₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (16974, 17723, 32667, 11438, 10957, 30957) ||class="entry q0 g0"| 16974₆ ||class="entry q1 g0"| 17723₈ ||class="entry q1 g0"| 32667₁₂ ||class="entry q0 g0"| 11438₈ ||class="entry q1 g0"| 10957₈ ||class="entry q1 g0"| 30957₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (17454, 17243, 31227, 10958, 11437, 32397) ||class="entry q0 g0"| 17454₆ ||class="entry q1 g0"| 17243₈ ||class="entry q1 g0"| 31227₁₂ ||class="entry q0 g0"| 10958₈ ||class="entry q1 g0"| 11437₈ ||class="entry q1 g0"| 32397₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4806, 32039, 4665, 31782, 4817, 5455) ||class="entry q0 g0"| 4806₆ ||class="entry q1 g0"| 32039₁₀ ||class="entry q1 g0"| 4665₆ ||class="entry q0 g0"| 31782₈ ||class="entry q1 g0"| 4817₆ ||class="entry q1 g0"| 5455₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4820, 10095, 18531, 31796, 18585, 20245) ||class="entry q0 g0"| 4820₆ ||class="entry q1 g0"| 10095₁₀ ||class="entry q1 g0"| 18531₆ ||class="entry q0 g0"| 31796₈ ||class="entry q1 g0"| 18585₆ ||class="entry q1 g0"| 20245₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5286, 31559, 5209, 31302, 5297, 4911) ||class="entry q0 g0"| 5286₆ ||class="entry q1 g0"| 31559₁₀ ||class="entry q1 g0"| 5209₆ ||class="entry q0 g0"| 31302₈ ||class="entry q1 g0"| 5297₆ ||class="entry q1 g0"| 4911₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5298, 18287, 10341, 31314, 10393, 12051) ||class="entry q0 g0"| 5298₆ ||class="entry q1 g0"| 18287₁₀ ||class="entry q1 g0"| 10341₆ ||class="entry q0 g0"| 31314₈ ||class="entry q1 g0"| 10393₆ ||class="entry q1 g0"| 12051₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6726, 29879, 5449, 29862, 6977, 4671) ||class="entry q0 g0"| 6726₆ ||class="entry q1 g0"| 29879₁₀ ||class="entry q1 g0"| 5449₆ ||class="entry q0 g0"| 29862₈ ||class="entry q1 g0"| 6977₆ ||class="entry q1 g0"| 4671₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6978, 18047, 9861, 30114, 10633, 8691) ||class="entry q0 g0"| 6978₆ ||class="entry q1 g0"| 18047₁₀ ||class="entry q1 g0"| 9861₆ ||class="entry q0 g0"| 30114₈ ||class="entry q1 g0"| 10633₆ ||class="entry q1 g0"| 8691₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (7206, 29399, 4905, 29382, 7457, 5215) ||class="entry q0 g0"| 7206₆ ||class="entry q1 g0"| 29399₁₀ ||class="entry q1 g0"| 4905₆ ||class="entry q0 g0"| 29382₈ ||class="entry q1 g0"| 7457₆ ||class="entry q1 g0"| 5215₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (7460, 9855, 18051, 29636, 18825, 16885) ||class="entry q0 g0"| 7460₆ ||class="entry q1 g0"| 9855₁₀ ||class="entry q1 g0"| 18051₆ ||class="entry q0 g0"| 29636₈ ||class="entry q1 g0"| 18825₆ ||class="entry q1 g0"| 16885₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9034, 6133, 18993, 19882, 30723, 19783) ||class="entry q0 g0"| 9034₆ ||class="entry q1 g0"| 6133₁₀ ||class="entry q1 g0"| 18993₆ ||class="entry q0 g0"| 19882₈ ||class="entry q1 g0"| 30723₆ ||class="entry q1 g0"| 19783₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9048, 19901, 4203, 19896, 8779, 5917) ||class="entry q0 g0"| 9048₆ ||class="entry q1 g0"| 19901₁₀ ||class="entry q1 g0"| 4203₆ ||class="entry q0 g0"| 19896₈ ||class="entry q1 g0"| 8779₆ ||class="entry q1 g0"| 5917₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9274, 8053, 17313, 19162, 28803, 17623) ||class="entry q0 g0"| 9274₆ ||class="entry q1 g0"| 8053₁₀ ||class="entry q1 g0"| 17313₆ ||class="entry q0 g0"| 19162₈ ||class="entry q1 g0"| 28803₆ ||class="entry q1 g0"| 17623₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9528, 19421, 5643, 19416, 9259, 4477) ||class="entry q0 g0"| 9528₆ ||class="entry q1 g0"| 19421₁₀ ||class="entry q1 g0"| 5643₆ ||class="entry q0 g0"| 19416₈ ||class="entry q1 g0"| 9259₆ ||class="entry q1 g0"| 4477₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10382, 18299, 5197, 18030, 10381, 4923) ||class="entry q0 g0"| 10382₆ ||class="entry q1 g0"| 18299₁₀ ||class="entry q1 g0"| 5197₆ ||class="entry q0 g0"| 18030₈ ||class="entry q1 g0"| 10381₆ ||class="entry q1 g0"| 4923₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10394, 31571, 10353, 18042, 5285, 12039) ||class="entry q0 g0"| 10394₆ ||class="entry q1 g0"| 31571₁₀ ||class="entry q1 g0"| 10353₆ ||class="entry q0 g0"| 18042₈ ||class="entry q1 g0"| 5285₆ ||class="entry q1 g0"| 12039₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10634, 30131, 10113, 18282, 6725, 8439) ||class="entry q0 g0"| 10634₆ ||class="entry q1 g0"| 30131₁₀ ||class="entry q1 g0"| 10113₆ ||class="entry q0 g0"| 18282₈ ||class="entry q1 g0"| 6725₆ ||class="entry q1 g0"| 8439₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (16988, 8051, 9665, 11452, 28805, 8887) ||class="entry q0 g0"| 16988₆ ||class="entry q1 g0"| 8051₁₀ ||class="entry q1 g0"| 9665₆ ||class="entry q0 g0"| 11452₈ ||class="entry q1 g0"| 28805₆ ||class="entry q1 g0"| 8887₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17240, 11707, 5645, 11704, 16973, 4475) ||class="entry q0 g0"| 17240₆ ||class="entry q1 g0"| 11707₁₀ ||class="entry q1 g0"| 5645₆ ||class="entry q0 g0"| 11704₈ ||class="entry q1 g0"| 16973₆ ||class="entry q1 g0"| 4475₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17708, 6131, 11345, 11212, 30725, 11047) ||class="entry q0 g0"| 17708₆ ||class="entry q1 g0"| 6131₁₀ ||class="entry q1 g0"| 11345₆ ||class="entry q0 g0"| 11212₈ ||class="entry q1 g0"| 30725₆ ||class="entry q1 g0"| 11047₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17720, 11227, 4205, 11224, 17453, 5915) ||class="entry q0 g0"| 17720₆ ||class="entry q1 g0"| 11227₁₀ ||class="entry q1 g0"| 4205₆ ||class="entry q0 g0"| 11224₈ ||class="entry q1 g0"| 17453₆ ||class="entry q1 g0"| 5915₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18574, 10109, 4651, 9838, 18571, 5469) ||class="entry q0 g0"| 18574₆ ||class="entry q1 g0"| 10109₁₀ ||class="entry q1 g0"| 4651₆ ||class="entry q0 g0"| 9838₈ ||class="entry q1 g0"| 18571₆ ||class="entry q1 g0"| 5469₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18588, 32053, 18545, 9852, 4803, 20231) ||class="entry q0 g0"| 18588₆ ||class="entry q1 g0"| 32053₁₀ ||class="entry q1 g0"| 18545₆ ||class="entry q0 g0"| 9852₈ ||class="entry q1 g0"| 4803₆ ||class="entry q1 g0"| 20231₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18828, 29653, 18305, 10092, 7203, 16631) ||class="entry q0 g0"| 18828₆ ||class="entry q1 g0"| 29653₁₀ ||class="entry q1 g0"| 18305₆ ||class="entry q0 g0"| 10092₈ ||class="entry q1 g0"| 7203₆ ||class="entry q1 g0"| 16631₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (28818, 19167, 17059, 7794, 9513, 17877) ||class="entry q0 g0"| 28818₆ ||class="entry q1 g0"| 19167₁₀ ||class="entry q1 g0"| 17059₆ ||class="entry q0 g0"| 7794₈ ||class="entry q1 g0"| 9513₆ ||class="entry q1 g0"| 17877₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (28820, 11455, 9413, 7796, 17225, 9139) ||class="entry q0 g0"| 28820₆ ||class="entry q1 g0"| 11455₁₀ ||class="entry q1 g0"| 9413₆ ||class="entry q0 g0"| 7796₈ ||class="entry q1 g0"| 17225₆ ||class="entry q1 g0"| 9139₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (29072, 7799, 5897, 8048, 29057, 4223) ||class="entry q0 g0"| 29072₆ ||class="entry q1 g0"| 7799₁₀ ||class="entry q1 g0"| 5897₆ ||class="entry q0 g0"| 8048₈ ||class="entry q1 g0"| 29057₆ ||class="entry q1 g0"| 4223₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (30978, 19887, 18979, 6114, 8793, 19797) ||class="entry q0 g0"| 30978₆ ||class="entry q1 g0"| 19887₁₀ ||class="entry q1 g0"| 18979₆ ||class="entry q0 g0"| 6114₈ ||class="entry q1 g0"| 8793₆ ||class="entry q1 g0"| 19797₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (30980, 11215, 11333, 6116, 17465, 11059) ||class="entry q0 g0"| 30980₆ ||class="entry q1 g0"| 11215₁₀ ||class="entry q1 g0"| 11333₆ ||class="entry q0 g0"| 6116₈ ||class="entry q1 g0"| 17465₆ ||class="entry q1 g0"| 11059₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (30992, 6119, 4217, 6128, 30737, 5903) ||class="entry q0 g0"| 30992₆ ||class="entry q1 g0"| 6119₁₀ ||class="entry q1 g0"| 4217₆ ||class="entry q0 g0"| 6128₈ ||class="entry q1 g0"| 30737₆ ||class="entry q1 g0"| 5903₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (732, 24305, 12523, 27708, 12551, 14237) ||class="entry q0 g0"| 732₆ ||class="entry q1 g0"| 24305₁₀ ||class="entry q1 g0"| 12523₈ ||class="entry q0 g0"| 27708₈ ||class="entry q1 g0"| 12551₆ ||class="entry q1 g0"| 14237₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (1210, 16113, 20717, 27226, 20743, 22427) ||class="entry q0 g0"| 1210₆ ||class="entry q1 g0"| 16113₁₀ ||class="entry q1 g0"| 20717₈ ||class="entry q0 g0"| 27226₈ ||class="entry q1 g0"| 20743₆ ||class="entry q1 g0"| 22427₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (2890, 16353, 24077, 26026, 20503, 22907) ||class="entry q0 g0"| 2890₆ ||class="entry q1 g0"| 16353₁₀ ||class="entry q1 g0"| 24077₈ ||class="entry q0 g0"| 26026₈ ||class="entry q1 g0"| 20503₆ ||class="entry q1 g0"| 22907₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (3372, 24545, 15883, 25548, 12311, 14717) ||class="entry q0 g0"| 3372₆ ||class="entry q1 g0"| 24545₁₀ ||class="entry q1 g0"| 15883₈ ||class="entry q0 g0"| 25548₈ ||class="entry q1 g0"| 12311₆ ||class="entry q1 g0"| 14717₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (13122, 28267, 12985, 23970, 413, 13775) ||class="entry q0 g0"| 13122₆ ||class="entry q1 g0"| 28267₁₀ ||class="entry q1 g0"| 12985₈ ||class="entry q0 g0"| 23970₈ ||class="entry q1 g0"| 413₆ ||class="entry q1 g0"| 13775₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (13362, 26347, 15145, 23250, 2333, 15455) ||class="entry q0 g0"| 13362₆ ||class="entry q1 g0"| 26347₁₀ ||class="entry q1 g0"| 15145₈ ||class="entry q0 g0"| 23250₈ ||class="entry q1 g0"| 2333₆ ||class="entry q1 g0"| 15455₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (14470, 16101, 27845, 22118, 20755, 27571) ||class="entry q0 g0"| 14470₆ ||class="entry q1 g0"| 16101₁₀ ||class="entry q1 g0"| 27845₈ ||class="entry q0 g0"| 22118₈ ||class="entry q1 g0"| 20755₆ ||class="entry q1 g0"| 27571₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (21076, 26349, 23881, 15540, 2331, 23103) ||class="entry q0 g0"| 21076₆ ||class="entry q1 g0"| 26349₁₀ ||class="entry q1 g0"| 23881₈ ||class="entry q0 g0"| 15540₈ ||class="entry q1 g0"| 2331₆ ||class="entry q1 g0"| 23103₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (21796, 28269, 21721, 15300, 411, 21423) ||class="entry q0 g0"| 21796₆ ||class="entry q1 g0"| 28269₁₀ ||class="entry q1 g0"| 21721₈ ||class="entry q0 g0"| 15300₈ ||class="entry q1 g0"| 411₆ ||class="entry q1 g0"| 21423₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (22662, 24291, 27299, 13926, 12565, 28117) ||class="entry q0 g0"| 22662₆ ||class="entry q1 g0"| 24291₁₀ ||class="entry q1 g0"| 27299₈ ||class="entry q0 g0"| 13926₈ ||class="entry q1 g0"| 12565₆ ||class="entry q1 g0"| 28117₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (24984, 26601, 28545, 3960, 2079, 26871) ||class="entry q0 g0"| 24984₆ ||class="entry q1 g0"| 26601₁₀ ||class="entry q1 g0"| 28545₈ ||class="entry q0 g0"| 3960₈ ||class="entry q1 g0"| 2079₆ ||class="entry q1 g0"| 26871₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (26904, 28281, 26865, 2040, 399, 28551) ||class="entry q0 g0"| 26904₆ ||class="entry q1 g0"| 28281₁₀ ||class="entry q1 g0"| 26865₈ ||class="entry q0 g0"| 2040₈ ||class="entry q1 g0"| 399₆ ||class="entry q1 g0"| 28551₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (5058, 20463, 8693, 32034, 8217, 9859) ||class="entry q0 g0"| 5058₆ ||class="entry q1 g0"| 20463₁₂ ||class="entry q1 g0"| 8693₈ ||class="entry q0 g0"| 32034₈ ||class="entry q1 g0"| 8217₄ ||class="entry q1 g0"| 9859₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (5540, 12271, 16883, 31556, 16409, 18053) ||class="entry q0 g0"| 5540₆ ||class="entry q1 g0"| 12271₁₂ ||class="entry q1 g0"| 16883₈ ||class="entry q0 g0"| 31556₈ ||class="entry q1 g0"| 16409₄ ||class="entry q1 g0"| 18053₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (6740, 12031, 20243, 29876, 16649, 18533) ||class="entry q0 g0"| 6740₆ ||class="entry q1 g0"| 12031₁₂ ||class="entry q1 g0"| 20243₈ ||class="entry q0 g0"| 29876₈ ||class="entry q1 g0"| 16649₄ ||class="entry q1 g0"| 18533₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (7218, 20223, 12053, 29394, 8457, 10339) ||class="entry q0 g0"| 7218₆ ||class="entry q1 g0"| 20223₁₂ ||class="entry q1 g0"| 12053₈ ||class="entry q0 g0"| 29394₈ ||class="entry q1 g0"| 8457₄ ||class="entry q1 g0"| 10339₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (8796, 32629, 9127, 19644, 4227, 9425) ||class="entry q0 g0"| 8796₆ ||class="entry q1 g0"| 32629₁₂ ||class="entry q1 g0"| 9127₈ ||class="entry q0 g0"| 19644₈ ||class="entry q1 g0"| 4227₄ ||class="entry q1 g0"| 9425₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (9516, 30709, 10807, 19404, 6147, 11585) ||class="entry q0 g0"| 9516₆ ||class="entry q1 g0"| 30709₁₂ ||class="entry q1 g0"| 10807₈ ||class="entry q0 g0"| 19404₈ ||class="entry q1 g0"| 6147₄ ||class="entry q1 g0"| 11585₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (17226, 30707, 19543, 11690, 6149, 19233) ||class="entry q0 g0"| 17226₆ ||class="entry q1 g0"| 30707₁₂ ||class="entry q1 g0"| 19543₈ ||class="entry q0 g0"| 11690₈ ||class="entry q1 g0"| 6149₄ ||class="entry q1 g0"| 19233₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (17466, 32627, 17863, 10970, 4229, 17073) ||class="entry q0 g0"| 17466₆ ||class="entry q1 g0"| 32627₁₂ ||class="entry q1 g0"| 17863₈ ||class="entry q0 g0"| 10970₈ ||class="entry q1 g0"| 4229₄ ||class="entry q1 g0"| 17073₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (10648, 12283, 32219, 18296, 16397, 31405) ||class="entry q0 g0"| 10648₆ ||class="entry q1 g0"| 12283₁₂ ||class="entry q1 g0"| 32219₁₂ ||class="entry q0 g0"| 18296₈ ||class="entry q1 g0"| 16397₄ ||class="entry q1 g0"| 31405₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (18840, 20477, 31677, 10104, 8203, 31947) ||class="entry q0 g0"| 18840₆ ||class="entry q1 g0"| 20477₁₂ ||class="entry q1 g0"| 31677₁₂ ||class="entry q0 g0"| 10104₈ ||class="entry q1 g0"| 8203₄ ||class="entry q1 g0"| 31947₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (28806, 30455, 32415, 7782, 6401, 31209) ||class="entry q0 g0"| 28806₆ ||class="entry q1 g0"| 30455₁₂ ||class="entry q1 g0"| 32415₁₂ ||class="entry q0 g0"| 7782₈ ||class="entry q1 g0"| 6401₄ ||class="entry q1 g0"| 31209₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (30726, 32615, 31215, 5862, 4241, 32409) ||class="entry q0 g0"| 30726₆ ||class="entry q1 g0"| 32615₁₂ ||class="entry q1 g0"| 31215₁₂ ||class="entry q0 g0"| 5862₈ ||class="entry q1 g0"| 4241₄ ||class="entry q1 g0"| 32409₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (14742, 12293, 25397, 22390, 24563, 25667) ||class="entry q0 g0"| 14742₈ ||class="entry q1 g0"| 12293₄ ||class="entry q1 g0"| 25397₈ ||class="entry q0 g0"| 22390₁₀ ||class="entry q1 g0"| 24563₁₂ ||class="entry q1 g0"| 25667₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (22934, 20483, 25939, 14198, 16373, 25125) ||class="entry q0 g0"| 22934₈ ||class="entry q1 g0"| 20483₄ ||class="entry q1 g0"| 25939₈ ||class="entry q0 g0"| 14198₁₀ ||class="entry q1 g0"| 16373₁₂ ||class="entry q1 g0"| 25125₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (24990, 393, 2535, 3966, 28287, 3729) ||class="entry q0 g0"| 24990₈ ||class="entry q1 g0"| 393₄ ||class="entry q1 g0"| 2535₈ ||class="entry q0 g0"| 3966₁₀ ||class="entry q1 g0"| 28287₁₂ ||class="entry q1 g0"| 3729₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (26910, 2073, 3735, 2046, 26607, 2529) ||class="entry q0 g0"| 26910₈ ||class="entry q1 g0"| 2073₄ ||class="entry q1 g0"| 3735₈ ||class="entry q0 g0"| 2046₁₀ ||class="entry q1 g0"| 26607₁₂ ||class="entry q1 g0"| 2529₆ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (990, 2649, 25921, 27966, 26031, 25143) ||class="entry q0 g0"| 990₈ ||class="entry q1 g0"| 2649₆ ||class="entry q1 g0"| 25921₆ ||class="entry q0 g0"| 27966₁₀ ||class="entry q1 g0"| 26031₁₀ ||class="entry q1 g0"| 25143₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (1470, 3129, 25377, 27486, 25551, 25687) ||class="entry q0 g0"| 1470₈ ||class="entry q1 g0"| 3129₆ ||class="entry q1 g0"| 25377₆ ||class="entry q0 g0"| 27486₁₀ ||class="entry q1 g0"| 25551₁₀ ||class="entry q1 g0"| 25687₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (2910, 969, 25137, 26046, 27711, 25927) ||class="entry q0 g0"| 2910₈ ||class="entry q1 g0"| 969₆ ||class="entry q1 g0"| 25137₆ ||class="entry q0 g0"| 26046₁₀ ||class="entry q1 g0"| 27711₁₀ ||class="entry q1 g0"| 25927₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (3390, 1449, 25681, 25566, 27231, 25383) ||class="entry q0 g0"| 3390₈ ||class="entry q1 g0"| 1449₆ ||class="entry q1 g0"| 25681₆ ||class="entry q0 g0"| 25566₁₀ ||class="entry q1 g0"| 27231₁₀ ||class="entry q1 g0"| 25383₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (13142, 21059, 3717, 23990, 15797, 2547) ||class="entry q0 g0"| 13142₈ ||class="entry q1 g0"| 21059₆ ||class="entry q1 g0"| 3717₆ ||class="entry q0 g0"| 23990₁₀ ||class="entry q1 g0"| 15797₁₀ ||class="entry q1 g0"| 2547₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (13622, 21539, 2277, 23510, 15317, 3987) ||class="entry q0 g0"| 13622₈ ||class="entry q1 g0"| 21539₆ ||class="entry q1 g0"| 2277₆ ||class="entry q0 g0"| 23510₁₀ ||class="entry q1 g0"| 15317₁₀ ||class="entry q1 g0"| 3987₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (21334, 12869, 2275, 15798, 23987, 3989) ||class="entry q0 g0"| 21334₈ ||class="entry q1 g0"| 12869₆ ||class="entry q1 g0"| 2275₆ ||class="entry q0 g0"| 15798₁₀ ||class="entry q1 g0"| 23987₁₀ ||class="entry q1 g0"| 3989₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (21814, 13349, 3715, 15318, 23507, 2549) ||class="entry q0 g0"| 21814₈ ||class="entry q1 g0"| 13349₆ ||class="entry q1 g0"| 3715₆ ||class="entry q0 g0"| 15318₁₀ ||class="entry q1 g0"| 23507₁₀ ||class="entry q1 g0"| 2549₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (10654, 18843, 7101, 18302, 9837, 7371) ||class="entry q0 g0"| 10654₈ ||class="entry q1 g0"| 18843₈ ||class="entry q1 g0"| 7101₁₀ ||class="entry q0 g0"| 18302₁₀ ||class="entry q1 g0"| 9837₈ ||class="entry q1 g0"| 7371₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (18846, 10653, 7643, 10110, 18027, 6829) ||class="entry q0 g0"| 18846₈ ||class="entry q1 g0"| 10653₈ ||class="entry q1 g0"| 7643₁₀ ||class="entry q0 g0"| 10110₁₀ ||class="entry q1 g0"| 18027₈ ||class="entry q1 g0"| 6829₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (29078, 30743, 29039, 8054, 6113, 30233) ||class="entry q0 g0"| 29078₈ ||class="entry q1 g0"| 30743₈ ||class="entry q1 g0"| 29039₁₀ ||class="entry q0 g0"| 8054₁₀ ||class="entry q1 g0"| 6113₈ ||class="entry q1 g0"| 30233₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (30998, 29063, 30239, 6134, 7793, 29033) ||class="entry q0 g0"| 30998₈ ||class="entry q1 g0"| 29063₈ ||class="entry q1 g0"| 30239₁₀ ||class="entry q0 g0"| 6134₁₀ ||class="entry q1 g0"| 7793₈ ||class="entry q1 g0"| 29033₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (5078, 29639, 7625, 32054, 7217, 6847) ||class="entry q0 g0"| 5078₈ ||class="entry q1 g0"| 29639₁₀ ||class="entry q1 g0"| 7625₈ ||class="entry q0 g0"| 32054₁₀ ||class="entry q1 g0"| 7217₆ ||class="entry q1 g0"| 6847₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (5558, 30119, 7081, 31574, 6737, 7391) ||class="entry q0 g0"| 5558₈ ||class="entry q1 g0"| 30119₁₀ ||class="entry q1 g0"| 7081₈ ||class="entry q0 g0"| 31574₁₀ ||class="entry q1 g0"| 6737₆ ||class="entry q1 g0"| 7391₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6998, 31319, 6841, 30134, 5537, 7631) ||class="entry q0 g0"| 6998₈ ||class="entry q1 g0"| 31319₁₀ ||class="entry q1 g0"| 6841₈ ||class="entry q0 g0"| 30134₁₀ ||class="entry q1 g0"| 5537₆ ||class="entry q1 g0"| 7631₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (7478, 31799, 7385, 29654, 5057, 7087) ||class="entry q0 g0"| 7478₈ ||class="entry q1 g0"| 31799₁₀ ||class="entry q1 g0"| 7385₈ ||class="entry q0 g0"| 29654₁₀ ||class="entry q1 g0"| 5057₆ ||class="entry q1 g0"| 7087₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (9054, 11229, 30221, 19902, 17451, 29051) ||class="entry q0 g0"| 9054₈ ||class="entry q1 g0"| 11229₁₀ ||class="entry q1 g0"| 30221₈ ||class="entry q0 g0"| 19902₁₀ ||class="entry q1 g0"| 17451₆ ||class="entry q1 g0"| 29051₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (9534, 11709, 28781, 19422, 16971, 30491) ||class="entry q0 g0"| 9534₈ ||class="entry q1 g0"| 11709₁₀ ||class="entry q1 g0"| 28781₈ ||class="entry q0 g0"| 19422₁₀ ||class="entry q1 g0"| 16971₆ ||class="entry q1 g0"| 30491₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (17246, 19419, 28779, 11710, 9261, 30493) ||class="entry q0 g0"| 17246₈ ||class="entry q1 g0"| 19419₁₀ ||class="entry q1 g0"| 28779₈ ||class="entry q0 g0"| 11710₁₀ ||class="entry q1 g0"| 9261₆ ||class="entry q1 g0"| 30493₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (17726, 19899, 30219, 11230, 8781, 29053) ||class="entry q0 g0"| 17726₈ ||class="entry q1 g0"| 19899₁₀ ||class="entry q1 g0"| 30219₈ ||class="entry q0 g0"| 11230₁₀ ||class="entry q1 g0"| 8781₆ ||class="entry q1 g0"| 29053₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (15298, 26619, 13769, 21794, 2061, 12991) ||class="entry q0 g0"| 15298₈ ||class="entry q1 g0"| 26619₁₂ ||class="entry q1 g0"| 13769₈ ||class="entry q0 g0"| 21794₆ ||class="entry q1 g1"| 2061₄ ||class="entry q1 g0"| 12991₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (15538, 28539, 15449, 21074, 141, 15151) ||class="entry q0 g0"| 15538₈ ||class="entry q1 g0"| 28539₁₂ ||class="entry q1 g0"| 15449₈ ||class="entry q0 g0"| 21074₆ ||class="entry q1 g1"| 141₄ ||class="entry q1 g0"| 15151₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (23252, 28541, 23097, 13364, 139, 23887) ||class="entry q0 g0"| 23252₈ ||class="entry q1 g0"| 28541₁₂ ||class="entry q1 g0"| 23097₈ ||class="entry q0 g0"| 13364₆ ||class="entry q1 g1"| 139₄ ||class="entry q1 g0"| 23887₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (23972, 26621, 21417, 13124, 2059, 21727) ||class="entry q0 g0"| 23972₈ ||class="entry q1 g0"| 26621₁₂ ||class="entry q1 g0"| 21417₈ ||class="entry q0 g0"| 13124₆ ||class="entry q1 g1"| 2059₄ ||class="entry q1 g0"| 21727₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (25308, 16119, 13965, 3132, 20737, 12795) ||class="entry q0 g0"| 25308₈ ||class="entry q1 g0"| 16119₁₂ ||class="entry q1 g0"| 13965₈ ||class="entry q0 g0"| 3132₆ ||class="entry q1 g1"| 20737₄ ||class="entry q1 g0"| 12795₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (25786, 24311, 22155, 2650, 12545, 20989) ||class="entry q0 g0"| 25786₈ ||class="entry q1 g0"| 24311₁₂ ||class="entry q1 g0"| 22155₈ ||class="entry q0 g0"| 2650₆ ||class="entry q1 g1"| 12545₄ ||class="entry q1 g0"| 20989₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (27466, 24551, 22635, 1450, 12305, 24349) ||class="entry q0 g0"| 27466₈ ||class="entry q1 g0"| 24551₁₂ ||class="entry q1 g0"| 22635₈ ||class="entry q0 g0"| 1450₆ ||class="entry q1 g1"| 12305₄ ||class="entry q1 g0"| 24349₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (27948, 16359, 14445, 972, 20497, 16155) ||class="entry q0 g0"| 27948₈ ||class="entry q1 g0"| 16359₁₂ ||class="entry q1 g0"| 14445₈ ||class="entry q0 g0"| 972₆ ||class="entry q1 g1"| 20497₄ ||class="entry q1 g0"| 16155₁₀ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16116, 25293, 12543, 20500, 3387, 14217) ||class="entry q0 g0"| 16116₁₀ ||class="entry q1 g0"| 25293₈ ||class="entry q1 g0"| 12543₁₀ ||class="entry q0 g1"| 20500₄ ||class="entry q1 g0"| 3387₈ ||class="entry q1 g0"| 14217₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16356, 27693, 16143, 20740, 987, 14457) ||class="entry q0 g0"| 16356₁₀ ||class="entry q1 g0"| 27693₈ ||class="entry q1 g0"| 16143₁₀ ||class="entry q0 g1"| 20740₄ ||class="entry q1 g0"| 987₈ ||class="entry q1 g0"| 14457₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24306, 25771, 20735, 12306, 2909, 22409) ||class="entry q0 g0"| 24306₁₀ ||class="entry q1 g0"| 25771₈ ||class="entry q1 g0"| 20735₁₀ ||class="entry q0 g1"| 12306₄ ||class="entry q1 g0"| 2909₈ ||class="entry q1 g0"| 22409₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24546, 27211, 24335, 12546, 1469, 22649) ||class="entry q0 g0"| 24546₁₀ ||class="entry q1 g0"| 27211₈ ||class="entry q1 g0"| 24335₁₀ ||class="entry q0 g1"| 12546₄ ||class="entry q1 g0"| 1469₈ ||class="entry q1 g0"| 22649₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (26602, 15297, 13243, 2314, 21559, 13517) ||class="entry q0 g0"| 26602₁₀ ||class="entry q1 g0"| 15297₈ ||class="entry q1 g0"| 13243₁₀ ||class="entry q0 g1"| 2314₄ ||class="entry q1 g0"| 21559₈ ||class="entry q1 g0"| 13517₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (26604, 23969, 21981, 2316, 12887, 21163) ||class="entry q0 g0"| 26604₁₀ ||class="entry q1 g0"| 23969₈ ||class="entry q1 g0"| 21981₁₀ ||class="entry q0 g1"| 2316₄ ||class="entry q1 g0"| 12887₈ ||class="entry q1 g0"| 21163₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (28282, 15537, 15163, 154, 21319, 15437) ||class="entry q0 g0"| 28282₁₀ ||class="entry q1 g0"| 15537₈ ||class="entry q1 g0"| 15163₁₀ ||class="entry q0 g1"| 154₄ ||class="entry q1 g0"| 21319₈ ||class="entry q1 g0"| 15437₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (28284, 23249, 23901, 156, 13607, 23083) ||class="entry q0 g0"| 28284₁₀ ||class="entry q1 g0"| 23249₈ ||class="entry q1 g0"| 23901₁₀ ||class="entry q0 g1"| 156₄ ||class="entry q1 g0"| 13607₈ ||class="entry q1 g0"| 23083₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (16096, 24293, 3267, 20480, 12563, 2997) ||class="entry q0 g0"| 16096₈ ||class="entry q1 g0"| 24293₁₀ ||class="entry q1 g0"| 3267₆ ||class="entry q0 g1"| 20480₂ ||class="entry q1 g1"| 12563₆ ||class="entry q1 g0"| 2997₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (24288, 16099, 2725, 12288, 20757, 3539) ||class="entry q0 g0"| 24288₈ ||class="entry q1 g0"| 16099₁₀ ||class="entry q1 g0"| 2725₆ ||class="entry q0 g1"| 12288₂ ||class="entry q1 g1"| 20757₆ ||class="entry q1 g0"| 3539₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (26344, 28521, 26129, 2056, 159, 24935) ||class="entry q0 g0"| 26344₈ ||class="entry q1 g0"| 28521₁₀ ||class="entry q1 g0"| 26129₆ ||class="entry q0 g1"| 2056₂ ||class="entry q1 g1"| 159₆ ||class="entry q1 g0"| 24935₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (28264, 26361, 24929, 136, 2319, 26135) ||class="entry q0 g0"| 28264₈ ||class="entry q1 g0"| 26361₁₀ ||class="entry q1 g0"| 24929₆ ||class="entry q0 g1"| 136₂ ||class="entry q1 g1"| 2319₆ ||class="entry q1 g0"| 26135₈ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (16368, 20485, 819, 20752, 16371, 1093) ||class="entry q0 g0"| 16368₁₀ ||class="entry q1 g1"| 20485₄ ||class="entry q1 g0"| 819₆ ||class="entry q0 g1"| 20752₄ ||class="entry q1 g0"| 16371₁₂ ||class="entry q1 g0"| 1093₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (24560, 12291, 1365, 12560, 24565, 547) ||class="entry q0 g0"| 24560₁₀ ||class="entry q1 g1"| 12291₄ ||class="entry q1 g0"| 1365₆ ||class="entry q0 g1"| 12560₄ ||class="entry q1 g0"| 24565₁₂ ||class="entry q1 g0"| 547₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (26350, 2313, 119, 2062, 26367, 1793) ||class="entry q0 g0"| 26350₁₀ ||class="entry q1 g1"| 2313₄ ||class="entry q1 g0"| 119₆ ||class="entry q0 g1"| 2062₄ ||class="entry q1 g0"| 26367₁₂ ||class="entry q1 g0"| 1793₄ |- |class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (28270, 153, 1799, 142, 28527, 113) ||class="entry q0 g0"| 28270₁₀ ||class="entry q1 g1"| 153₄ ||class="entry q1 g0"| 1799₆ ||class="entry q0 g1"| 142₄ ||class="entry q1 g0"| 28527₁₂ ||class="entry q1 g0"| 113₄ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (32864, 62990, 36758, 37246, 59152, 40584) ||class="entry q2 g1"| 32864₃ ||class="entry q2 g1"| 62990₉ ||class="entry q2 g1"| 36758₉ ||class="entry q2 g1"| 37246₉ ||class="entry q2 g1"| 59152₇ ||class="entry q2 g1"| 40584₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (33312, 40408, 58710, 37694, 36038, 62536) ||class="entry q2 g1"| 33312₃ ||class="entry q2 g1"| 40408₉ ||class="entry q2 g1"| 58710₉ ||class="entry q2 g1"| 37694₉ ||class="entry q2 g1"| 36038₇ ||class="entry q2 g1"| 62536₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (33856, 39864, 58166, 38238, 35494, 61992) ||class="entry q2 g1"| 33856₃ ||class="entry q2 g1"| 39864₉ ||class="entry q2 g1"| 58166₉ ||class="entry q2 g1"| 38238₉ ||class="entry q2 g1"| 35494₇ ||class="entry q2 g1"| 61992₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 9, 9, 9, 7, 7) ||class="c"| (34304, 61550, 35318, 38686, 57712, 39144) ||class="entry q2 g1"| 34304₃ ||class="entry q2 g1"| 61550₉ ||class="entry q2 g1"| 35318₉ ||class="entry q2 g1"| 38686₉ ||class="entry q2 g1"| 57712₇ ||class="entry q2 g1"| 39144₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (35488, 37960, 57894, 39870, 34134, 62264) ||class="entry q2 g1"| 35488₅ ||class="entry q2 g1"| 37960₅ ||class="entry q2 g1"| 57894₇ ||class="entry q2 g1"| 39870₁₁ ||class="entry q2 g1"| 34134₇ ||class="entry q2 g1"| 62264₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (36032, 37416, 58438, 40414, 33590, 62808) ||class="entry q2 g1"| 36032₅ ||class="entry q2 g1"| 37416₅ ||class="entry q2 g1"| 58438₇ ||class="entry q2 g1"| 40414₁₁ ||class="entry q2 g1"| 33590₇ ||class="entry q2 g1"| 62808₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (57440, 38408, 35312, 61822, 34582, 39150) ||class="entry q2 g1"| 57440₅ ||class="entry q2 g1"| 38408₅ ||class="entry q2 g1"| 35312₇ ||class="entry q2 g1"| 61822₁₁ ||class="entry q2 g1"| 34582₇ ||class="entry q2 g1"| 39150₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (58880, 36968, 36752, 63262, 33142, 40590) ||class="entry q2 g1"| 58880₅ ||class="entry q2 g1"| 36968₅ ||class="entry q2 g1"| 36752₇ ||class="entry q2 g1"| 63262₁₁ ||class="entry q2 g1"| 33142₇ ||class="entry q2 g1"| 40590₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (32870, 36974, 59888, 37240, 33136, 63726) ||class="entry q2 g1"| 32870₅ ||class="entry q2 g1"| 36974₇ ||class="entry q2 g1"| 59888₉ ||class="entry q2 g1"| 37240₇ ||class="entry q2 g1"| 33136₅ ||class="entry q2 g1"| 63726₁₁ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (33584, 37688, 60070, 37422, 33318, 64440) ||class="entry q2 g1"| 33584₅ ||class="entry q2 g1"| 37688₇ ||class="entry q2 g1"| 60070₉ ||class="entry q2 g1"| 37422₇ ||class="entry q2 g1"| 33318₅ ||class="entry q2 g1"| 64440₁₁ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (34128, 38232, 60614, 37966, 33862, 64984) ||class="entry q2 g1"| 34128₅ ||class="entry q2 g1"| 38232₇ ||class="entry q2 g1"| 60614₉ ||class="entry q2 g1"| 37966₇ ||class="entry q2 g1"| 33862₅ ||class="entry q2 g1"| 64984₁₁ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (34310, 38414, 61328, 38680, 34576, 65166) ||class="entry q2 g1"| 34310₅ ||class="entry q2 g1"| 38414₇ ||class="entry q2 g1"| 61328₉ ||class="entry q2 g1"| 38680₇ ||class="entry q2 g1"| 34576₅ ||class="entry q2 g1"| 65166₁₁ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32882, 44102, 54732, 37228, 48472, 50386) ||class="entry q2 g1"| 32882₅ ||class="entry q2 g1"| 44102₇ ||class="entry q2 g1"| 54732₉ ||class="entry q2 g1"| 37228₇ ||class="entry q2 g1"| 48472₉ ||class="entry q2 g1"| 50386₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32884, 51750, 45994, 37226, 56120, 41652) ||class="entry q2 g1"| 32884₅ ||class="entry q2 g1"| 51750₇ ||class="entry q2 g1"| 45994₉ ||class="entry q2 g1"| 37226₇ ||class="entry q2 g1"| 56120₉ ||class="entry q2 g1"| 41652₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33122, 41638, 55868, 36988, 46008, 52002) ||class="entry q2 g1"| 33122₅ ||class="entry q2 g1"| 41638₇ ||class="entry q2 g1"| 55868₉ ||class="entry q2 g1"| 36988₇ ||class="entry q2 g1"| 46008₉ ||class="entry q2 g1"| 52002₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33124, 50374, 48218, 36986, 54744, 44356) ||class="entry q2 g1"| 33124₅ ||class="entry q2 g1"| 50374₇ ||class="entry q2 g1"| 48218₉ ||class="entry q2 g1"| 36986₇ ||class="entry q2 g1"| 54744₉ ||class="entry q2 g1"| 44356₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33330, 51088, 48908, 37676, 54926, 44562) ||class="entry q2 g1"| 33330₅ ||class="entry q2 g1"| 51088₇ ||class="entry q2 g1"| 48908₉ ||class="entry q2 g1"| 37676₇ ||class="entry q2 g1"| 54926₉ ||class="entry q2 g1"| 44562₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33332, 41456, 55658, 37674, 45294, 51316) ||class="entry q2 g1"| 33332₅ ||class="entry q2 g1"| 41456₇ ||class="entry q2 g1"| 55658₉ ||class="entry q2 g1"| 37674₇ ||class="entry q2 g1"| 45294₉ ||class="entry q2 g1"| 51316₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33570, 51568, 45308, 37436, 55406, 41442) ||class="entry q2 g1"| 33570₅ ||class="entry q2 g1"| 51568₇ ||class="entry q2 g1"| 45308₉ ||class="entry q2 g1"| 37436₇ ||class="entry q2 g1"| 55406₉ ||class="entry q2 g1"| 41442₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33572, 44816, 54938, 37434, 48654, 51076) ||class="entry q2 g1"| 33572₅ ||class="entry q2 g1"| 44816₇ ||class="entry q2 g1"| 54938₉ ||class="entry q2 g1"| 37434₇ ||class="entry q2 g1"| 48654₉ ||class="entry q2 g1"| 51076₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33874, 49648, 47468, 38220, 53486, 43122) ||class="entry q2 g1"| 33874₅ ||class="entry q2 g1"| 49648₇ ||class="entry q2 g1"| 47468₉ ||class="entry q2 g1"| 38220₇ ||class="entry q2 g1"| 53486₉ ||class="entry q2 g1"| 43122₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33876, 42896, 57098, 38218, 46734, 52756) ||class="entry q2 g1"| 33876₅ ||class="entry q2 g1"| 42896₇ ||class="entry q2 g1"| 57098₉ ||class="entry q2 g1"| 38218₇ ||class="entry q2 g1"| 46734₉ ||class="entry q2 g1"| 52756₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34114, 53008, 46748, 37980, 56846, 42882) ||class="entry q2 g1"| 34114₅ ||class="entry q2 g1"| 53008₇ ||class="entry q2 g1"| 46748₉ ||class="entry q2 g1"| 37980₇ ||class="entry q2 g1"| 56846₉ ||class="entry q2 g1"| 42882₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34116, 43376, 53498, 37978, 47214, 49636) ||class="entry q2 g1"| 34116₅ ||class="entry q2 g1"| 43376₇ ||class="entry q2 g1"| 53498₉ ||class="entry q2 g1"| 37978₇ ||class="entry q2 g1"| 47214₉ ||class="entry q2 g1"| 49636₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34322, 43558, 54188, 38668, 47928, 49842) ||class="entry q2 g1"| 34322₅ ||class="entry q2 g1"| 43558₇ ||class="entry q2 g1"| 54188₉ ||class="entry q2 g1"| 38668₇ ||class="entry q2 g1"| 47928₉ ||class="entry q2 g1"| 49842₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34324, 52294, 46538, 38666, 56664, 42196) ||class="entry q2 g1"| 34324₅ ||class="entry q2 g1"| 52294₇ ||class="entry q2 g1"| 46538₉ ||class="entry q2 g1"| 38666₇ ||class="entry q2 g1"| 56664₉ ||class="entry q2 g1"| 42196₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34562, 42182, 56412, 38428, 46552, 52546) ||class="entry q2 g1"| 34562₅ ||class="entry q2 g1"| 42182₇ ||class="entry q2 g1"| 56412₉ ||class="entry q2 g1"| 38428₇ ||class="entry q2 g1"| 46552₉ ||class="entry q2 g1"| 52546₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34564, 49830, 47674, 38426, 54200, 43812) ||class="entry q2 g1"| 34564₅ ||class="entry q2 g1"| 49830₇ ||class="entry q2 g1"| 47674₉ ||class="entry q2 g1"| 38426₇ ||class="entry q2 g1"| 54200₉ ||class="entry q2 g1"| 43812₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (41184, 55178, 40154, 45566, 50836, 36292) ||class="entry q2 g1"| 41184₅ ||class="entry q2 g1"| 55178₉ ||class="entry q2 g1"| 40154₉ ||class="entry q2 g1"| 45566₁₁ ||class="entry q2 g1"| 50836₇ ||class="entry q2 g1"| 36292₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (41632, 48220, 63002, 46014, 44354, 59140) ||class="entry q2 g1"| 41632₅ ||class="entry q2 g1"| 48220₉ ||class="entry q2 g1"| 63002₉ ||class="entry q2 g1"| 46014₁₁ ||class="entry q2 g1"| 44354₇ ||class="entry q2 g1"| 59140₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (42176, 47676, 61562, 46558, 43810, 57700) ||class="entry q2 g1"| 42176₅ ||class="entry q2 g1"| 47676₉ ||class="entry q2 g1"| 61562₉ ||class="entry q2 g1"| 46558₁₁ ||class="entry q2 g1"| 43810₇ ||class="entry q2 g1"| 57700₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (42624, 53738, 39610, 47006, 49396, 35748) ||class="entry q2 g1"| 42624₅ ||class="entry q2 g1"| 53738₉ ||class="entry q2 g1"| 39610₉ ||class="entry q2 g1"| 47006₁₁ ||class="entry q2 g1"| 49396₇ ||class="entry q2 g1"| 35748₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43104, 56858, 39850, 47486, 52996, 35508) ||class="entry q2 g1"| 43104₅ ||class="entry q2 g1"| 56858₉ ||class="entry q2 g1"| 39850₉ ||class="entry q2 g1"| 47486₁₁ ||class="entry q2 g1"| 52996₇ ||class="entry q2 g1"| 35508₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43552, 46540, 61802, 47934, 42194, 57460) ||class="entry q2 g1"| 43552₅ ||class="entry q2 g1"| 46540₉ ||class="entry q2 g1"| 61802₉ ||class="entry q2 g1"| 47934₁₁ ||class="entry q2 g1"| 42194₇ ||class="entry q2 g1"| 57460₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (44096, 45996, 63242, 48478, 41650, 58900) ||class="entry q2 g1"| 44096₅ ||class="entry q2 g1"| 45996₉ ||class="entry q2 g1"| 63242₉ ||class="entry q2 g1"| 48478₁₁ ||class="entry q2 g1"| 41650₇ ||class="entry q2 g1"| 58900₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (44544, 55418, 40394, 48926, 51556, 36052) ||class="entry q2 g1"| 44544₅ ||class="entry q2 g1"| 55418₉ ||class="entry q2 g1"| 40394₉ ||class="entry q2 g1"| 48926₁₁ ||class="entry q2 g1"| 51556₇ ||class="entry q2 g1"| 36052₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (49376, 46988, 39612, 53758, 42642, 35746) ||class="entry q2 g1"| 49376₅ ||class="entry q2 g1"| 46988₉ ||class="entry q2 g1"| 39612₉ ||class="entry q2 g1"| 53758₁₁ ||class="entry q2 g1"| 42642₇ ||class="entry q2 g1"| 35746₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (49824, 56410, 61564, 54206, 52548, 57698) ||class="entry q2 g1"| 49824₅ ||class="entry q2 g1"| 56410₉ ||class="entry q2 g1"| 61564₉ ||class="entry q2 g1"| 54206₁₁ ||class="entry q2 g1"| 52548₇ ||class="entry q2 g1"| 57698₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (50368, 55866, 63004, 54750, 52004, 59138) ||class="entry q2 g1"| 50368₅ ||class="entry q2 g1"| 55866₉ ||class="entry q2 g1"| 63004₉ ||class="entry q2 g1"| 54750₁₁ ||class="entry q2 g1"| 52004₇ ||class="entry q2 g1"| 59138₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (50816, 45548, 40156, 55198, 41202, 36290) ||class="entry q2 g1"| 50816₅ ||class="entry q2 g1"| 45548₉ ||class="entry q2 g1"| 40156₉ ||class="entry q2 g1"| 55198₁₁ ||class="entry q2 g1"| 41202₇ ||class="entry q2 g1"| 36290₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (51296, 48668, 40396, 55678, 44802, 36050) ||class="entry q2 g1"| 51296₅ ||class="entry q2 g1"| 48668₉ ||class="entry q2 g1"| 40396₉ ||class="entry q2 g1"| 55678₁₁ ||class="entry q2 g1"| 44802₇ ||class="entry q2 g1"| 36050₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (51744, 54730, 63244, 56126, 50388, 58898) ||class="entry q2 g1"| 51744₅ ||class="entry q2 g1"| 54730₉ ||class="entry q2 g1"| 63244₉ ||class="entry q2 g1"| 56126₁₁ ||class="entry q2 g1"| 50388₇ ||class="entry q2 g1"| 58898₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (52288, 54186, 61804, 56670, 49844, 57458) ||class="entry q2 g1"| 52288₅ ||class="entry q2 g1"| 54186₉ ||class="entry q2 g1"| 61804₉ ||class="entry q2 g1"| 56670₁₁ ||class="entry q2 g1"| 49844₇ ||class="entry q2 g1"| 57458₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (52736, 47228, 39852, 57118, 43362, 35506) ||class="entry q2 g1"| 52736₅ ||class="entry q2 g1"| 47228₉ ||class="entry q2 g1"| 39852₉ ||class="entry q2 g1"| 57118₁₁ ||class="entry q2 g1"| 43362₇ ||class="entry q2 g1"| 35506₇ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (35040, 65438, 35046, 39422, 61056, 39416) ||class="entry q2 g1"| 35040₅ ||class="entry q2 g1"| 65438₁₃ ||class="entry q2 g1"| 35046₇ ||class="entry q2 g1"| 39422₁₁ ||class="entry q2 g1"| 61056₇ ||class="entry q2 g1"| 39416₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (36480, 63998, 36486, 40862, 59616, 40856) ||class="entry q2 g1"| 36480₅ ||class="entry q2 g1"| 63998₁₃ ||class="entry q2 g1"| 36486₇ ||class="entry q2 g1"| 40862₁₁ ||class="entry q2 g1"| 59616₇ ||class="entry q2 g1"| 40856₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (57888, 64990, 58160, 62270, 60608, 61998) ||class="entry q2 g1"| 57888₅ ||class="entry q2 g1"| 64990₁₃ ||class="entry q2 g1"| 58160₇ ||class="entry q2 g1"| 62270₁₁ ||class="entry q2 g1"| 60608₇ ||class="entry q2 g1"| 61998₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 13, 7, 11, 7, 9) ||class="c"| (58432, 64446, 58704, 62814, 60064, 62542) ||class="entry q2 g1"| 58432₅ ||class="entry q2 g1"| 64446₁₃ ||class="entry q2 g1"| 58704₇ ||class="entry q2 g1"| 62814₁₁ ||class="entry q2 g1"| 60064₇ ||class="entry q2 g1"| 62542₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45288, 44564, 58450, 41462, 48906, 62796) ||class="entry q2 g1"| 45288₇ ||class="entry q2 g1"| 44564₇ ||class="entry q2 g1"| 58450₇ ||class="entry q2 g1"| 41462₉ ||class="entry q2 g1"| 48906₉ ||class="entry q2 g1"| 62796₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45736, 50626, 36498, 41910, 54492, 40844) ||class="entry q2 g1"| 45736₇ ||class="entry q2 g1"| 50626₇ ||class="entry q2 g1"| 36498₇ ||class="entry q2 g1"| 41910₉ ||class="entry q2 g1"| 54492₉ ||class="entry q2 g1"| 40844₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (46280, 50082, 35058, 42454, 53948, 39404) ||class="entry q2 g1"| 46280₇ ||class="entry q2 g1"| 50082₇ ||class="entry q2 g1"| 35058₇ ||class="entry q2 g1"| 42454₉ ||class="entry q2 g1"| 53948₉ ||class="entry q2 g1"| 39404₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (46728, 43124, 57906, 42902, 47466, 62252) ||class="entry q2 g1"| 46728₇ ||class="entry q2 g1"| 43124₇ ||class="entry q2 g1"| 57906₇ ||class="entry q2 g1"| 42902₉ ||class="entry q2 g1"| 47466₉ ||class="entry q2 g1"| 62252₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (47208, 42884, 58146, 43382, 46746, 62012) ||class="entry q2 g1"| 47208₇ ||class="entry q2 g1"| 42884₇ ||class="entry q2 g1"| 58146₇ ||class="entry q2 g1"| 43382₉ ||class="entry q2 g1"| 46746₉ ||class="entry q2 g1"| 62012₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (47656, 52306, 35298, 43830, 56652, 39164) ||class="entry q2 g1"| 47656₇ ||class="entry q2 g1"| 52306₇ ||class="entry q2 g1"| 35298₇ ||class="entry q2 g1"| 43830₉ ||class="entry q2 g1"| 56652₉ ||class="entry q2 g1"| 39164₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (48200, 51762, 36738, 44374, 56108, 40604) ||class="entry q2 g1"| 48200₇ ||class="entry q2 g1"| 51762₇ ||class="entry q2 g1"| 36738₇ ||class="entry q2 g1"| 44374₉ ||class="entry q2 g1"| 56108₉ ||class="entry q2 g1"| 40604₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (48648, 41444, 58690, 44822, 45306, 62556) ||class="entry q2 g1"| 48648₇ ||class="entry q2 g1"| 41444₇ ||class="entry q2 g1"| 58690₇ ||class="entry q2 g1"| 44822₉ ||class="entry q2 g1"| 45306₉ ||class="entry q2 g1"| 62556₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (53480, 52754, 57908, 49654, 57100, 62250) ||class="entry q2 g1"| 53480₇ ||class="entry q2 g1"| 52754₇ ||class="entry q2 g1"| 57908₇ ||class="entry q2 g1"| 49654₉ ||class="entry q2 g1"| 57100₉ ||class="entry q2 g1"| 62250₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (53928, 42436, 35060, 50102, 46298, 39402) ||class="entry q2 g1"| 53928₇ ||class="entry q2 g1"| 42436₇ ||class="entry q2 g1"| 35060₇ ||class="entry q2 g1"| 50102₉ ||class="entry q2 g1"| 46298₉ ||class="entry q2 g1"| 39402₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54472, 41892, 36500, 50646, 45754, 40842) ||class="entry q2 g1"| 54472₇ ||class="entry q2 g1"| 41892₇ ||class="entry q2 g1"| 36500₇ ||class="entry q2 g1"| 50646₉ ||class="entry q2 g1"| 45754₉ ||class="entry q2 g1"| 40842₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54920, 51314, 58452, 51094, 55660, 62794) ||class="entry q2 g1"| 54920₇ ||class="entry q2 g1"| 51314₇ ||class="entry q2 g1"| 58452₇ ||class="entry q2 g1"| 51094₉ ||class="entry q2 g1"| 55660₉ ||class="entry q2 g1"| 62794₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (55400, 51074, 58692, 51574, 54940, 62554) ||class="entry q2 g1"| 55400₇ ||class="entry q2 g1"| 51074₇ ||class="entry q2 g1"| 58692₇ ||class="entry q2 g1"| 51574₉ ||class="entry q2 g1"| 54940₉ ||class="entry q2 g1"| 62554₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (55848, 44116, 36740, 52022, 48458, 40602) ||class="entry q2 g1"| 55848₇ ||class="entry q2 g1"| 44116₇ ||class="entry q2 g1"| 36740₇ ||class="entry q2 g1"| 52022₉ ||class="entry q2 g1"| 48458₉ ||class="entry q2 g1"| 40602₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (56392, 43572, 35300, 52566, 47914, 39162) ||class="entry q2 g1"| 56392₇ ||class="entry q2 g1"| 43572₇ ||class="entry q2 g1"| 35300₇ ||class="entry q2 g1"| 52566₉ ||class="entry q2 g1"| 47914₉ ||class="entry q2 g1"| 39162₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (56840, 49634, 58148, 53014, 53500, 62010) ||class="entry q2 g1"| 56840₇ ||class="entry q2 g1"| 49634₇ ||class="entry q2 g1"| 58148₇ ||class="entry q2 g1"| 53014₉ ||class="entry q2 g1"| 53500₉ ||class="entry q2 g1"| 62010₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (35760, 39592, 60886, 39598, 35766, 64712) ||class="entry q2 g1"| 35760₇ ||class="entry q2 g1"| 39592₇ ||class="entry q2 g1"| 60886₁₁ ||class="entry q2 g1"| 39598₉ ||class="entry q2 g1"| 35766₉ ||class="entry q2 g1"| 64712₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (36304, 40136, 60342, 40142, 36310, 64168) ||class="entry q2 g1"| 36304₇ ||class="entry q2 g1"| 40136₇ ||class="entry q2 g1"| 60342₁₁ ||class="entry q2 g1"| 40142₉ ||class="entry q2 g1"| 36310₉ ||class="entry q2 g1"| 64168₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (57446, 61544, 61334, 61816, 57718, 65160) ||class="entry q2 g1"| 57446₇ ||class="entry q2 g1"| 61544₇ ||class="entry q2 g1"| 61334₁₁ ||class="entry q2 g1"| 61816₉ ||class="entry q2 g1"| 57718₉ ||class="entry q2 g1"| 65160₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (58886, 62984, 59894, 63256, 59158, 63720) ||class="entry q2 g1"| 58886₇ ||class="entry q2 g1"| 62984₇ ||class="entry q2 g1"| 59894₁₁ ||class="entry q2 g1"| 63256₉ ||class="entry q2 g1"| 59158₉ ||class="entry q2 g1"| 63720₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41204, 60322, 41190, 45546, 64188, 45560) ||class="entry q2 g1"| 41204₇ ||class="entry q2 g1"| 60322₉ ||class="entry q2 g1"| 41190₇ ||class="entry q2 g1"| 45546₉ ||class="entry q2 g1"| 64188₁₁ ||class="entry q2 g1"| 45560₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41890, 59636, 41904, 45756, 63978, 45742) ||class="entry q2 g1"| 41890₇ ||class="entry q2 g1"| 59636₉ ||class="entry q2 g1"| 41904₇ ||class="entry q2 g1"| 45756₉ ||class="entry q2 g1"| 63978₁₁ ||class="entry q2 g1"| 45742₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42434, 61076, 42448, 46300, 65418, 46286) ||class="entry q2 g1"| 42434₇ ||class="entry q2 g1"| 61076₉ ||class="entry q2 g1"| 42448₇ ||class="entry q2 g1"| 46300₉ ||class="entry q2 g1"| 65418₁₁ ||class="entry q2 g1"| 46286₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42644, 60866, 42630, 46986, 64732, 47000) ||class="entry q2 g1"| 42644₇ ||class="entry q2 g1"| 60866₉ ||class="entry q2 g1"| 42630₇ ||class="entry q2 g1"| 46986₉ ||class="entry q2 g1"| 64732₁₁ ||class="entry q2 g1"| 47000₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43364, 60626, 43110, 47226, 64972, 47480) ||class="entry q2 g1"| 43364₇ ||class="entry q2 g1"| 60626₉ ||class="entry q2 g1"| 43110₇ ||class="entry q2 g1"| 47226₉ ||class="entry q2 g1"| 64972₁₁ ||class="entry q2 g1"| 47480₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43570, 61316, 43824, 47916, 65178, 47662) ||class="entry q2 g1"| 43570₇ ||class="entry q2 g1"| 61316₉ ||class="entry q2 g1"| 43824₇ ||class="entry q2 g1"| 47916₉ ||class="entry q2 g1"| 65178₁₁ ||class="entry q2 g1"| 47662₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (44114, 59876, 44368, 48460, 63738, 48206) ||class="entry q2 g1"| 44114₇ ||class="entry q2 g1"| 59876₉ ||class="entry q2 g1"| 44368₇ ||class="entry q2 g1"| 48460₉ ||class="entry q2 g1"| 63738₁₁ ||class="entry q2 g1"| 48206₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (44804, 60082, 44550, 48666, 64428, 48920) ||class="entry q2 g1"| 44804₇ ||class="entry q2 g1"| 60082₉ ||class="entry q2 g1"| 44550₇ ||class="entry q2 g1"| 48666₉ ||class="entry q2 g1"| 64428₁₁ ||class="entry q2 g1"| 48920₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (49394, 60868, 49382, 53740, 64730, 53752) ||class="entry q2 g1"| 49394₇ ||class="entry q2 g1"| 60868₉ ||class="entry q2 g1"| 49382₇ ||class="entry q2 g1"| 53740₉ ||class="entry q2 g1"| 64730₁₁ ||class="entry q2 g1"| 53752₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50084, 61074, 50096, 53946, 65420, 53934) ||class="entry q2 g1"| 50084₇ ||class="entry q2 g1"| 61074₉ ||class="entry q2 g1"| 50096₇ ||class="entry q2 g1"| 53946₉ ||class="entry q2 g1"| 65420₁₁ ||class="entry q2 g1"| 53934₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50628, 59634, 50640, 54490, 63980, 54478) ||class="entry q2 g1"| 50628₇ ||class="entry q2 g1"| 59634₉ ||class="entry q2 g1"| 50640₇ ||class="entry q2 g1"| 54490₉ ||class="entry q2 g1"| 63980₁₁ ||class="entry q2 g1"| 54478₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50834, 60324, 50822, 55180, 64186, 55192) ||class="entry q2 g1"| 50834₇ ||class="entry q2 g1"| 60324₉ ||class="entry q2 g1"| 50822₇ ||class="entry q2 g1"| 55180₉ ||class="entry q2 g1"| 64186₁₁ ||class="entry q2 g1"| 55192₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (51554, 60084, 51302, 55420, 64426, 55672) ||class="entry q2 g1"| 51554₇ ||class="entry q2 g1"| 60084₉ ||class="entry q2 g1"| 51302₇ ||class="entry q2 g1"| 55420₉ ||class="entry q2 g1"| 64426₁₁ ||class="entry q2 g1"| 55672₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (51764, 59874, 52016, 56106, 63740, 55854) ||class="entry q2 g1"| 51764₇ ||class="entry q2 g1"| 59874₉ ||class="entry q2 g1"| 52016₇ ||class="entry q2 g1"| 56106₉ ||class="entry q2 g1"| 63740₁₁ ||class="entry q2 g1"| 55854₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (52308, 61314, 52560, 56650, 65180, 56398) ||class="entry q2 g1"| 52308₇ ||class="entry q2 g1"| 61314₉ ||class="entry q2 g1"| 52560₇ ||class="entry q2 g1"| 56650₉ ||class="entry q2 g1"| 65180₁₁ ||class="entry q2 g1"| 56398₉ |- |class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (52994, 60628, 52742, 56860, 64970, 57112) ||class="entry q2 g1"| 52994₇ ||class="entry q2 g1"| 60628₉ ||class="entry q2 g1"| 52742₇ ||class="entry q2 g1"| 56860₉ ||class="entry q2 g1"| 64970₁₁ ||class="entry q2 g1"| 57112₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (32904, 40807, 36737, 63766, 59375, 40841) ||class="entry q2 g1"| 32904₃ ||class="entry q3 g1"| 40807₁₁ ||class="entry q3 g1"| 36737₇ ||class="entry q2 g1"| 63766₉ ||class="entry q3 g1"| 59375₁₃ ||class="entry q3 g1"| 40841₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (34824, 38647, 35057, 61846, 61055, 39161) ||class="entry q2 g1"| 34824₃ ||class="entry q3 g1"| 38647₁₁ ||class="entry q3 g1"| 35057₇ ||class="entry q2 g1"| 61846₉ ||class="entry q3 g1"| 61055₁₃ ||class="entry q3 g1"| 39161₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (45056, 51069, 58437, 51614, 49141, 62541) ||class="entry q2 g1"| 45056₃ ||class="entry q3 g1"| 51069₁₁ ||class="entry q3 g1"| 58437₇ ||class="entry q2 g1"| 51614₉ ||class="entry q3 g1"| 49141₁₃ ||class="entry q3 g1"| 62541₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 7, 9, 13, 9) ||class="c"| (53248, 42875, 57891, 43422, 57331, 61995) ||class="entry q2 g1"| 53248₃ ||class="entry q3 g1"| 42875₁₁ ||class="entry q3 g1"| 57891₇ ||class="entry q2 g1"| 43422₉ ||class="entry q3 g1"| 57331₁₃ ||class="entry q3 g1"| 61995₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (36992, 59129, 63241, 59678, 40561, 59137) ||class="entry q2 g1"| 36992₃ ||class="entry q3 g1"| 59129₁₁ ||class="entry q3 g1"| 63241₉ ||class="entry q2 g1"| 59678₉ ||class="entry q3 g1"| 40561₉ ||class="entry q3 g1"| 59137₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (38912, 61289, 61561, 57758, 38881, 57457) ||class="entry q2 g1"| 38912₃ ||class="entry q3 g1"| 61289₁₁ ||class="entry q3 g1"| 61561₉ ||class="entry q2 g1"| 57758₉ ||class="entry q3 g1"| 38881₉ ||class="entry q3 g1"| 57457₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (40968, 48867, 40141, 55702, 50795, 36037) ||class="entry q2 g1"| 40968₃ ||class="entry q3 g1"| 48867₁₁ ||class="entry q3 g1"| 40141₉ ||class="entry q2 g1"| 55702₉ ||class="entry q3 g1"| 50795₉ ||class="entry q3 g1"| 36037₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 9, 9, 9, 7) ||class="c"| (49160, 57061, 39595, 47510, 42605, 35491) ||class="entry q2 g1"| 49160₃ ||class="entry q3 g1"| 57061₁₁ ||class="entry q3 g1"| 39595₉ ||class="entry q2 g1"| 47510₉ ||class="entry q3 g1"| 42605₉ ||class="entry q3 g1"| 35491₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (36998, 32921, 37231, 59672, 63505, 33127) ||class="entry q2 g1"| 36998₅ ||class="entry q3 g1"| 32921₅ ||class="entry q3 g1"| 37231₉ ||class="entry q2 g1"| 59672₇ ||class="entry q3 g1"| 63505₇ ||class="entry q3 g1"| 33127₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (38918, 35081, 38431, 57752, 61825, 34327) ||class="entry q2 g1"| 38918₅ ||class="entry q3 g1"| 35081₅ ||class="entry q3 g1"| 38431₉ ||class="entry q2 g1"| 57752₇ ||class="entry q3 g1"| 61825₇ ||class="entry q3 g1"| 34327₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (41240, 45059, 37693, 55430, 51339, 33589) ||class="entry q2 g1"| 41240₅ ||class="entry q3 g1"| 45059₅ ||class="entry q3 g1"| 37693₉ ||class="entry q2 g1"| 55430₇ ||class="entry q3 g1"| 51339₇ ||class="entry q3 g1"| 33589₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 9, 7, 7, 7) ||class="c"| (49432, 53253, 38235, 47238, 43149, 34131) ||class="entry q2 g1"| 49432₅ ||class="entry q3 g1"| 53253₅ ||class="entry q3 g1"| 38235₉ ||class="entry q2 g1"| 47238₇ ||class="entry q3 g1"| 43149₇ ||class="entry q3 g1"| 34131₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (33176, 37255, 32881, 63494, 59663, 36985) ||class="entry q2 g1"| 33176₅ ||class="entry q3 g1"| 37255₇ ||class="entry q3 g1"| 32881₅ ||class="entry q2 g1"| 63494₇ ||class="entry q3 g1"| 59663₉ ||class="entry q3 g1"| 36985₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (35096, 38935, 34561, 61574, 57503, 38665) ||class="entry q2 g1"| 35096₅ ||class="entry q3 g1"| 38935₇ ||class="entry q3 g1"| 34561₅ ||class="entry q2 g1"| 61574₇ ||class="entry q3 g1"| 57503₉ ||class="entry q3 g1"| 38665₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (45062, 41245, 33315, 51608, 55701, 37419) ||class="entry q2 g1"| 45062₅ ||class="entry q3 g1"| 41245₇ ||class="entry q3 g1"| 33315₅ ||class="entry q2 g1"| 51608₇ ||class="entry q3 g1"| 55701₉ ||class="entry q3 g1"| 37419₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 5, 7, 9, 7) ||class="c"| (53254, 49435, 33861, 43416, 47507, 37965) ||class="entry q2 g1"| 53254₅ ||class="entry q3 g1"| 49435₇ ||class="entry q3 g1"| 33861₅ ||class="entry q2 g1"| 43416₇ ||class="entry q3 g1"| 47507₉ ||class="entry q3 g1"| 37965₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (37250, 45649, 41635, 59420, 51929, 45739) ||class="entry q2 g1"| 37250₅ ||class="entry q3 g1"| 45649₇ ||class="entry q3 g1"| 41635₇ ||class="entry q2 g1"| 59420₇ ||class="entry q3 g1"| 51929₉ ||class="entry q3 g1"| 45739₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (37252, 54321, 50373, 59418, 44217, 54477) ||class="entry q2 g1"| 37252₅ ||class="entry q3 g1"| 54321₇ ||class="entry q3 g1"| 50373₇ ||class="entry q2 g1"| 59418₇ ||class="entry q3 g1"| 44217₉ ||class="entry q3 g1"| 54477₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (38930, 46369, 43555, 57740, 52649, 47659) ||class="entry q2 g1"| 38930₅ ||class="entry q3 g1"| 46369₇ ||class="entry q3 g1"| 43555₇ ||class="entry q2 g1"| 57740₇ ||class="entry q3 g1"| 52649₉ ||class="entry q3 g1"| 47659₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (38932, 54081, 52293, 57738, 43977, 56397) ||class="entry q2 g1"| 38932₅ ||class="entry q3 g1"| 54081₇ ||class="entry q3 g1"| 52293₇ ||class="entry q2 g1"| 57738₇ ||class="entry q3 g1"| 43977₉ ||class="entry q3 g1"| 56397₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (40988, 33483, 41201, 55682, 64067, 45305) ||class="entry q2 g1"| 40988₅ ||class="entry q3 g1"| 33483₇ ||class="entry q3 g1"| 41201₇ ||class="entry q2 g1"| 55682₇ ||class="entry q3 g1"| 64067₉ ||class="entry q3 g1"| 45305₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41228, 35883, 44801, 55442, 62627, 48905) ||class="entry q2 g1"| 41228₅ ||class="entry q3 g1"| 35883₇ ||class="entry q3 g1"| 44801₇ ||class="entry q2 g1"| 55442₇ ||class="entry q3 g1"| 62627₉ ||class="entry q3 g1"| 48905₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49178, 33965, 49393, 47492, 64549, 53497) ||class="entry q2 g1"| 49178₅ ||class="entry q3 g1"| 33965₇ ||class="entry q3 g1"| 49393₇ ||class="entry q2 g1"| 47492₇ ||class="entry q3 g1"| 64549₉ ||class="entry q3 g1"| 53497₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49418, 35405, 52993, 47252, 62149, 57097) ||class="entry q2 g1"| 49418₅ ||class="entry q3 g1"| 35405₇ ||class="entry q3 g1"| 52993₇ ||class="entry q2 g1"| 47252₇ ||class="entry q3 g1"| 62149₉ ||class="entry q3 g1"| 57097₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (37264, 59417, 63737, 59406, 37009, 59633) ||class="entry q2 g1"| 37264₅ ||class="entry q3 g1"| 59417₇ ||class="entry q3 g1"| 63737₁₁ ||class="entry q2 g1"| 59406₇ ||class="entry q3 g1"| 37009₅ ||class="entry q3 g1"| 59633₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (39184, 57737, 65417, 57486, 39169, 61313) ||class="entry q2 g1"| 39184₅ ||class="entry q3 g1"| 57737₇ ||class="entry q3 g1"| 65417₁₁ ||class="entry q2 g1"| 57486₇ ||class="entry q3 g1"| 39169₅ ||class="entry q3 g1"| 61313₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (40974, 55427, 64171, 55696, 40971, 60067) ||class="entry q2 g1"| 40974₅ ||class="entry q3 g1"| 55427₇ ||class="entry q3 g1"| 64171₁₁ ||class="entry q2 g1"| 55696₇ ||class="entry q3 g1"| 40971₅ ||class="entry q3 g1"| 60067₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (49166, 47237, 64717, 47504, 49165, 60613) ||class="entry q2 g1"| 49166₅ ||class="entry q3 g1"| 47237₇ ||class="entry q3 g1"| 64717₁₁ ||class="entry q2 g1"| 47504₇ ||class="entry q3 g1"| 49165₅ ||class="entry q3 g1"| 60613₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33480, 62641, 58689, 64342, 35897, 62793) ||class="entry q2 g1"| 33480₅ ||class="entry q3 g1"| 62641₉ ||class="entry q3 g1"| 58689₇ ||class="entry q2 g1"| 64342₁₁ ||class="entry q3 g1"| 35897₇ ||class="entry q3 g1"| 62793₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33960, 62161, 58145, 64822, 35417, 62249) ||class="entry q2 g1"| 33960₅ ||class="entry q3 g1"| 62161₉ ||class="entry q3 g1"| 58145₇ ||class="entry q2 g1"| 64822₁₁ ||class="entry q3 g1"| 35417₇ ||class="entry q3 g1"| 62249₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (35400, 64801, 57905, 62422, 34217, 62009) ||class="entry q2 g1"| 35400₅ ||class="entry q3 g1"| 64801₉ ||class="entry q3 g1"| 57905₇ ||class="entry q2 g1"| 62422₁₁ ||class="entry q3 g1"| 34217₇ ||class="entry q3 g1"| 62009₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (35880, 64321, 58449, 62902, 33737, 62553) ||class="entry q2 g1"| 35880₅ ||class="entry q3 g1"| 64321₉ ||class="entry q3 g1"| 58449₇ ||class="entry q2 g1"| 62902₁₁ ||class="entry q3 g1"| 33737₇ ||class="entry q3 g1"| 62553₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (45632, 44203, 36485, 52190, 54307, 40589) ||class="entry q2 g1"| 45632₅ ||class="entry q3 g1"| 44203₉ ||class="entry q3 g1"| 36485₇ ||class="entry q2 g1"| 52190₁₁ ||class="entry q3 g1"| 54307₇ ||class="entry q3 g1"| 40589₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (46112, 43723, 35045, 52670, 53827, 39149) ||class="entry q2 g1"| 46112₅ ||class="entry q3 g1"| 43723₉ ||class="entry q3 g1"| 35045₇ ||class="entry q2 g1"| 52670₁₁ ||class="entry q3 g1"| 53827₇ ||class="entry q3 g1"| 39149₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (53824, 52397, 35043, 43998, 46117, 39147) ||class="entry q2 g1"| 53824₅ ||class="entry q3 g1"| 52397₉ ||class="entry q3 g1"| 35043₇ ||class="entry q2 g1"| 43998₁₁ ||class="entry q3 g1"| 46117₇ ||class="entry q3 g1"| 39147₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (54304, 51917, 36483, 44478, 45637, 40587) ||class="entry q2 g1"| 54304₅ ||class="entry q3 g1"| 51917₉ ||class="entry q3 g1"| 36483₇ ||class="entry q2 g1"| 44478₁₁ ||class="entry q3 g1"| 45637₇ ||class="entry q3 g1"| 40587₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (37010, 48305, 44371, 59660, 50233, 48475) ||class="entry q2 g1"| 37010₅ ||class="entry q3 g1"| 48305₉ ||class="entry q3 g1"| 44371₉ ||class="entry q2 g1"| 59660₇ ||class="entry q3 g1"| 50233₇ ||class="entry q3 g1"| 48475₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (37012, 56017, 52021, 59658, 41561, 56125) ||class="entry q2 g1"| 37012₅ ||class="entry q3 g1"| 56017₉ ||class="entry q3 g1"| 52021₉ ||class="entry q2 g1"| 59658₇ ||class="entry q3 g1"| 41561₇ ||class="entry q3 g1"| 56125₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (39170, 48065, 42451, 57500, 49993, 46555) ||class="entry q2 g1"| 39170₅ ||class="entry q3 g1"| 48065₉ ||class="entry q3 g1"| 42451₉ ||class="entry q2 g1"| 57500₇ ||class="entry q3 g1"| 49993₇ ||class="entry q3 g1"| 46555₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (39172, 56737, 50101, 57498, 42281, 54205) ||class="entry q2 g1"| 39172₅ ||class="entry q3 g1"| 56737₉ ||class="entry q3 g1"| 50101₉ ||class="entry q2 g1"| 57498₇ ||class="entry q3 g1"| 42281₇ ||class="entry q3 g1"| 54205₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (40986, 58539, 50839, 55684, 39971, 54943) ||class="entry q2 g1"| 40986₅ ||class="entry q3 g1"| 58539₉ ||class="entry q3 g1"| 50839₉ ||class="entry q2 g1"| 55684₇ ||class="entry q3 g1"| 39971₇ ||class="entry q3 g1"| 54943₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (41226, 59979, 51559, 55444, 37571, 55663) ||class="entry q2 g1"| 41226₅ ||class="entry q3 g1"| 59979₉ ||class="entry q3 g1"| 51559₉ ||class="entry q2 g1"| 55444₇ ||class="entry q3 g1"| 37571₇ ||class="entry q3 g1"| 55663₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49180, 58061, 42647, 47490, 39493, 46751) ||class="entry q2 g1"| 49180₅ ||class="entry q3 g1"| 58061₉ ||class="entry q3 g1"| 42647₉ ||class="entry q2 g1"| 47490₇ ||class="entry q3 g1"| 39493₇ ||class="entry q3 g1"| 46751₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49420, 60461, 43367, 47250, 38053, 47471) ||class="entry q2 g1"| 49420₅ ||class="entry q3 g1"| 60461₉ ||class="entry q3 g1"| 43367₉ ||class="entry q2 g1"| 47250₇ ||class="entry q3 g1"| 38053₇ ||class="entry q3 g1"| 47471₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (37568, 36143, 40393, 60254, 62887, 36289) ||class="entry q2 g1"| 37568₅ ||class="entry q3 g1"| 36143₉ ||class="entry q3 g1"| 40393₉ ||class="entry q2 g1"| 60254₁₁ ||class="entry q3 g1"| 62887₁₁ ||class="entry q3 g1"| 36289₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38048, 35663, 39849, 60734, 62407, 35745) ||class="entry q2 g1"| 38048₅ ||class="entry q3 g1"| 35663₉ ||class="entry q3 g1"| 39849₉ ||class="entry q2 g1"| 60734₁₁ ||class="entry q3 g1"| 62407₁₁ ||class="entry q3 g1"| 35745₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (39488, 33983, 39609, 58334, 64567, 35505) ||class="entry q2 g1"| 39488₅ ||class="entry q3 g1"| 33983₉ ||class="entry q3 g1"| 39609₉ ||class="entry q2 g1"| 58334₁₁ ||class="entry q3 g1"| 64567₁₁ ||class="entry q3 g1"| 35505₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (39968, 33503, 40153, 58814, 64087, 36049) ||class="entry q2 g1"| 39968₅ ||class="entry q3 g1"| 33503₉ ||class="entry q3 g1"| 40153₉ ||class="entry q2 g1"| 58814₁₁ ||class="entry q3 g1"| 64087₁₁ ||class="entry q3 g1"| 36049₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (41544, 54581, 62989, 56278, 44477, 58885) ||class="entry q2 g1"| 41544₅ ||class="entry q3 g1"| 54581₉ ||class="entry q3 g1"| 62989₉ ||class="entry q2 g1"| 56278₁₁ ||class="entry q3 g1"| 44477₁₁ ||class="entry q3 g1"| 58885₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (42024, 54101, 61549, 56758, 43997, 57445) ||class="entry q2 g1"| 42024₅ ||class="entry q3 g1"| 54101₉ ||class="entry q3 g1"| 61549₉ ||class="entry q2 g1"| 56758₁₁ ||class="entry q3 g1"| 43997₁₁ ||class="entry q3 g1"| 57445₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (49736, 46387, 61547, 48086, 52667, 57443) ||class="entry q2 g1"| 49736₅ ||class="entry q3 g1"| 46387₉ ||class="entry q3 g1"| 61547₉ ||class="entry q2 g1"| 48086₁₁ ||class="entry q3 g1"| 52667₁₁ ||class="entry q3 g1"| 57443₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (50216, 45907, 62987, 48566, 52187, 58883) ||class="entry q2 g1"| 50216₅ ||class="entry q3 g1"| 45907₉ ||class="entry q3 g1"| 62987₉ ||class="entry q2 g1"| 48566₁₁ ||class="entry q3 g1"| 52187₁₁ ||class="entry q3 g1"| 58883₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (32910, 63751, 59879, 63760, 33167, 63983) ||class="entry q2 g1"| 32910₅ ||class="entry q3 g1"| 63751₉ ||class="entry q3 g1"| 59879₁₁ ||class="entry q2 g1"| 63760₇ ||class="entry q3 g1"| 33167₇ ||class="entry q3 g1"| 63983₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (34830, 61591, 61079, 61840, 34847, 65183) ||class="entry q2 g1"| 34830₅ ||class="entry q3 g1"| 61591₉ ||class="entry q3 g1"| 61079₁₁ ||class="entry q2 g1"| 61840₇ ||class="entry q3 g1"| 34847₇ ||class="entry q3 g1"| 65183₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (45328, 51613, 60341, 51342, 45333, 64445) ||class="entry q2 g1"| 45328₅ ||class="entry q3 g1"| 51613₉ ||class="entry q3 g1"| 60341₁₁ ||class="entry q2 g1"| 51342₇ ||class="entry q3 g1"| 45333₇ ||class="entry q3 g1"| 64445₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (53520, 43419, 60883, 43150, 53523, 64987) ||class="entry q2 g1"| 53520₅ ||class="entry q3 g1"| 43419₉ ||class="entry q3 g1"| 60883₁₁ ||class="entry q2 g1"| 43150₇ ||class="entry q3 g1"| 53523₇ ||class="entry q3 g1"| 64987₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32922, 50479, 54747, 63748, 48551, 50643) ||class="entry q2 g1"| 32922₅ ||class="entry q3 g1"| 50479₉ ||class="entry q3 g1"| 54747₁₁ ||class="entry q2 g1"| 63748₇ ||class="entry q3 g1"| 48551₁₁ ||class="entry q3 g1"| 50643₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32924, 41807, 46013, 63746, 56263, 41909) ||class="entry q2 g1"| 32924₅ ||class="entry q3 g1"| 41807₉ ||class="entry q3 g1"| 46013₁₁ ||class="entry q2 g1"| 63746₇ ||class="entry q3 g1"| 56263₁₁ ||class="entry q3 g1"| 41909₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35082, 49759, 56667, 61588, 47831, 52563) ||class="entry q2 g1"| 35082₅ ||class="entry q3 g1"| 49759₉ ||class="entry q3 g1"| 56667₁₁ ||class="entry q2 g1"| 61588₇ ||class="entry q3 g1"| 47831₁₁ ||class="entry q3 g1"| 52563₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35084, 42047, 47933, 61586, 56503, 43829) ||class="entry q2 g1"| 35084₅ ||class="entry q3 g1"| 42047₉ ||class="entry q3 g1"| 47933₁₁ ||class="entry q2 g1"| 61586₇ ||class="entry q3 g1"| 56503₁₁ ||class="entry q3 g1"| 43829₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (45074, 40245, 48671, 51596, 58813, 44567) ||class="entry q2 g1"| 45074₅ ||class="entry q3 g1"| 40245₉ ||class="entry q3 g1"| 48671₁₁ ||class="entry q2 g1"| 51596₇ ||class="entry q3 g1"| 58813₁₁ ||class="entry q3 g1"| 44567₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (45314, 37845, 45551, 51356, 60253, 41447) ||class="entry q2 g1"| 45314₅ ||class="entry q3 g1"| 37845₉ ||class="entry q3 g1"| 45551₁₁ ||class="entry q2 g1"| 51356₇ ||class="entry q3 g1"| 60253₁₁ ||class="entry q3 g1"| 41447₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (53268, 39763, 56863, 43402, 58331, 52759) ||class="entry q2 g1"| 53268₅ ||class="entry q3 g1"| 39763₉ ||class="entry q3 g1"| 56863₁₁ ||class="entry q2 g1"| 43402₇ ||class="entry q3 g1"| 58331₁₁ ||class="entry q3 g1"| 52759₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (53508, 38323, 53743, 43162, 60731, 49639) ||class="entry q2 g1"| 53508₅ ||class="entry q3 g1"| 38323₉ ||class="entry q3 g1"| 53743₁₁ ||class="entry q2 g1"| 43162₇ ||class="entry q3 g1"| 60731₁₁ ||class="entry q3 g1"| 49639₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33162, 52175, 55851, 63508, 45895, 51747) ||class="entry q2 g1"| 33162₅ ||class="entry q3 g1"| 52175₁₁ ||class="entry q3 g1"| 55851₉ ||class="entry q2 g1"| 63508₇ ||class="entry q3 g1"| 45895₉ ||class="entry q3 g1"| 51747₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33164, 44463, 48205, 63506, 54567, 44101) ||class="entry q2 g1"| 33164₅ ||class="entry q3 g1"| 44463₁₁ ||class="entry q3 g1"| 48205₉ ||class="entry q2 g1"| 63506₇ ||class="entry q3 g1"| 54567₉ ||class="entry q3 g1"| 44101₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (34842, 52415, 53931, 61828, 46135, 49827) ||class="entry q2 g1"| 34842₅ ||class="entry q3 g1"| 52415₁₁ ||class="entry q3 g1"| 53931₉ ||class="entry q2 g1"| 61828₇ ||class="entry q3 g1"| 46135₉ ||class="entry q3 g1"| 49827₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (34844, 43743, 46285, 61826, 53847, 42181) ||class="entry q2 g1"| 34844₅ ||class="entry q3 g1"| 43743₁₁ ||class="entry q3 g1"| 46285₉ ||class="entry q2 g1"| 61826₇ ||class="entry q3 g1"| 53847₉ ||class="entry q3 g1"| 42181₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (45076, 64341, 55417, 51594, 33757, 51313) ||class="entry q2 g1"| 45076₅ ||class="entry q3 g1"| 64341₁₁ ||class="entry q3 g1"| 55417₉ ||class="entry q2 g1"| 51594₇ ||class="entry q3 g1"| 33757₉ ||class="entry q3 g1"| 51313₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (45316, 62901, 55177, 51354, 36157, 51073) ||class="entry q2 g1"| 45316₅ ||class="entry q3 g1"| 62901₁₁ ||class="entry q3 g1"| 55177₉ ||class="entry q2 g1"| 51354₇ ||class="entry q3 g1"| 36157₉ ||class="entry q3 g1"| 51073₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (53266, 64819, 47225, 43404, 34235, 43121) ||class="entry q2 g1"| 53266₅ ||class="entry q3 g1"| 64819₁₁ ||class="entry q3 g1"| 47225₉ ||class="entry q2 g1"| 43404₇ ||class="entry q3 g1"| 34235₉ ||class="entry q3 g1"| 43121₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (53506, 62419, 46985, 43164, 35675, 42881) ||class="entry q2 g1"| 53506₅ ||class="entry q3 g1"| 62419₁₁ ||class="entry q3 g1"| 46985₉ ||class="entry q2 g1"| 43164₇ ||class="entry q3 g1"| 35675₉ ||class="entry q3 g1"| 42881₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (47232, 52973, 58165, 49438, 46693, 62269) ||class="entry q2 g1"| 47232₅ ||class="entry q3 g1"| 52973₁₁ ||class="entry q3 g1"| 58165₉ ||class="entry q2 g1"| 49438₇ ||class="entry q3 g1"| 46693₉ ||class="entry q3 g1"| 62269₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (55424, 44779, 58707, 41246, 54883, 62811) ||class="entry q2 g1"| 55424₅ ||class="entry q3 g1"| 44779₁₁ ||class="entry q3 g1"| 58707₉ ||class="entry q2 g1"| 41246₇ ||class="entry q3 g1"| 54883₉ ||class="entry q3 g1"| 62811₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (57480, 65377, 35303, 39190, 34793, 39407) ||class="entry q2 g1"| 57480₅ ||class="entry q3 g1"| 65377₁₁ ||class="entry q3 g1"| 35303₉ ||class="entry q2 g1"| 39190₇ ||class="entry q3 g1"| 34793₉ ||class="entry q3 g1"| 39407₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (59400, 63217, 36503, 37270, 36473, 40607) ||class="entry q2 g1"| 59400₅ ||class="entry q3 g1"| 63217₁₁ ||class="entry q3 g1"| 36503₉ ||class="entry q2 g1"| 37270₇ ||class="entry q3 g1"| 36473₉ ||class="entry q3 g1"| 40607₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (43144, 46963, 39869, 53526, 53243, 35765) ||class="entry q2 g1"| 43144₅ ||class="entry q3 g1"| 46963₁₁ ||class="entry q3 g1"| 39869₁₁ ||class="entry q2 g1"| 53526₇ ||class="entry q3 g1"| 53243₁₃ ||class="entry q3 g1"| 35765₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (51336, 55157, 40411, 45334, 45053, 36307) ||class="entry q2 g1"| 51336₅ ||class="entry q3 g1"| 55157₁₁ ||class="entry q3 g1"| 40411₁₁ ||class="entry q2 g1"| 45334₇ ||class="entry q3 g1"| 45053₁₃ ||class="entry q3 g1"| 36307₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (61568, 34559, 61807, 35102, 65143, 57703) ||class="entry q2 g1"| 61568₅ ||class="entry q3 g1"| 34559₁₁ ||class="entry q3 g1"| 61807₁₁ ||class="entry q2 g1"| 35102₇ ||class="entry q3 g1"| 65143₁₃ ||class="entry q3 g1"| 57703₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (63488, 36719, 63007, 33182, 63463, 58903) ||class="entry q2 g1"| 63488₅ ||class="entry q3 g1"| 36719₁₁ ||class="entry q3 g1"| 63007₁₁ ||class="entry q2 g1"| 33182₇ ||class="entry q3 g1"| 63463₁₃ ||class="entry q3 g1"| 58903₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (33738, 40985, 45291, 64084, 55441, 41187) ||class="entry q2 g1"| 33738₇ ||class="entry q3 g1"| 40985₅ ||class="entry q3 g1"| 45291₉ ||class="entry q2 g1"| 64084₉ ||class="entry q3 g1"| 55441₇ ||class="entry q3 g1"| 41187₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (34220, 49177, 53485, 64562, 47249, 49381) ||class="entry q2 g1"| 34220₇ ||class="entry q3 g1"| 49177₅ ||class="entry q3 g1"| 53485₉ ||class="entry q2 g1"| 64562₉ ||class="entry q3 g1"| 47249₇ ||class="entry q3 g1"| 49381₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (35420, 49417, 56845, 62402, 47489, 52741) ||class="entry q2 g1"| 35420₇ ||class="entry q3 g1"| 49417₅ ||class="entry q3 g1"| 56845₉ ||class="entry q2 g1"| 62402₉ ||class="entry q3 g1"| 47489₇ ||class="entry q3 g1"| 52741₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (35898, 41225, 48651, 62884, 55681, 44547) ||class="entry q2 g1"| 35898₇ ||class="entry q3 g1"| 41225₅ ||class="entry q3 g1"| 48651₉ ||class="entry q2 g1"| 62884₉ ||class="entry q3 g1"| 55681₇ ||class="entry q3 g1"| 44547₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (45652, 36995, 45753, 52170, 59403, 41649) ||class="entry q2 g1"| 45652₇ ||class="entry q3 g1"| 36995₅ ||class="entry q3 g1"| 45753₉ ||class="entry q2 g1"| 52170₉ ||class="entry q3 g1"| 59403₇ ||class="entry q3 g1"| 41649₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (46372, 38915, 47913, 52410, 57483, 43809) ||class="entry q2 g1"| 46372₇ ||class="entry q3 g1"| 38915₅ ||class="entry q3 g1"| 47913₉ ||class="entry q2 g1"| 52410₉ ||class="entry q3 g1"| 57483₇ ||class="entry q3 g1"| 43809₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (54082, 38917, 56649, 43740, 57485, 52545) ||class="entry q2 g1"| 54082₇ ||class="entry q3 g1"| 38917₅ ||class="entry q3 g1"| 56649₉ ||class="entry q2 g1"| 43740₉ ||class="entry q3 g1"| 57485₇ ||class="entry q3 g1"| 52545₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (54322, 36997, 54489, 44460, 59405, 50385) ||class="entry q2 g1"| 54322₇ ||class="entry q3 g1"| 36997₅ ||class="entry q3 g1"| 54489₉ ||class="entry q2 g1"| 44460₉ ||class="entry q3 g1"| 59405₇ ||class="entry q3 g1"| 50385₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33486, 37585, 33575, 64336, 59993, 37679) ||class="entry q2 g1"| 33486₇ ||class="entry q3 g1"| 37585₇ ||class="entry q3 g1"| 33575₇ ||class="entry q2 g1"| 64336₉ ||class="entry q3 g1"| 59993₉ ||class="entry q3 g1"| 37679₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33966, 38065, 34119, 64816, 60473, 38223) ||class="entry q2 g1"| 33966₇ ||class="entry q3 g1"| 38065₇ ||class="entry q3 g1"| 34119₇ ||class="entry q2 g1"| 64816₉ ||class="entry q3 g1"| 60473₉ ||class="entry q3 g1"| 38223₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (35406, 39745, 33879, 62416, 58313, 37983) ||class="entry q2 g1"| 35406₇ ||class="entry q3 g1"| 39745₇ ||class="entry q3 g1"| 33879₇ ||class="entry q2 g1"| 62416₉ ||class="entry q3 g1"| 58313₉ ||class="entry q3 g1"| 37983₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (35886, 40225, 33335, 62896, 58793, 37439) ||class="entry q2 g1"| 35886₇ ||class="entry q3 g1"| 40225₇ ||class="entry q3 g1"| 33335₇ ||class="entry q2 g1"| 62896₉ ||class="entry q3 g1"| 58793₉ ||class="entry q3 g1"| 37439₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45904, 41547, 33141, 51918, 56003, 37245) ||class="entry q2 g1"| 45904₇ ||class="entry q3 g1"| 41547₇ ||class="entry q3 g1"| 33141₇ ||class="entry q2 g1"| 51918₉ ||class="entry q3 g1"| 56003₉ ||class="entry q3 g1"| 37245₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (46384, 42027, 34581, 52398, 56483, 38685) ||class="entry q2 g1"| 46384₇ ||class="entry q3 g1"| 42027₇ ||class="entry q3 g1"| 34581₇ ||class="entry q2 g1"| 52398₉ ||class="entry q3 g1"| 56483₉ ||class="entry q3 g1"| 38685₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54096, 49741, 34579, 43726, 47813, 38683) ||class="entry q2 g1"| 54096₇ ||class="entry q3 g1"| 49741₇ ||class="entry q3 g1"| 34579₇ ||class="entry q2 g1"| 43726₉ ||class="entry q3 g1"| 47813₉ ||class="entry q3 g1"| 38683₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54576, 50221, 33139, 44206, 48293, 37243) ||class="entry q2 g1"| 54576₇ ||class="entry q3 g1"| 50221₇ ||class="entry q3 g1"| 33139₇ ||class="entry q2 g1"| 44206₉ ||class="entry q3 g1"| 48293₉ ||class="entry q3 g1"| 37243₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (34536, 39175, 35297, 65398, 57743, 39401) ||class="entry q2 g1"| 34536₇ ||class="entry q3 g1"| 39175₇ ||class="entry q3 g1"| 35297₇ ||class="entry q2 g1"| 65398₁₃ ||class="entry q3 g1"| 57743₉ ||class="entry q3 g1"| 39401₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (36456, 37015, 36497, 63478, 59423, 40601) ||class="entry q2 g1"| 36456₇ ||class="entry q3 g1"| 37015₇ ||class="entry q3 g1"| 36497₇ ||class="entry q2 g1"| 63478₁₃ ||class="entry q3 g1"| 59423₉ ||class="entry q3 g1"| 40601₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (46688, 49437, 57893, 53246, 47509, 61997) ||class="entry q2 g1"| 46688₇ ||class="entry q3 g1"| 49437₇ ||class="entry q3 g1"| 57893₇ ||class="entry q2 g1"| 53246₁₃ ||class="entry q3 g1"| 47509₉ ||class="entry q3 g1"| 61997₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (54880, 41243, 58435, 45054, 55699, 62539) ||class="entry q2 g1"| 54880₇ ||class="entry q3 g1"| 41243₇ ||class="entry q3 g1"| 58435₇ ||class="entry q2 g1"| 45054₁₃ ||class="entry q3 g1"| 55699₉ ||class="entry q3 g1"| 62539₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (37588, 45319, 41461, 60234, 51599, 45565) ||class="entry q2 g1"| 37588₇ ||class="entry q3 g1"| 45319₇ ||class="entry q3 g1"| 41461₉ ||class="entry q2 g1"| 60234₉ ||class="entry q3 g1"| 51599₉ ||class="entry q3 g1"| 45565₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38066, 53511, 49651, 60716, 43407, 53755) ||class="entry q2 g1"| 38066₇ ||class="entry q3 g1"| 53511₇ ||class="entry q3 g1"| 49651₉ ||class="entry q2 g1"| 60716₉ ||class="entry q3 g1"| 43407₉ ||class="entry q3 g1"| 53755₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (39746, 53271, 53011, 58076, 43167, 57115) ||class="entry q2 g1"| 39746₇ ||class="entry q3 g1"| 53271₇ ||class="entry q3 g1"| 53011₉ ||class="entry q2 g1"| 58076₉ ||class="entry q3 g1"| 43167₉ ||class="entry q3 g1"| 57115₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (40228, 45079, 44821, 58554, 51359, 48925) ||class="entry q2 g1"| 40228₇ ||class="entry q3 g1"| 45079₇ ||class="entry q3 g1"| 44821₉ ||class="entry q2 g1"| 58554₉ ||class="entry q3 g1"| 51359₉ ||class="entry q3 g1"| 48925₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (41802, 33181, 41895, 56020, 63765, 45999) ||class="entry q2 g1"| 41802₇ ||class="entry q3 g1"| 33181₇ ||class="entry q3 g1"| 41895₉ ||class="entry q2 g1"| 56020₉ ||class="entry q3 g1"| 63765₉ ||class="entry q3 g1"| 45999₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (42042, 35101, 43575, 56740, 61845, 47679) ||class="entry q2 g1"| 42042₇ ||class="entry q3 g1"| 35101₇ ||class="entry q3 g1"| 43575₉ ||class="entry q2 g1"| 56740₉ ||class="entry q3 g1"| 61845₉ ||class="entry q3 g1"| 47679₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (49756, 35099, 52311, 48066, 61843, 56415) ||class="entry q2 g1"| 49756₇ ||class="entry q3 g1"| 35099₇ ||class="entry q3 g1"| 52311₉ ||class="entry q2 g1"| 48066₉ ||class="entry q3 g1"| 61843₉ ||class="entry q3 g1"| 56415₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (50476, 33179, 50631, 48306, 63763, 54735) ||class="entry q2 g1"| 50476₇ ||class="entry q3 g1"| 33179₇ ||class="entry q3 g1"| 50631₉ ||class="entry q2 g1"| 48306₉ ||class="entry q3 g1"| 63763₉ ||class="entry q3 g1"| 54735₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (38624, 57497, 61801, 61310, 38929, 57697) ||class="entry q2 g1"| 38624₇ ||class="entry q3 g1"| 57497₇ ||class="entry q3 g1"| 61801₉ ||class="entry q2 g1"| 61310₁₃ ||class="entry q3 g1"| 38929₅ ||class="entry q3 g1"| 57697₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (40544, 59657, 63001, 59390, 37249, 58897) ||class="entry q2 g1"| 40544₇ ||class="entry q3 g1"| 59657₇ ||class="entry q3 g1"| 63001₉ ||class="entry q2 g1"| 59390₁₃ ||class="entry q3 g1"| 37249₅ ||class="entry q3 g1"| 58897₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (42600, 47235, 39597, 57334, 49163, 35493) ||class="entry q2 g1"| 42600₇ ||class="entry q3 g1"| 47235₇ ||class="entry q3 g1"| 39597₉ ||class="entry q2 g1"| 57334₁₃ ||class="entry q3 g1"| 49163₅ ||class="entry q3 g1"| 35493₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (50792, 55429, 40139, 49142, 40973, 36035) ||class="entry q2 g1"| 50792₇ ||class="entry q3 g1"| 55429₇ ||class="entry q3 g1"| 40139₉ ||class="entry q2 g1"| 49142₁₃ ||class="entry q3 g1"| 40973₅ ||class="entry q3 g1"| 36035₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (33500, 51353, 55677, 64322, 45073, 51573) ||class="entry q2 g1"| 33500₇ ||class="entry q3 g1"| 51353₇ ||class="entry q3 g1"| 55677₁₁ ||class="entry q2 g1"| 64322₉ ||class="entry q3 g1"| 45073₅ ||class="entry q3 g1"| 51573₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (33978, 43161, 47483, 64804, 53265, 43379) ||class="entry q2 g1"| 33978₇ ||class="entry q3 g1"| 43161₇ ||class="entry q3 g1"| 47483₁₁ ||class="entry q2 g1"| 64804₉ ||class="entry q3 g1"| 53265₅ ||class="entry q3 g1"| 43379₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (35658, 43401, 47003, 62164, 53505, 42899) ||class="entry q2 g1"| 35658₇ ||class="entry q3 g1"| 43401₇ ||class="entry q3 g1"| 47003₁₁ ||class="entry q2 g1"| 62164₉ ||class="entry q3 g1"| 53505₅ ||class="entry q3 g1"| 42899₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (36140, 51593, 55197, 62642, 45313, 51093) ||class="entry q2 g1"| 36140₇ ||class="entry q3 g1"| 51593₇ ||class="entry q3 g1"| 55197₁₁ ||class="entry q2 g1"| 62642₉ ||class="entry q3 g1"| 45313₅ ||class="entry q3 g1"| 51093₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (45890, 63491, 56111, 51932, 32907, 52007) ||class="entry q2 g1"| 45890₇ ||class="entry q3 g1"| 63491₇ ||class="entry q3 g1"| 56111₁₁ ||class="entry q2 g1"| 51932₉ ||class="entry q3 g1"| 32907₅ ||class="entry q3 g1"| 52007₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (46130, 61571, 53951, 52652, 34827, 49847) ||class="entry q2 g1"| 46130₇ ||class="entry q3 g1"| 61571₇ ||class="entry q3 g1"| 53951₁₁ ||class="entry q2 g1"| 52652₉ ||class="entry q3 g1"| 34827₅ ||class="entry q3 g1"| 49847₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (53844, 61573, 46303, 43978, 34829, 42199) ||class="entry q2 g1"| 53844₇ ||class="entry q3 g1"| 61573₇ ||class="entry q3 g1"| 46303₁₁ ||class="entry q2 g1"| 43978₉ ||class="entry q3 g1"| 34829₅ ||class="entry q3 g1"| 42199₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (54564, 63493, 48463, 44218, 32909, 44359) ||class="entry q2 g1"| 54564₇ ||class="entry q3 g1"| 63493₇ ||class="entry q3 g1"| 48463₁₁ ||class="entry q2 g1"| 44218₉ ||class="entry q3 g1"| 32909₅ ||class="entry q3 g1"| 44359₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (37826, 55687, 51299, 59996, 41231, 55403) ||class="entry q2 g1"| 37826₇ ||class="entry q3 g1"| 55687₉ ||class="entry q3 g1"| 51299₇ ||class="entry q2 g1"| 59996₉ ||class="entry q3 g1"| 41231₇ ||class="entry q3 g1"| 55403₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (38308, 47495, 43109, 60474, 49423, 47213) ||class="entry q2 g1"| 38308₇ ||class="entry q3 g1"| 47495₉ ||class="entry q3 g1"| 43109₇ ||class="entry q2 g1"| 60474₉ ||class="entry q3 g1"| 49423₇ ||class="entry q3 g1"| 47213₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39508, 47255, 42629, 58314, 49183, 46733) ||class="entry q2 g1"| 39508₇ ||class="entry q3 g1"| 47255₉ ||class="entry q3 g1"| 42629₇ ||class="entry q2 g1"| 58314₉ ||class="entry q3 g1"| 49183₇ ||class="entry q3 g1"| 46733₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39986, 55447, 50819, 58796, 40991, 54923) ||class="entry q2 g1"| 39986₇ ||class="entry q3 g1"| 55447₉ ||class="entry q3 g1"| 50819₇ ||class="entry q2 g1"| 58796₉ ||class="entry q3 g1"| 40991₇ ||class="entry q3 g1"| 54923₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (41564, 59677, 51761, 56258, 37269, 55865) ||class="entry q2 g1"| 41564₇ ||class="entry q3 g1"| 59677₉ ||class="entry q3 g1"| 51761₇ ||class="entry q2 g1"| 56258₉ ||class="entry q3 g1"| 37269₇ ||class="entry q3 g1"| 55865₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42284, 57757, 50081, 56498, 39189, 54185) ||class="entry q2 g1"| 42284₇ ||class="entry q3 g1"| 57757₉ ||class="entry q3 g1"| 50081₇ ||class="entry q2 g1"| 56498₉ ||class="entry q3 g1"| 39189₇ ||class="entry q3 g1"| 54185₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (49994, 57755, 42433, 47828, 39187, 46537) ||class="entry q2 g1"| 49994₇ ||class="entry q3 g1"| 57755₉ ||class="entry q3 g1"| 42433₇ ||class="entry q2 g1"| 47828₉ ||class="entry q3 g1"| 39187₇ ||class="entry q3 g1"| 46537₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (50234, 59675, 44113, 48548, 37267, 48217) ||class="entry q2 g1"| 50234₇ ||class="entry q3 g1"| 59675₉ ||class="entry q3 g1"| 44113₇ ||class="entry q2 g1"| 48548₉ ||class="entry q3 g1"| 37267₇ ||class="entry q3 g1"| 48217₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (37840, 33743, 37433, 59982, 64327, 33329) ||class="entry q2 g1"| 37840₇ ||class="entry q3 g1"| 33743₉ ||class="entry q3 g1"| 37433₇ ||class="entry q2 g1"| 59982₉ ||class="entry q3 g1"| 64327₁₁ ||class="entry q3 g1"| 33329₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38320, 34223, 37977, 60462, 64807, 33873) ||class="entry q2 g1"| 38320₇ ||class="entry q3 g1"| 34223₉ ||class="entry q3 g1"| 37977₇ ||class="entry q2 g1"| 60462₉ ||class="entry q3 g1"| 64807₁₁ ||class="entry q3 g1"| 33873₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (39760, 35423, 38217, 58062, 62167, 34113) ||class="entry q2 g1"| 39760₇ ||class="entry q3 g1"| 35423₉ ||class="entry q3 g1"| 38217₇ ||class="entry q2 g1"| 58062₉ ||class="entry q3 g1"| 62167₁₁ ||class="entry q3 g1"| 34113₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (40240, 35903, 37673, 58542, 62647, 33569) ||class="entry q2 g1"| 40240₇ ||class="entry q3 g1"| 35903₉ ||class="entry q3 g1"| 37673₇ ||class="entry q2 g1"| 58542₉ ||class="entry q3 g1"| 62647₁₁ ||class="entry q3 g1"| 33569₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (41550, 45909, 36971, 56272, 52189, 32867) ||class="entry q2 g1"| 41550₇ ||class="entry q3 g1"| 45909₉ ||class="entry q3 g1"| 36971₇ ||class="entry q2 g1"| 56272₉ ||class="entry q3 g1"| 52189₁₁ ||class="entry q3 g1"| 32867₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (42030, 46389, 38411, 56752, 52669, 34307) ||class="entry q2 g1"| 42030₇ ||class="entry q3 g1"| 46389₉ ||class="entry q3 g1"| 38411₇ ||class="entry q2 g1"| 56752₉ ||class="entry q3 g1"| 52669₁₁ ||class="entry q3 g1"| 34307₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (49742, 54099, 38413, 48080, 43995, 34309) ||class="entry q2 g1"| 49742₇ ||class="entry q3 g1"| 54099₉ ||class="entry q3 g1"| 38413₇ ||class="entry q2 g1"| 48080₉ ||class="entry q3 g1"| 43995₁₁ ||class="entry q3 g1"| 34309₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (50222, 54579, 36973, 48560, 44475, 32869) ||class="entry q2 g1"| 50222₇ ||class="entry q3 g1"| 54579₉ ||class="entry q3 g1"| 36973₇ ||class="entry q2 g1"| 48560₉ ||class="entry q3 g1"| 44475₁₁ ||class="entry q3 g1"| 32869₅ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (33752, 64081, 60081, 64070, 33497, 64185) ||class="entry q2 g1"| 33752₇ ||class="entry q3 g1"| 64081₉ ||class="entry q3 g1"| 60081₉ ||class="entry q2 g1"| 64070₉ ||class="entry q3 g1"| 33497₇ ||class="entry q3 g1"| 64185₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (34232, 64561, 60625, 64550, 33977, 64729) ||class="entry q2 g1"| 34232₇ ||class="entry q3 g1"| 64561₉ ||class="entry q3 g1"| 60625₉ ||class="entry q2 g1"| 64550₉ ||class="entry q3 g1"| 33977₇ ||class="entry q3 g1"| 64729₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (35672, 62401, 60865, 62150, 35657, 64969) ||class="entry q2 g1"| 35672₇ ||class="entry q3 g1"| 62401₉ ||class="entry q3 g1"| 60865₉ ||class="entry q2 g1"| 62150₉ ||class="entry q3 g1"| 35657₇ ||class="entry q3 g1"| 64969₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (36152, 62881, 60321, 62630, 36137, 64425) ||class="entry q2 g1"| 36152₇ ||class="entry q3 g1"| 62881₉ ||class="entry q3 g1"| 60321₉ ||class="entry q2 g1"| 62630₉ ||class="entry q3 g1"| 36137₇ ||class="entry q3 g1"| 64425₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (45638, 51915, 59619, 52184, 45635, 63723) ||class="entry q2 g1"| 45638₇ ||class="entry q3 g1"| 51915₉ ||class="entry q3 g1"| 59619₉ ||class="entry q2 g1"| 52184₉ ||class="entry q3 g1"| 45635₇ ||class="entry q3 g1"| 63723₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (46118, 52395, 61059, 52664, 46115, 65163) ||class="entry q2 g1"| 46118₇ ||class="entry q3 g1"| 52395₉ ||class="entry q3 g1"| 61059₉ ||class="entry q2 g1"| 52664₉ ||class="entry q3 g1"| 46115₇ ||class="entry q3 g1"| 65163₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (53830, 43725, 61061, 43992, 53829, 65165) ||class="entry q2 g1"| 53830₇ ||class="entry q3 g1"| 43725₉ ||class="entry q3 g1"| 61061₉ ||class="entry q2 g1"| 43992₉ ||class="entry q3 g1"| 53829₇ ||class="entry q3 g1"| 65165₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (54310, 44205, 59621, 44472, 54309, 63725) ||class="entry q2 g1"| 54310₇ ||class="entry q3 g1"| 44205₉ ||class="entry q3 g1"| 59621₉ ||class="entry q2 g1"| 44472₉ ||class="entry q3 g1"| 54309₇ ||class="entry q3 g1"| 63725₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (33740, 50809, 54925, 64082, 48881, 50821) ||class="entry q2 g1"| 33740₇ ||class="entry q3 g1"| 50809₉ ||class="entry q3 g1"| 54925₉ ||class="entry q2 g1"| 64082₉ ||class="entry q3 g1"| 48881₁₁ ||class="entry q3 g1"| 50821₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (34218, 42617, 46731, 64564, 57073, 42627) ||class="entry q2 g1"| 34218₇ ||class="entry q3 g1"| 42617₉ ||class="entry q3 g1"| 46731₉ ||class="entry q2 g1"| 64564₉ ||class="entry q3 g1"| 57073₁₁ ||class="entry q3 g1"| 42627₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (35418, 42857, 47211, 62404, 57313, 43107) ||class="entry q2 g1"| 35418₇ ||class="entry q3 g1"| 42857₉ ||class="entry q3 g1"| 47211₉ ||class="entry q2 g1"| 62404₉ ||class="entry q3 g1"| 57313₁₁ ||class="entry q3 g1"| 43107₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (35900, 51049, 55405, 62882, 49121, 51301) ||class="entry q2 g1"| 35900₇ ||class="entry q3 g1"| 51049₉ ||class="entry q3 g1"| 55405₉ ||class="entry q2 g1"| 62882₉ ||class="entry q3 g1"| 49121₁₁ ||class="entry q3 g1"| 51301₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (45892, 40547, 48457, 51930, 59115, 44353) ||class="entry q2 g1"| 45892₇ ||class="entry q3 g1"| 40547₉ ||class="entry q3 g1"| 48457₉ ||class="entry q2 g1"| 51930₉ ||class="entry q3 g1"| 59115₁₁ ||class="entry q3 g1"| 44353₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (46132, 38627, 46297, 52650, 61035, 42193) ||class="entry q2 g1"| 46132₇ ||class="entry q3 g1"| 38627₉ ||class="entry q3 g1"| 46297₉ ||class="entry q2 g1"| 52650₉ ||class="entry q3 g1"| 61035₁₁ ||class="entry q3 g1"| 42193₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (53842, 38629, 53945, 43980, 61037, 49841) ||class="entry q2 g1"| 53842₇ ||class="entry q3 g1"| 38629₉ ||class="entry q3 g1"| 53945₉ ||class="entry q2 g1"| 43980₉ ||class="entry q3 g1"| 61037₁₁ ||class="entry q3 g1"| 49841₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (54562, 40549, 56105, 44220, 59117, 52001) ||class="entry q2 g1"| 54562₇ ||class="entry q3 g1"| 40549₉ ||class="entry q3 g1"| 56105₉ ||class="entry q2 g1"| 44220₉ ||class="entry q3 g1"| 59117₁₁ ||class="entry q3 g1"| 52001₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (47808, 42299, 35317, 50014, 56755, 39421) ||class="entry q2 g1"| 47808₇ ||class="entry q3 g1"| 42299₉ ||class="entry q3 g1"| 35317₉ ||class="entry q2 g1"| 50014₉ ||class="entry q3 g1"| 56755₁₁ ||class="entry q3 g1"| 39421₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (48288, 41819, 36757, 50494, 56275, 40861) ||class="entry q2 g1"| 48288₇ ||class="entry q3 g1"| 41819₉ ||class="entry q3 g1"| 36757₉ ||class="entry q2 g1"| 50494₉ ||class="entry q3 g1"| 56275₁₁ ||class="entry q3 g1"| 40861₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (56000, 50493, 36755, 41822, 48565, 40859) ||class="entry q2 g1"| 56000₇ ||class="entry q3 g1"| 50493₉ ||class="entry q3 g1"| 36755₉ ||class="entry q2 g1"| 41822₉ ||class="entry q3 g1"| 48565₁₁ ||class="entry q3 g1"| 40859₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (56480, 50013, 35315, 42302, 48085, 39419) ||class="entry q2 g1"| 56480₇ ||class="entry q3 g1"| 50013₉ ||class="entry q3 g1"| 35315₉ ||class="entry q2 g1"| 42302₉ ||class="entry q3 g1"| 48085₁₁ ||class="entry q3 g1"| 39419₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (58056, 38071, 58151, 39766, 60479, 62255) ||class="entry q2 g1"| 58056₇ ||class="entry q3 g1"| 38071₉ ||class="entry q3 g1"| 58151₉ ||class="entry q2 g1"| 39766₉ ||class="entry q3 g1"| 60479₁₁ ||class="entry q3 g1"| 62255₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (58536, 37591, 58695, 40246, 59999, 62799) ||class="entry q2 g1"| 58536₇ ||class="entry q3 g1"| 37591₉ ||class="entry q3 g1"| 58695₉ ||class="entry q2 g1"| 40246₉ ||class="entry q3 g1"| 59999₁₁ ||class="entry q3 g1"| 62799₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (59976, 40231, 58455, 37846, 58799, 62559) ||class="entry q2 g1"| 59976₇ ||class="entry q3 g1"| 40231₉ ||class="entry q3 g1"| 58455₉ ||class="entry q2 g1"| 37846₉ ||class="entry q3 g1"| 58799₁₁ ||class="entry q3 g1"| 62559₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (60456, 39751, 57911, 38326, 58319, 62015) ||class="entry q2 g1"| 60456₇ ||class="entry q3 g1"| 39751₉ ||class="entry q3 g1"| 57911₉ ||class="entry q2 g1"| 38326₉ ||class="entry q3 g1"| 58319₁₁ ||class="entry q3 g1"| 62015₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (43720, 56485, 61821, 54102, 42029, 57717) ||class="entry q2 g1"| 43720₇ ||class="entry q3 g1"| 56485₉ ||class="entry q3 g1"| 61821₁₁ ||class="entry q2 g1"| 54102₉ ||class="entry q3 g1"| 42029₇ ||class="entry q3 g1"| 57717₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (44200, 56005, 63261, 54582, 41549, 59157) ||class="entry q2 g1"| 44200₇ ||class="entry q3 g1"| 56005₉ ||class="entry q3 g1"| 63261₁₁ ||class="entry q2 g1"| 54582₉ ||class="entry q3 g1"| 41549₇ ||class="entry q3 g1"| 59157₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (51912, 48291, 63259, 45910, 50219, 59155) ||class="entry q2 g1"| 51912₇ ||class="entry q3 g1"| 48291₉ ||class="entry q3 g1"| 63259₁₁ ||class="entry q2 g1"| 45910₉ ||class="entry q3 g1"| 50219₇ ||class="entry q3 g1"| 59155₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (52392, 47811, 61819, 46390, 49739, 57715) ||class="entry q2 g1"| 52392₇ ||class="entry q3 g1"| 47811₉ ||class="entry q3 g1"| 61819₁₁ ||class="entry q2 g1"| 46390₉ ||class="entry q3 g1"| 49739₇ ||class="entry q3 g1"| 57715₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (62144, 60713, 39855, 35678, 38305, 35751) ||class="entry q2 g1"| 62144₇ ||class="entry q3 g1"| 60713₉ ||class="entry q3 g1"| 39855₁₁ ||class="entry q2 g1"| 35678₉ ||class="entry q3 g1"| 38305₇ ||class="entry q3 g1"| 35751₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (62624, 60233, 40399, 36158, 37825, 36295) ||class="entry q2 g1"| 62624₇ ||class="entry q3 g1"| 60233₉ ||class="entry q3 g1"| 40399₁₁ ||class="entry q2 g1"| 36158₉ ||class="entry q3 g1"| 37825₇ ||class="entry q3 g1"| 36295₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64064, 58553, 40159, 33758, 39985, 36055) ||class="entry q2 g1"| 64064₇ ||class="entry q3 g1"| 58553₉ ||class="entry q3 g1"| 40159₁₁ ||class="entry q2 g1"| 33758₉ ||class="entry q3 g1"| 39985₇ ||class="entry q3 g1"| 36055₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64544, 58073, 39615, 34238, 39505, 35511) ||class="entry q2 g1"| 64544₇ ||class="entry q3 g1"| 58073₉ ||class="entry q3 g1"| 39615₁₁ ||class="entry q2 g1"| 34238₉ ||class="entry q3 g1"| 39505₇ ||class="entry q3 g1"| 35511₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (37586, 55143, 51091, 60236, 45039, 55195) ||class="entry q2 g1"| 37586₇ ||class="entry q3 g1"| 55143₁₁ ||class="entry q3 g1"| 51091₉ ||class="entry q2 g1"| 60236₉ ||class="entry q3 g1"| 45039₁₃ ||class="entry q3 g1"| 55195₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (38068, 46951, 42901, 60714, 53231, 47005) ||class="entry q2 g1"| 38068₇ ||class="entry q3 g1"| 46951₁₁ ||class="entry q3 g1"| 42901₉ ||class="entry q2 g1"| 60714₉ ||class="entry q3 g1"| 53231₁₃ ||class="entry q3 g1"| 47005₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (39748, 46711, 43381, 58074, 52991, 47485) ||class="entry q2 g1"| 39748₇ ||class="entry q3 g1"| 46711₁₁ ||class="entry q3 g1"| 43381₉ ||class="entry q2 g1"| 58074₉ ||class="entry q3 g1"| 52991₁₃ ||class="entry q3 g1"| 47485₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (40226, 54903, 51571, 58556, 44799, 55675) ||class="entry q2 g1"| 40226₇ ||class="entry q3 g1"| 54903₁₁ ||class="entry q3 g1"| 51571₉ ||class="entry q2 g1"| 58556₉ ||class="entry q3 g1"| 44799₁₃ ||class="entry q3 g1"| 55675₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (41562, 36733, 44119, 56260, 63477, 48223) ||class="entry q2 g1"| 41562₇ ||class="entry q3 g1"| 36733₁₁ ||class="entry q3 g1"| 44119₉ ||class="entry q2 g1"| 56260₉ ||class="entry q3 g1"| 63477₁₃ ||class="entry q3 g1"| 48223₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (42282, 34813, 42439, 56500, 65397, 46543) ||class="entry q2 g1"| 42282₇ ||class="entry q3 g1"| 34813₁₁ ||class="entry q3 g1"| 42439₉ ||class="entry q2 g1"| 56500₉ ||class="entry q3 g1"| 65397₁₃ ||class="entry q3 g1"| 46543₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (49996, 34811, 50087, 47826, 65395, 54191) ||class="entry q2 g1"| 49996₇ ||class="entry q3 g1"| 34811₁₁ ||class="entry q3 g1"| 50087₉ ||class="entry q2 g1"| 47826₉ ||class="entry q3 g1"| 65395₁₃ ||class="entry q3 g1"| 54191₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (50236, 36731, 51767, 48546, 63475, 55871) ||class="entry q2 g1"| 50236₇ ||class="entry q3 g1"| 36731₁₁ ||class="entry q3 g1"| 51767₉ ||class="entry q2 g1"| 48546₉ ||class="entry q3 g1"| 63475₁₃ ||class="entry q3 g1"| 55871₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33498, 44793, 48923, 64324, 54897, 44819) ||class="entry q2 g1"| 33498₇ ||class="entry q3 g1"| 44793₁₁ ||class="entry q3 g1"| 48923₁₁ ||class="entry q2 g1"| 64324₉ ||class="entry q3 g1"| 54897₉ ||class="entry q3 g1"| 44819₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33980, 52985, 57117, 64802, 46705, 53013) ||class="entry q2 g1"| 33980₇ ||class="entry q3 g1"| 52985₁₁ ||class="entry q3 g1"| 57117₁₁ ||class="entry q2 g1"| 64802₉ ||class="entry q3 g1"| 46705₉ ||class="entry q3 g1"| 53013₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (35660, 53225, 53757, 62162, 46945, 49653) ||class="entry q2 g1"| 35660₇ ||class="entry q3 g1"| 53225₁₁ ||class="entry q3 g1"| 53757₁₁ ||class="entry q2 g1"| 62162₉ ||class="entry q3 g1"| 46945₉ ||class="entry q3 g1"| 49653₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (36138, 45033, 45563, 62644, 55137, 41459) ||class="entry q2 g1"| 36138₇ ||class="entry q3 g1"| 45033₁₁ ||class="entry q3 g1"| 45563₁₁ ||class="entry q2 g1"| 62644₉ ||class="entry q3 g1"| 55137₉ ||class="entry q3 g1"| 41459₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (45650, 63203, 54495, 52172, 36459, 50391) ||class="entry q2 g1"| 45650₇ ||class="entry q3 g1"| 63203₁₁ ||class="entry q3 g1"| 54495₁₁ ||class="entry q2 g1"| 52172₉ ||class="entry q3 g1"| 36459₉ ||class="entry q3 g1"| 50391₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (46370, 65123, 56655, 52412, 34539, 52551) ||class="entry q2 g1"| 46370₇ ||class="entry q3 g1"| 65123₁₁ ||class="entry q3 g1"| 56655₁₁ ||class="entry q2 g1"| 52412₉ ||class="entry q3 g1"| 34539₉ ||class="entry q3 g1"| 52551₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54084, 65125, 47919, 43738, 34541, 43815) ||class="entry q2 g1"| 54084₇ ||class="entry q3 g1"| 65125₁₁ ||class="entry q3 g1"| 47919₁₁ ||class="entry q2 g1"| 43738₉ ||class="entry q3 g1"| 34541₉ ||class="entry q3 g1"| 43815₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54324, 63205, 45759, 44458, 36461, 41655) ||class="entry q2 g1"| 54324₇ ||class="entry q3 g1"| 63205₁₁ ||class="entry q3 g1"| 45759₁₁ ||class="entry q2 g1"| 44458₉ ||class="entry q3 g1"| 36461₉ ||class="entry q3 g1"| 41655₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (37574, 60239, 64431, 60248, 37831, 60327) ||class="entry q2 g1"| 37574₇ ||class="entry q3 g1"| 60239₁₁ ||class="entry q3 g1"| 64431₁₃ ||class="entry q2 g1"| 60248₉ ||class="entry q3 g1"| 37831₉ ||class="entry q3 g1"| 60327₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (38054, 60719, 64975, 60728, 38311, 60871) ||class="entry q2 g1"| 38054₇ ||class="entry q3 g1"| 60719₁₁ ||class="entry q3 g1"| 64975₁₃ ||class="entry q2 g1"| 60728₉ ||class="entry q3 g1"| 38311₉ ||class="entry q3 g1"| 60871₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39494, 58079, 64735, 58328, 39511, 60631) ||class="entry q2 g1"| 39494₇ ||class="entry q3 g1"| 58079₁₁ ||class="entry q3 g1"| 64735₁₃ ||class="entry q2 g1"| 58328₉ ||class="entry q3 g1"| 39511₉ ||class="entry q3 g1"| 60631₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39974, 58559, 64191, 58808, 39991, 60087) ||class="entry q2 g1"| 39974₇ ||class="entry q3 g1"| 58559₁₁ ||class="entry q3 g1"| 64191₁₃ ||class="entry q2 g1"| 58808₉ ||class="entry q3 g1"| 39991₉ ||class="entry q3 g1"| 60087₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (41816, 56277, 63997, 56006, 41821, 59893) ||class="entry q2 g1"| 41816₇ ||class="entry q3 g1"| 56277₁₁ ||class="entry q3 g1"| 63997₁₃ ||class="entry q2 g1"| 56006₉ ||class="entry q3 g1"| 41821₉ ||class="entry q3 g1"| 59893₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (42296, 56757, 65437, 56486, 42301, 61333) ||class="entry q2 g1"| 42296₇ ||class="entry q3 g1"| 56757₁₁ ||class="entry q3 g1"| 65437₁₃ ||class="entry q2 g1"| 56486₉ ||class="entry q3 g1"| 42301₉ ||class="entry q3 g1"| 61333₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50008, 48083, 65435, 47814, 50011, 61331) ||class="entry q2 g1"| 50008₇ ||class="entry q3 g1"| 48083₁₁ ||class="entry q3 g1"| 65435₁₃ ||class="entry q2 g1"| 47814₉ ||class="entry q3 g1"| 50011₉ ||class="entry q3 g1"| 61331₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50488, 48563, 63995, 48294, 50491, 59891) ||class="entry q2 g1"| 50488₇ ||class="entry q3 g1"| 48563₁₁ ||class="entry q3 g1"| 63995₁₃ ||class="entry q2 g1"| 48294₉ ||class="entry q3 g1"| 50491₉ ||class="entry q3 g1"| 59891₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (37828, 49127, 44549, 59994, 51055, 48653) ||class="entry q2 g1"| 37828₇ ||class="entry q3 g1"| 49127₁₃ ||class="entry q3 g1"| 44549₇ ||class="entry q2 g1"| 59994₉ ||class="entry q3 g1"| 51055₁₁ ||class="entry q3 g1"| 48653₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38306, 57319, 52739, 60476, 42863, 56843) ||class="entry q2 g1"| 38306₇ ||class="entry q3 g1"| 57319₁₃ ||class="entry q3 g1"| 52739₇ ||class="entry q2 g1"| 60476₉ ||class="entry q3 g1"| 42863₁₁ ||class="entry q3 g1"| 56843₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (39506, 57079, 49379, 58316, 42623, 53483) ||class="entry q2 g1"| 39506₇ ||class="entry q3 g1"| 57079₁₃ ||class="entry q3 g1"| 49379₇ ||class="entry q2 g1"| 58316₉ ||class="entry q3 g1"| 42623₁₁ ||class="entry q3 g1"| 53483₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (39988, 48887, 41189, 58794, 50815, 45293) ||class="entry q2 g1"| 39988₇ ||class="entry q3 g1"| 48887₁₃ ||class="entry q3 g1"| 41189₇ ||class="entry q2 g1"| 58794₉ ||class="entry q3 g1"| 50815₁₁ ||class="entry q3 g1"| 45293₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (41804, 59389, 50625, 56018, 40821, 54729) ||class="entry q2 g1"| 41804₇ ||class="entry q3 g1"| 59389₁₃ ||class="entry q3 g1"| 50625₇ ||class="entry q2 g1"| 56018₉ ||class="entry q3 g1"| 40821₁₁ ||class="entry q3 g1"| 54729₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (42044, 61309, 52305, 56738, 38901, 56409) ||class="entry q2 g1"| 42044₇ ||class="entry q3 g1"| 61309₁₃ ||class="entry q3 g1"| 52305₇ ||class="entry q2 g1"| 56738₉ ||class="entry q3 g1"| 38901₁₁ ||class="entry q3 g1"| 56409₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (49754, 61307, 43569, 48068, 38899, 47673) ||class="entry q2 g1"| 49754₇ ||class="entry q3 g1"| 61307₁₃ ||class="entry q3 g1"| 43569₇ ||class="entry q2 g1"| 48068₉ ||class="entry q3 g1"| 38899₁₁ ||class="entry q3 g1"| 47673₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (50474, 59387, 41889, 48308, 40819, 45993) ||class="entry q2 g1"| 50474₇ ||class="entry q3 g1"| 59387₁₃ ||class="entry q3 g1"| 41889₇ ||class="entry q2 g1"| 48308₉ ||class="entry q3 g1"| 40819₁₁ ||class="entry q3 g1"| 45993₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38882, 46129, 42179, 61052, 52409, 46283) ||class="entry q2 g1"| 38882₉ ||class="entry q3 g1"| 46129₇ ||class="entry q3 g1"| 42179₇ ||class="entry q2 g1"| 61052₁₁ ||class="entry q3 g1"| 52409₉ ||class="entry q3 g1"| 46283₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38884, 53841, 49829, 61050, 43737, 53933) ||class="entry q2 g1"| 38884₉ ||class="entry q3 g1"| 53841₇ ||class="entry q3 g1"| 49829₇ ||class="entry q2 g1"| 61050₁₁ ||class="entry q3 g1"| 43737₉ ||class="entry q3 g1"| 53933₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40562, 45889, 44099, 59372, 52169, 48203) ||class="entry q2 g1"| 40562₉ ||class="entry q3 g1"| 45889₇ ||class="entry q3 g1"| 44099₇ ||class="entry q2 g1"| 59372₁₁ ||class="entry q3 g1"| 52169₉ ||class="entry q3 g1"| 48203₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40564, 54561, 51749, 59370, 44457, 55853) ||class="entry q2 g1"| 40564₉ ||class="entry q3 g1"| 54561₇ ||class="entry q3 g1"| 51749₇ ||class="entry q2 g1"| 59370₁₁ ||class="entry q3 g1"| 44457₉ ||class="entry q3 g1"| 55853₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (42620, 33963, 42641, 57314, 64547, 46745) ||class="entry q2 g1"| 42620₉ ||class="entry q3 g1"| 33963₇ ||class="entry q3 g1"| 42641₇ ||class="entry q2 g1"| 57314₁₁ ||class="entry q3 g1"| 64547₉ ||class="entry q3 g1"| 46745₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (42860, 35403, 43361, 57074, 62147, 47465) ||class="entry q2 g1"| 42860₉ ||class="entry q3 g1"| 35403₇ ||class="entry q3 g1"| 43361₇ ||class="entry q2 g1"| 57074₁₁ ||class="entry q3 g1"| 62147₉ ||class="entry q3 g1"| 47465₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (50810, 33485, 50833, 49124, 64069, 54937) ||class="entry q2 g1"| 50810₉ ||class="entry q3 g1"| 33485₇ ||class="entry q3 g1"| 50833₇ ||class="entry q2 g1"| 49124₁₁ ||class="entry q3 g1"| 64069₉ ||class="entry q3 g1"| 54937₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (51050, 35885, 51553, 48884, 62629, 55657) ||class="entry q2 g1"| 51050₉ ||class="entry q3 g1"| 35885₇ ||class="entry q3 g1"| 51553₇ ||class="entry q2 g1"| 48884₁₁ ||class="entry q3 g1"| 62629₉ ||class="entry q3 g1"| 55657₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (48864, 51341, 58709, 51070, 45061, 62813) ||class="entry q2 g1"| 48864₉ ||class="entry q3 g1"| 51341₇ ||class="entry q3 g1"| 58709₉ ||class="entry q2 g1"| 51070₁₁ ||class="entry q3 g1"| 45061₅ ||class="entry q3 g1"| 62813₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (57056, 43147, 58163, 42878, 53251, 62267) ||class="entry q2 g1"| 57056₉ ||class="entry q3 g1"| 43147₇ ||class="entry q3 g1"| 58163₉ ||class="entry q2 g1"| 42878₁₁ ||class="entry q3 g1"| 53251₅ ||class="entry q3 g1"| 62267₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (59112, 63745, 36743, 40822, 33161, 40847) ||class="entry q2 g1"| 59112₉ ||class="entry q3 g1"| 63745₇ ||class="entry q3 g1"| 36743₉ ||class="entry q2 g1"| 40822₁₁ ||class="entry q3 g1"| 33161₅ ||class="entry q3 g1"| 40847₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (61032, 61585, 35063, 38902, 34841, 39167) ||class="entry q2 g1"| 61032₉ ||class="entry q3 g1"| 61585₇ ||class="entry q3 g1"| 35063₉ ||class="entry q2 g1"| 38902₁₁ ||class="entry q3 g1"| 34841₅ ||class="entry q3 g1"| 39167₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (44776, 45331, 40413, 55158, 51611, 36309) ||class="entry q2 g1"| 44776₉ ||class="entry q3 g1"| 45331₇ ||class="entry q3 g1"| 40413₁₁ ||class="entry q2 g1"| 55158₁₁ ||class="entry q3 g1"| 51611₉ ||class="entry q3 g1"| 36309₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (52968, 53525, 39867, 46966, 43421, 35763) ||class="entry q2 g1"| 52968₉ ||class="entry q3 g1"| 53525₇ ||class="entry q3 g1"| 39867₁₁ ||class="entry q2 g1"| 46966₁₁ ||class="entry q3 g1"| 43421₉ ||class="entry q3 g1"| 35763₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (63200, 32927, 63247, 36734, 63511, 59143) ||class="entry q2 g1"| 63200₉ ||class="entry q3 g1"| 32927₇ ||class="entry q3 g1"| 63247₁₁ ||class="entry q2 g1"| 36734₁₁ ||class="entry q3 g1"| 63511₉ ||class="entry q3 g1"| 59143₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (65120, 35087, 61567, 34814, 61831, 57463) ||class="entry q2 g1"| 65120₉ ||class="entry q3 g1"| 35087₇ ||class="entry q3 g1"| 61567₁₁ ||class="entry q2 g1"| 34814₁₁ ||class="entry q3 g1"| 61831₉ ||class="entry q3 g1"| 57463₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38642, 47825, 43827, 61292, 49753, 47931) ||class="entry q2 g1"| 38642₉ ||class="entry q3 g1"| 47825₉ ||class="entry q3 g1"| 43827₉ ||class="entry q2 g1"| 61292₁₁ ||class="entry q3 g1"| 49753₇ ||class="entry q3 g1"| 47931₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38644, 56497, 52565, 61290, 42041, 56669) ||class="entry q2 g1"| 38644₉ ||class="entry q3 g1"| 56497₉ ||class="entry q3 g1"| 52565₉ ||class="entry q2 g1"| 61290₁₁ ||class="entry q3 g1"| 42041₇ ||class="entry q3 g1"| 56669₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40802, 48545, 41907, 59132, 50473, 46011) ||class="entry q2 g1"| 40802₉ ||class="entry q3 g1"| 48545₉ ||class="entry q3 g1"| 41907₉ ||class="entry q2 g1"| 59132₁₁ ||class="entry q3 g1"| 50473₇ ||class="entry q3 g1"| 46011₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40804, 56257, 50645, 59130, 41801, 54749) ||class="entry q2 g1"| 40804₉ ||class="entry q3 g1"| 56257₉ ||class="entry q3 g1"| 50645₉ ||class="entry q2 g1"| 59130₁₁ ||class="entry q3 g1"| 41801₇ ||class="entry q3 g1"| 54749₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (42618, 58059, 49399, 57316, 39491, 53503) ||class="entry q2 g1"| 42618₉ ||class="entry q3 g1"| 58059₉ ||class="entry q3 g1"| 49399₉ ||class="entry q2 g1"| 57316₁₁ ||class="entry q3 g1"| 39491₇ ||class="entry q3 g1"| 53503₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (42858, 60459, 52999, 57076, 38051, 57103) ||class="entry q2 g1"| 42858₉ ||class="entry q3 g1"| 60459₉ ||class="entry q3 g1"| 52999₉ ||class="entry q2 g1"| 57076₁₁ ||class="entry q3 g1"| 38051₇ ||class="entry q3 g1"| 57103₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (50812, 58541, 41207, 49122, 39973, 45311) ||class="entry q2 g1"| 50812₉ ||class="entry q3 g1"| 58541₉ ||class="entry q3 g1"| 41207₉ ||class="entry q2 g1"| 49122₁₁ ||class="entry q3 g1"| 39973₇ ||class="entry q3 g1"| 45311₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (51052, 59981, 44807, 48882, 37573, 48911) ||class="entry q2 g1"| 51052₉ ||class="entry q3 g1"| 59981₉ ||class="entry q3 g1"| 44807₉ ||class="entry q2 g1"| 48882₁₁ ||class="entry q3 g1"| 37573₇ ||class="entry q3 g1"| 48911₁₁ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (38630, 34553, 38671, 61304, 65137, 34567) ||class="entry q2 g1"| 38630₉ ||class="entry q3 g1"| 34553₉ ||class="entry q3 g1"| 38671₉ ||class="entry q2 g1"| 61304₁₁ ||class="entry q3 g1"| 65137₁₁ ||class="entry q3 g1"| 34567₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (40550, 36713, 36991, 59384, 63457, 32887) ||class="entry q2 g1"| 40550₉ ||class="entry q3 g1"| 36713₉ ||class="entry q3 g1"| 36991₉ ||class="entry q2 g1"| 59384₁₁ ||class="entry q3 g1"| 63457₁₁ ||class="entry q3 g1"| 32887₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (42872, 46691, 38237, 57062, 52971, 34133) ||class="entry q2 g1"| 42872₉ ||class="entry q3 g1"| 46691₉ ||class="entry q3 g1"| 38237₉ ||class="entry q2 g1"| 57062₁₁ ||class="entry q3 g1"| 52971₁₁ ||class="entry q3 g1"| 34133₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (51064, 54885, 37691, 48870, 44781, 33587) ||class="entry q2 g1"| 51064₉ ||class="entry q3 g1"| 54885₉ ||class="entry q3 g1"| 37691₉ ||class="entry q2 g1"| 48870₁₁ ||class="entry q3 g1"| 44781₁₁ ||class="entry q3 g1"| 33587₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (34554, 49999, 54203, 65380, 48071, 50099) ||class="entry q2 g1"| 34554₉ ||class="entry q3 g1"| 49999₉ ||class="entry q3 g1"| 54203₁₁ ||class="entry q2 g1"| 65380₁₁ ||class="entry q3 g1"| 48071₁₁ ||class="entry q3 g1"| 50099₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (34556, 42287, 46557, 65378, 56743, 42453) ||class="entry q2 g1"| 34556₉ ||class="entry q3 g1"| 42287₉ ||class="entry q3 g1"| 46557₁₁ ||class="entry q2 g1"| 65378₁₁ ||class="entry q3 g1"| 56743₁₁ ||class="entry q3 g1"| 42453₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (36714, 50239, 56123, 63220, 48311, 52019) ||class="entry q2 g1"| 36714₉ ||class="entry q3 g1"| 50239₉ ||class="entry q3 g1"| 56123₁₁ ||class="entry q2 g1"| 63220₁₁ ||class="entry q3 g1"| 48311₁₁ ||class="entry q3 g1"| 52019₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (36716, 41567, 48477, 63218, 56023, 44373) ||class="entry q2 g1"| 36716₉ ||class="entry q3 g1"| 41567₉ ||class="entry q3 g1"| 48477₁₁ ||class="entry q2 g1"| 63218₁₁ ||class="entry q3 g1"| 56023₁₁ ||class="entry q3 g1"| 44373₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46706, 39765, 47231, 53228, 58333, 43127) ||class="entry q2 g1"| 46706₉ ||class="entry q3 g1"| 39765₉ ||class="entry q3 g1"| 47231₁₁ ||class="entry q2 g1"| 53228₁₁ ||class="entry q3 g1"| 58333₁₁ ||class="entry q3 g1"| 43127₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46946, 38325, 46991, 52988, 60733, 42887) ||class="entry q2 g1"| 46946₉ ||class="entry q3 g1"| 38325₉ ||class="entry q3 g1"| 46991₁₁ ||class="entry q2 g1"| 52988₁₁ ||class="entry q3 g1"| 60733₁₁ ||class="entry q3 g1"| 42887₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54900, 40243, 55423, 45034, 58811, 51319) ||class="entry q2 g1"| 54900₉ ||class="entry q3 g1"| 40243₉ ||class="entry q3 g1"| 55423₁₁ ||class="entry q2 g1"| 45034₁₁ ||class="entry q3 g1"| 58811₁₁ ||class="entry q3 g1"| 51319₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (55140, 37843, 55183, 44794, 60251, 51079) ||class="entry q2 g1"| 55140₉ ||class="entry q3 g1"| 37843₉ ||class="entry q3 g1"| 55183₁₁ ||class="entry q2 g1"| 44794₁₁ ||class="entry q3 g1"| 60251₁₁ ||class="entry q3 g1"| 51079₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (34808, 38887, 34321, 65126, 61295, 38425) ||class="entry q2 g1"| 34808₉ ||class="entry q3 g1"| 38887₁₁ ||class="entry q3 g1"| 34321₅ ||class="entry q2 g1"| 65126₁₁ ||class="entry q3 g1"| 61295₁₃ ||class="entry q3 g1"| 38425₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (36728, 40567, 33121, 63206, 59135, 37225) ||class="entry q2 g1"| 36728₉ ||class="entry q3 g1"| 40567₁₁ ||class="entry q3 g1"| 33121₅ ||class="entry q2 g1"| 63206₁₁ ||class="entry q3 g1"| 59135₁₃ ||class="entry q3 g1"| 37225₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (46694, 42877, 33859, 53240, 57333, 37963) ||class="entry q2 g1"| 46694₉ ||class="entry q3 g1"| 42877₁₁ ||class="entry q3 g1"| 33859₅ ||class="entry q2 g1"| 53240₁₁ ||class="entry q3 g1"| 57333₁₃ ||class="entry q3 g1"| 37963₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54886, 51067, 33317, 45048, 49139, 37421) ||class="entry q2 g1"| 54886₉ ||class="entry q3 g1"| 51067₁₁ ||class="entry q3 g1"| 33317₅ ||class="entry q2 g1"| 45048₁₁ ||class="entry q3 g1"| 49139₁₃ ||class="entry q3 g1"| 37421₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (34794, 52655, 56395, 65140, 46375, 52291) ||class="entry q2 g1"| 34794₉ ||class="entry q3 g1"| 52655₁₁ ||class="entry q3 g1"| 56395₉ ||class="entry q2 g1"| 65140₁₁ ||class="entry q3 g1"| 46375₉ ||class="entry q3 g1"| 52291₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (34796, 43983, 47661, 65138, 54087, 43557) ||class="entry q2 g1"| 34796₉ ||class="entry q3 g1"| 43983₁₁ ||class="entry q3 g1"| 47661₉ ||class="entry q2 g1"| 65138₁₁ ||class="entry q3 g1"| 54087₉ ||class="entry q3 g1"| 43557₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (36474, 51935, 54475, 63460, 45655, 50371) ||class="entry q2 g1"| 36474₉ ||class="entry q3 g1"| 51935₁₁ ||class="entry q3 g1"| 54475₉ ||class="entry q2 g1"| 63460₁₁ ||class="entry q3 g1"| 45655₉ ||class="entry q3 g1"| 50371₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (36476, 44223, 45741, 63458, 54327, 41637) ||class="entry q2 g1"| 36476₉ ||class="entry q3 g1"| 44223₁₁ ||class="entry q3 g1"| 45741₉ ||class="entry q2 g1"| 63458₁₁ ||class="entry q3 g1"| 54327₉ ||class="entry q3 g1"| 41637₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46708, 64821, 56857, 53226, 34237, 52753) ||class="entry q2 g1"| 46708₉ ||class="entry q3 g1"| 64821₁₁ ||class="entry q3 g1"| 56857₉ ||class="entry q2 g1"| 53226₁₁ ||class="entry q3 g1"| 34237₉ ||class="entry q3 g1"| 52753₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46948, 62421, 53737, 52986, 35677, 49633) ||class="entry q2 g1"| 46948₉ ||class="entry q3 g1"| 62421₁₁ ||class="entry q3 g1"| 53737₉ ||class="entry q2 g1"| 52986₁₁ ||class="entry q3 g1"| 35677₉ ||class="entry q3 g1"| 49633₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54898, 64339, 48665, 45036, 33755, 44561) ||class="entry q2 g1"| 54898₉ ||class="entry q3 g1"| 64339₁₁ ||class="entry q3 g1"| 48665₉ ||class="entry q2 g1"| 45036₁₁ ||class="entry q3 g1"| 33755₉ ||class="entry q3 g1"| 44561₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (55138, 62899, 45545, 44796, 36155, 41441) ||class="entry q2 g1"| 55138₉ ||class="entry q3 g1"| 62899₁₁ ||class="entry q3 g1"| 45545₉ ||class="entry q2 g1"| 44796₁₁ ||class="entry q3 g1"| 36155₉ ||class="entry q3 g1"| 41441₇ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38896, 61049, 65177, 61038, 38641, 61073) ||class="entry q2 g1"| 38896₉ ||class="entry q3 g1"| 61049₁₁ ||class="entry q3 g1"| 65177₁₁ ||class="entry q2 g1"| 61038₁₁ ||class="entry q3 g1"| 38641₉ ||class="entry q3 g1"| 61073₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (40816, 59369, 63977, 59118, 40801, 59873) ||class="entry q2 g1"| 40816₉ ||class="entry q3 g1"| 59369₁₁ ||class="entry q3 g1"| 63977₁₁ ||class="entry q2 g1"| 59118₁₁ ||class="entry q3 g1"| 40801₉ ||class="entry q3 g1"| 59873₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (42606, 57059, 64715, 57328, 42603, 60611) ||class="entry q2 g1"| 42606₉ ||class="entry q3 g1"| 57059₁₁ ||class="entry q3 g1"| 64715₁₁ ||class="entry q2 g1"| 57328₁₁ ||class="entry q3 g1"| 42603₉ ||class="entry q3 g1"| 60611₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (50798, 48869, 64173, 49136, 50797, 60069) ||class="entry q2 g1"| 50798₉ ||class="entry q3 g1"| 48869₁₁ ||class="entry q3 g1"| 64173₁₁ ||class="entry q2 g1"| 49136₁₁ ||class="entry q3 g1"| 50797₉ ||class="entry q3 g1"| 60069₉ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (34542, 65383, 61319, 65392, 34799, 65423) ||class="entry q2 g1"| 34542₉ ||class="entry q3 g1"| 65383₁₃ ||class="entry q3 g1"| 61319₁₁ ||class="entry q2 g1"| 65392₁₁ ||class="entry q3 g1"| 34799₁₁ ||class="entry q3 g1"| 65423₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (36462, 63223, 59639, 63472, 36479, 63743) ||class="entry q2 g1"| 36462₉ ||class="entry q3 g1"| 63223₁₃ ||class="entry q3 g1"| 59639₁₁ ||class="entry q2 g1"| 63472₁₁ ||class="entry q3 g1"| 36479₁₁ ||class="entry q3 g1"| 63743₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (46960, 53245, 60885, 52974, 46965, 64989) ||class="entry q2 g1"| 46960₉ ||class="entry q3 g1"| 53245₁₃ ||class="entry q3 g1"| 60885₁₁ ||class="entry q2 g1"| 52974₁₁ ||class="entry q3 g1"| 46965₁₁ ||class="entry q3 g1"| 64989₁₃ |- |class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (55152, 45051, 60339, 44782, 55155, 64443) ||class="entry q2 g1"| 55152₉ ||class="entry q3 g1"| 45051₁₃ ||class="entry q3 g1"| 60339₁₁ ||class="entry q2 g1"| 44782₁₁ ||class="entry q3 g1"| 55155₁₁ ||class="entry q3 g1"| 64443₁₃ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (72, 18522, 23040, 4608, 23058, 18504) ||class="entry q0 g0"| 72₂ ||class="entry q0 g0"| 18522₆ ||class="entry q0 g0"| 23040₄ ||class="entry q0 g0"| 4608₂ ||class="entry q0 g0"| 23058₆ ||class="entry q0 g0"| 18504₄ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (520, 9100, 12480, 4160, 12740, 8840) ||class="entry q0 g0"| 520₂ ||class="entry q0 g0"| 9100₆ ||class="entry q0 g0"| 12480₄ ||class="entry q0 g0"| 4160₂ ||class="entry q0 g0"| 12740₆ ||class="entry q0 g0"| 8840₄ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (2248, 16842, 23920, 6784, 21378, 20280) ||class="entry q0 g0"| 2248₄ ||class="entry q0 g0"| 16842₆ ||class="entry q0 g0"| 23920₈ ||class="entry q0 g0"| 6784₄ ||class="entry q0 g0"| 21378₆ ||class="entry q0 g0"| 20280₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (2696, 10780, 14256, 6336, 14420, 9720) ||class="entry q0 g0"| 2696₄ ||class="entry q0 g0"| 10780₆ ||class="entry q0 g0"| 14256₈ ||class="entry q0 g0"| 6336₄ ||class="entry q0 g0"| 14420₆ ||class="entry q0 g0"| 9720₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (10312, 24654, 20028, 14848, 29190, 23668) ||class="entry q0 g0"| 10312₄ ||class="entry q0 g0"| 24654₆ ||class="entry q0 g0"| 20028₈ ||class="entry q0 g0"| 14848₄ ||class="entry q0 g0"| 29190₆ ||class="entry q0 g0"| 23668₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (10760, 2968, 9468, 14400, 6608, 14004) ||class="entry q0 g0"| 10760₄ ||class="entry q0 g0"| 2968₆ ||class="entry q0 g0"| 9468₈ ||class="entry q0 g0"| 14400₄ ||class="entry q0 g0"| 6608₆ ||class="entry q0 g0"| 14004₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (16584, 2520, 20266, 21120, 7056, 23906) ||class="entry q0 g0"| 16584₄ ||class="entry q0 g0"| 2520₆ ||class="entry q0 g0"| 20266₈ ||class="entry q0 g0"| 21120₄ ||class="entry q0 g0"| 7056₆ ||class="entry q0 g0"| 23906₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (17032, 25102, 9706, 20672, 28742, 14242) ||class="entry q0 g0"| 17032₄ ||class="entry q0 g0"| 25102₆ ||class="entry q0 g0"| 9706₈ ||class="entry q0 g0"| 20672₄ ||class="entry q0 g0"| 28742₆ ||class="entry q0 g0"| 14242₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (24648, 10332, 23654, 29184, 14868, 20014) ||class="entry q0 g0"| 24648₄ ||class="entry q0 g0"| 10332₆ ||class="entry q0 g0"| 23654₈ ||class="entry q0 g0"| 29184₄ ||class="entry q0 g0"| 14868₆ ||class="entry q0 g0"| 20014₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (25096, 17290, 13990, 28736, 20930, 9454) ||class="entry q0 g0"| 25096₄ ||class="entry q0 g0"| 17290₆ ||class="entry q0 g0"| 13990₈ ||class="entry q0 g0"| 28736₄ ||class="entry q0 g0"| 20930₆ ||class="entry q0 g0"| 9454₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (1128, 9708, 13984, 5664, 14244, 9448) ||class="entry q0 g0"| 1128₄ ||class="entry q0 g0"| 9708₈ ||class="entry q0 g0"| 13984₆ ||class="entry q0 g0"| 5664₄ ||class="entry q0 g0"| 14244₈ ||class="entry q0 g0"| 9448₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (1576, 20026, 23648, 5216, 23666, 20008) ||class="entry q0 g0"| 1576₄ ||class="entry q0 g0"| 20026₈ ||class="entry q0 g0"| 23648₆ ||class="entry q0 g0"| 5216₄ ||class="entry q0 g0"| 23666₈ ||class="entry q0 g0"| 20008₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (78, 11834, 15462, 4614, 15474, 11822) ||class="entry q0 g0"| 78₄ ||class="entry q0 g0"| 11834₈ ||class="entry q0 g0"| 15462₈ ||class="entry q0 g0"| 4614₄ ||class="entry q0 g0"| 15474₈ ||class="entry q0 g0"| 11822₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (92, 29810, 26172, 4628, 26170, 29812) ||class="entry q0 g0"| 92₄ ||class="entry q0 g0"| 29810₈ ||class="entry q0 g0"| 26172₈ ||class="entry q0 g0"| 4628₄ ||class="entry q0 g0"| 26170₈ ||class="entry q0 g0"| 29812₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (330, 7410, 4010, 4866, 3770, 7650) ||class="entry q0 g0"| 330₄ ||class="entry q0 g0"| 7410₈ ||class="entry q0 g0"| 4010₈ ||class="entry q0 g0"| 4866₄ ||class="entry q0 g0"| 3770₈ ||class="entry q0 g0"| 7650₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (332, 31378, 27084, 4868, 26842, 31620) ||class="entry q0 g0"| 332₄ ||class="entry q0 g0"| 31378₈ ||class="entry q0 g0"| 27084₈ ||class="entry q0 g0"| 4868₄ ||class="entry q0 g0"| 26842₈ ||class="entry q0 g0"| 31620₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (344, 18106, 22000, 4880, 21746, 18360) ||class="entry q0 g0"| 344₄ ||class="entry q0 g0"| 18106₈ ||class="entry q0 g0"| 22000₈ ||class="entry q0 g0"| 4880₄ ||class="entry q0 g0"| 21746₈ ||class="entry q0 g0"| 18360₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (526, 17900, 22182, 4166, 22436, 17646) ||class="entry q0 g0"| 526₄ ||class="entry q0 g0"| 17900₈ ||class="entry q0 g0"| 22182₈ ||class="entry q0 g0"| 4166₄ ||class="entry q0 g0"| 22436₈ ||class="entry q0 g0"| 17646₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (538, 31172, 27290, 4178, 27532, 30930) ||class="entry q0 g0"| 538₄ ||class="entry q0 g0"| 31172₈ ||class="entry q0 g0"| 27290₈ ||class="entry q0 g0"| 4178₄ ||class="entry q0 g0"| 27532₈ ||class="entry q0 g0"| 30930₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (540, 8100, 3324, 4180, 3564, 7860) ||class="entry q0 g0"| 540₄ ||class="entry q0 g0"| 8100₈ ||class="entry q0 g0"| 3324₈ ||class="entry q0 g0"| 4180₄ ||class="entry q0 g0"| 3564₈ ||class="entry q0 g0"| 7860₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (778, 30500, 25962, 4418, 25964, 30498) ||class="entry q0 g0"| 778₄ ||class="entry q0 g0"| 30500₈ ||class="entry q0 g0"| 25962₈ ||class="entry q0 g0"| 4418₄ ||class="entry q0 g0"| 25964₈ ||class="entry q0 g0"| 30498₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (792, 11628, 16176, 4432, 16164, 11640) ||class="entry q0 g0"| 792₄ ||class="entry q0 g0"| 11628₈ ||class="entry q0 g0"| 16176₈ ||class="entry q0 g0"| 4432₄ ||class="entry q0 g0"| 16164₈ ||class="entry q0 g0"| 11640₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (8392, 27102, 18764, 12928, 31638, 23300) ||class="entry q0 g0"| 8392₄ ||class="entry q0 g0"| 27102₁₀ ||class="entry q0 g0"| 18764₆ ||class="entry q0 g0"| 12928₄ ||class="entry q0 g0"| 31638₁₀ ||class="entry q0 g0"| 23300₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (18952, 27550, 8858, 22592, 31190, 12498) ||class="entry q0 g0"| 18952₄ ||class="entry q0 g0"| 27550₁₀ ||class="entry q0 g0"| 8858₆ ||class="entry q0 g0"| 22592₄ ||class="entry q0 g0"| 31190₁₀ ||class="entry q0 g0"| 12498₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1134, 17292, 20678, 5670, 20932, 17038) ||class="entry q0 g0"| 1134₆ ||class="entry q0 g0"| 17292₆ ||class="entry q0 g0"| 20678₆ ||class="entry q0 g0"| 5670₆ ||class="entry q0 g0"| 20932₆ ||class="entry q0 g0"| 17038₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1148, 6596, 2716, 5684, 2956, 6356) ||class="entry q0 g0"| 1148₆ ||class="entry q0 g0"| 6596₆ ||class="entry q0 g0"| 2716₆ ||class="entry q0 g0"| 5684₆ ||class="entry q0 g0"| 2956₆ ||class="entry q0 g0"| 6356₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1386, 28996, 25354, 5922, 25356, 28994) ||class="entry q0 g0"| 1386₆ ||class="entry q0 g0"| 28996₆ ||class="entry q0 g0"| 25354₆ ||class="entry q0 g0"| 5922₆ ||class="entry q0 g0"| 25356₆ ||class="entry q0 g0"| 28994₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1400, 11020, 14672, 5936, 14660, 11032) ||class="entry q0 g0"| 1400₆ ||class="entry q0 g0"| 11020₆ ||class="entry q0 g0"| 14672₆ ||class="entry q0 g0"| 5936₆ ||class="entry q0 g0"| 14660₆ ||class="entry q0 g0"| 11032₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1582, 10330, 14854, 5222, 14866, 10318) ||class="entry q0 g0"| 1582₆ ||class="entry q0 g0"| 10330₆ ||class="entry q0 g0"| 14854₆ ||class="entry q0 g0"| 5222₆ ||class="entry q0 g0"| 14866₆ ||class="entry q0 g0"| 10318₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1596, 29202, 24668, 5236, 24666, 29204) ||class="entry q0 g0"| 1596₆ ||class="entry q0 g0"| 29202₆ ||class="entry q0 g0"| 24668₆ ||class="entry q0 g0"| 5236₆ ||class="entry q0 g0"| 24666₆ ||class="entry q0 g0"| 29204₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1834, 6802, 2506, 5474, 2266, 7042) ||class="entry q0 g0"| 1834₆ ||class="entry q0 g0"| 6802₆ ||class="entry q0 g0"| 2506₆ ||class="entry q0 g0"| 5474₆ ||class="entry q0 g0"| 2266₆ ||class="entry q0 g0"| 7042₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1848, 16602, 21392, 5488, 21138, 16856) ||class="entry q0 g0"| 1848₆ ||class="entry q0 g0"| 16602₆ ||class="entry q0 g0"| 21392₆ ||class="entry q0 g0"| 5488₆ ||class="entry q0 g0"| 21138₆ ||class="entry q0 g0"| 16856₆ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (350, 8410, 13206, 4886, 12946, 8670) ||class="entry q0 g0"| 350₆ ||class="entry q0 g0"| 8410₆ ||class="entry q0 g0"| 13206₈ ||class="entry q0 g0"| 4886₆ ||class="entry q0 g0"| 12946₆ ||class="entry q0 g0"| 8670₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (798, 19212, 22870, 4438, 22852, 19230) ||class="entry q0 g0"| 798₆ ||class="entry q0 g0"| 19212₆ ||class="entry q0 g0"| 22870₈ ||class="entry q0 g0"| 4438₆ ||class="entry q0 g0"| 22852₆ ||class="entry q0 g0"| 19230₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (8846, 25704, 17898, 12486, 30240, 22434) ||class="entry q0 g0"| 8846₆ ||class="entry q0 g0"| 25704₆ ||class="entry q0 g0"| 17898₈ ||class="entry q0 g0"| 12486₆ ||class="entry q0 g0"| 30240₆ ||class="entry q0 g0"| 22434₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (8860, 15904, 8112, 12500, 11368, 3576) ||class="entry q0 g0"| 8860₆ ||class="entry q0 g0"| 15904₆ ||class="entry q0 g0"| 8112₈ ||class="entry q0 g0"| 12500₆ ||class="entry q0 g0"| 11368₆ ||class="entry q0 g0"| 3576₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (9098, 22176, 30246, 12738, 17640, 25710) ||class="entry q0 g0"| 9098₆ ||class="entry q0 g0"| 22176₆ ||class="entry q0 g0"| 30246₈ ||class="entry q0 g0"| 12738₆ ||class="entry q0 g0"| 17640₆ ||class="entry q0 g0"| 25710₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (9112, 3304, 11388, 12752, 7840, 15924) ||class="entry q0 g0"| 9112₆ ||class="entry q0 g0"| 3304₆ ||class="entry q0 g0"| 11388₈ ||class="entry q0 g0"| 12752₆ ||class="entry q0 g0"| 7840₆ ||class="entry q0 g0"| 15924₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (18510, 26152, 11836, 23046, 29792, 15476) ||class="entry q0 g0"| 18510₆ ||class="entry q0 g0"| 26152₆ ||class="entry q0 g0"| 11836₈ ||class="entry q0 g0"| 23046₆ ||class="entry q0 g0"| 29792₆ ||class="entry q0 g0"| 15476₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (18524, 15456, 29798, 23060, 11816, 26158) ||class="entry q0 g0"| 18524₆ ||class="entry q0 g0"| 15456₆ ||class="entry q0 g0"| 29798₈ ||class="entry q0 g0"| 23060₆ ||class="entry q0 g0"| 11816₆ ||class="entry q0 g0"| 26158₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (18762, 21728, 7664, 23298, 18088, 4024) ||class="entry q0 g0"| 18762₆ ||class="entry q0 g0"| 21728₆ ||class="entry q0 g0"| 7664₈ ||class="entry q0 g0"| 23298₆ ||class="entry q0 g0"| 18088₆ ||class="entry q0 g0"| 4024₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (18776, 3752, 18346, 23312, 7392, 21986) ||class="entry q0 g0"| 18776₆ ||class="entry q0 g0"| 3752₆ ||class="entry q0 g0"| 18346₈ ||class="entry q0 g0"| 23312₆ ||class="entry q0 g0"| 7392₆ ||class="entry q0 g0"| 21986₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (26824, 8652, 23318, 31360, 13188, 18782) ||class="entry q0 g0"| 26824₆ ||class="entry q0 g0"| 8652₆ ||class="entry q0 g0"| 23318₈ ||class="entry q0 g0"| 31360₆ ||class="entry q0 g0"| 13188₆ ||class="entry q0 g0"| 18782₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (27272, 18970, 12758, 30912, 22610, 9118) ||class="entry q0 g0"| 27272₆ ||class="entry q0 g0"| 18970₆ ||class="entry q0 g0"| 12758₈ ||class="entry q0 g0"| 30912₆ ||class="entry q0 g0"| 22610₆ ||class="entry q0 g0"| 9118₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (2508, 29442, 28348, 7044, 24906, 31988) ||class="entry q0 g0"| 2508₆ ||class="entry q0 g0"| 29442₆ ||class="entry q0 g0"| 28348₁₀ ||class="entry q0 g0"| 7044₆ ||class="entry q0 g0"| 24906₆ ||class="entry q0 g0"| 31988₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (2714, 28756, 28138, 6354, 25116, 32674) ||class="entry q0 g0"| 2714₆ ||class="entry q0 g0"| 28756₆ ||class="entry q0 g0"| 28138₁₀ ||class="entry q0 g0"| 6354₆ ||class="entry q0 g0"| 25116₆ ||class="entry q0 g0"| 32674₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (10572, 21126, 32240, 15108, 16590, 28600) ||class="entry q0 g0"| 10572₆ ||class="entry q0 g0"| 21126₆ ||class="entry q0 g0"| 32240₁₀ ||class="entry q0 g0"| 15108₆ ||class="entry q0 g0"| 16590₆ ||class="entry q0 g0"| 28600₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (10778, 20944, 32422, 14418, 17304, 27886) ||class="entry q0 g0"| 10778₆ ||class="entry q0 g0"| 20944₆ ||class="entry q0 g0"| 32422₁₀ ||class="entry q0 g0"| 14418₆ ||class="entry q0 g0"| 17304₆ ||class="entry q0 g0"| 27886₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (16844, 15120, 31974, 21380, 10584, 28334) ||class="entry q0 g0"| 16844₆ ||class="entry q0 g0"| 15120₆ ||class="entry q0 g0"| 31974₁₀ ||class="entry q0 g0"| 21380₆ ||class="entry q0 g0"| 10584₆ ||class="entry q0 g0"| 28334₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (17050, 14406, 32688, 20690, 10766, 28152) ||class="entry q0 g0"| 17050₆ ||class="entry q0 g0"| 14406₆ ||class="entry q0 g0"| 32688₁₀ ||class="entry q0 g0"| 20690₆ ||class="entry q0 g0"| 10766₆ ||class="entry q0 g0"| 28152₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (24908, 6804, 28586, 29444, 2268, 32226) ||class="entry q0 g0"| 24908₆ ||class="entry q0 g0"| 6804₆ ||class="entry q0 g0"| 28586₁₀ ||class="entry q0 g0"| 29444₆ ||class="entry q0 g0"| 2268₆ ||class="entry q0 g0"| 32226₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (25114, 6594, 27900, 28754, 2954, 32436) ||class="entry q0 g0"| 25114₆ ||class="entry q0 g0"| 6594₆ ||class="entry q0 g0"| 27900₁₀ ||class="entry q0 g0"| 28754₆ ||class="entry q0 g0"| 2954₆ ||class="entry q0 g0"| 32436₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (2254, 10154, 15126, 6790, 13794, 10590) ||class="entry q0 g0"| 2254₆ ||class="entry q0 g0"| 10154₈ ||class="entry q0 g0"| 15126₈ ||class="entry q0 g0"| 6790₆ ||class="entry q0 g0"| 13794₈ ||class="entry q0 g0"| 10590₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (2702, 19580, 20950, 6342, 24116, 17310) ||class="entry q0 g0"| 2702₆ ||class="entry q0 g0"| 19580₈ ||class="entry q0 g0"| 20950₈ ||class="entry q0 g0"| 6342₆ ||class="entry q0 g0"| 24116₈ ||class="entry q0 g0"| 17310₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (10570, 13542, 7062, 15106, 9902, 2526) ||class="entry q0 g0"| 10570₆ ||class="entry q0 g0"| 13542₈ ||class="entry q0 g0"| 7062₈ ||class="entry q0 g0"| 15106₆ ||class="entry q0 g0"| 9902₈ ||class="entry q0 g0"| 2526₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (11018, 24368, 29014, 14658, 19832, 25374) ||class="entry q0 g0"| 11018₆ ||class="entry q0 g0"| 24368₈ ||class="entry q0 g0"| 29014₈ ||class="entry q0 g0"| 14658₆ ||class="entry q0 g0"| 19832₈ ||class="entry q0 g0"| 25374₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (16604, 13808, 29462, 21140, 10168, 24926) ||class="entry q0 g0"| 16604₆ ||class="entry q0 g0"| 13808₈ ||class="entry q0 g0"| 29462₈ ||class="entry q0 g0"| 21140₆ ||class="entry q0 g0"| 10168₈ ||class="entry q0 g0"| 24926₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (17052, 24102, 6614, 20692, 19566, 2974) ||class="entry q0 g0"| 17052₆ ||class="entry q0 g0"| 24102₈ ||class="entry q0 g0"| 6614₈ ||class="entry q0 g0"| 20692₆ ||class="entry q0 g0"| 19566₈ ||class="entry q0 g0"| 2974₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (24920, 9916, 21398, 29456, 13556, 16862) ||class="entry q0 g0"| 24920₆ ||class="entry q0 g0"| 9916₈ ||class="entry q0 g0"| 21398₈ ||class="entry q0 g0"| 29456₆ ||class="entry q0 g0"| 13556₈ ||class="entry q0 g0"| 16862₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (25368, 19818, 14678, 29008, 24354, 11038) ||class="entry q0 g0"| 25368₆ ||class="entry q0 g0"| 19818₈ ||class="entry q0 g0"| 14678₈ ||class="entry q0 g0"| 29008₆ ||class="entry q0 g0"| 24354₈ ||class="entry q0 g0"| 11038₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (8398, 4030, 12074, 12934, 7670, 15714) ||class="entry q0 g0"| 8398₆ ||class="entry q0 g0"| 4030₁₀ ||class="entry q0 g0"| 12074₈ ||class="entry q0 g0"| 12934₆ ||class="entry q0 g0"| 7670₁₀ ||class="entry q0 g0"| 15714₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (8412, 22006, 30064, 12948, 18366, 26424) ||class="entry q0 g0"| 8412₆ ||class="entry q0 g0"| 22006₁₀ ||class="entry q0 g0"| 30064₈ ||class="entry q0 g0"| 12948₆ ||class="entry q0 g0"| 18366₁₀ ||class="entry q0 g0"| 26424₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (8650, 15734, 7398, 13186, 12094, 3758) ||class="entry q0 g0"| 8650₆ ||class="entry q0 g0"| 15734₁₀ ||class="entry q0 g0"| 7398₈ ||class="entry q0 g0"| 13186₆ ||class="entry q0 g0"| 12094₁₀ ||class="entry q0 g0"| 3758₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (8664, 26430, 18108, 13200, 30070, 21748) ||class="entry q0 g0"| 8664₆ ||class="entry q0 g0"| 26430₁₀ ||class="entry q0 g0"| 18108₈ ||class="entry q0 g0"| 13200₆ ||class="entry q0 g0"| 30070₁₀ ||class="entry q0 g0"| 21748₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (18958, 3582, 17660, 22598, 8118, 22196) ||class="entry q0 g0"| 18958₆ ||class="entry q0 g0"| 3582₁₀ ||class="entry q0 g0"| 17660₈ ||class="entry q0 g0"| 22598₆ ||class="entry q0 g0"| 8118₁₀ ||class="entry q0 g0"| 22196₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (18972, 22454, 7846, 22612, 17918, 3310) ||class="entry q0 g0"| 18972₆ ||class="entry q0 g0"| 22454₁₀ ||class="entry q0 g0"| 7846₈ ||class="entry q0 g0"| 22612₆ ||class="entry q0 g0"| 17918₁₀ ||class="entry q0 g0"| 3310₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (19210, 16182, 30512, 22850, 11646, 25976) ||class="entry q0 g0"| 19210₆ ||class="entry q0 g0"| 16182₁₀ ||class="entry q0 g0"| 30512₈ ||class="entry q0 g0"| 22850₆ ||class="entry q0 g0"| 11646₁₀ ||class="entry q0 g0"| 25976₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (19224, 25982, 11626, 22864, 30518, 16162) ||class="entry q0 g0"| 19224₆ ||class="entry q0 g0"| 25982₁₀ ||class="entry q0 g0"| 11626₈ ||class="entry q0 g0"| 22864₆ ||class="entry q0 g0"| 30518₁₀ ||class="entry q0 g0"| 16162₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (1146, 32676, 27898, 5682, 28140, 32434) ||class="entry q0 g0"| 1146₆ ||class="entry q0 g0"| 32676₁₀ ||class="entry q0 g0"| 27898₁₀ ||class="entry q0 g0"| 5682₆ ||class="entry q0 g0"| 28140₁₀ ||class="entry q0 g0"| 32434₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (1836, 31986, 28588, 5476, 28346, 32228) ||class="entry q0 g0"| 1836₆ ||class="entry q0 g0"| 31986₁₀ ||class="entry q0 g0"| 28588₁₀ ||class="entry q0 g0"| 5476₆ ||class="entry q0 g0"| 28346₁₀ ||class="entry q0 g0"| 32228₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (9896, 28606, 20268, 13536, 32246, 23908) ||class="entry q0 g0"| 9896₆ ||class="entry q0 g0"| 28606₁₂ ||class="entry q0 g0"| 20268₈ ||class="entry q0 g0"| 13536₆ ||class="entry q0 g0"| 32246₁₂ ||class="entry q0 g0"| 23908₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (19560, 28158, 9466, 24096, 32694, 14002) ||class="entry q0 g0"| 19560₆ ||class="entry q0 g0"| 28158₁₂ ||class="entry q0 g0"| 9466₈ ||class="entry q0 g0"| 24096₆ ||class="entry q0 g0"| 32694₁₂ ||class="entry q0 g0"| 14002₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (3322, 30260, 27530, 7858, 25724, 31170) ||class="entry q0 g0"| 3322₈ ||class="entry q0 g0"| 30260₈ ||class="entry q0 g0"| 27530₈ ||class="entry q0 g0"| 7858₈ ||class="entry q0 g0"| 25724₈ ||class="entry q0 g0"| 31170₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (3562, 30932, 25722, 8098, 27292, 30258) ||class="entry q0 g0"| 3562₈ ||class="entry q0 g0"| 30932₈ ||class="entry q0 g0"| 25722₈ ||class="entry q0 g0"| 8098₈ ||class="entry q0 g0"| 27292₈ ||class="entry q0 g0"| 30258₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (3772, 31618, 26412, 7412, 27082, 30052) ||class="entry q0 g0"| 3772₈ ||class="entry q0 g0"| 31618₈ ||class="entry q0 g0"| 26412₈ ||class="entry q0 g0"| 7412₈ ||class="entry q0 g0"| 27082₈ ||class="entry q0 g0"| 30052₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (4012, 30050, 26844, 7652, 26410, 31380) ||class="entry q0 g0"| 4012₈ ||class="entry q0 g0"| 30050₈ ||class="entry q0 g0"| 26844₈ ||class="entry q0 g0"| 7652₈ ||class="entry q0 g0"| 26410₈ ||class="entry q0 g0"| 31380₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (11374, 27544, 17658, 15910, 31184, 22194) ||class="entry q0 g0"| 11374₈ ||class="entry q0 g0"| 27544₈ ||class="entry q0 g0"| 17658₈ ||class="entry q0 g0"| 15910₈ ||class="entry q0 g0"| 31184₈ ||class="entry q0 g0"| 22194₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (11386, 22448, 30918, 15922, 17912, 27278) ||class="entry q0 g0"| 11386₈ ||class="entry q0 g0"| 22448₈ ||class="entry q0 g0"| 30918₈ ||class="entry q0 g0"| 15922₈ ||class="entry q0 g0"| 17912₈ ||class="entry q0 g0"| 27278₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (12076, 21734, 31632, 15716, 18094, 27096) ||class="entry q0 g0"| 12076₈ ||class="entry q0 g0"| 21734₈ ||class="entry q0 g0"| 31632₈ ||class="entry q0 g0"| 15716₈ ||class="entry q0 g0"| 18094₈ ||class="entry q0 g0"| 27096₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (12088, 26830, 18348, 15728, 31366, 21988) ||class="entry q0 g0"| 12088₈ ||class="entry q0 g0"| 26830₈ ||class="entry q0 g0"| 18348₈ ||class="entry q0 g0"| 15728₈ ||class="entry q0 g0"| 31366₈ ||class="entry q0 g0"| 21988₈ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (1406, 19820, 24374, 5942, 24356, 19838) ||class="entry q0 g0"| 1406₈ ||class="entry q0 g0"| 19820₈ ||class="entry q0 g0"| 24374₁₀ ||class="entry q0 g0"| 5942₈ ||class="entry q0 g0"| 24356₈ ||class="entry q0 g0"| 19838₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (1854, 9914, 13814, 5494, 13554, 10174) ||class="entry q0 g0"| 1854₈ ||class="entry q0 g0"| 9914₈ ||class="entry q0 g0"| 13814₁₀ ||class="entry q0 g0"| 5494₈ ||class="entry q0 g0"| 13554₈ ||class="entry q0 g0"| 10174₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (27880, 19578, 14262, 32416, 24114, 9726) ||class="entry q0 g0"| 27880₈ ||class="entry q0 g0"| 19578₈ ||class="entry q0 g0"| 14262₁₀ ||class="entry q0 g0"| 32416₈ ||class="entry q0 g0"| 24114₈ ||class="entry q0 g0"| 9726₁₀ |- |class="f"| 4680 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (28328, 10156, 23926, 31968, 13796, 20286) ||class="entry q0 g0"| 28328₈ ||class="entry q0 g0"| 10156₈ ||class="entry q0 g0"| 23926₁₀ ||class="entry q0 g0"| 31968₈ ||class="entry q0 g0"| 13796₈ ||class="entry q0 g0"| 20286₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (1152, 19589, 14007, 32328, 14171, 9705) ||class="entry q0 g0"| 1152₂ ||class="entry q1 g0"| 19589₆ ||class="entry q1 g0"| 14007₁₀ ||class="entry q0 g0"| 32328₈ ||class="entry q1 g0"| 14171₁₀ ||class="entry q1 g0"| 9705₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (2080, 10403, 23911, 29416, 21373, 20025) ||class="entry q0 g0"| 2080₂ ||class="entry q1 g0"| 10403₆ ||class="entry q1 g0"| 23911₁₀ ||class="entry q0 g0"| 29416₈ ||class="entry q1 g0"| 21373₁₀ ||class="entry q1 g0"| 20025₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (9216, 27905, 9723, 24264, 5855, 13989) ||class="entry q0 g0"| 9216₂ ||class="entry q1 g0"| 27905₆ ||class="entry q1 g0"| 9723₁₀ ||class="entry q0 g0"| 24264₈ ||class="entry q1 g0"| 5855₁₀ ||class="entry q1 g0"| 13989₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (16416, 24753, 20285, 15080, 7023, 23651) ||class="entry q0 g0"| 16416₂ ||class="entry q1 g0"| 24753₆ ||class="entry q1 g0"| 20285₁₀ ||class="entry q0 g0"| 15080₈ ||class="entry q1 g0"| 7023₁₀ ||class="entry q1 g0"| 23651₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (11392, 25745, 8843, 22088, 8015, 12757) ||class="entry q0 g0"| 11392₄ ||class="entry q1 g0"| 25745₆ ||class="entry q1 g0"| 8843₆ ||class="entry q0 g0"| 22088₆ ||class="entry q1 g0"| 8015₁₀ ||class="entry q1 g0"| 12757₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (18592, 26913, 18509, 12904, 4863, 23315) ||class="entry q0 g0"| 18592₄ ||class="entry q1 g0"| 26913₆ ||class="entry q1 g0"| 18509₆ ||class="entry q0 g0"| 12904₆ ||class="entry q1 g0"| 4863₁₀ ||class="entry q1 g0"| 23315₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (25728, 11395, 12497, 7752, 22365, 9103) ||class="entry q0 g0"| 25728₄ ||class="entry q1 g0"| 11395₆ ||class="entry q1 g0"| 12497₆ ||class="entry q0 g0"| 7752₆ ||class="entry q1 g0"| 22365₁₀ ||class="entry q1 g0"| 9103₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (26656, 18597, 23297, 4840, 13179, 18527) ||class="entry q0 g0"| 26656₄ ||class="entry q1 g0"| 18597₆ ||class="entry q1 g0"| 23297₆ ||class="entry q0 g0"| 4840₆ ||class="entry q1 g0"| 13179₁₀ ||class="entry q1 g0"| 18527₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (10400, 2343, 20011, 21096, 29433, 23925) ||class="entry q0 g0"| 10400₄ ||class="entry q1 g0"| 2343₆ ||class="entry q1 g0"| 20011₈ ||class="entry q0 g0"| 21096₆ ||class="entry q1 g0"| 29433₁₀ ||class="entry q1 g0"| 23925₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (19584, 1175, 9453, 13896, 32585, 14259) ||class="entry q0 g0"| 19584₄ ||class="entry q1 g0"| 1175₆ ||class="entry q1 g0"| 9453₈ ||class="entry q0 g0"| 13896₆ ||class="entry q1 g0"| 32585₁₀ ||class="entry q1 g0"| 14259₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (24736, 16693, 23665, 6760, 15083, 20271) ||class="entry q0 g0"| 24736₄ ||class="entry q1 g0"| 16693₆ ||class="entry q1 g0"| 23665₈ ||class="entry q0 g0"| 6760₆ ||class="entry q1 g0"| 15083₁₀ ||class="entry q1 g0"| 20271₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (27648, 9491, 14241, 5832, 24269, 9471) ||class="entry q0 g0"| 27648₄ ||class="entry q1 g0"| 9491₆ ||class="entry q1 g0"| 14241₈ ||class="entry q0 g0"| 5832₆ ||class="entry q1 g0"| 24269₁₀ ||class="entry q1 g0"| 9471₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (1410, 6189, 25373, 32586, 25587, 28739) ||class="entry q0 g0"| 1410₄ ||class="entry q1 g0"| 6189₆ ||class="entry q1 g0"| 25373₈ ||class="entry q0 g0"| 32586₁₀ ||class="entry q1 g0"| 25587₁₀ ||class="entry q1 g0"| 28739₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (1424, 16997, 14663, 32600, 14779, 10777) ||class="entry q0 g0"| 1424₄ ||class="entry q1 g0"| 16997₆ ||class="entry q1 g0"| 14663₈ ||class="entry q0 g0"| 32600₁₀ ||class="entry q1 g0"| 14779₁₀ ||class="entry q1 g0"| 10777₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (2100, 5259, 24923, 29436, 28501, 29189) ||class="entry q0 g0"| 2100₄ ||class="entry q1 g0"| 5259₆ ||class="entry q1 g0"| 24923₈ ||class="entry q0 g0"| 29436₁₀ ||class="entry q1 g0"| 28501₁₀ ||class="entry q1 g0"| 29189₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (2352, 9795, 21143, 29688, 23965, 16841) ||class="entry q0 g0"| 2352₄ ||class="entry q1 g0"| 9795₆ ||class="entry q1 g0"| 21143₈ ||class="entry q0 g0"| 29688₁₀ ||class="entry q1 g0"| 23965₁₀ ||class="entry q1 g0"| 16841₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (9222, 2913, 17309, 24270, 28863, 20675) ||class="entry q0 g0"| 9222₄ ||class="entry q1 g0"| 2913₆ ||class="entry q1 g0"| 17309₈ ||class="entry q0 g0"| 24270₁₀ ||class="entry q1 g0"| 28863₁₀ ||class="entry q1 g0"| 20675₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (9236, 20777, 6599, 24284, 10999, 2713) ||class="entry q0 g0"| 9236₄ ||class="entry q1 g0"| 20777₆ ||class="entry q1 g0"| 6599₈ ||class="entry q0 g0"| 24284₁₀ ||class="entry q1 g0"| 10999₁₀ ||class="entry q1 g0"| 2713₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16422, 1745, 10587, 15086, 32015, 14853) ||class="entry q0 g0"| 16422₄ ||class="entry q1 g0"| 1745₆ ||class="entry q1 g0"| 10587₈ ||class="entry q0 g0"| 15086₁₀ ||class="entry q1 g0"| 32015₁₀ ||class="entry q1 g0"| 14853₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16674, 13337, 6807, 15338, 20423, 2505) ||class="entry q0 g0"| 16674₄ ||class="entry q1 g0"| 13337₆ ||class="entry q1 g0"| 6807₈ ||class="entry q0 g0"| 15338₁₀ ||class="entry q1 g0"| 20423₁₀ ||class="entry q1 g0"| 2505₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (1158, 10981, 20689, 32334, 20795, 17295) ||class="entry q0 g0"| 1158₄ ||class="entry q1 g0"| 10981₈ ||class="entry q1 g0"| 20689₆ ||class="entry q0 g0"| 32334₁₀ ||class="entry q1 g0"| 20795₈ ||class="entry q1 g0"| 17295₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (1172, 28845, 2699, 32348, 2931, 6613) ||class="entry q0 g0"| 1172₄ ||class="entry q1 g0"| 28845₈ ||class="entry q1 g0"| 2699₆ ||class="entry q0 g0"| 32348₁₀ ||class="entry q1 g0"| 2931₈ ||class="entry q1 g0"| 6613₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (2086, 20163, 15105, 29422, 13597, 10335) ||class="entry q0 g0"| 2086₄ ||class="entry q1 g0"| 20163₈ ||class="entry q1 g0"| 15105₆ ||class="entry q0 g0"| 29422₁₀ ||class="entry q1 g0"| 13597₈ ||class="entry q1 g0"| 10335₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (2338, 31755, 2253, 29674, 2005, 7059) ||class="entry q0 g0"| 2338₄ ||class="entry q1 g0"| 31755₈ ||class="entry q1 g0"| 2253₆ ||class="entry q0 g0"| 29674₁₀ ||class="entry q1 g0"| 2005₈ ||class="entry q1 g0"| 7059₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (9474, 14761, 28753, 24522, 17015, 25359) ||class="entry q0 g0"| 9474₄ ||class="entry q1 g0"| 14761₈ ||class="entry q1 g0"| 28753₆ ||class="entry q0 g0"| 24522₁₀ ||class="entry q1 g0"| 17015₈ ||class="entry q1 g0"| 25359₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (9488, 25569, 10763, 24536, 6207, 14677) ||class="entry q0 g0"| 9488₄ ||class="entry q1 g0"| 25569₈ ||class="entry q1 g0"| 10763₆ ||class="entry q0 g0"| 24536₁₀ ||class="entry q1 g0"| 6207₈ ||class="entry q1 g0"| 14677₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (16436, 23705, 29441, 15100, 10055, 24671) ||class="entry q0 g0"| 16436₄ ||class="entry q1 g0"| 23705₈ ||class="entry q1 g0"| 29441₆ ||class="entry q0 g0"| 15100₁₀ ||class="entry q1 g0"| 10055₈ ||class="entry q1 g0"| 24671₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (16688, 28241, 16589, 15352, 5519, 21395) ||class="entry q0 g0"| 16688₄ ||class="entry q1 g0"| 28241₈ ||class="entry q1 g0"| 16589₆ ||class="entry q0 g0"| 15352₁₀ ||class="entry q1 g0"| 5519₈ ||class="entry q1 g0"| 21395₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (736, 19173, 12503, 30760, 12603, 9097) ||class="entry q0 g0"| 736₄ ||class="entry q1 g0"| 19173₈ ||class="entry q1 g0"| 12503₈ ||class="entry q0 g0"| 30760₆ ||class="entry q1 g0"| 12603₈ ||class="entry q1 g0"| 9097₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (3648, 11971, 23303, 29832, 21789, 18521) ||class="entry q0 g0"| 3648₄ ||class="entry q1 g0"| 11971₈ ||class="entry q1 g0"| 23303₈ ||class="entry q0 g0"| 29832₆ ||class="entry q1 g0"| 21789₈ ||class="entry q1 g0"| 18521₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (4264, 22701, 8863, 27232, 9075, 12737) ||class="entry q0 g0"| 4264₄ ||class="entry q1 g0"| 22701₈ ||class="entry q1 g0"| 8863₈ ||class="entry q0 g0"| 27232₆ ||class="entry q1 g0"| 9075₈ ||class="entry q1 g0"| 12737₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (7176, 15499, 18767, 26304, 18261, 23057) ||class="entry q0 g0"| 7176₄ ||class="entry q1 g0"| 15499₈ ||class="entry q1 g0"| 18767₈ ||class="entry q0 g0"| 26304₆ ||class="entry q1 g0"| 18261₈ ||class="entry q1 g0"| 23057₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (8800, 27489, 9115, 22696, 4287, 12485) ||class="entry q0 g0"| 8800₄ ||class="entry q1 g0"| 27489₈ ||class="entry q1 g0"| 9115₈ ||class="entry q0 g0"| 22696₆ ||class="entry q1 g0"| 4287₈ ||class="entry q1 g0"| 12485₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (12328, 31017, 12755, 19168, 759, 8845) ||class="entry q0 g0"| 12328₄ ||class="entry q1 g0"| 31017₈ ||class="entry q1 g0"| 12755₈ ||class="entry q0 g0"| 19168₆ ||class="entry q1 g0"| 759₈ ||class="entry q1 g0"| 8845₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (17984, 26321, 18781, 15496, 7439, 23043) ||class="entry q0 g0"| 17984₄ ||class="entry q1 g0"| 26321₈ ||class="entry q1 g0"| 18781₈ ||class="entry q0 g0"| 15496₆ ||class="entry q1 g0"| 7439₈ ||class="entry q1 g0"| 23043₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (21512, 29849, 23317, 11968, 3911, 18507) ||class="entry q0 g0"| 21512₄ ||class="entry q1 g0"| 29849₈ ||class="entry q1 g0"| 23317₈ ||class="entry q0 g0"| 11968₆ ||class="entry q1 g0"| 3911₈ ||class="entry q1 g0"| 18507₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (1728, 10067, 23671, 31752, 23693, 20265) ||class="entry q0 g0"| 1728₄ ||class="entry q1 g0"| 10067₈ ||class="entry q1 g0"| 23671₁₀ ||class="entry q0 g0"| 31752₆ ||class="entry q1 g0"| 23693₈ ||class="entry q1 g0"| 20265₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (2656, 17269, 14247, 28840, 14507, 9465) ||class="entry q0 g0"| 2656₄ ||class="entry q1 g0"| 17269₈ ||class="entry q1 g0"| 14247₁₀ ||class="entry q0 g0"| 28840₆ ||class="entry q1 g0"| 14507₈ ||class="entry q1 g0"| 9465₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (5256, 13595, 20031, 28224, 20165, 23905) ||class="entry q0 g0"| 5256₄ ||class="entry q1 g0"| 13595₈ ||class="entry q1 g0"| 20031₁₀ ||class="entry q0 g0"| 28224₆ ||class="entry q1 g0"| 20165₈ ||class="entry q1 g0"| 23905₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (6184, 20797, 9711, 25312, 10979, 14001) ||class="entry q0 g0"| 6184₄ ||class="entry q1 g0"| 20797₈ ||class="entry q1 g0"| 9711₁₀ ||class="entry q0 g0"| 25312₆ ||class="entry q1 g0"| 10979₈ ||class="entry q1 g0"| 14001₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (9792, 1751, 20283, 23688, 32009, 23653) ||class="entry q0 g0"| 9792₄ ||class="entry q1 g0"| 1751₈ ||class="entry q1 g0"| 20283₁₀ ||class="entry q0 g0"| 23688₆ ||class="entry q1 g0"| 32009₈ ||class="entry q1 g0"| 23653₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (13320, 5279, 23923, 20160, 28481, 20013) ||class="entry q0 g0"| 13320₄ ||class="entry q1 g0"| 5279₈ ||class="entry q1 g0"| 23923₁₀ ||class="entry q0 g0"| 20160₆ ||class="entry q1 g0"| 28481₈ ||class="entry q1 g0"| 20013₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (16992, 2919, 9725, 14504, 28857, 13987) ||class="entry q0 g0"| 16992₄ ||class="entry q1 g0"| 2919₈ ||class="entry q1 g0"| 9725₁₀ ||class="entry q0 g0"| 14504₆ ||class="entry q1 g0"| 28857₈ ||class="entry q1 g0"| 13987₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (20520, 6447, 14261, 10976, 25329, 9451) ||class="entry q0 g0"| 20520₄ ||class="entry q1 g0"| 6447₈ ||class="entry q1 g0"| 14261₁₀ ||class="entry q0 g0"| 10976₆ ||class="entry q1 g0"| 25329₈ ||class="entry q1 g0"| 9451₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (1170, 5837, 27885, 32346, 27923, 32691) ||class="entry q0 g0"| 1170₄ ||class="entry q1 g0"| 5837₈ ||class="entry q1 g0"| 27885₁₀ ||class="entry q0 g0"| 32346₁₀ ||class="entry q1 g0"| 27923₈ ||class="entry q1 g0"| 32691₁₂ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (2340, 6763, 28331, 29676, 25013, 32245) ||class="entry q0 g0"| 2340₄ ||class="entry q1 g0"| 6763₈ ||class="entry q1 g0"| 28331₁₀ ||class="entry q0 g0"| 29676₁₀ ||class="entry q1 g0"| 25013₈ ||class="entry q1 g0"| 32245₁₂ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (9234, 14153, 32673, 24282, 19607, 27903) ||class="entry q0 g0"| 9234₄ ||class="entry q1 g0"| 14153₈ ||class="entry q1 g0"| 32673₁₀ ||class="entry q0 g0"| 24282₁₀ ||class="entry q1 g0"| 19607₈ ||class="entry q1 g0"| 27903₁₂ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (16676, 21113, 31985, 15340, 10663, 28591) ||class="entry q0 g0"| 16676₄ ||class="entry q1 g0"| 21113₈ ||class="entry q1 g0"| 31985₁₀ ||class="entry q0 g0"| 15340₁₀ ||class="entry q1 g0"| 10663₈ ||class="entry q1 g0"| 28591₁₂ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (1412, 32333, 1403, 32588, 1427, 5669) ||class="entry q0 g0"| 1412₄ ||class="entry q1 g0"| 32333₁₀ ||class="entry q1 g0"| 1403₈ ||class="entry q0 g0"| 32588₁₀ ||class="entry q1 g0"| 1427₆ ||class="entry q1 g0"| 5669₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (2098, 29419, 1853, 29434, 2357, 5219) ||class="entry q0 g0"| 2098₄ ||class="entry q1 g0"| 29419₁₀ ||class="entry q1 g0"| 1853₈ ||class="entry q0 g0"| 29434₁₀ ||class="entry q1 g0"| 2357₆ ||class="entry q1 g0"| 5219₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (9476, 24521, 5687, 24524, 9239, 1385) ||class="entry q0 g0"| 9476₄ ||class="entry q1 g0"| 24521₁₀ ||class="entry q1 g0"| 5687₈ ||class="entry q0 g0"| 24524₁₀ ||class="entry q1 g0"| 9239₆ ||class="entry q1 g0"| 1385₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (16434, 15097, 5479, 15098, 16679, 1593) ||class="entry q0 g0"| 16434₄ ||class="entry q1 g0"| 15097₁₀ ||class="entry q1 g0"| 5479₈ ||class="entry q0 g0"| 15098₁₀ ||class="entry q1 g0"| 16679₆ ||class="entry q1 g0"| 1593₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (418, 30107, 4029, 31594, 3653, 7395) ||class="entry q0 g0"| 418₄ ||class="entry q1 g0"| 30107₁₀ ||class="entry q1 g0"| 4029₁₀ ||class="entry q0 g0"| 31594₁₀ ||class="entry q1 g0"| 3653₆ ||class="entry q1 g0"| 7395₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (420, 5115, 27099, 31596, 26661, 31365) ||class="entry q0 g0"| 420₄ ||class="entry q1 g0"| 5115₁₀ ||class="entry q1 g0"| 27099₁₀ ||class="entry q0 g0"| 31596₁₀ ||class="entry q1 g0"| 26661₆ ||class="entry q1 g0"| 31365₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (432, 12243, 21991, 31608, 21517, 18105) ||class="entry q0 g0"| 432₄ ||class="entry q1 g0"| 12243₁₀ ||class="entry q1 g0"| 21991₁₀ ||class="entry q0 g0"| 31608₁₀ ||class="entry q1 g0"| 21517₆ ||class="entry q1 g0"| 18105₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (3090, 8029, 27549, 30426, 25731, 30915) ||class="entry q0 g0"| 3090₄ ||class="entry q1 g0"| 8029₁₀ ||class="entry q1 g0"| 27549₁₀ ||class="entry q0 g0"| 30426₁₀ ||class="entry q1 g0"| 25731₆ ||class="entry q1 g0"| 30915₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (3092, 31037, 3579, 30428, 739, 7845) ||class="entry q0 g0"| 3092₄ ||class="entry q1 g0"| 31037₁₀ ||class="entry q1 g0"| 3579₁₀ ||class="entry q0 g0"| 30428₁₀ ||class="entry q1 g0"| 739₆ ||class="entry q1 g0"| 7845₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (3344, 19445, 15927, 30680, 12331, 11625) ||class="entry q0 g0"| 3344₄ ||class="entry q1 g0"| 19445₁₀ ||class="entry q1 g0"| 15927₁₀ ||class="entry q0 g0"| 30680₁₀ ||class="entry q1 g0"| 12331₆ ||class="entry q1 g0"| 11625₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8230, 26327, 12093, 23278, 7433, 15459) ||class="entry q0 g0"| 8230₄ ||class="entry q1 g0"| 26327₁₀ ||class="entry q1 g0"| 12093₁₀ ||class="entry q0 g0"| 23278₁₀ ||class="entry q1 g0"| 7433₆ ||class="entry q1 g0"| 15459₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8244, 15519, 30055, 23292, 18241, 26169) ||class="entry q0 g0"| 8244₄ ||class="entry q1 g0"| 15519₁₀ ||class="entry q1 g0"| 30055₁₀ ||class="entry q0 g0"| 23292₁₀ ||class="entry q1 g0"| 18241₆ ||class="entry q1 g0"| 26169₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8484, 12927, 31383, 23532, 18849, 27081) ||class="entry q0 g0"| 8484₄ ||class="entry q1 g0"| 12927₁₀ ||class="entry q1 g0"| 31383₁₀ ||class="entry q0 g0"| 23532₁₀ ||class="entry q1 g0"| 18849₆ ||class="entry q1 g0"| 27081₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (17414, 27495, 17915, 16078, 4281, 22181) ||class="entry q0 g0"| 17414₄ ||class="entry q1 g0"| 27495₁₀ ||class="entry q1 g0"| 17915₁₀ ||class="entry q0 g0"| 16078₁₀ ||class="entry q1 g0"| 4281₆ ||class="entry q1 g0"| 22181₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (17426, 22351, 31175, 16090, 11409, 27289) ||class="entry q0 g0"| 17426₄ ||class="entry q1 g0"| 22351₁₀ ||class="entry q1 g0"| 31175₁₀ ||class="entry q0 g0"| 16090₁₀ ||class="entry q1 g0"| 11409₆ ||class="entry q1 g0"| 27289₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (17666, 22959, 30263, 16330, 8817, 25961) ||class="entry q0 g0"| 17666₄ ||class="entry q1 g0"| 22959₁₀ ||class="entry q1 g0"| 30263₁₀ ||class="entry q0 g0"| 16330₁₀ ||class="entry q1 g0"| 8817₆ ||class="entry q1 g0"| 25961₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (1430, 9221, 24353, 32606, 24539, 19583) ||class="entry q0 g0"| 1430₆ ||class="entry q1 g0"| 9221₄ ||class="entry q1 g0"| 24353₈ ||class="entry q0 g0"| 32606₁₂ ||class="entry q1 g0"| 24539₁₂ ||class="entry q1 g0"| 19583₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (2358, 16419, 13553, 29694, 15357, 10159) ||class="entry q0 g0"| 2358₆ ||class="entry q1 g0"| 16419₄ ||class="entry q1 g0"| 13553₈ ||class="entry q0 g0"| 29694₁₂ ||class="entry q1 g0"| 15357₁₂ ||class="entry q1 g0"| 10159₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (9494, 1409, 19565, 24542, 32351, 24371) ||class="entry q0 g0"| 9494₆ ||class="entry q1 g0"| 1409₄ ||class="entry q1 g0"| 19565₈ ||class="entry q0 g0"| 24542₁₂ ||class="entry q1 g0"| 32351₁₂ ||class="entry q1 g0"| 24371₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (16694, 2097, 9899, 15358, 29679, 13813) ||class="entry q0 g0"| 16694₆ ||class="entry q1 g0"| 2097₄ ||class="entry q1 g0"| 9899₈ ||class="entry q0 g0"| 15358₁₂ ||class="entry q1 g0"| 29679₁₂ ||class="entry q1 g0"| 13813₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (1734, 16691, 14865, 31758, 15085, 10575) ||class="entry q0 g0"| 1734₆ ||class="entry q1 g0"| 16691₆ ||class="entry q1 g0"| 14865₆ ||class="entry q0 g0"| 31758₈ ||class="entry q1 g0"| 15085₁₀ ||class="entry q1 g0"| 10575₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (2662, 9493, 20929, 28846, 24267, 17055) ||class="entry q0 g0"| 2662₆ ||class="entry q1 g0"| 9493₆ ||class="entry q1 g0"| 20929₆ ||class="entry q0 g0"| 28846₈ ||class="entry q1 g0"| 24267₁₀ ||class="entry q1 g0"| 17055₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (5276, 2355, 29187, 28244, 29421, 24925) ||class="entry q0 g0"| 5276₆ ||class="entry q1 g0"| 2355₆ ||class="entry q1 g0"| 29187₆ ||class="entry q0 g0"| 28244₈ ||class="entry q1 g0"| 29421₁₀ ||class="entry q1 g0"| 24925₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (6442, 1429, 28741, 25570, 32331, 25371) ||class="entry q0 g0"| 6442₆ ||class="entry q1 g0"| 1429₆ ||class="entry q1 g0"| 28741₆ ||class="entry q0 g0"| 25570₈ ||class="entry q1 g0"| 32331₁₀ ||class="entry q1 g0"| 25371₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (10064, 2103, 16587, 23960, 29673, 21397) ||class="entry q0 g0"| 10064₆ ||class="entry q1 g0"| 2103₆ ||class="entry q1 g0"| 16587₆ ||class="entry q0 g0"| 23960₈ ||class="entry q1 g0"| 29673₁₀ ||class="entry q1 g0"| 21397₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (13578, 16439, 2265, 20418, 15337, 7047) ||class="entry q0 g0"| 13578₆ ||class="entry q1 g0"| 16439₆ ||class="entry q1 g0"| 2265₆ ||class="entry q0 g0"| 20418₈ ||class="entry q1 g0"| 15337₁₀ ||class="entry q1 g0"| 7047₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (17264, 1415, 10765, 14776, 32345, 14675) ||class="entry q0 g0"| 17264₆ ||class="entry q1 g0"| 1415₆ ||class="entry q1 g0"| 10765₆ ||class="entry q0 g0"| 14776₈ ||class="entry q1 g0"| 32345₁₀ ||class="entry q1 g0"| 14675₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (20540, 9479, 2953, 10996, 24281, 6359) ||class="entry q0 g0"| 20540₆ ||class="entry q1 g0"| 9479₆ ||class="entry q1 g0"| 2953₆ ||class="entry q0 g0"| 10996₈ ||class="entry q1 g0"| 24281₁₀ ||class="entry q1 g0"| 6359₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (742, 11397, 22193, 30766, 22363, 17903) ||class="entry q0 g0"| 742₆ ||class="entry q1 g0"| 11397₆ ||class="entry q1 g0"| 22193₈ ||class="entry q0 g0"| 30766₈ ||class="entry q1 g0"| 22363₁₀ ||class="entry q1 g0"| 17903₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (754, 4269, 27277, 30778, 27507, 31187) ||class="entry q0 g0"| 754₆ ||class="entry q1 g0"| 4269₆ ||class="entry q1 g0"| 27277₈ ||class="entry q0 g0"| 30778₈ ||class="entry q1 g0"| 27507₁₀ ||class="entry q1 g0"| 31187₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3654, 18595, 15713, 29838, 13181, 11839) ||class="entry q0 g0"| 3654₆ ||class="entry q1 g0"| 18595₆ ||class="entry q1 g0"| 15713₈ ||class="entry q0 g0"| 29838₈ ||class="entry q1 g0"| 13181₁₀ ||class="entry q1 g0"| 11839₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3908, 7179, 26827, 30092, 26581, 31637) ||class="entry q0 g0"| 3908₆ ||class="entry q1 g0"| 7179₆ ||class="entry q1 g0"| 26827₈ ||class="entry q0 g0"| 30092₈ ||class="entry q1 g0"| 26581₁₀ ||class="entry q1 g0"| 31637₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4282, 741, 30917, 27250, 31035, 27547) ||class="entry q0 g0"| 4282₆ ||class="entry q1 g0"| 741₆ ||class="entry q1 g0"| 30917₈ ||class="entry q0 g0"| 27250₈ ||class="entry q1 g0"| 31035₁₀ ||class="entry q1 g0"| 27547₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4284, 25733, 7843, 27252, 8027, 3581) ||class="entry q0 g0"| 4284₆ ||class="entry q1 g0"| 25733₆ ||class="entry q1 g0"| 7843₈ ||class="entry q0 g0"| 27252₈ ||class="entry q1 g0"| 8027₁₀ ||class="entry q1 g0"| 3581₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7434, 26659, 7397, 26562, 5117, 4027) ||class="entry q0 g0"| 7434₆ ||class="entry q1 g0"| 26659₆ ||class="entry q1 g0"| 7397₈ ||class="entry q0 g0"| 26562₈ ||class="entry q1 g0"| 5117₁₀ ||class="entry q1 g0"| 4027₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7436, 3651, 31363, 26564, 30109, 27101) ||class="entry q0 g0"| 7436₆ ||class="entry q1 g0"| 3651₆ ||class="entry q1 g0"| 31363₈ ||class="entry q0 g0"| 26564₈ ||class="entry q1 g0"| 30109₁₀ ||class="entry q1 g0"| 27101₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (8818, 12585, 31169, 22714, 19191, 27295) ||class="entry q0 g0"| 8818₆ ||class="entry q1 g0"| 12585₆ ||class="entry q1 g0"| 31169₈ ||class="entry q0 g0"| 22714₈ ||class="entry q1 g0"| 19191₁₀ ||class="entry q1 g0"| 27295₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (9072, 25985, 11371, 22968, 7775, 16181) ||class="entry q0 g0"| 9072₆ ||class="entry q1 g0"| 25985₆ ||class="entry q1 g0"| 11371₈ ||class="entry q0 g0"| 22968₈ ||class="entry q1 g0"| 7775₁₀ ||class="entry q1 g0"| 16181₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (11398, 753, 17645, 22094, 31023, 22451) ||class="entry q0 g0"| 11398₆ ||class="entry q1 g0"| 753₆ ||class="entry q1 g0"| 17645₈ ||class="entry q0 g0"| 22094₈ ||class="entry q1 g0"| 31023₁₀ ||class="entry q1 g0"| 22451₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (11650, 12345, 30497, 22346, 19431, 25727) ||class="entry q0 g0"| 11650₆ ||class="entry q1 g0"| 12345₆ ||class="entry q1 g0"| 30497₈ ||class="entry q0 g0"| 22346₈ ||class="entry q1 g0"| 19431₁₀ ||class="entry q1 g0"| 25727₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12346, 9057, 27529, 19186, 22719, 30935) ||class="entry q0 g0"| 12346₆ ||class="entry q1 g0"| 9057₆ ||class="entry q1 g0"| 27529₈ ||class="entry q0 g0"| 19186₈ ||class="entry q1 g0"| 22719₁₀ ||class="entry q1 g0"| 30935₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12586, 11649, 25721, 19426, 22111, 30503) ||class="entry q0 g0"| 12586₆ ||class="entry q1 g0"| 11649₆ ||class="entry q1 g0"| 25721₈ ||class="entry q0 g0"| 19426₈ ||class="entry q1 g0"| 22111₁₀ ||class="entry q1 g0"| 30503₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18244, 21529, 31377, 15756, 12231, 27087) ||class="entry q0 g0"| 18244₆ ||class="entry q1 g0"| 21529₆ ||class="entry q1 g0"| 31377₈ ||class="entry q0 g0"| 15756₈ ||class="entry q1 g0"| 12231₁₀ ||class="entry q1 g0"| 27087₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18256, 26673, 18093, 15768, 5103, 22003) ||class="entry q0 g0"| 18256₆ ||class="entry q1 g0"| 26673₆ ||class="entry q1 g0"| 18093₈ ||class="entry q0 g0"| 15768₈ ||class="entry q1 g0"| 5103₁₀ ||class="entry q1 g0"| 22003₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18598, 3905, 11819, 12910, 29855, 15733) ||class="entry q0 g0"| 18598₆ ||class="entry q1 g0"| 3905₆ ||class="entry q1 g0"| 11819₈ ||class="entry q0 g0"| 12910₈ ||class="entry q1 g0"| 29855₁₀ ||class="entry q1 g0"| 15733₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18612, 21769, 29809, 12924, 11991, 26415) ||class="entry q0 g0"| 18612₆ ||class="entry q1 g0"| 21769₆ ||class="entry q1 g0"| 29809₈ ||class="entry q0 g0"| 12924₈ ||class="entry q1 g0"| 11991₁₀ ||class="entry q1 g0"| 26415₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21532, 18609, 26409, 11988, 13167, 29815) ||class="entry q0 g0"| 21532₆ ||class="entry q1 g0"| 18609₆ ||class="entry q1 g0"| 26409₈ ||class="entry q0 g0"| 11988₈ ||class="entry q1 g0"| 13167₁₀ ||class="entry q1 g0"| 29815₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21772, 18001, 26841, 12228, 15759, 31623) ||class="entry q0 g0"| 21772₆ ||class="entry q1 g0"| 18001₆ ||class="entry q1 g0"| 26841₈ ||class="entry q0 g0"| 12228₈ ||class="entry q1 g0"| 15759₁₀ ||class="entry q1 g0"| 31623₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25748, 4267, 3309, 7772, 27509, 8115) ||class="entry q0 g0"| 25748₆ ||class="entry q1 g0"| 4267₆ ||class="entry q1 g0"| 3309₈ ||class="entry q0 g0"| 7772₈ ||class="entry q1 g0"| 27509₁₀ ||class="entry q1 g0"| 8115₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26000, 8803, 16161, 8024, 22973, 11391) ||class="entry q0 g0"| 26000₆ ||class="entry q1 g0"| 8803₆ ||class="entry q1 g0"| 16161₈ ||class="entry q0 g0"| 8024₈ ||class="entry q1 g0"| 22973₁₀ ||class="entry q1 g0"| 11391₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26914, 7181, 3755, 5098, 26579, 7669) ||class="entry q0 g0"| 26914₆ ||class="entry q1 g0"| 7181₆ ||class="entry q1 g0"| 3755₈ ||class="entry q0 g0"| 5098₈ ||class="entry q1 g0"| 26579₁₀ ||class="entry q1 g0"| 7669₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26928, 17989, 21745, 5112, 15771, 18351) ||class="entry q0 g0"| 26928₆ ||class="entry q1 g0"| 17989₆ ||class="entry q1 g0"| 21745₈ ||class="entry q0 g0"| 5112₈ ||class="entry q1 g0"| 15771₁₀ ||class="entry q1 g0"| 18351₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (996, 30765, 795, 31020, 1011, 4165) ||class="entry q0 g0"| 996₆ ||class="entry q1 g0"| 30765₈ ||class="entry q1 g0"| 795₆ ||class="entry q0 g0"| 31020₈ ||class="entry q1 g0"| 1011₈ ||class="entry q1 g0"| 4165₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (3666, 29835, 349, 29850, 3925, 4611) ||class="entry q0 g0"| 3666₆ ||class="entry q1 g0"| 29835₈ ||class="entry q1 g0"| 349₆ ||class="entry q0 g0"| 29850₈ ||class="entry q1 g0"| 3925₈ ||class="entry q1 g0"| 4611₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (4524, 27237, 4435, 27492, 4539, 525) ||class="entry q0 g0"| 4524₆ ||class="entry q1 g0"| 27237₈ ||class="entry q1 g0"| 4435₆ ||class="entry q0 g0"| 27492₈ ||class="entry q1 g0"| 4539₈ ||class="entry q1 g0"| 525₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (7194, 26307, 4885, 26322, 7453, 75) ||class="entry q0 g0"| 7194₆ ||class="entry q1 g0"| 26307₈ ||class="entry q1 g0"| 4885₆ ||class="entry q0 g0"| 26322₈ ||class="entry q1 g0"| 7453₈ ||class="entry q1 g0"| 75₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (9060, 22953, 4183, 22956, 8823, 777) ||class="entry q0 g0"| 9060₆ ||class="entry q1 g0"| 22953₈ ||class="entry q1 g0"| 4183₆ ||class="entry q0 g0"| 22956₈ ||class="entry q1 g0"| 8823₈ ||class="entry q1 g0"| 777₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (11652, 22105, 4423, 22348, 11655, 537) ||class="entry q0 g0"| 11652₆ ||class="entry q1 g0"| 22105₈ ||class="entry q1 g0"| 4423₆ ||class="entry q0 g0"| 22348₈ ||class="entry q1 g0"| 11655₈ ||class="entry q1 g0"| 537₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (12588, 19425, 543, 19428, 12351, 4417) ||class="entry q0 g0"| 12588₆ ||class="entry q1 g0"| 19425₈ ||class="entry q1 g0"| 543₆ ||class="entry q0 g0"| 19428₈ ||class="entry q1 g0"| 12351₈ ||class="entry q1 g0"| 4417₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (18002, 15513, 4871, 15514, 18247, 89) ||class="entry q0 g0"| 18002₆ ||class="entry q1 g0"| 15513₈ ||class="entry q1 g0"| 4871₆ ||class="entry q0 g0"| 15514₈ ||class="entry q1 g0"| 18247₈ ||class="entry q1 g0"| 89₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (18610, 13161, 4631, 12922, 18615, 329) ||class="entry q0 g0"| 18610₆ ||class="entry q1 g0"| 13161₈ ||class="entry q1 g0"| 4631₆ ||class="entry q0 g0"| 12922₈ ||class="entry q1 g0"| 18615₈ ||class="entry q1 g0"| 329₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (21530, 11985, 335, 11986, 21775, 4625) ||class="entry q0 g0"| 21530₆ ||class="entry q1 g0"| 11985₈ ||class="entry q1 g0"| 335₆ ||class="entry q0 g0"| 11986₈ ||class="entry q1 g0"| 21775₈ ||class="entry q1 g0"| 4625₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (25988, 7755, 797, 8012, 26005, 4163) ||class="entry q0 g0"| 25988₆ ||class="entry q1 g0"| 7755₈ ||class="entry q1 g0"| 797₆ ||class="entry q0 g0"| 8012₈ ||class="entry q1 g0"| 26005₈ ||class="entry q1 g0"| 4163₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (26674, 4845, 347, 4858, 26931, 4613) ||class="entry q0 g0"| 26674₆ ||class="entry q1 g0"| 4845₈ ||class="entry q1 g0"| 347₆ ||class="entry q0 g0"| 4858₈ ||class="entry q1 g0"| 26931₈ ||class="entry q1 g0"| 4613₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (438, 18867, 13185, 31614, 12909, 8415) ||class="entry q0 g0"| 438₆ ||class="entry q1 g0"| 18867₈ ||class="entry q1 g0"| 13185₆ ||class="entry q0 g0"| 31614₁₂ ||class="entry q1 g0"| 12909₈ ||class="entry q1 g0"| 8415₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (3350, 11669, 22609, 30686, 22091, 19215) ||class="entry q0 g0"| 3350₆ ||class="entry q1 g0"| 11669₈ ||class="entry q1 g0"| 22609₆ ||class="entry q0 g0"| 30686₁₂ ||class="entry q1 g0"| 22091₈ ||class="entry q1 g0"| 19215₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (8502, 26679, 8397, 23550, 5097, 13203) ||class="entry q0 g0"| 8502₆ ||class="entry q1 g0"| 26679₈ ||class="entry q1 g0"| 8397₆ ||class="entry q0 g0"| 23550₁₂ ||class="entry q1 g0"| 5097₈ ||class="entry q1 g0"| 13203₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (17686, 25991, 18955, 16350, 7769, 22869) ||class="entry q0 g0"| 17686₆ ||class="entry q1 g0"| 25991₈ ||class="entry q1 g0"| 18955₆ ||class="entry q0 g0"| 16350₁₂ ||class="entry q1 g0"| 7769₈ ||class="entry q1 g0"| 22869₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (2000, 10675, 21383, 32024, 21101, 16601) ||class="entry q0 g0"| 2000₆ ||class="entry q1 g0"| 10675₈ ||class="entry q1 g0"| 21383₈ ||class="entry q0 g0"| 32024₈ ||class="entry q1 g0"| 21101₈ ||class="entry q1 g0"| 16601₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (2928, 19861, 14423, 29112, 13899, 11017) ||class="entry q0 g0"| 2928₆ ||class="entry q1 g0"| 19861₈ ||class="entry q1 g0"| 14423₈ ||class="entry q0 g0"| 29112₈ ||class="entry q1 g0"| 13899₈ ||class="entry q1 g0"| 11017₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (5514, 25011, 7061, 28482, 6765, 2251) ||class="entry q0 g0"| 5514₆ ||class="entry q1 g0"| 25011₈ ||class="entry q1 g0"| 7061₈ ||class="entry q0 g0"| 28482₈ ||class="entry q1 g0"| 6765₈ ||class="entry q1 g0"| 2251₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6204, 27925, 6611, 25332, 5835, 2701) ||class="entry q0 g0"| 6204₆ ||class="entry q1 g0"| 27925₈ ||class="entry q1 g0"| 6611₈ ||class="entry q0 g0"| 25332₈ ||class="entry q1 g0"| 5835₈ ||class="entry q1 g0"| 2701₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9798, 24759, 10589, 23694, 7017, 14851) ||class="entry q0 g0"| 9798₆ ||class="entry q1 g0"| 24759₈ ||class="entry q1 g0"| 10589₈ ||class="entry q0 g0"| 23694₈ ||class="entry q1 g0"| 7017₈ ||class="entry q1 g0"| 14851₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (10420, 13583, 29207, 21116, 20177, 24905) ||class="entry q0 g0"| 10420₆ ||class="entry q1 g0"| 13583₈ ||class="entry q1 g0"| 29207₈ ||class="entry q0 g0"| 21116₈ ||class="entry q1 g0"| 20177₈ ||class="entry q1 g0"| 24905₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (10672, 1991, 16859, 21368, 31769, 21125) ||class="entry q0 g0"| 10672₆ ||class="entry q1 g0"| 1991₈ ||class="entry q1 g0"| 16859₈ ||class="entry q0 g0"| 21368₈ ||class="entry q1 g0"| 31769₈ ||class="entry q1 g0"| 21125₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (13340, 10423, 24911, 20180, 21353, 29201) ||class="entry q0 g0"| 13340₆ ||class="entry q1 g0"| 10423₈ ||class="entry q1 g0"| 24911₈ ||class="entry q0 g0"| 20180₈ ||class="entry q1 g0"| 21353₈ ||class="entry q1 g0"| 29201₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (16998, 27911, 17307, 14510, 5849, 20677) ||class="entry q0 g0"| 16998₆ ||class="entry q1 g0"| 27911₈ ||class="entry q1 g0"| 17307₈ ||class="entry q0 g0"| 14510₈ ||class="entry q1 g0"| 5849₈ ||class="entry q1 g0"| 20677₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (19842, 20543, 28999, 14154, 11233, 25113) ||class="entry q0 g0"| 19842₆ ||class="entry q1 g0"| 20543₈ ||class="entry q1 g0"| 28999₈ ||class="entry q0 g0"| 14154₈ ||class="entry q1 g0"| 11233₈ ||class="entry q1 g0"| 25113₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (19856, 2679, 11037, 14168, 29097, 14403) ||class="entry q0 g0"| 19856₆ ||class="entry q1 g0"| 2679₈ ||class="entry q1 g0"| 11037₈ ||class="entry q0 g0"| 14168₈ ||class="entry q1 g0"| 29097₈ ||class="entry q1 g0"| 14403₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (20778, 19847, 25119, 11234, 13913, 28993) ||class="entry q0 g0"| 20778₆ ||class="entry q1 g0"| 19847₈ ||class="entry q1 g0"| 25119₈ ||class="entry q0 g0"| 11234₈ ||class="entry q1 g0"| 13913₈ ||class="entry q1 g0"| 28993₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (24742, 10069, 14871, 6766, 23691, 10569) ||class="entry q0 g0"| 24742₆ ||class="entry q1 g0"| 10069₈ ||class="entry q1 g0"| 14871₈ ||class="entry q0 g0"| 6766₈ ||class="entry q1 g0"| 23691₈ ||class="entry q1 g0"| 10569₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (24994, 5533, 2523, 7018, 28227, 6789) ||class="entry q0 g0"| 24994₆ ||class="entry q1 g0"| 5533₈ ||class="entry q1 g0"| 2523₈ ||class="entry q0 g0"| 7018₈ ||class="entry q1 g0"| 28227₈ ||class="entry q1 g0"| 6789₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27654, 17267, 20935, 5838, 14509, 17049) ||class="entry q0 g0"| 27654₆ ||class="entry q1 g0"| 17267₈ ||class="entry q1 g0"| 20935₈ ||class="entry q0 g0"| 5838₈ ||class="entry q1 g0"| 14509₈ ||class="entry q1 g0"| 17049₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27668, 6459, 2973, 5852, 25317, 6339) ||class="entry q0 g0"| 27668₆ ||class="entry q1 g0"| 6459₈ ||class="entry q1 g0"| 2973₈ ||class="entry q0 g0"| 5852₈ ||class="entry q1 g0"| 25317₈ ||class="entry q1 g0"| 6339₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (994, 7757, 25981, 31018, 26003, 30243) ||class="entry q0 g0"| 994₆ ||class="entry q1 g0"| 7757₈ ||class="entry q1 g0"| 25981₁₀ ||class="entry q0 g0"| 31018₈ ||class="entry q1 g0"| 26003₈ ||class="entry q1 g0"| 30243₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3668, 4843, 26427, 29852, 26933, 29797) ||class="entry q0 g0"| 3668₆ ||class="entry q1 g0"| 4843₈ ||class="entry q1 g0"| 26427₁₀ ||class="entry q0 g0"| 29852₈ ||class="entry q1 g0"| 26933₈ ||class="entry q1 g0"| 29797₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (4536, 22093, 11631, 27504, 11667, 15921) ||class="entry q0 g0"| 4536₆ ||class="entry q1 g0"| 22093₈ ||class="entry q1 g0"| 11631₁₀ ||class="entry q0 g0"| 27504₈ ||class="entry q1 g0"| 11667₈ ||class="entry q1 g0"| 15921₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (7448, 12907, 18111, 26576, 18869, 21985) ||class="entry q0 g0"| 7448₆ ||class="entry q1 g0"| 12907₈ ||class="entry q1 g0"| 18111₁₀ ||class="entry q0 g0"| 26576₈ ||class="entry q1 g0"| 18869₈ ||class="entry q1 g0"| 21985₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (8820, 22345, 8103, 22716, 11415, 3321) ||class="entry q0 g0"| 8820₆ ||class="entry q1 g0"| 22345₈ ||class="entry q1 g0"| 8103₁₀ ||class="entry q0 g0"| 22716₈ ||class="entry q1 g0"| 11415₈ ||class="entry q1 g0"| 3321₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (11412, 22713, 7863, 22108, 9063, 3561) ||class="entry q0 g0"| 11412₆ ||class="entry q1 g0"| 22713₈ ||class="entry q1 g0"| 7863₁₀ ||class="entry q0 g0"| 22108₈ ||class="entry q1 g0"| 9063₈ ||class="entry q1 g0"| 3561₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (11664, 27249, 11643, 22360, 4527, 15909) ||class="entry q0 g0"| 11664₆ ||class="entry q1 g0"| 27249₈ ||class="entry q1 g0"| 11643₁₀ ||class="entry q0 g0"| 22360₈ ||class="entry q1 g0"| 4527₈ ||class="entry q1 g0"| 15909₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (12334, 8009, 22453, 19174, 25751, 17643) ||class="entry q0 g0"| 12334₆ ||class="entry q1 g0"| 8009₈ ||class="entry q1 g0"| 22453₁₀ ||class="entry q0 g0"| 19174₈ ||class="entry q1 g0"| 25751₈ ||class="entry q1 g0"| 17643₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (18242, 12921, 7415, 15754, 18855, 4009) ||class="entry q0 g0"| 18242₆ ||class="entry q1 g0"| 12921₈ ||class="entry q1 g0"| 7415₁₀ ||class="entry q0 g0"| 15754₈ ||class="entry q1 g0"| 18855₈ ||class="entry q1 g0"| 4009₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (18850, 15753, 7655, 13162, 18007, 3769) ||class="entry q0 g0"| 18850₆ ||class="entry q1 g0"| 15753₈ ||class="entry q1 g0"| 7655₁₀ ||class="entry q0 g0"| 13162₈ ||class="entry q1 g0"| 18007₈ ||class="entry q1 g0"| 3769₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (18864, 26561, 18365, 13176, 7199, 21731) ||class="entry q0 g0"| 18864₆ ||class="entry q1 g0"| 26561₈ ||class="entry q1 g0"| 18365₁₀ ||class="entry q0 g0"| 13176₈ ||class="entry q1 g0"| 7199₈ ||class="entry q1 g0"| 21731₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (21518, 4857, 15731, 11974, 26919, 11821) ||class="entry q0 g0"| 21518₆ ||class="entry q1 g0"| 4857₈ ||class="entry q1 g0"| 15731₁₀ ||class="entry q0 g0"| 11974₈ ||class="entry q1 g0"| 26919₈ ||class="entry q1 g0"| 11821₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (25734, 19171, 22199, 7758, 12605, 17897) ||class="entry q0 g0"| 25734₆ ||class="entry q1 g0"| 19171₈ ||class="entry q1 g0"| 22199₁₀ ||class="entry q0 g0"| 7758₈ ||class="entry q1 g0"| 12605₈ ||class="entry q1 g0"| 17897₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (25986, 30763, 25979, 8010, 1013, 30245) ||class="entry q0 g0"| 25986₆ ||class="entry q1 g0"| 30763₈ ||class="entry q1 g0"| 25979₁₀ ||class="entry q0 g0"| 8010₈ ||class="entry q1 g0"| 1013₈ ||class="entry q1 g0"| 30245₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26662, 11973, 15719, 4846, 21787, 11833) ||class="entry q0 g0"| 26662₆ ||class="entry q1 g0"| 11973₈ ||class="entry q1 g0"| 15719₁₀ ||class="entry q0 g0"| 4846₈ ||class="entry q1 g0"| 21787₈ ||class="entry q1 g0"| 11833₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26676, 29837, 26429, 4860, 3923, 29795) ||class="entry q0 g0"| 26676₆ ||class="entry q1 g0"| 29837₈ ||class="entry q1 g0"| 26429₁₀ ||class="entry q0 g0"| 4860₈ ||class="entry q1 g0"| 3923₈ ||class="entry q1 g0"| 29795₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (1988, 5531, 28603, 32012, 28229, 31973) ||class="entry q0 g0"| 1988₆ ||class="entry q1 g0"| 5531₈ ||class="entry q1 g0"| 28603₁₂ ||class="entry q0 g0"| 32012₈ ||class="entry q1 g0"| 28229₈ ||class="entry q1 g0"| 31973₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (2674, 6461, 28157, 28858, 25315, 32419) ||class="entry q0 g0"| 2674₆ ||class="entry q1 g0"| 6461₈ ||class="entry q1 g0"| 28157₁₂ ||class="entry q0 g0"| 28858₈ ||class="entry q1 g0"| 25315₈ ||class="entry q1 g0"| 32419₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (5516, 2003, 32243, 28484, 31757, 28333) ||class="entry q0 g0"| 5516₆ ||class="entry q1 g0"| 2003₈ ||class="entry q1 g0"| 32243₁₂ ||class="entry q0 g0"| 28484₈ ||class="entry q1 g0"| 31757₈ ||class="entry q1 g0"| 28333₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (6202, 2933, 32693, 25330, 28843, 27883) ||class="entry q0 g0"| 6202₆ ||class="entry q1 g0"| 2933₈ ||class="entry q1 g0"| 32693₁₂ ||class="entry q0 g0"| 25330₈ ||class="entry q1 g0"| 28843₈ ||class="entry q1 g0"| 27883₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (10052, 13343, 31991, 23948, 20417, 28585) ||class="entry q0 g0"| 10052₆ ||class="entry q1 g0"| 13343₈ ||class="entry q1 g0"| 31991₁₂ ||class="entry q0 g0"| 23948₈ ||class="entry q1 g0"| 20417₈ ||class="entry q1 g0"| 28585₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (13580, 9815, 28351, 20420, 23945, 32225) ||class="entry q0 g0"| 13580₆ ||class="entry q1 g0"| 9815₈ ||class="entry q1 g0"| 28351₁₂ ||class="entry q0 g0"| 20420₈ ||class="entry q1 g0"| 23945₈ ||class="entry q1 g0"| 32225₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (17010, 20783, 32679, 14522, 10993, 27897) ||class="entry q0 g0"| 17010₆ ||class="entry q1 g0"| 20783₈ ||class="entry q1 g0"| 32679₁₂ ||class="entry q0 g0"| 14522₈ ||class="entry q1 g0"| 10993₈ ||class="entry q1 g0"| 27897₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (20538, 17255, 28143, 10994, 14521, 32433) ||class="entry q0 g0"| 20538₆ ||class="entry q1 g0"| 17255₈ ||class="entry q1 g0"| 28143₁₂ ||class="entry q0 g0"| 10994₈ ||class="entry q1 g0"| 14521₈ ||class="entry q1 g0"| 32433₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1746, 32027, 1581, 31770, 1733, 5491) ||class="entry q0 g0"| 1746₆ ||class="entry q1 g0"| 32027₁₀ ||class="entry q1 g0"| 1581₆ ||class="entry q0 g0"| 31770₈ ||class="entry q1 g0"| 1733₆ ||class="entry q1 g0"| 5491₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1748, 7035, 24651, 31772, 24741, 29461) ||class="entry q0 g0"| 1748₆ ||class="entry q1 g0"| 7035₁₀ ||class="entry q1 g0"| 24651₆ ||class="entry q0 g0"| 31772₈ ||class="entry q1 g0"| 24741₆ ||class="entry q1 g0"| 29461₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (2914, 6109, 25101, 29098, 27651, 29011) ||class="entry q0 g0"| 2914₆ ||class="entry q1 g0"| 6109₁₀ ||class="entry q1 g0"| 25101₆ ||class="entry q0 g0"| 29098₈ ||class="entry q1 g0"| 27651₆ ||class="entry q1 g0"| 29011₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (2916, 29117, 1131, 29100, 2659, 5941) ||class="entry q0 g0"| 2916₆ ||class="entry q1 g0"| 29117₁₀ ||class="entry q1 g0"| 1131₆ ||class="entry q0 g0"| 29100₈ ||class="entry q1 g0"| 2659₆ ||class="entry q1 g0"| 5941₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5262, 21371, 10329, 28230, 10405, 15111) ||class="entry q0 g0"| 5262₆ ||class="entry q1 g0"| 21371₁₀ ||class="entry q1 g0"| 10329₆ ||class="entry q0 g0"| 28230₈ ||class="entry q1 g0"| 10405₆ ||class="entry q1 g0"| 15111₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5274, 28499, 5221, 28242, 5261, 1851) ||class="entry q0 g0"| 5274₆ ||class="entry q1 g0"| 28499₁₀ ||class="entry q1 g0"| 5221₆ ||class="entry q0 g0"| 28242₈ ||class="entry q1 g0"| 5261₆ ||class="entry q1 g0"| 1851₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6190, 14173, 17289, 25318, 19587, 20695) ||class="entry q0 g0"| 6190₆ ||class="entry q1 g0"| 14173₁₀ ||class="entry q1 g0"| 17289₆ ||class="entry q0 g0"| 25318₈ ||class="entry q1 g0"| 19587₆ ||class="entry q1 g0"| 20695₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6444, 25589, 5667, 25572, 6187, 1405) ||class="entry q0 g0"| 6444₆ ||class="entry q1 g0"| 25589₁₀ ||class="entry q1 g0"| 5667₆ ||class="entry q0 g0"| 25572₈ ||class="entry q1 g0"| 6187₆ ||class="entry q1 g0"| 1405₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9810, 23711, 5473, 23706, 10049, 1599) ||class="entry q0 g0"| 9810₆ ||class="entry q1 g0"| 23711₁₀ ||class="entry q1 g0"| 5473₆ ||class="entry q0 g0"| 23706₈ ||class="entry q1 g0"| 10049₆ ||class="entry q1 g0"| 1599₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10050, 21119, 6801, 23946, 10657, 2511) ||class="entry q0 g0"| 10050₆ ||class="entry q1 g0"| 21119₁₀ ||class="entry q1 g0"| 6801₆ ||class="entry q0 g0"| 23946₈ ||class="entry q1 g0"| 10657₆ ||class="entry q1 g0"| 2511₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10406, 28487, 10317, 21102, 5273, 15123) ||class="entry q0 g0"| 10406₆ ||class="entry q1 g0"| 28487₁₀ ||class="entry q1 g0"| 10317₆ ||class="entry q0 g0"| 21102₈ ||class="entry q1 g0"| 5273₆ ||class="entry q1 g0"| 15123₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10418, 21359, 5233, 21114, 10417, 1839) ||class="entry q0 g0"| 10418₆ ||class="entry q1 g0"| 21359₁₀ ||class="entry q1 g0"| 5233₆ ||class="entry q0 g0"| 21114₈ ||class="entry q1 g0"| 10417₆ ||class="entry q1 g0"| 1839₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10658, 23951, 7041, 21354, 9809, 2271) ||class="entry q0 g0"| 10658₆ ||class="entry q1 g0"| 23951₁₀ ||class="entry q1 g0"| 7041₆ ||class="entry q0 g0"| 21354₈ ||class="entry q1 g0"| 9809₆ ||class="entry q1 g0"| 2271₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (13338, 20183, 1833, 20178, 13577, 5239) ||class="entry q0 g0"| 13338₆ ||class="entry q1 g0"| 20183₁₀ ||class="entry q1 g0"| 1833₆ ||class="entry q0 g0"| 20178₈ ||class="entry q1 g0"| 13577₆ ||class="entry q1 g0"| 5239₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (13592, 6783, 21123, 20432, 24993, 16861) ||class="entry q0 g0"| 13592₆ ||class="entry q1 g0"| 6783₁₀ ||class="entry q1 g0"| 21123₆ ||class="entry q0 g0"| 20432₈ ||class="entry q1 g0"| 24993₆ ||class="entry q1 g0"| 16861₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17012, 14159, 6593, 14524, 19601, 2719) ||class="entry q0 g0"| 17012₆ ||class="entry q1 g0"| 14159₁₀ ||class="entry q1 g0"| 6593₆ ||class="entry q0 g0"| 14524₈ ||class="entry q1 g0"| 19601₆ ||class="entry q1 g0"| 2719₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17252, 14767, 5681, 14764, 17009, 1391) ||class="entry q0 g0"| 17252₆ ||class="entry q1 g0"| 14767₁₀ ||class="entry q1 g0"| 5681₆ ||class="entry q0 g0"| 14764₈ ||class="entry q1 g0"| 17009₆ ||class="entry q1 g0"| 1391₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (19590, 25335, 17035, 13902, 6441, 20949) ||class="entry q0 g0"| 19590₆ ||class="entry q1 g0"| 25335₁₀ ||class="entry q1 g0"| 17035₆ ||class="entry q0 g0"| 13902₈ ||class="entry q1 g0"| 6441₆ ||class="entry q1 g0"| 20949₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (19604, 14527, 6353, 13916, 17249, 2959) ||class="entry q0 g0"| 19604₆ ||class="entry q1 g0"| 14527₁₀ ||class="entry q1 g0"| 6353₆ ||class="entry q0 g0"| 13916₈ ||class="entry q1 g0"| 17249₆ ||class="entry q1 g0"| 2959₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (19844, 13919, 5921, 14156, 19841, 1151) ||class="entry q0 g0"| 19844₆ ||class="entry q1 g0"| 13919₁₀ ||class="entry q1 g0"| 5921₆ ||class="entry q0 g0"| 14156₈ ||class="entry q1 g0"| 19841₆ ||class="entry q1 g0"| 1151₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (20780, 11239, 1145, 11236, 20537, 5927) ||class="entry q0 g0"| 20780₆ ||class="entry q1 g0"| 11239₁₀ ||class="entry q1 g0"| 1145₆ ||class="entry q0 g0"| 11236₈ ||class="entry q1 g0"| 20537₆ ||class="entry q1 g0"| 5927₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (20792, 6095, 14405, 11248, 27665, 11035) ||class="entry q0 g0"| 20792₆ ||class="entry q1 g0"| 6095₁₀ ||class="entry q1 g0"| 14405₆ ||class="entry q0 g0"| 11248₈ ||class="entry q1 g0"| 27665₆ ||class="entry q1 g0"| 11035₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (24754, 7037, 1579, 6778, 24739, 5493) ||class="entry q0 g0"| 24754₆ ||class="entry q1 g0"| 7037₁₀ ||class="entry q1 g0"| 1579₆ ||class="entry q0 g0"| 6778₈ ||class="entry q1 g0"| 24739₆ ||class="entry q1 g0"| 5493₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (24756, 32029, 24653, 6780, 1731, 29459) ||class="entry q0 g0"| 24756₆ ||class="entry q1 g0"| 32029₁₀ ||class="entry q1 g0"| 24653₆ ||class="entry q0 g0"| 6780₈ ||class="entry q1 g0"| 1731₆ ||class="entry q1 g0"| 29459₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (25008, 20437, 21377, 7032, 13323, 16607) ||class="entry q0 g0"| 25008₆ ||class="entry q1 g0"| 20437₁₀ ||class="entry q1 g0"| 21377₆ ||class="entry q0 g0"| 7032₈ ||class="entry q1 g0"| 13323₆ ||class="entry q1 g0"| 16607₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27906, 29115, 25099, 6090, 2661, 29013) ||class="entry q0 g0"| 27906₆ ||class="entry q1 g0"| 29115₁₀ ||class="entry q1 g0"| 25099₆ ||class="entry q0 g0"| 6090₈ ||class="entry q1 g0"| 2661₆ ||class="entry q1 g0"| 29013₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27908, 6107, 1133, 6092, 27653, 5939) ||class="entry q0 g0"| 27908₆ ||class="entry q1 g0"| 6107₁₀ ||class="entry q1 g0"| 1133₆ ||class="entry q0 g0"| 6092₈ ||class="entry q1 g0"| 27653₆ ||class="entry q1 g0"| 5939₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27920, 11251, 14417, 6104, 20525, 11023) ||class="entry q0 g0"| 27920₆ ||class="entry q1 g0"| 11251₁₀ ||class="entry q1 g0"| 14417₆ ||class="entry q0 g0"| 6104₈ ||class="entry q1 g0"| 20525₆ ||class="entry q1 g0"| 11023₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (756, 30413, 3307, 30780, 3347, 8117) ||class="entry q0 g0"| 756₆ ||class="entry q1 g0"| 30413₁₀ ||class="entry q1 g0"| 3307₈ ||class="entry q0 g0"| 30780₈ ||class="entry q1 g0"| 3347₆ ||class="entry q1 g0"| 8117₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (3906, 31339, 3757, 30090, 437, 7667) ||class="entry q0 g0"| 3906₆ ||class="entry q1 g0"| 31339₁₀ ||class="entry q1 g0"| 3757₈ ||class="entry q0 g0"| 30090₈ ||class="entry q1 g0"| 437₆ ||class="entry q1 g0"| 7667₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (4270, 16077, 17657, 27238, 17683, 22439) ||class="entry q0 g0"| 4270₆ ||class="entry q1 g0"| 16077₁₀ ||class="entry q1 g0"| 17657₈ ||class="entry q0 g0"| 27238₈ ||class="entry q1 g0"| 17683₆ ||class="entry q1 g0"| 22439₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (7182, 23275, 12073, 26310, 8501, 15479) ||class="entry q0 g0"| 7182₆ ||class="entry q1 g0"| 23275₁₀ ||class="entry q1 g0"| 12073₈ ||class="entry q0 g0"| 26310₈ ||class="entry q1 g0"| 8501₆ ||class="entry q1 g0"| 15479₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (9058, 16329, 30257, 22954, 17431, 25967) ||class="entry q0 g0"| 9058₆ ||class="entry q1 g0"| 16329₁₀ ||class="entry q1 g0"| 30257₈ ||class="entry q0 g0"| 22954₈ ||class="entry q1 g0"| 17431₆ ||class="entry q1 g0"| 25967₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (11410, 16089, 30929, 22106, 17671, 27535) ||class="entry q0 g0"| 11410₆ ||class="entry q1 g0"| 16089₁₀ ||class="entry q1 g0"| 30929₈ ||class="entry q0 g0"| 22106₈ ||class="entry q1 g0"| 17671₆ ||class="entry q1 g0"| 27535₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (12600, 30665, 15907, 19440, 3095, 11645) ||class="entry q0 g0"| 12600₆ ||class="entry q1 g0"| 30665₁₀ ||class="entry q1 g0"| 15907₈ ||class="entry q0 g0"| 19440₈ ||class="entry q1 g0"| 3095₆ ||class="entry q1 g0"| 11645₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (18004, 23289, 30049, 15516, 8487, 26175) ||class="entry q0 g0"| 18004₆ ||class="entry q1 g0"| 23289₁₀ ||class="entry q1 g0"| 30049₈ ||class="entry q0 g0"| 15516₈ ||class="entry q1 g0"| 8487₆ ||class="entry q1 g0"| 26175₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (18852, 23529, 31617, 13164, 8247, 26847) ||class="entry q0 g0"| 18852₆ ||class="entry q1 g0"| 23529₁₀ ||class="entry q1 g0"| 31617₈ ||class="entry q0 g0"| 13164₈ ||class="entry q1 g0"| 8247₆ ||class="entry q1 g0"| 26847₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (21784, 31353, 21733, 12240, 423, 18363) ||class="entry q0 g0"| 21784₆ ||class="entry q1 g0"| 31353₁₀ ||class="entry q1 g0"| 21733₈ ||class="entry q0 g0"| 12240₈ ||class="entry q1 g0"| 423₆ ||class="entry q1 g0"| 18363₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (25746, 30411, 27275, 7770, 3349, 31189) ||class="entry q0 g0"| 25746₆ ||class="entry q1 g0"| 30411₁₀ ||class="entry q1 g0"| 27275₈ ||class="entry q0 g0"| 7770₈ ||class="entry q1 g0"| 3349₆ ||class="entry q1 g0"| 31189₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (26916, 31341, 26829, 5100, 435, 31635) ||class="entry q0 g0"| 26916₆ ||class="entry q1 g0"| 31341₁₀ ||class="entry q1 g0"| 26829₈ ||class="entry q0 g0"| 5100₈ ||class="entry q1 g0"| 435₆ ||class="entry q1 g0"| 31635₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (1986, 29691, 2525, 32010, 2085, 6787) ||class="entry q0 g0"| 1986₆ ||class="entry q1 g0"| 29691₁₂ ||class="entry q1 g0"| 2525₈ ||class="entry q0 g0"| 32010₈ ||class="entry q1 g0"| 2085₄ ||class="entry q1 g0"| 6787₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (2676, 32605, 2971, 28860, 1155, 6341) ||class="entry q0 g0"| 2676₆ ||class="entry q1 g0"| 32605₁₂ ||class="entry q1 g0"| 2971₈ ||class="entry q0 g0"| 28860₈ ||class="entry q1 g0"| 1155₄ ||class="entry q1 g0"| 6341₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (5528, 15355, 16847, 28496, 16421, 21137) ||class="entry q0 g0"| 5528₆ ||class="entry q1 g0"| 15355₁₂ ||class="entry q1 g0"| 16847₈ ||class="entry q0 g0"| 28496₈ ||class="entry q1 g0"| 16421₄ ||class="entry q1 g0"| 21137₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (6456, 24541, 10783, 25584, 9219, 14657) ||class="entry q0 g0"| 6456₆ ||class="entry q1 g0"| 24541₁₂ ||class="entry q1 g0"| 10783₈ ||class="entry q0 g0"| 25584₈ ||class="entry q1 g0"| 9219₄ ||class="entry q1 g0"| 14657₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (9812, 15103, 29447, 23708, 16673, 24665) ||class="entry q0 g0"| 9812₆ ||class="entry q1 g0"| 15103₁₂ ||class="entry q1 g0"| 29447₈ ||class="entry q0 g0"| 23708₈ ||class="entry q1 g0"| 16673₄ ||class="entry q1 g0"| 24665₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (13326, 29439, 15125, 20166, 2337, 10315) ||class="entry q0 g0"| 13326₆ ||class="entry q1 g0"| 29439₁₂ ||class="entry q1 g0"| 15125₈ ||class="entry q0 g0"| 20166₈ ||class="entry q1 g0"| 2337₄ ||class="entry q1 g0"| 10315₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (17250, 24527, 28759, 14762, 9233, 25353) ||class="entry q0 g0"| 17250₆ ||class="entry q1 g0"| 24527₁₂ ||class="entry q1 g0"| 28759₈ ||class="entry q0 g0"| 14762₈ ||class="entry q1 g0"| 9233₄ ||class="entry q1 g0"| 25353₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (20526, 32591, 20947, 10982, 1169, 17037) ||class="entry q0 g0"| 20526₆ ||class="entry q1 g0"| 32591₁₂ ||class="entry q1 g0"| 20947₈ ||class="entry q0 g0"| 10982₈ ||class="entry q1 g0"| 1169₄ ||class="entry q1 g0"| 17037₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (10660, 15343, 32231, 21356, 16433, 28345) ||class="entry q0 g0"| 10660₆ ||class="entry q1 g0"| 15343₁₂ ||class="entry q1 g0"| 32231₁₂ ||class="entry q0 g0"| 21356₈ ||class="entry q1 g0"| 16433₄ ||class="entry q1 g0"| 28345₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (19602, 24287, 32439, 13914, 9473, 28137) ||class="entry q0 g0"| 19602₆ ||class="entry q1 g0"| 24287₁₂ ||class="entry q1 g0"| 32439₁₂ ||class="entry q0 g0"| 13914₈ ||class="entry q1 g0"| 9473₄ ||class="entry q1 g0"| 28137₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (24996, 29693, 28605, 7020, 2083, 31971) ||class="entry q0 g0"| 24996₆ ||class="entry q1 g0"| 29693₁₂ ||class="entry q1 g0"| 28605₁₂ ||class="entry q0 g0"| 7020₈ ||class="entry q1 g0"| 2083₄ ||class="entry q1 g0"| 31971₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (27666, 32603, 28155, 5850, 1157, 32421) ||class="entry q0 g0"| 27666₆ ||class="entry q1 g0"| 32603₁₂ ||class="entry q1 g0"| 28155₁₂ ||class="entry q0 g0"| 5850₈ ||class="entry q1 g0"| 1157₄ ||class="entry q1 g0"| 32421₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (11670, 3089, 19229, 22366, 30671, 22595) ||class="entry q0 g0"| 11670₈ ||class="entry q1 g0"| 3089₄ ||class="entry q1 g0"| 19229₈ ||class="entry q0 g0"| 22366₁₀ ||class="entry q1 g0"| 30671₁₂ ||class="entry q1 g0"| 22595₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (18870, 417, 8667, 13182, 31359, 12933) ||class="entry q0 g0"| 18870₈ ||class="entry q1 g0"| 417₄ ||class="entry q1 g0"| 8667₈ ||class="entry q0 g0"| 13182₁₀ ||class="entry q1 g0"| 31359₁₂ ||class="entry q1 g0"| 12933₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (26006, 17411, 22855, 8030, 16349, 18969) ||class="entry q0 g0"| 26006₈ ||class="entry q1 g0"| 17411₄ ||class="entry q1 g0"| 22855₈ ||class="entry q0 g0"| 8030₁₀ ||class="entry q1 g0"| 16349₁₂ ||class="entry q1 g0"| 18969₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (26934, 8229, 12951, 5118, 23547, 8649) ||class="entry q0 g0"| 26934₈ ||class="entry q1 g0"| 8229₄ ||class="entry q1 g0"| 12951₈ ||class="entry q0 g0"| 5118₁₀ ||class="entry q1 g0"| 23547₁₂ ||class="entry q1 g0"| 8649₆ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (1014, 8805, 22849, 31038, 22971, 18975) ||class="entry q0 g0"| 1014₈ ||class="entry q1 g0"| 8805₆ ||class="entry q1 g0"| 22849₆ ||class="entry q0 g0"| 31038₁₀ ||class="entry q1 g0"| 22971₁₀ ||class="entry q1 g0"| 18975₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (3926, 17987, 12945, 30110, 15773, 8655) ||class="entry q0 g0"| 3926₈ ||class="entry q1 g0"| 17987₆ ||class="entry q1 g0"| 12945₆ ||class="entry q0 g0"| 30110₁₀ ||class="entry q1 g0"| 15773₁₀ ||class="entry q1 g0"| 8655₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (4542, 12333, 19209, 27510, 19443, 22615) ||class="entry q0 g0"| 4542₈ ||class="entry q1 g0"| 12333₆ ||class="entry q1 g0"| 19209₆ ||class="entry q0 g0"| 27510₁₀ ||class="entry q1 g0"| 19443₁₀ ||class="entry q1 g0"| 22615₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (7454, 21515, 8409, 26582, 12245, 13191) ||class="entry q0 g0"| 7454₈ ||class="entry q1 g0"| 21515₆ ||class="entry q1 g0"| 8409₆ ||class="entry q0 g0"| 26582₁₀ ||class="entry q1 g0"| 12245₁₀ ||class="entry q1 g0"| 13191₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (9078, 993, 18957, 22974, 30783, 22867) ||class="entry q0 g0"| 9078₈ ||class="entry q1 g0"| 993₆ ||class="entry q1 g0"| 18957₆ ||class="entry q0 g0"| 22974₁₀ ||class="entry q1 g0"| 30783₁₀ ||class="entry q1 g0"| 22867₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (12606, 4521, 22597, 19446, 27255, 19227) ||class="entry q0 g0"| 12606₈ ||class="entry q1 g0"| 4521₆ ||class="entry q1 g0"| 22597₆ ||class="entry q0 g0"| 19446₁₀ ||class="entry q1 g0"| 27255₁₀ ||class="entry q1 g0"| 19227₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (18262, 3665, 8395, 15774, 30095, 13205) ||class="entry q0 g0"| 18262₈ ||class="entry q1 g0"| 3665₆ ||class="entry q1 g0"| 8395₆ ||class="entry q0 g0"| 15774₁₀ ||class="entry q1 g0"| 30095₁₀ ||class="entry q1 g0"| 13205₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (21790, 7193, 12931, 12246, 26567, 8669) ||class="entry q0 g0"| 21790₈ ||class="entry q1 g0"| 7193₆ ||class="entry q1 g0"| 12931₆ ||class="entry q0 g0"| 12246₁₀ ||class="entry q1 g0"| 26567₁₀ ||class="entry q1 g0"| 8669₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (10678, 24999, 10173, 21374, 6777, 13539) ||class="entry q0 g0"| 10678₈ ||class="entry q1 g0"| 24999₈ ||class="entry q1 g0"| 10173₁₀ ||class="entry q0 g0"| 21374₁₀ ||class="entry q1 g0"| 6777₈ ||class="entry q1 g0"| 13539₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (19862, 27671, 19835, 14174, 6089, 24101) ||class="entry q0 g0"| 19862₈ ||class="entry q1 g0"| 27671₈ ||class="entry q1 g0"| 19835₁₀ ||class="entry q0 g0"| 14174₁₀ ||class="entry q1 g0"| 6089₈ ||class="entry q1 g0"| 24101₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (25014, 10677, 13799, 7038, 21099, 9913) ||class="entry q0 g0"| 25014₈ ||class="entry q1 g0"| 10677₈ ||class="entry q1 g0"| 13799₁₀ ||class="entry q0 g0"| 7038₁₀ ||class="entry q1 g0"| 21099₈ ||class="entry q1 g0"| 9913₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (27926, 19859, 24119, 6110, 13901, 19817) ||class="entry q0 g0"| 27926₈ ||class="entry q1 g0"| 19859₈ ||class="entry q1 g0"| 24119₁₀ ||class="entry q0 g0"| 6110₁₀ ||class="entry q1 g0"| 13901₈ ||class="entry q1 g0"| 19817₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (2006, 20435, 13793, 32030, 13325, 9919) ||class="entry q0 g0"| 2006₈ ||class="entry q1 g0"| 20435₁₀ ||class="entry q1 g0"| 13793₈ ||class="entry q0 g0"| 32030₁₀ ||class="entry q1 g0"| 13325₆ ||class="entry q1 g0"| 9919₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (2934, 11253, 24113, 29118, 20523, 19823) ||class="entry q0 g0"| 2934₈ ||class="entry q1 g0"| 11253₁₀ ||class="entry q1 g0"| 24113₈ ||class="entry q0 g0"| 29118₁₀ ||class="entry q1 g0"| 20523₆ ||class="entry q1 g0"| 19823₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (5534, 23963, 10153, 28502, 9797, 13559) ||class="entry q0 g0"| 5534₈ ||class="entry q1 g0"| 23963₁₀ ||class="entry q1 g0"| 10153₈ ||class="entry q0 g0"| 28502₁₀ ||class="entry q1 g0"| 9797₆ ||class="entry q1 g0"| 13559₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6462, 14781, 19577, 25590, 16995, 24359) ||class="entry q0 g0"| 6462₈ ||class="entry q1 g0"| 14781₁₀ ||class="entry q1 g0"| 19577₈ ||class="entry q0 g0"| 25590₁₀ ||class="entry q1 g0"| 16995₆ ||class="entry q1 g0"| 24359₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (10070, 28247, 9901, 23966, 5513, 13811) ||class="entry q0 g0"| 10070₈ ||class="entry q1 g0"| 28247₁₀ ||class="entry q1 g0"| 9901₈ ||class="entry q0 g0"| 23966₁₀ ||class="entry q1 g0"| 5513₆ ||class="entry q1 g0"| 13811₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (13598, 31775, 13541, 20438, 1985, 10171) ||class="entry q0 g0"| 13598₈ ||class="entry q1 g0"| 31775₁₀ ||class="entry q1 g0"| 13541₈ ||class="entry q0 g0"| 20438₁₀ ||class="entry q1 g0"| 1985₆ ||class="entry q1 g0"| 10171₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (17270, 25575, 19563, 14782, 6201, 24373) ||class="entry q0 g0"| 17270₈ ||class="entry q1 g0"| 25575₁₀ ||class="entry q1 g0"| 19563₈ ||class="entry q0 g0"| 14782₁₀ ||class="entry q1 g0"| 6201₆ ||class="entry q1 g0"| 24373₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (20798, 29103, 24099, 11254, 2673, 19837) ||class="entry q0 g0"| 20798₈ ||class="entry q1 g0"| 29103₁₀ ||class="entry q1 g0"| 24099₈ ||class="entry q0 g0"| 11254₁₀ ||class="entry q1 g0"| 2673₆ ||class="entry q1 g0"| 19837₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (12226, 23535, 7649, 21770, 8241, 3775) ||class="entry q0 g0"| 12226₈ ||class="entry q1 g0"| 23535₁₂ ||class="entry q1 g0"| 7649₈ ||class="entry q0 g0"| 21770₆ ||class="entry q1 g1"| 8241₄ ||class="entry q1 g0"| 3775₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (15502, 31599, 15461, 17990, 177, 12091) ||class="entry q0 g0"| 15502₈ ||class="entry q1 g0"| 31599₁₂ ||class="entry q1 g0"| 15461₈ ||class="entry q0 g0"| 17990₆ ||class="entry q1 g1"| 177₄ ||class="entry q1 g0"| 12091₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (19188, 16095, 7857, 12348, 17665, 3567) ||class="entry q0 g0"| 19188₈ ||class="entry q1 g0"| 16095₁₂ ||class="entry q1 g0"| 7857₈ ||class="entry q0 g0"| 12348₆ ||class="entry q1 g1"| 17665₄ ||class="entry q1 g0"| 3567₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (22702, 30431, 22179, 8806, 3329, 17917) ||class="entry q0 g0"| 22702₈ ||class="entry q1 g0"| 30431₁₂ ||class="entry q1 g0"| 22179₈ ||class="entry q0 g0"| 8806₆ ||class="entry q1 g1"| 3329₄ ||class="entry q1 g0"| 17917₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (26324, 31613, 26157, 7196, 163, 30067) ||class="entry q0 g0"| 26324₈ ||class="entry q1 g0"| 31613₁₂ ||class="entry q1 g0"| 26157₈ ||class="entry q0 g0"| 7196₆ ||class="entry q1 g1"| 163₄ ||class="entry q1 g0"| 30067₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (27490, 30683, 25707, 4522, 3077, 30517) ||class="entry q0 g0"| 27490₈ ||class="entry q1 g0"| 30683₁₂ ||class="entry q1 g0"| 25707₈ ||class="entry q0 g0"| 4522₆ ||class="entry q1 g1"| 3077₄ ||class="entry q1 g0"| 30517₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (30104, 23549, 18345, 3920, 8227, 21751) ||class="entry q0 g0"| 30104₈ ||class="entry q1 g0"| 23549₁₂ ||class="entry q1 g0"| 18345₈ ||class="entry q0 g0"| 3920₆ ||class="entry q1 g1"| 8227₄ ||class="entry q1 g0"| 21751₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (31032, 16347, 11385, 1008, 17413, 16167) ||class="entry q0 g0"| 31032₈ ||class="entry q1 g0"| 16347₁₂ ||class="entry q1 g0"| 11385₈ ||class="entry q0 g0"| 1008₆ ||class="entry q1 g1"| 17413₄ ||class="entry q1 g0"| 16167₁₀ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16092, 19185, 3327, 17428, 12591, 8097) ||class="entry q0 g0"| 16092₁₀ ||class="entry q1 g0"| 19185₈ ||class="entry q1 g0"| 3327₁₀ ||class="entry q0 g1"| 17428₄ ||class="entry q1 g0"| 12591₈ ||class="entry q1 g0"| 8097₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16344, 30777, 16179, 17680, 999, 11373) ||class="entry q0 g0"| 16344₁₀ ||class="entry q1 g0"| 30777₈ ||class="entry q1 g0"| 16179₁₀ ||class="entry q0 g1"| 17680₄ ||class="entry q1 g0"| 999₈ ||class="entry q1 g0"| 11373₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (23530, 12225, 4015, 8482, 21535, 7409) ||class="entry q0 g0"| 23530₁₀ ||class="entry q1 g0"| 12225₈ ||class="entry q1 g0"| 4015₁₀ ||class="entry q0 g1"| 8482₄ ||class="entry q1 g0"| 21535₈ ||class="entry q1 g0"| 7409₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (23544, 30089, 22005, 8496, 3671, 18091) ||class="entry q0 g0"| 23544₁₀ ||class="entry q1 g0"| 30089₈ ||class="entry q1 g0"| 22005₁₀ ||class="entry q0 g1"| 8496₄ ||class="entry q1 g0"| 3671₈ ||class="entry q1 g0"| 18091₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30414, 22699, 17663, 3078, 9077, 22433) ||class="entry q0 g0"| 30414₁₀ ||class="entry q1 g0"| 22699₈ ||class="entry q1 g0"| 17663₁₀ ||class="entry q0 g1"| 3078₄ ||class="entry q1 g0"| 9077₈ ||class="entry q1 g0"| 22433₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30666, 27235, 30515, 3330, 4541, 25709) ||class="entry q0 g0"| 30666₁₀ ||class="entry q1 g0"| 27235₈ ||class="entry q1 g0"| 30515₁₀ ||class="entry q0 g1"| 3330₄ ||class="entry q1 g0"| 4541₈ ||class="entry q1 g0"| 25709₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31342, 15501, 12079, 166, 18259, 15473) ||class="entry q0 g0"| 31342₁₀ ||class="entry q1 g0"| 15501₈ ||class="entry q1 g0"| 12079₁₀ ||class="entry q0 g1"| 166₄ ||class="entry q1 g0"| 18259₈ ||class="entry q1 g0"| 15473₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31356, 26309, 30069, 180, 7451, 26155) ||class="entry q0 g0"| 31356₁₀ ||class="entry q1 g0"| 26309₈ ||class="entry q1 g0"| 30069₁₀ ||class="entry q0 g1"| 180₄ ||class="entry q1 g0"| 7451₈ ||class="entry q1 g0"| 26155₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (16072, 30425, 12483, 17408, 3335, 9117) ||class="entry q0 g0"| 16072₈ ||class="entry q1 g0"| 30425₁₀ ||class="entry q1 g0"| 12483₆ ||class="entry q0 g1"| 17408₂ ||class="entry q1 g1"| 3335₆ ||class="entry q1 g0"| 9117₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (23272, 31593, 23045, 8224, 183, 18779) ||class="entry q0 g0"| 23272₈ ||class="entry q1 g0"| 31593₁₀ ||class="entry q1 g0"| 23045₆ ||class="entry q0 g1"| 8224₂ ||class="entry q1 g1"| 183₆ ||class="entry q1 g0"| 18779₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (30408, 16075, 8857, 3072, 17685, 12743) ||class="entry q0 g0"| 30408₈ ||class="entry q1 g0"| 16075₁₀ ||class="entry q1 g0"| 8857₆ ||class="entry q0 g1"| 3072₂ ||class="entry q1 g1"| 17685₆ ||class="entry q1 g0"| 12743₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (31336, 23277, 18761, 160, 8499, 23063) ||class="entry q0 g0"| 31336₈ ||class="entry q1 g0"| 23277₁₀ ||class="entry q1 g0"| 18761₆ ||class="entry q0 g1"| 160₂ ||class="entry q1 g1"| 8499₆ ||class="entry q1 g0"| 23063₈ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (16332, 17425, 783, 17668, 16335, 4177) ||class="entry q0 g0"| 16332₁₀ ||class="entry q1 g1"| 17425₄ ||class="entry q1 g0"| 783₆ ||class="entry q0 g1"| 17668₄ ||class="entry q1 g0"| 16335₁₂ ||class="entry q1 g0"| 4177₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (23290, 8481, 95, 8242, 23295, 4865) ||class="entry q0 g0"| 23290₁₀ ||class="entry q1 g1"| 8481₄ ||class="entry q1 g0"| 95₆ ||class="entry q0 g1"| 8242₄ ||class="entry q1 g0"| 23295₁₂ ||class="entry q1 g0"| 4865₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (30668, 3075, 4437, 3332, 30685, 523) ||class="entry q0 g0"| 30668₁₀ ||class="entry q1 g1"| 3075₄ ||class="entry q1 g0"| 4437₆ ||class="entry q0 g1"| 3332₄ ||class="entry q1 g0"| 30685₁₂ ||class="entry q1 g0"| 523₄ |- |class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (31354, 165, 4883, 178, 31611, 77) ||class="entry q0 g0"| 31354₁₀ ||class="entry q1 g1"| 165₄ ||class="entry q1 g0"| 4883₆ ||class="entry q0 g1"| 178₄ ||class="entry q1 g0"| 31611₁₂ ||class="entry q1 g0"| 77₄ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (40, 10300, 15360, 5120, 15380, 10280) ||class="entry q0 g0"| 40₂ ||class="entry q0 g0"| 10300₆ ||class="entry q0 g0"| 15360₄ ||class="entry q0 g0"| 5120₂ ||class="entry q0 g0"| 15380₆ ||class="entry q0 g0"| 10280₄ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (1032, 17802, 20640, 4128, 20898, 17544) ||class="entry q0 g0"| 1032₂ ||class="entry q0 g0"| 17802₆ ||class="entry q0 g0"| 20640₄ ||class="entry q0 g0"| 4128₂ ||class="entry q0 g0"| 20898₆ ||class="entry q0 g0"| 17544₄ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (2216, 8620, 15216, 7296, 13700, 12120) ||class="entry q0 g0"| 2216₄ ||class="entry q0 g0"| 8620₆ ||class="entry q0 g0"| 15216₈ ||class="entry q0 g0"| 7296₄ ||class="entry q0 g0"| 13700₆ ||class="entry q0 g0"| 12120₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (3208, 19482, 22480, 6304, 22578, 17400) ||class="entry q0 g0"| 3208₄ ||class="entry q0 g0"| 19482₆ ||class="entry q0 g0"| 22480₈ ||class="entry q0 g0"| 6304₄ ||class="entry q0 g0"| 22578₆ ||class="entry q0 g0"| 17400₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (8360, 2488, 12108, 13440, 7568, 15204) ||class="entry q0 g0"| 8360₄ ||class="entry q0 g0"| 2488₆ ||class="entry q0 g0"| 12108₈ ||class="entry q0 g0"| 13440₄ ||class="entry q0 g0"| 7568₆ ||class="entry q0 g0"| 15204₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (9352, 25614, 17388, 12448, 28710, 22468) ||class="entry q0 g0"| 9352₄ ||class="entry q0 g0"| 25614₆ ||class="entry q0 g0"| 17388₈ ||class="entry q0 g0"| 12448₄ ||class="entry q0 g0"| 28710₆ ||class="entry q0 g0"| 22468₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (18472, 24622, 11866, 23552, 29702, 14962) ||class="entry q0 g0"| 18472₄ ||class="entry q0 g0"| 24622₆ ||class="entry q0 g0"| 11866₈ ||class="entry q0 g0"| 23552₄ ||class="entry q0 g0"| 29702₆ ||class="entry q0 g0"| 14962₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (19464, 3480, 17146, 22560, 6576, 22226) ||class="entry q0 g0"| 19464₄ ||class="entry q0 g0"| 3480₆ ||class="entry q0 g0"| 17146₈ ||class="entry q0 g0"| 22560₄ ||class="entry q0 g0"| 6576₆ ||class="entry q0 g0"| 22226₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (24616, 18490, 14950, 29696, 23570, 11854) ||class="entry q0 g0"| 24616₄ ||class="entry q0 g0"| 18490₆ ||class="entry q0 g0"| 14950₈ ||class="entry q0 g0"| 29696₄ ||class="entry q0 g0"| 23570₆ ||class="entry q0 g0"| 11854₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (25608, 9612, 22214, 28704, 12708, 17134) ||class="entry q0 g0"| 25608₄ ||class="entry q0 g0"| 9612₆ ||class="entry q0 g0"| 22214₈ ||class="entry q0 g0"| 28704₄ ||class="entry q0 g0"| 12708₆ ||class="entry q0 g0"| 17134₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (616, 17386, 22208, 5696, 22466, 17128) ||class="entry q0 g0"| 616₄ ||class="entry q0 g0"| 17386₈ ||class="entry q0 g0"| 22208₆ ||class="entry q0 g0"| 5696₄ ||class="entry q0 g0"| 22466₈ ||class="entry q0 g0"| 17128₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (1608, 11868, 14944, 4704, 14964, 11848) ||class="entry q0 g0"| 1608₄ ||class="entry q0 g0"| 11868₈ ||class="entry q0 g0"| 14944₆ ||class="entry q0 g0"| 4704₄ ||class="entry q0 g0"| 14964₈ ||class="entry q0 g0"| 11848₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (46, 20060, 23142, 5126, 23156, 20046) ||class="entry q0 g0"| 46₄ ||class="entry q0 g0"| 20060₈ ||class="entry q0 g0"| 23142₈ ||class="entry q0 g0"| 5126₄ ||class="entry q0 g0"| 23156₈ ||class="entry q0 g0"| 20046₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (58, 29300, 26202, 5138, 26204, 29298) ||class="entry q0 g0"| 58₄ ||class="entry q0 g0"| 29300₈ ||class="entry q0 g0"| 26202₈ ||class="entry q0 g0"| 5138₄ ||class="entry q0 g0"| 26204₈ ||class="entry q0 g0"| 29298₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (298, 31892, 27050, 5378, 26812, 32130) ||class="entry q0 g0"| 298₄ ||class="entry q0 g0"| 31892₈ ||class="entry q0 g0"| 27050₈ ||class="entry q0 g0"| 5378₄ ||class="entry q0 g0"| 26812₈ ||class="entry q0 g0"| 32130₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (300, 6900, 4044, 5380, 3804, 7140) ||class="entry q0 g0"| 300₄ ||class="entry q0 g0"| 6900₈ ||class="entry q0 g0"| 4044₈ ||class="entry q0 g0"| 5380₄ ||class="entry q0 g0"| 3804₈ ||class="entry q0 g0"| 7140₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (312, 9948, 13296, 5392, 13044, 10200) ||class="entry q0 g0"| 312₄ ||class="entry q0 g0"| 9948₈ ||class="entry q0 g0"| 13296₈ ||class="entry q0 g0"| 5392₄ ||class="entry q0 g0"| 13044₈ ||class="entry q0 g0"| 10200₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1038, 9194, 14022, 4134, 14274, 8942) ||class="entry q0 g0"| 1038₄ ||class="entry q0 g0"| 9194₈ ||class="entry q0 g0"| 14022₈ ||class="entry q0 g0"| 4134₄ ||class="entry q0 g0"| 14274₈ ||class="entry q0 g0"| 8942₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1050, 8130, 2810, 4146, 3050, 7890) ||class="entry q0 g0"| 1050₄ ||class="entry q0 g0"| 8130₈ ||class="entry q0 g0"| 2810₈ ||class="entry q0 g0"| 4146₄ ||class="entry q0 g0"| 3050₈ ||class="entry q0 g0"| 7890₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1052, 31138, 27804, 4148, 28042, 30900) ||class="entry q0 g0"| 1052₄ ||class="entry q0 g0"| 31138₈ ||class="entry q0 g0"| 27804₈ ||class="entry q0 g0"| 4148₄ ||class="entry q0 g0"| 28042₈ ||class="entry q0 g0"| 30900₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1292, 30530, 25452, 4388, 25450, 30532) ||class="entry q0 g0"| 1292₄ ||class="entry q0 g0"| 30530₈ ||class="entry q0 g0"| 25452₈ ||class="entry q0 g0"| 4388₄ ||class="entry q0 g0"| 25450₈ ||class="entry q0 g0"| 30532₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1304, 19306, 24400, 4400, 24386, 19320) ||class="entry q0 g0"| 1304₄ ||class="entry q0 g0"| 19306₈ ||class="entry q0 g0"| 24400₈ ||class="entry q0 g0"| 4400₄ ||class="entry q0 g0"| 24386₈ ||class="entry q0 g0"| 19320₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (11272, 28062, 17564, 14368, 31158, 20660) ||class="entry q0 g0"| 11272₄ ||class="entry q0 g0"| 28062₁₀ ||class="entry q0 g0"| 17564₆ ||class="entry q0 g0"| 14368₄ ||class="entry q0 g0"| 31158₁₀ ||class="entry q0 g0"| 20660₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 6, 4, 10, 6) ||class="c"| (16552, 27070, 10538, 21632, 32150, 15618) ||class="entry q0 g0"| 16552₄ ||class="entry q0 g0"| 27070₁₀ ||class="entry q0 g0"| 10538₆ ||class="entry q0 g0"| 21632₄ ||class="entry q0 g0"| 32150₁₀ ||class="entry q0 g0"| 15618₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (622, 9610, 12454, 5702, 12706, 9358) ||class="entry q0 g0"| 622₆ ||class="entry q0 g0"| 9610₆ ||class="entry q0 g0"| 12454₆ ||class="entry q0 g0"| 5702₆ ||class="entry q0 g0"| 12706₆ ||class="entry q0 g0"| 9358₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (634, 6562, 3226, 5714, 3466, 6322) ||class="entry q0 g0"| 634₆ ||class="entry q0 g0"| 6562₆ ||class="entry q0 g0"| 3226₆ ||class="entry q0 g0"| 5714₆ ||class="entry q0 g0"| 3466₆ ||class="entry q0 g0"| 6322₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (876, 28962, 25868, 5956, 25866, 28964) ||class="entry q0 g0"| 876₆ ||class="entry q0 g0"| 28962₆ ||class="entry q0 g0"| 25868₆ ||class="entry q0 g0"| 5956₆ ||class="entry q0 g0"| 25866₆ ||class="entry q0 g0"| 28964₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (888, 19722, 22832, 5968, 22818, 19736) ||class="entry q0 g0"| 888₆ ||class="entry q0 g0"| 19722₆ ||class="entry q0 g0"| 22832₆ ||class="entry q0 g0"| 5968₆ ||class="entry q0 g0"| 22818₆ ||class="entry q0 g0"| 19736₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1614, 18492, 23558, 4710, 23572, 18478) ||class="entry q0 g0"| 1614₆ ||class="entry q0 g0"| 18492₆ ||class="entry q0 g0"| 23558₆ ||class="entry q0 g0"| 4710₆ ||class="entry q0 g0"| 23572₆ ||class="entry q0 g0"| 18478₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1626, 29716, 24634, 4722, 24636, 29714) ||class="entry q0 g0"| 1626₆ ||class="entry q0 g0"| 29716₆ ||class="entry q0 g0"| 24634₆ ||class="entry q0 g0"| 4722₆ ||class="entry q0 g0"| 24636₆ ||class="entry q0 g0"| 29714₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1868, 7316, 2476, 4964, 2236, 7556) ||class="entry q0 g0"| 1868₆ ||class="entry q0 g0"| 7316₆ ||class="entry q0 g0"| 2476₆ ||class="entry q0 g0"| 4964₆ ||class="entry q0 g0"| 2236₆ ||class="entry q0 g0"| 7556₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 6, 6, 6) ||class="c"| (1880, 8380, 13712, 4976, 13460, 8632) ||class="entry q0 g0"| 1880₆ ||class="entry q0 g0"| 8380₆ ||class="entry q0 g0"| 13712₆ ||class="entry q0 g0"| 4976₆ ||class="entry q0 g0"| 13460₆ ||class="entry q0 g0"| 8632₆ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (318, 16572, 21910, 5398, 21652, 16830) ||class="entry q0 g0"| 318₆ ||class="entry q0 g0"| 16572₆ ||class="entry q0 g0"| 21910₈ ||class="entry q0 g0"| 5398₆ ||class="entry q0 g0"| 21652₆ ||class="entry q0 g0"| 16830₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (1310, 11530, 14646, 4406, 14626, 11550) ||class="entry q0 g0"| 1310₆ ||class="entry q0 g0"| 11530₆ ||class="entry q0 g0"| 14646₈ ||class="entry q0 g0"| 4406₆ ||class="entry q0 g0"| 14626₆ ||class="entry q0 g0"| 11550₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (10286, 26184, 20058, 15366, 29280, 23154) ||class="entry q0 g0"| 10286₆ ||class="entry q0 g0"| 26184₆ ||class="entry q0 g0"| 20058₈ ||class="entry q0 g0"| 15366₆ ||class="entry q0 g0"| 29280₆ ||class="entry q0 g0"| 23154₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (10298, 23136, 29286, 15378, 20040, 26190) ||class="entry q0 g0"| 10298₆ ||class="entry q0 g0"| 23136₆ ||class="entry q0 g0"| 29286₈ ||class="entry q0 g0"| 15378₆ ||class="entry q0 g0"| 20040₆ ||class="entry q0 g0"| 26190₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (10540, 13024, 7152, 15620, 9928, 4056) ||class="entry q0 g0"| 10540₆ ||class="entry q0 g0"| 13024₆ ||class="entry q0 g0"| 7152₈ ||class="entry q0 g0"| 15620₆ ||class="entry q0 g0"| 9928₆ ||class="entry q0 g0"| 4056₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (10552, 3784, 10188, 15632, 6880, 13284) ||class="entry q0 g0"| 10552₆ ||class="entry q0 g0"| 3784₆ ||class="entry q0 g0"| 10188₈ ||class="entry q0 g0"| 15632₆ ||class="entry q0 g0"| 6880₆ ||class="entry q0 g0"| 13284₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (17550, 25192, 9196, 20646, 30272, 14276) ||class="entry q0 g0"| 17550₆ ||class="entry q0 g0"| 25192₆ ||class="entry q0 g0"| 9196₈ ||class="entry q0 g0"| 20646₆ ||class="entry q0 g0"| 30272₆ ||class="entry q0 g0"| 14276₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (17562, 24128, 8144, 20658, 19048, 3064) ||class="entry q0 g0"| 17562₆ ||class="entry q0 g0"| 24128₆ ||class="entry q0 g0"| 8144₈ ||class="entry q0 g0"| 20658₆ ||class="entry q0 g0"| 19048₆ ||class="entry q0 g0"| 3064₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (17804, 14016, 30278, 20900, 8936, 25198) ||class="entry q0 g0"| 17804₆ ||class="entry q0 g0"| 14016₆ ||class="entry q0 g0"| 30278₈ ||class="entry q0 g0"| 20900₆ ||class="entry q0 g0"| 8936₆ ||class="entry q0 g0"| 25198₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (17816, 2792, 19066, 20912, 7872, 24146) ||class="entry q0 g0"| 17816₆ ||class="entry q0 g0"| 2792₆ ||class="entry q0 g0"| 19066₈ ||class="entry q0 g0"| 20912₆ ||class="entry q0 g0"| 7872₆ ||class="entry q0 g0"| 24146₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (26792, 16810, 15638, 31872, 21890, 10558) ||class="entry q0 g0"| 26792₆ ||class="entry q0 g0"| 16810₆ ||class="entry q0 g0"| 15638₈ ||class="entry q0 g0"| 31872₆ ||class="entry q0 g0"| 21890₆ ||class="entry q0 g0"| 10558₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 6, 6, 8) ||class="c"| (27784, 11292, 20918, 30880, 14388, 17822) ||class="entry q0 g0"| 27784₆ ||class="entry q0 g0"| 11292₆ ||class="entry q0 g0"| 20918₈ ||class="entry q0 g0"| 30880₆ ||class="entry q0 g0"| 14388₆ ||class="entry q0 g0"| 17822₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (2474, 29956, 28378, 7554, 24876, 31474) ||class="entry q0 g0"| 2474₆ ||class="entry q0 g0"| 29956₆ ||class="entry q0 g0"| 28378₁₀ ||class="entry q0 g0"| 7554₆ ||class="entry q0 g0"| 24876₆ ||class="entry q0 g0"| 31474₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (3228, 28722, 27628, 6324, 25626, 32708) ||class="entry q0 g0"| 3228₆ ||class="entry q0 g0"| 28722₆ ||class="entry q0 g0"| 27628₁₀ ||class="entry q0 g0"| 6324₆ ||class="entry q0 g0"| 25626₆ ||class="entry q0 g0"| 32708₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (8618, 23824, 31462, 13698, 18744, 28366) ||class="entry q0 g0"| 8618₆ ||class="entry q0 g0"| 23824₆ ||class="entry q0 g0"| 31462₁₀ ||class="entry q0 g0"| 13698₆ ||class="entry q0 g0"| 18744₆ ||class="entry q0 g0"| 28366₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (9372, 22566, 32720, 12468, 19470, 27640) ||class="entry q0 g0"| 9372₆ ||class="entry q0 g0"| 22566₆ ||class="entry q0 g0"| 32720₁₀ ||class="entry q0 g0"| 12468₆ ||class="entry q0 g0"| 19470₆ ||class="entry q0 g0"| 27640₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (18730, 13446, 31728, 23810, 8366, 28632) ||class="entry q0 g0"| 18730₆ ||class="entry q0 g0"| 13446₆ ||class="entry q0 g0"| 31728₁₀ ||class="entry q0 g0"| 23810₆ ||class="entry q0 g0"| 8366₆ ||class="entry q0 g0"| 28632₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (19484, 12720, 32454, 22580, 9624, 27374) ||class="entry q0 g0"| 19484₆ ||class="entry q0 g0"| 12720₆ ||class="entry q0 g0"| 32454₁₀ ||class="entry q0 g0"| 22580₆ ||class="entry q0 g0"| 9624₆ ||class="entry q0 g0"| 27374₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (24874, 7314, 28620, 29954, 2234, 31716) ||class="entry q0 g0"| 24874₆ ||class="entry q0 g0"| 7314₆ ||class="entry q0 g0"| 28620₁₀ ||class="entry q0 g0"| 29954₆ ||class="entry q0 g0"| 2234₆ ||class="entry q0 g0"| 31716₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 10, 6, 6, 10) ||class="c"| (25628, 6564, 27386, 28724, 3468, 32466) ||class="entry q0 g0"| 25628₆ ||class="entry q0 g0"| 6564₆ ||class="entry q0 g0"| 27386₁₀ ||class="entry q0 g0"| 28724₆ ||class="entry q0 g0"| 3468₆ ||class="entry q0 g0"| 32466₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (2222, 18380, 23830, 7302, 21476, 18750) ||class="entry q0 g0"| 2222₆ ||class="entry q0 g0"| 18380₈ ||class="entry q0 g0"| 23830₈ ||class="entry q0 g0"| 7302₆ ||class="entry q0 g0"| 21476₈ ||class="entry q0 g0"| 18750₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (3214, 10874, 12726, 6310, 15954, 9630) ||class="entry q0 g0"| 3214₆ ||class="entry q0 g0"| 10874₈ ||class="entry q0 g0"| 12726₈ ||class="entry q0 g0"| 6310₆ ||class="entry q0 g0"| 15954₈ ||class="entry q0 g0"| 9630₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (8378, 21488, 29974, 13458, 18392, 24894) ||class="entry q0 g0"| 8378₆ ||class="entry q0 g0"| 21488₈ ||class="entry q0 g0"| 29974₈ ||class="entry q0 g0"| 13458₆ ||class="entry q0 g0"| 18392₈ ||class="entry q0 g0"| 24894₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (9370, 15942, 6582, 12466, 10862, 3486) ||class="entry q0 g0"| 9370₆ ||class="entry q0 g0"| 15942₈ ||class="entry q0 g0"| 6582₈ ||class="entry q0 g0"| 12466₆ ||class="entry q0 g0"| 10862₈ ||class="entry q0 g0"| 3486₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (18732, 21222, 7574, 23812, 18126, 2494) ||class="entry q0 g0"| 18732₆ ||class="entry q0 g0"| 21222₈ ||class="entry q0 g0"| 7574₈ ||class="entry q0 g0"| 23812₆ ||class="entry q0 g0"| 18126₈ ||class="entry q0 g0"| 2494₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (19724, 16208, 28982, 22820, 11128, 25886) ||class="entry q0 g0"| 19724₆ ||class="entry q0 g0"| 16208₈ ||class="entry q0 g0"| 28982₈ ||class="entry q0 g0"| 22820₆ ||class="entry q0 g0"| 11128₈ ||class="entry q0 g0"| 25886₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (24888, 18138, 13718, 29968, 21234, 8638) ||class="entry q0 g0"| 24888₆ ||class="entry q0 g0"| 18138₈ ||class="entry q0 g0"| 13718₈ ||class="entry q0 g0"| 29968₆ ||class="entry q0 g0"| 21234₈ ||class="entry q0 g0"| 8638₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 6, 8, 8) ||class="c"| (25880, 11116, 22838, 28976, 16196, 19742) ||class="entry q0 g0"| 25880₆ ||class="entry q0 g0"| 11116₈ ||class="entry q0 g0"| 22838₈ ||class="entry q0 g0"| 28976₆ ||class="entry q0 g0"| 16196₈ ||class="entry q0 g0"| 19742₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (11278, 3070, 8954, 14374, 8150, 14034) ||class="entry q0 g0"| 11278₆ ||class="entry q0 g0"| 3070₁₀ ||class="entry q0 g0"| 8954₈ ||class="entry q0 g0"| 14374₆ ||class="entry q0 g0"| 8150₁₀ ||class="entry q0 g0"| 14034₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (11290, 14294, 7878, 14386, 9214, 2798) ||class="entry q0 g0"| 11290₆ ||class="entry q0 g0"| 14294₁₀ ||class="entry q0 g0"| 7878₈ ||class="entry q0 g0"| 14386₆ ||class="entry q0 g0"| 9214₁₀ ||class="entry q0 g0"| 2798₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (11532, 24406, 30544, 14628, 19326, 25464) ||class="entry q0 g0"| 11532₆ ||class="entry q0 g0"| 24406₁₀ ||class="entry q0 g0"| 30544₈ ||class="entry q0 g0"| 14628₆ ||class="entry q0 g0"| 19326₁₀ ||class="entry q0 g0"| 25464₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (11544, 25470, 19308, 14640, 30550, 24388) ||class="entry q0 g0"| 11544₆ ||class="entry q0 g0"| 25470₁₀ ||class="entry q0 g0"| 19308₈ ||class="entry q0 g0"| 14640₆ ||class="entry q0 g0"| 30550₁₀ ||class="entry q0 g0"| 24388₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (16558, 4062, 20300, 21638, 7158, 23396) ||class="entry q0 g0"| 16558₆ ||class="entry q0 g0"| 4062₁₀ ||class="entry q0 g0"| 20300₈ ||class="entry q0 g0"| 21638₆ ||class="entry q0 g0"| 7158₁₀ ||class="entry q0 g0"| 23396₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (16570, 13302, 29552, 21650, 10206, 26456) ||class="entry q0 g0"| 16570₆ ||class="entry q0 g0"| 13302₁₀ ||class="entry q0 g0"| 29552₈ ||class="entry q0 g0"| 21650₆ ||class="entry q0 g0"| 10206₁₀ ||class="entry q0 g0"| 26456₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (16812, 23414, 6886, 21892, 20318, 3790) ||class="entry q0 g0"| 16812₆ ||class="entry q0 g0"| 23414₁₀ ||class="entry q0 g0"| 6886₈ ||class="entry q0 g0"| 21892₆ ||class="entry q0 g0"| 20318₁₀ ||class="entry q0 g0"| 3790₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 6, 10, 8) ||class="c"| (16824, 26462, 9946, 21904, 29558, 13042) ||class="entry q0 g0"| 16824₆ ||class="entry q0 g0"| 26462₁₀ ||class="entry q0 g0"| 9946₈ ||class="entry q0 g0"| 21904₆ ||class="entry q0 g0"| 29558₁₀ ||class="entry q0 g0"| 13042₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (636, 32706, 27388, 5716, 27626, 32468) ||class="entry q0 g0"| 636₆ ||class="entry q0 g0"| 32706₁₀ ||class="entry q0 g0"| 27388₁₀ ||class="entry q0 g0"| 5716₆ ||class="entry q0 g0"| 27626₁₀ ||class="entry q0 g0"| 32468₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 10, 6, 10, 10) ||class="c"| (1866, 31476, 28618, 4962, 28380, 31714) ||class="entry q0 g0"| 1866₆ ||class="entry q0 g0"| 31476₁₀ ||class="entry q0 g0"| 28618₁₀ ||class="entry q0 g0"| 4962₆ ||class="entry q0 g0"| 28380₁₀ ||class="entry q0 g0"| 31714₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (10856, 27646, 17148, 15936, 32726, 22228) ||class="entry q0 g0"| 10856₆ ||class="entry q0 g0"| 27646₁₂ ||class="entry q0 g0"| 17148₈ ||class="entry q0 g0"| 15936₆ ||class="entry q0 g0"| 32726₁₂ ||class="entry q0 g0"| 22228₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 6, 12, 8) ||class="c"| (18120, 28638, 12106, 21216, 31734, 15202) ||class="entry q0 g0"| 18120₆ ||class="entry q0 g0"| 28638₁₂ ||class="entry q0 g0"| 12106₈ ||class="entry q0 g0"| 21216₆ ||class="entry q0 g0"| 31734₁₂ ||class="entry q0 g0"| 15202₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (2812, 30290, 28044, 7892, 25210, 31140) ||class="entry q0 g0"| 2812₈ ||class="entry q0 g0"| 30290₈ ||class="entry q0 g0"| 28044₈ ||class="entry q0 g0"| 7892₈ ||class="entry q0 g0"| 25210₈ ||class="entry q0 g0"| 31140₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (3052, 30898, 25212, 8132, 27802, 30292) ||class="entry q0 g0"| 3052₈ ||class="entry q0 g0"| 30898₈ ||class="entry q0 g0"| 25212₈ ||class="entry q0 g0"| 8132₈ ||class="entry q0 g0"| 27802₈ ||class="entry q0 g0"| 30292₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (3802, 32132, 26442, 6898, 27052, 29538) ||class="entry q0 g0"| 3802₈ ||class="entry q0 g0"| 32132₈ ||class="entry q0 g0"| 26442₈ ||class="entry q0 g0"| 6898₈ ||class="entry q0 g0"| 27052₈ ||class="entry q0 g0"| 29538₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (4042, 29540, 26810, 7138, 26444, 31890) ||class="entry q0 g0"| 4042₈ ||class="entry q0 g0"| 29540₈ ||class="entry q0 g0"| 26810₈ ||class="entry q0 g0"| 7138₈ ||class="entry q0 g0"| 26444₈ ||class="entry q0 g0"| 31890₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (8956, 24134, 31152, 14036, 19054, 28056) ||class="entry q0 g0"| 8956₈ ||class="entry q0 g0"| 24134₈ ||class="entry q0 g0"| 31152₈ ||class="entry q0 g0"| 14036₈ ||class="entry q0 g0"| 19054₈ ||class="entry q0 g0"| 28056₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (9208, 27790, 19068, 14288, 30886, 24148) ||class="entry q0 g0"| 9208₈ ||class="entry q0 g0"| 27790₈ ||class="entry q0 g0"| 19068₈ ||class="entry q0 g0"| 14288₈ ||class="entry q0 g0"| 30886₈ ||class="entry q0 g0"| 24148₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (9934, 27064, 20298, 13030, 32144, 23394) ||class="entry q0 g0"| 9934₈ ||class="entry q0 g0"| 27064₈ ||class="entry q0 g0"| 20298₈ ||class="entry q0 g0"| 13030₈ ||class="entry q0 g0"| 32144₈ ||class="entry q0 g0"| 23394₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 8, 8, 8, 8) ||class="c"| (10186, 23408, 31878, 13282, 20312, 26798) ||class="entry q0 g0"| 10186₈ ||class="entry q0 g0"| 23408₈ ||class="entry q0 g0"| 31878₈ ||class="entry q0 g0"| 13282₈ ||class="entry q0 g0"| 20312₈ ||class="entry q0 g0"| 26798₈ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (894, 11114, 16214, 5974, 16194, 11134) ||class="entry q0 g0"| 894₈ ||class="entry q0 g0"| 11114₈ ||class="entry q0 g0"| 16214₁₀ ||class="entry q0 g0"| 5974₈ ||class="entry q0 g0"| 16194₈ ||class="entry q0 g0"| 11134₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (1886, 18140, 21494, 4982, 21236, 18398) ||class="entry q0 g0"| 1886₈ ||class="entry q0 g0"| 18140₈ ||class="entry q0 g0"| 21494₁₀ ||class="entry q0 g0"| 4982₈ ||class="entry q0 g0"| 21236₈ ||class="entry q0 g0"| 18398₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (27368, 10876, 22486, 32448, 15956, 17406) ||class="entry q0 g0"| 27368₈ ||class="entry q0 g0"| 10876₈ ||class="entry q0 g0"| 22486₁₀ ||class="entry q0 g0"| 32448₈ ||class="entry q0 g0"| 15956₈ ||class="entry q0 g0"| 17406₁₀ |- |class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 8, 8, 10) ||class="c"| (28360, 18378, 15222, 31456, 21474, 12126) ||class="entry q0 g0"| 28360₈ ||class="entry q0 g0"| 18378₈ ||class="entry q0 g0"| 15222₁₀ ||class="entry q0 g0"| 31456₈ ||class="entry q0 g0"| 21474₈ ||class="entry q0 g0"| 12126₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (640, 10883, 22231, 32296, 22333, 17385) ||class="entry q0 g0"| 640₂ ||class="entry q1 g0"| 10883₆ ||class="entry q1 g0"| 22231₁₀ ||class="entry q0 g0"| 32296₈ ||class="entry q1 g0"| 22333₁₀ ||class="entry q1 g0"| 17385₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (2112, 18629, 15207, 29928, 13691, 11865) ||class="entry q0 g0"| 2112₂ ||class="entry q1 g0"| 18629₆ ||class="entry q1 g0"| 15207₁₀ ||class="entry q0 g0"| 29928₈ ||class="entry q1 g0"| 13691₁₀ ||class="entry q1 g0"| 11865₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (8256, 24785, 12123, 23784, 7535, 14949) ||class="entry q0 g0"| 8256₂ ||class="entry q1 g0"| 24785₆ ||class="entry q1 g0"| 12123₁₀ ||class="entry q0 g0"| 23784₈ ||class="entry q1 g0"| 7535₁₀ ||class="entry q1 g0"| 14949₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 10, 8, 10, 8) ||class="c"| (16896, 27393, 17405, 16040, 5823, 22211) ||class="entry q0 g0"| 16896₂ ||class="entry q1 g0"| 27393₆ ||class="entry q1 g0"| 17405₁₀ ||class="entry q0 g0"| 16040₈ ||class="entry q1 g0"| 5823₁₀ ||class="entry q1 g0"| 22211₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (10432, 26945, 10283, 21608, 5375, 15637) ||class="entry q0 g0"| 10432₄ ||class="entry q1 g0"| 26945₆ ||class="entry q1 g0"| 10283₆ ||class="entry q0 g0"| 21608₆ ||class="entry q1 g0"| 5375₁₀ ||class="entry q1 g0"| 15637₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (19072, 25233, 17549, 13864, 7983, 20915) ||class="entry q0 g0"| 19072₄ ||class="entry q1 g0"| 25233₆ ||class="entry q1 g0"| 17549₆ ||class="entry q0 g0"| 13864₆ ||class="entry q1 g0"| 7983₁₀ ||class="entry q1 g0"| 20915₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (25216, 19077, 20657, 7720, 14139, 17807) ||class="entry q0 g0"| 25216₄ ||class="entry q1 g0"| 19077₆ ||class="entry q1 g0"| 20657₆ ||class="entry q0 g0"| 7720₆ ||class="entry q1 g0"| 14139₁₀ ||class="entry q1 g0"| 17807₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 6, 6, 10, 8) ||class="c"| (26688, 10435, 15617, 5352, 21885, 10303) ||class="entry q0 g0"| 26688₄ ||class="entry q1 g0"| 10435₆ ||class="entry q1 g0"| 15617₆ ||class="entry q0 g0"| 5352₆ ||class="entry q1 g0"| 21885₁₀ ||class="entry q1 g0"| 10303₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (10880, 663, 17131, 22056, 32553, 22485) ||class="entry q0 g0"| 10880₄ ||class="entry q1 g0"| 663₆ ||class="entry q1 g0"| 17131₈ ||class="entry q0 g0"| 22056₆ ||class="entry q1 g0"| 32553₁₀ ||class="entry q1 g0"| 22485₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (18624, 2375, 11853, 13416, 29945, 15219) ||class="entry q0 g0"| 18624₄ ||class="entry q1 g0"| 2375₆ ||class="entry q1 g0"| 11853₈ ||class="entry q0 g0"| 13416₆ ||class="entry q1 g0"| 29945₁₀ ||class="entry q1 g0"| 15219₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (24768, 8531, 14961, 7272, 23789, 12111) ||class="entry q0 g0"| 24768₄ ||class="entry q1 g0"| 8531₆ ||class="entry q1 g0"| 14961₈ ||class="entry q0 g0"| 7272₆ ||class="entry q1 g0"| 23789₁₀ ||class="entry q1 g0"| 12111₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 6, 10, 10) ||class="c"| (27136, 17173, 22465, 5800, 16043, 17151) ||class="entry q0 g0"| 27136₄ ||class="entry q1 g0"| 17173₆ ||class="entry q1 g0"| 22465₈ ||class="entry q0 g0"| 5800₆ ||class="entry q1 g0"| 16043₁₀ ||class="entry q1 g0"| 17151₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (900, 6219, 25883, 32556, 26101, 28709) ||class="entry q0 g0"| 900₄ ||class="entry q1 g0"| 6219₆ ||class="entry q1 g0"| 25883₈ ||class="entry q0 g0"| 32556₁₀ ||class="entry q1 g0"| 26101₁₀ ||class="entry q1 g0"| 28709₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (912, 9315, 22823, 32568, 23005, 19481) ||class="entry q0 g0"| 912₄ ||class="entry q1 g0"| 9315₆ ||class="entry q1 g0"| 22823₈ ||class="entry q0 g0"| 32568₁₀ ||class="entry q1 g0"| 23005₁₀ ||class="entry q1 g0"| 19481₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (2130, 4749, 24893, 29946, 28467, 29699) ||class="entry q0 g0"| 2130₄ ||class="entry q1 g0"| 4749₆ ||class="entry q1 g0"| 24893₈ ||class="entry q0 g0"| 29946₁₀ ||class="entry q1 g0"| 28467₁₀ ||class="entry q1 g0"| 29699₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (2384, 17957, 13463, 30200, 15259, 8617) ||class="entry q0 g0"| 2384₄ ||class="entry q1 g0"| 17957₆ ||class="entry q1 g0"| 13463₈ ||class="entry q0 g0"| 30200₁₀ ||class="entry q1 g0"| 15259₁₀ ||class="entry q1 g0"| 8617₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (8262, 1713, 18749, 23790, 31503, 23555) ||class="entry q0 g0"| 8262₄ ||class="entry q1 g0"| 1713₆ ||class="entry q1 g0"| 18749₈ ||class="entry q0 g0"| 23790₁₀ ||class="entry q1 g0"| 31503₁₀ ||class="entry q1 g0"| 23555₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (8516, 21017, 7319, 24044, 12199, 2473) ||class="entry q0 g0"| 8516₄ ||class="entry q1 g0"| 21017₆ ||class="entry q1 g0"| 7319₈ ||class="entry q0 g0"| 24044₁₀ ||class="entry q1 g0"| 12199₁₀ ||class="entry q1 g0"| 2473₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16902, 3425, 9627, 16046, 28895, 12453) ||class="entry q0 g0"| 16902₄ ||class="entry q1 g0"| 3425₆ ||class="entry q1 g0"| 9627₈ ||class="entry q0 g0"| 16046₁₀ ||class="entry q1 g0"| 28895₁₀ ||class="entry q1 g0"| 12453₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 10, 10, 6) ||class="c"| (16914, 12617, 6567, 16058, 19703, 3225) ||class="entry q0 g0"| 16914₄ ||class="entry q1 g0"| 12617₆ ||class="entry q1 g0"| 6567₈ ||class="entry q0 g0"| 16058₁₀ ||class="entry q1 g0"| 19703₁₀ ||class="entry q1 g0"| 3225₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (646, 19683, 12465, 32302, 12637, 9615) ||class="entry q0 g0"| 646₄ ||class="entry q1 g0"| 19683₈ ||class="entry q1 g0"| 12465₆ ||class="entry q0 g0"| 32302₁₀ ||class="entry q1 g0"| 12637₈ ||class="entry q1 g0"| 9615₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (658, 28875, 3213, 32314, 3445, 6579) ||class="entry q0 g0"| 658₄ ||class="entry q1 g0"| 28875₈ ||class="entry q1 g0"| 3213₆ ||class="entry q0 g0"| 32314₁₀ ||class="entry q1 g0"| 3445₈ ||class="entry q1 g0"| 6579₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (2118, 11941, 23809, 29934, 21275, 18495) ||class="entry q0 g0"| 2118₄ ||class="entry q1 g0"| 11941₈ ||class="entry q1 g0"| 23809₆ ||class="entry q0 g0"| 29934₁₀ ||class="entry q1 g0"| 21275₈ ||class="entry q1 g0"| 18495₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (2372, 31245, 2219, 30188, 1971, 7573) ||class="entry q0 g0"| 2372₄ ||class="entry q1 g0"| 31245₈ ||class="entry q1 g0"| 2219₆ ||class="entry q0 g0"| 30188₁₀ ||class="entry q1 g0"| 1971₈ ||class="entry q1 g0"| 7573₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (8274, 15001, 29953, 23802, 18215, 24639) ||class="entry q0 g0"| 8274₄ ||class="entry q1 g0"| 15001₈ ||class="entry q1 g0"| 29953₆ ||class="entry q0 g0"| 23802₁₀ ||class="entry q1 g0"| 18215₈ ||class="entry q1 g0"| 24639₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (8528, 28209, 8363, 24056, 5007, 13717) ||class="entry q0 g0"| 8528₄ ||class="entry q1 g0"| 28209₈ ||class="entry q1 g0"| 8363₆ ||class="entry q0 g0"| 24056₁₀ ||class="entry q1 g0"| 5007₈ ||class="entry q1 g0"| 13717₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (17156, 22985, 28721, 16300, 9335, 25871) ||class="entry q0 g0"| 17156₄ ||class="entry q1 g0"| 22985₈ ||class="entry q1 g0"| 28721₆ ||class="entry q0 g0"| 16300₁₀ ||class="entry q1 g0"| 9335₈ ||class="entry q1 g0"| 25871₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 10, 8, 8) ||class="c"| (17168, 26081, 19469, 16312, 6239, 22835) ||class="entry q0 g0"| 17168₄ ||class="entry q1 g0"| 26081₈ ||class="entry q1 g0"| 19469₆ ||class="entry q0 g0"| 16312₁₀ ||class="entry q1 g0"| 6239₈ ||class="entry q1 g0"| 22835₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (1248, 11491, 20663, 30792, 20829, 17801) ||class="entry q0 g0"| 1248₄ ||class="entry q1 g0"| 11491₈ ||class="entry q1 g0"| 20663₈ ||class="entry q0 g0"| 30792₆ ||class="entry q1 g0"| 20829₈ ||class="entry q1 g0"| 17801₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (3616, 20133, 15623, 29320, 13083, 10297) ||class="entry q0 g0"| 3616₄ ||class="entry q1 g0"| 20133₈ ||class="entry q1 g0"| 15623₈ ||class="entry q0 g0"| 29320₆ ||class="entry q1 g0"| 13083₈ ||class="entry q1 g0"| 10297₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (4296, 14539, 17567, 27744, 17781, 20897) ||class="entry q0 g0"| 4296₄ ||class="entry q1 g0"| 14539₈ ||class="entry q1 g0"| 17567₈ ||class="entry q0 g0"| 27744₆ ||class="entry q1 g0"| 17781₈ ||class="entry q1 g0"| 20897₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (6664, 23181, 10543, 26272, 10035, 15377) ||class="entry q0 g0"| 6664₄ ||class="entry q1 g0"| 23181₈ ||class="entry q1 g0"| 10543₈ ||class="entry q0 g0"| 26272₆ ||class="entry q1 g0"| 10035₈ ||class="entry q1 g0"| 15377₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (9760, 26289, 10555, 23176, 6927, 15365) ||class="entry q0 g0"| 9760₄ ||class="entry q1 g0"| 26289₈ ||class="entry q1 g0"| 10555₈ ||class="entry q0 g0"| 23176₆ ||class="entry q1 g0"| 6927₈ ||class="entry q1 g0"| 15365₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (12808, 29337, 15635, 20128, 3879, 10285) ||class="entry q0 g0"| 12808₄ ||class="entry q1 g0"| 29337₈ ||class="entry q1 g0"| 15635₈ ||class="entry q0 g0"| 20128₆ ||class="entry q1 g0"| 3879₈ ||class="entry q1 g0"| 10285₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (17504, 28001, 17821, 14536, 4319, 20643) ||class="entry q0 g0"| 17504₄ ||class="entry q1 g0"| 28001₈ ||class="entry q1 g0"| 17821₈ ||class="entry q0 g0"| 14536₆ ||class="entry q1 g0"| 4319₈ ||class="entry q1 g0"| 20643₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (20552, 31049, 20917, 11488, 1271, 17547) ||class="entry q0 g0"| 20552₄ ||class="entry q1 g0"| 31049₈ ||class="entry q1 g0"| 20917₈ ||class="entry q0 g0"| 11488₆ ||class="entry q1 g0"| 1271₈ ||class="entry q1 g0"| 17547₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (1696, 18229, 14967, 31240, 14987, 12105) ||class="entry q0 g0"| 1696₄ ||class="entry q1 g0"| 18229₈ ||class="entry q1 g0"| 14967₁₀ ||class="entry q0 g0"| 31240₆ ||class="entry q1 g0"| 14987₈ ||class="entry q1 g0"| 12105₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (3168, 9587, 22471, 28872, 22733, 17145) ||class="entry q0 g0"| 3168₄ ||class="entry q1 g0"| 9587₈ ||class="entry q1 g0"| 22471₁₀ ||class="entry q0 g0"| 28872₆ ||class="entry q1 g0"| 22733₈ ||class="entry q1 g0"| 17145₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (4744, 21277, 11871, 28192, 11939, 15201) ||class="entry q0 g0"| 4744₄ ||class="entry q1 g0"| 21277₈ ||class="entry q1 g0"| 11871₁₀ ||class="entry q0 g0"| 28192₆ ||class="entry q1 g0"| 11939₈ ||class="entry q1 g0"| 15201₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (6216, 12635, 17391, 25824, 19685, 22225) ||class="entry q0 g0"| 6216₄ ||class="entry q1 g0"| 12635₈ ||class="entry q1 g0"| 17391₁₀ ||class="entry q0 g0"| 25824₆ ||class="entry q1 g0"| 19685₈ ||class="entry q1 g0"| 22225₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (9312, 3431, 17403, 22728, 28889, 22213) ||class="entry q0 g0"| 9312₄ ||class="entry q1 g0"| 3431₈ ||class="entry q1 g0"| 17403₁₀ ||class="entry q0 g0"| 22728₆ ||class="entry q1 g0"| 28889₈ ||class="entry q1 g0"| 22213₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (12360, 6479, 22483, 19680, 25841, 17133) ||class="entry q0 g0"| 12360₄ ||class="entry q1 g0"| 6479₈ ||class="entry q1 g0"| 22483₁₀ ||class="entry q0 g0"| 19680₆ ||class="entry q1 g0"| 25841₈ ||class="entry q1 g0"| 17133₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (17952, 1719, 12125, 14984, 31497, 14947) ||class="entry q0 g0"| 17952₄ ||class="entry q1 g0"| 1719₈ ||class="entry q1 g0"| 12125₁₀ ||class="entry q0 g0"| 14984₆ ||class="entry q1 g0"| 31497₈ ||class="entry q1 g0"| 14947₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (21000, 4767, 15221, 11936, 28449, 11851) ||class="entry q0 g0"| 21000₄ ||class="entry q1 g0"| 4767₈ ||class="entry q1 g0"| 15221₁₀ ||class="entry q0 g0"| 11936₆ ||class="entry q1 g0"| 28449₈ ||class="entry q1 g0"| 11851₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (660, 5803, 27371, 32316, 27413, 32725) ||class="entry q0 g0"| 660₄ ||class="entry q1 g0"| 5803₈ ||class="entry q1 g0"| 27371₁₀ ||class="entry q0 g0"| 32316₁₀ ||class="entry q1 g0"| 27413₈ ||class="entry q1 g0"| 32725₁₂ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (2370, 7277, 28365, 30186, 25043, 31731) ||class="entry q0 g0"| 2370₄ ||class="entry q1 g0"| 7277₈ ||class="entry q1 g0"| 28365₁₀ ||class="entry q0 g0"| 30186₁₀ ||class="entry q1 g0"| 25043₈ ||class="entry q1 g0"| 31731₁₂ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (8514, 13433, 31473, 24042, 18887, 28623) ||class="entry q0 g0"| 8514₄ ||class="entry q1 g0"| 13433₈ ||class="entry q1 g0"| 31473₁₀ ||class="entry q0 g0"| 24042₁₀ ||class="entry q1 g0"| 18887₈ ||class="entry q1 g0"| 28623₁₂ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 10, 8, 12) ||class="c"| (16916, 22313, 32705, 16060, 10903, 27391) ||class="entry q0 g0"| 16916₄ ||class="entry q1 g0"| 22313₈ ||class="entry q1 g0"| 32705₁₀ ||class="entry q0 g0"| 16060₁₀ ||class="entry q1 g0"| 10903₈ ||class="entry q1 g0"| 27391₁₂ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (898, 32299, 893, 32554, 917, 5699) ||class="entry q0 g0"| 898₄ ||class="entry q1 g0"| 32299₁₀ ||class="entry q1 g0"| 893₈ ||class="entry q0 g0"| 32554₁₀ ||class="entry q1 g0"| 917₆ ||class="entry q1 g0"| 5699₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (2132, 29933, 1883, 29948, 2387, 4709) ||class="entry q0 g0"| 2132₄ ||class="entry q1 g0"| 29933₁₀ ||class="entry q1 g0"| 1883₈ ||class="entry q0 g0"| 29948₁₀ ||class="entry q1 g0"| 2387₆ ||class="entry q1 g0"| 4709₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (8276, 23801, 4967, 23804, 8519, 1625) ||class="entry q0 g0"| 8276₄ ||class="entry q1 g0"| 23801₁₀ ||class="entry q1 g0"| 4967₈ ||class="entry q0 g0"| 23804₁₀ ||class="entry q1 g0"| 8519₆ ||class="entry q1 g0"| 1625₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 8, 10, 6, 6) ||class="c"| (17154, 16297, 5719, 16298, 16919, 873) ||class="entry q0 g0"| 17154₄ ||class="entry q1 g0"| 16297₁₀ ||class="entry q1 g0"| 5719₈ ||class="entry q0 g0"| 16298₁₀ ||class="entry q1 g0"| 16919₆ ||class="entry q1 g0"| 873₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (450, 5629, 27069, 32106, 26691, 31875) ||class="entry q0 g0"| 450₄ ||class="entry q1 g0"| 5629₁₀ ||class="entry q1 g0"| 27069₁₀ ||class="entry q0 g0"| 32106₁₀ ||class="entry q1 g0"| 26691₆ ||class="entry q1 g0"| 31875₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (452, 29597, 4059, 32108, 3619, 6885) ||class="entry q0 g0"| 452₄ ||class="entry q1 g0"| 29597₁₀ ||class="entry q1 g0"| 4059₁₀ ||class="entry q0 g0"| 32108₁₀ ||class="entry q1 g0"| 3619₆ ||class="entry q1 g0"| 6885₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (464, 20405, 13287, 32120, 12811, 9945) ||class="entry q0 g0"| 464₄ ||class="entry q1 g0"| 20405₁₀ ||class="entry q1 g0"| 13287₁₀ ||class="entry q0 g0"| 32120₁₀ ||class="entry q1 g0"| 12811₆ ||class="entry q1 g0"| 9945₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2578, 31067, 3069, 30394, 1253, 7875) ||class="entry q0 g0"| 2578₄ ||class="entry q1 g0"| 31067₁₀ ||class="entry q1 g0"| 3069₁₀ ||class="entry q0 g0"| 30394₁₀ ||class="entry q1 g0"| 1253₆ ||class="entry q1 g0"| 7875₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2580, 7995, 28059, 30396, 25221, 30885) ||class="entry q0 g0"| 2580₄ ||class="entry q1 g0"| 7995₁₀ ||class="entry q1 g0"| 28059₁₀ ||class="entry q0 g0"| 30396₁₀ ||class="entry q1 g0"| 25221₆ ||class="entry q1 g0"| 30885₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2832, 11763, 24151, 30648, 20557, 19305) ||class="entry q0 g0"| 2832₄ ||class="entry q1 g0"| 11763₁₀ ||class="entry q1 g0"| 24151₁₀ ||class="entry q0 g0"| 30648₁₀ ||class="entry q1 g0"| 20557₆ ||class="entry q1 g0"| 19305₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8710, 28007, 9213, 24238, 4313, 14019) ||class="entry q0 g0"| 8710₄ ||class="entry q1 g0"| 28007₁₀ ||class="entry q1 g0"| 9213₁₀ ||class="entry q0 g0"| 24238₁₀ ||class="entry q1 g0"| 4313₆ ||class="entry q1 g0"| 14019₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8724, 14127, 31143, 24252, 19089, 27801) ||class="entry q0 g0"| 8724₄ ||class="entry q1 g0"| 14127₁₀ ||class="entry q1 g0"| 31143₁₀ ||class="entry q0 g0"| 24252₁₀ ||class="entry q1 g0"| 19089₆ ||class="entry q1 g0"| 27801₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8964, 14799, 30295, 24492, 17521, 25449) ||class="entry q0 g0"| 8964₄ ||class="entry q1 g0"| 14799₁₀ ||class="entry q1 g0"| 30295₁₀ ||class="entry q0 g0"| 24492₁₀ ||class="entry q1 g0"| 17521₆ ||class="entry q1 g0"| 25449₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16454, 26295, 20315, 15598, 6921, 23141) ||class="entry q0 g0"| 16454₄ ||class="entry q1 g0"| 26295₁₀ ||class="entry q1 g0"| 20315₁₀ ||class="entry q0 g0"| 15598₁₀ ||class="entry q1 g0"| 6921₆ ||class="entry q1 g0"| 23141₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16466, 23199, 29543, 15610, 10017, 26201) ||class="entry q0 g0"| 16466₄ ||class="entry q1 g0"| 23199₁₀ ||class="entry q1 g0"| 29543₁₀ ||class="entry q0 g0"| 15610₁₀ ||class="entry q1 g0"| 10017₆ ||class="entry q1 g0"| 26201₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16706, 21631, 31895, 15850, 10689, 27049) ||class="entry q0 g0"| 16706₄ ||class="entry q1 g0"| 21631₁₀ ||class="entry q1 g0"| 31895₁₀ ||class="entry q0 g0"| 15850₁₀ ||class="entry q1 g0"| 10689₆ ||class="entry q1 g0"| 27049₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (918, 16899, 16193, 32574, 16317, 10879) ||class="entry q0 g0"| 918₆ ||class="entry q1 g0"| 16899₄ ||class="entry q1 g0"| 16193₈ ||class="entry q0 g0"| 32574₁₂ ||class="entry q1 g0"| 16317₁₂ ||class="entry q1 g0"| 10879₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (2390, 8261, 21233, 30206, 24059, 18383) ||class="entry q0 g0"| 2390₆ ||class="entry q1 g0"| 8261₄ ||class="entry q1 g0"| 21233₈ ||class="entry q0 g0"| 30206₁₂ ||class="entry q1 g0"| 24059₁₂ ||class="entry q1 g0"| 18383₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (8534, 2129, 18125, 24062, 30191, 21491) ||class="entry q0 g0"| 8534₆ ||class="entry q1 g0"| 2129₄ ||class="entry q1 g0"| 18125₈ ||class="entry q0 g0"| 24062₁₂ ||class="entry q1 g0"| 30191₁₂ ||class="entry q1 g0"| 21491₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (17174, 897, 10859, 16318, 32319, 16213) ||class="entry q0 g0"| 17174₆ ||class="entry q1 g0"| 897₄ ||class="entry q1 g0"| 10859₈ ||class="entry q0 g0"| 16318₁₂ ||class="entry q1 g0"| 32319₁₂ ||class="entry q1 g0"| 16213₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (1702, 8533, 23569, 31246, 23787, 18735) ||class="entry q0 g0"| 1702₆ ||class="entry q1 g0"| 8533₆ ||class="entry q1 g0"| 23569₆ ||class="entry q0 g0"| 31246₈ ||class="entry q1 g0"| 23787₁₀ ||class="entry q1 g0"| 18735₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (3174, 17171, 12705, 28878, 16045, 9375) ||class="entry q0 g0"| 3174₆ ||class="entry q1 g0"| 17171₆ ||class="entry q1 g0"| 12705₆ ||class="entry q0 g0"| 28878₈ ||class="entry q1 g0"| 16045₁₀ ||class="entry q1 g0"| 9375₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (4762, 2389, 29701, 28210, 29931, 24891) ||class="entry q0 g0"| 4762₆ ||class="entry q1 g0"| 2389₆ ||class="entry q1 g0"| 29701₆ ||class="entry q0 g0"| 28210₈ ||class="entry q1 g0"| 29931₁₀ ||class="entry q1 g0"| 24891₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (6476, 915, 28707, 26084, 32301, 25885) ||class="entry q0 g0"| 6476₆ ||class="entry q1 g0"| 915₆ ||class="entry q1 g0"| 28707₆ ||class="entry q0 g0"| 26084₈ ||class="entry q1 g0"| 32301₁₀ ||class="entry q1 g0"| 25885₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (9584, 903, 19467, 23000, 32313, 22837) ||class="entry q0 g0"| 9584₆ ||class="entry q1 g0"| 903₆ ||class="entry q1 g0"| 19467₆ ||class="entry q0 g0"| 23000₈ ||class="entry q1 g0"| 32313₁₀ ||class="entry q1 g0"| 22837₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (12378, 17159, 3465, 19698, 16057, 6327) ||class="entry q0 g0"| 12378₆ ||class="entry q1 g0"| 17159₆ ||class="entry q1 g0"| 3465₆ ||class="entry q0 g0"| 19698₈ ||class="entry q1 g0"| 16057₁₀ ||class="entry q1 g0"| 6327₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (18224, 2135, 8365, 15256, 30185, 13715) ||class="entry q0 g0"| 18224₆ ||class="entry q1 g0"| 2135₆ ||class="entry q1 g0"| 8365₆ ||class="entry q0 g0"| 15256₈ ||class="entry q1 g0"| 30185₁₀ ||class="entry q1 g0"| 13715₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (21260, 8279, 2233, 12196, 24041, 7559) ||class="entry q0 g0"| 21260₆ ||class="entry q1 g0"| 8279₆ ||class="entry q1 g0"| 2233₆ ||class="entry q0 g0"| 12196₈ ||class="entry q1 g0"| 24041₁₀ ||class="entry q1 g0"| 7559₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1254, 19075, 14033, 30798, 14141, 9199) ||class="entry q0 g0"| 1254₆ ||class="entry q1 g0"| 19075₆ ||class="entry q1 g0"| 14033₈ ||class="entry q0 g0"| 30798₈ ||class="entry q1 g0"| 14141₁₀ ||class="entry q1 g0"| 9199₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1268, 4299, 27787, 30812, 28021, 31157) ||class="entry q0 g0"| 1268₆ ||class="entry q1 g0"| 4299₆ ||class="entry q1 g0"| 27787₈ ||class="entry q0 g0"| 30812₈ ||class="entry q1 g0"| 28021₁₀ ||class="entry q1 g0"| 31157₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3622, 10437, 23393, 29326, 21883, 20063) ||class="entry q0 g0"| 3622₆ ||class="entry q1 g0"| 10437₆ ||class="entry q1 g0"| 23393₈ ||class="entry q0 g0"| 29326₈ ||class="entry q1 g0"| 21883₁₀ ||class="entry q1 g0"| 20063₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3874, 6669, 26797, 29578, 26547, 32147) ||class="entry q0 g0"| 3874₆ ||class="entry q1 g0"| 6669₆ ||class="entry q1 g0"| 26797₈ ||class="entry q0 g0"| 29578₈ ||class="entry q1 g0"| 26547₁₀ ||class="entry q1 g0"| 32147₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4314, 25219, 7877, 27762, 7997, 3067) ||class="entry q0 g0"| 4314₆ ||class="entry q1 g0"| 25219₆ ||class="entry q1 g0"| 7877₈ ||class="entry q0 g0"| 27762₈ ||class="entry q1 g0"| 7997₁₀ ||class="entry q1 g0"| 3067₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4316, 1251, 30883, 27764, 31069, 28061) ||class="entry q0 g0"| 4316₆ ||class="entry q1 g0"| 1251₆ ||class="entry q1 g0"| 30883₈ ||class="entry q0 g0"| 27764₈ ||class="entry q1 g0"| 31069₁₀ ||class="entry q1 g0"| 28061₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6922, 3621, 31877, 26530, 29595, 27067) ||class="entry q0 g0"| 6922₆ ||class="entry q1 g0"| 3621₆ ||class="entry q1 g0"| 31877₈ ||class="entry q0 g0"| 26530₈ ||class="entry q1 g0"| 29595₁₀ ||class="entry q1 g0"| 27067₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6924, 26693, 6883, 26532, 5627, 4061) ||class="entry q0 g0"| 6924₆ ||class="entry q1 g0"| 26693₆ ||class="entry q1 g0"| 6883₈ ||class="entry q0 g0"| 26532₈ ||class="entry q1 g0"| 5627₁₀ ||class="entry q1 g0"| 4061₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10018, 12825, 31889, 23434, 20391, 27055) ||class="entry q0 g0"| 10018₆ ||class="entry q1 g0"| 12825₆ ||class="entry q1 g0"| 31889₈ ||class="entry q0 g0"| 23434₈ ||class="entry q1 g0"| 20391₁₀ ||class="entry q1 g0"| 27055₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10032, 26705, 9931, 23448, 5615, 13301) ||class="entry q0 g0"| 10032₆ ||class="entry q1 g0"| 26705₆ ||class="entry q1 g0"| 9931₈ ||class="entry q0 g0"| 23448₈ ||class="entry q1 g0"| 5615₁₀ ||class="entry q1 g0"| 13301₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10438, 3873, 20045, 21614, 29343, 23411) ||class="entry q0 g0"| 10438₆ ||class="entry q1 g0"| 3873₆ ||class="entry q1 g0"| 20045₈ ||class="entry q0 g0"| 21614₈ ||class="entry q1 g0"| 29343₁₀ ||class="entry q1 g0"| 23411₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10450, 13065, 29297, 21626, 20151, 26447) ||class="entry q0 g0"| 10450₆ ||class="entry q1 g0"| 13065₆ ||class="entry q1 g0"| 29297₈ ||class="entry q0 g0"| 21626₈ ||class="entry q1 g0"| 20151₁₀ ||class="entry q1 g0"| 26447₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12826, 10449, 26441, 20146, 21871, 29303) ||class="entry q0 g0"| 12826₆ ||class="entry q1 g0"| 10449₆ ||class="entry q1 g0"| 26441₈ ||class="entry q0 g0"| 20146₈ ||class="entry q1 g0"| 21871₁₀ ||class="entry q1 g0"| 29303₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13066, 9777, 26809, 20386, 23439, 32135) ||class="entry q0 g0"| 13066₆ ||class="entry q1 g0"| 9777₆ ||class="entry q1 g0"| 26809₈ ||class="entry q0 g0"| 20386₈ ||class="entry q1 g0"| 23439₁₀ ||class="entry q1 g0"| 32135₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (17524, 20809, 31137, 14556, 11511, 27807) ||class="entry q0 g0"| 17524₆ ||class="entry q1 g0"| 20809₆ ||class="entry q1 g0"| 31137₈ ||class="entry q0 g0"| 14556₈ ||class="entry q1 g0"| 11511₁₀ ||class="entry q1 g0"| 27807₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (17776, 25473, 19053, 14808, 7743, 24403) ||class="entry q0 g0"| 17776₆ ||class="entry q1 g0"| 25473₆ ||class="entry q1 g0"| 19053₈ ||class="entry q0 g0"| 14808₈ ||class="entry q1 g0"| 7743₁₀ ||class="entry q1 g0"| 24403₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (19078, 1265, 8939, 13870, 31055, 14293) ||class="entry q0 g0"| 19078₆ ||class="entry q1 g0"| 1265₆ ||class="entry q1 g0"| 8939₈ ||class="entry q0 g0"| 13870₈ ||class="entry q1 g0"| 31055₁₀ ||class="entry q1 g0"| 14293₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (19332, 20569, 30529, 14124, 11751, 25215) ||class="entry q0 g0"| 19332₆ ||class="entry q1 g0"| 20569₆ ||class="entry q1 g0"| 30529₈ ||class="entry q0 g0"| 14124₈ ||class="entry q1 g0"| 11751₁₀ ||class="entry q1 g0"| 25215₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (20572, 17761, 28041, 11508, 14559, 30903) ||class="entry q0 g0"| 20572₆ ||class="entry q1 g0"| 17761₆ ||class="entry q1 g0"| 28041₈ ||class="entry q0 g0"| 11508₈ ||class="entry q1 g0"| 14559₁₀ ||class="entry q1 g0"| 30903₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (20812, 19329, 25209, 11748, 13887, 30535) ||class="entry q0 g0"| 20812₆ ||class="entry q1 g0"| 19329₆ ||class="entry q1 g0"| 25209₈ ||class="entry q0 g0"| 11748₈ ||class="entry q1 g0"| 13887₁₀ ||class="entry q1 g0"| 30535₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25234, 4301, 2795, 7738, 28019, 8149) ||class="entry q0 g0"| 25234₆ ||class="entry q1 g0"| 4301₆ ||class="entry q1 g0"| 2795₈ ||class="entry q0 g0"| 7738₈ ||class="entry q1 g0"| 28019₁₀ ||class="entry q1 g0"| 8149₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25488, 17509, 24385, 7992, 14811, 19071) ||class="entry q0 g0"| 25488₆ ||class="entry q1 g0"| 17509₆ ||class="entry q1 g0"| 24385₈ ||class="entry q0 g0"| 7992₈ ||class="entry q1 g0"| 14811₁₀ ||class="entry q1 g0"| 19071₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26948, 6667, 3789, 5612, 26549, 7155) ||class="entry q0 g0"| 26948₆ ||class="entry q1 g0"| 6667₆ ||class="entry q1 g0"| 3789₈ ||class="entry q0 g0"| 5612₈ ||class="entry q1 g0"| 26549₁₀ ||class="entry q1 g0"| 7155₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26960, 9763, 13041, 5624, 23453, 10191) ||class="entry q0 g0"| 26960₆ ||class="entry q1 g0"| 9763₆ ||class="entry q1 g0"| 13041₈ ||class="entry q0 g0"| 5624₈ ||class="entry q1 g0"| 23453₁₀ ||class="entry q1 g0"| 10191₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (1506, 30795, 1309, 31050, 1525, 4131) ||class="entry q0 g0"| 1506₆ ||class="entry q1 g0"| 30795₈ ||class="entry q1 g0"| 1309₆ ||class="entry q0 g0"| 31050₈ ||class="entry q1 g0"| 1525₈ ||class="entry q1 g0"| 4131₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (3636, 29325, 315, 29340, 3891, 5125) ||class="entry q0 g0"| 3636₆ ||class="entry q1 g0"| 29325₈ ||class="entry q1 g0"| 315₆ ||class="entry q0 g0"| 29340₈ ||class="entry q1 g0"| 3891₈ ||class="entry q1 g0"| 5125₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (4554, 27747, 4405, 28002, 4573, 1035) ||class="entry q0 g0"| 4554₆ ||class="entry q1 g0"| 27747₈ ||class="entry q1 g0"| 4405₆ ||class="entry q0 g0"| 28002₈ ||class="entry q1 g0"| 4573₈ ||class="entry q1 g0"| 1035₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (6684, 26277, 5395, 26292, 6939, 45) ||class="entry q0 g0"| 6684₆ ||class="entry q1 g0"| 26277₈ ||class="entry q1 g0"| 5395₆ ||class="entry q0 g0"| 26292₈ ||class="entry q1 g0"| 6939₈ ||class="entry q1 g0"| 45₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (9780, 23193, 5383, 23196, 10023, 57) ||class="entry q0 g0"| 9780₆ ||class="entry q1 g0"| 23193₈ ||class="entry q1 g0"| 5383₆ ||class="entry q0 g0"| 23196₈ ||class="entry q1 g0"| 10023₈ ||class="entry q1 g0"| 57₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (10452, 21865, 5143, 21628, 10455, 297) ||class="entry q0 g0"| 10452₆ ||class="entry q1 g0"| 21865₈ ||class="entry q1 g0"| 5143₆ ||class="entry q0 g0"| 21628₈ ||class="entry q1 g0"| 10455₈ ||class="entry q1 g0"| 297₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (12828, 20145, 303, 20148, 13071, 5137) ||class="entry q0 g0"| 12828₆ ||class="entry q1 g0"| 20145₈ ||class="entry q1 g0"| 303₆ ||class="entry q0 g0"| 20148₈ ||class="entry q1 g0"| 13071₈ ||class="entry q1 g0"| 5137₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (17762, 14793, 4151, 14794, 17527, 1289) ||class="entry q0 g0"| 17762₆ ||class="entry q1 g0"| 14793₈ ||class="entry q1 g0"| 4151₆ ||class="entry q0 g0"| 14794₈ ||class="entry q1 g0"| 17527₈ ||class="entry q1 g0"| 1289₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (19330, 13881, 4391, 14122, 19335, 1049) ||class="entry q0 g0"| 19330₆ ||class="entry q1 g0"| 13881₈ ||class="entry q1 g0"| 4391₆ ||class="entry q0 g0"| 14122₈ ||class="entry q1 g0"| 19335₈ ||class="entry q1 g0"| 1049₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (20810, 11745, 1055, 11746, 20575, 4385) ||class="entry q0 g0"| 20810₆ ||class="entry q1 g0"| 11745₈ ||class="entry q1 g0"| 1055₆ ||class="entry q0 g0"| 11746₈ ||class="entry q1 g0"| 20575₈ ||class="entry q1 g0"| 4385₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (25474, 7725, 1307, 7978, 25491, 4133) ||class="entry q0 g0"| 25474₆ ||class="entry q1 g0"| 7725₈ ||class="entry q1 g0"| 1307₆ ||class="entry q0 g0"| 7978₈ ||class="entry q1 g0"| 25491₈ ||class="entry q1 g0"| 4133₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (26708, 5355, 317, 5372, 26965, 5123) ||class="entry q0 g0"| 26708₆ ||class="entry q1 g0"| 5355₈ ||class="entry q1 g0"| 317₆ ||class="entry q0 g0"| 5372₈ ||class="entry q1 g0"| 26965₈ ||class="entry q1 g0"| 5123₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (470, 10709, 21889, 32126, 21611, 16575) ||class="entry q0 g0"| 470₆ ||class="entry q1 g0"| 10709₈ ||class="entry q1 g0"| 21889₆ ||class="entry q0 g0"| 32126₁₂ ||class="entry q1 g0"| 21611₈ ||class="entry q1 g0"| 16575₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (2838, 19347, 14385, 30654, 13869, 11535) ||class="entry q0 g0"| 2838₆ ||class="entry q1 g0"| 19347₈ ||class="entry q1 g0"| 14385₆ ||class="entry q0 g0"| 30654₁₂ ||class="entry q1 g0"| 13869₈ ||class="entry q1 g0"| 11535₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (8982, 25479, 11277, 24510, 7737, 14643) ||class="entry q0 g0"| 8982₆ ||class="entry q1 g0"| 25479₈ ||class="entry q1 g0"| 11277₆ ||class="entry q0 g0"| 24510₁₂ ||class="entry q1 g0"| 7737₈ ||class="entry q1 g0"| 14643₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (16726, 26711, 16555, 15870, 5609, 21909) ||class="entry q0 g0"| 16726₆ ||class="entry q1 g0"| 26711₈ ||class="entry q1 g0"| 16555₆ ||class="entry q0 g0"| 15870₁₂ ||class="entry q1 g0"| 5609₈ ||class="entry q1 g0"| 21909₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (1968, 18901, 13703, 31512, 13419, 8377) ||class="entry q0 g0"| 1968₆ ||class="entry q1 g0"| 18901₈ ||class="entry q1 g0"| 13703₈ ||class="entry q0 g0"| 31512₈ ||class="entry q1 g0"| 13419₈ ||class="entry q1 g0"| 8377₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (3440, 11155, 22583, 29144, 22061, 19721) ||class="entry q0 g0"| 3440₆ ||class="entry q1 g0"| 11155₈ ||class="entry q1 g0"| 22583₈ ||class="entry q0 g0"| 29144₈ ||class="entry q1 g0"| 22061₈ ||class="entry q1 g0"| 19721₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (5004, 25045, 7571, 28452, 7275, 2221) ||class="entry q0 g0"| 5004₆ ||class="entry q1 g0"| 25045₈ ||class="entry q1 g0"| 7571₈ ||class="entry q0 g0"| 28452₈ ||class="entry q1 g0"| 7275₈ ||class="entry q1 g0"| 2221₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6234, 27411, 6581, 25842, 5805, 3211) ||class="entry q0 g0"| 6234₆ ||class="entry q1 g0"| 27411₈ ||class="entry q1 g0"| 6581₈ ||class="entry q0 g0"| 25842₈ ||class="entry q1 g0"| 5805₈ ||class="entry q1 g0"| 3211₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9318, 27399, 9629, 22734, 5817, 12451) ||class="entry q0 g0"| 9318₆ ||class="entry q1 g0"| 27399₈ ||class="entry q1 g0"| 9629₈ ||class="entry q0 g0"| 22734₈ ||class="entry q1 g0"| 5817₈ ||class="entry q1 g0"| 12451₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (11140, 12383, 28967, 22316, 19937, 25625) ||class="entry q0 g0"| 11140₆ ||class="entry q1 g0"| 12383₈ ||class="entry q1 g0"| 28967₈ ||class="entry q0 g0"| 22316₈ ||class="entry q1 g0"| 19937₈ ||class="entry q1 g0"| 25625₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (11152, 3191, 19739, 22328, 29129, 22565) ||class="entry q0 g0"| 11152₆ ||class="entry q1 g0"| 3191₈ ||class="entry q1 g0"| 19739₈ ||class="entry q0 g0"| 22328₈ ||class="entry q1 g0"| 29129₈ ||class="entry q1 g0"| 22565₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (12620, 11143, 25631, 19940, 22073, 28961) ||class="entry q0 g0"| 12620₆ ||class="entry q1 g0"| 11143₈ ||class="entry q1 g0"| 25631₈ ||class="entry q0 g0"| 19940₈ ||class="entry q1 g0"| 22073₈ ||class="entry q1 g0"| 28961₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17958, 24791, 18747, 14990, 7529, 23557) ||class="entry q0 g0"| 17958₆ ||class="entry q1 g0"| 24791₈ ||class="entry q1 g0"| 18747₈ ||class="entry q0 g0"| 14990₈ ||class="entry q1 g0"| 7529₈ ||class="entry q1 g0"| 23557₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18642, 21263, 29719, 13434, 11953, 24873) ||class="entry q0 g0"| 18642₆ ||class="entry q1 g0"| 21263₈ ||class="entry q1 g0"| 29719₈ ||class="entry q0 g0"| 13434₈ ||class="entry q1 g0"| 11953₈ ||class="entry q1 g0"| 24873₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18896, 1959, 8637, 13688, 31257, 13443) ||class="entry q0 g0"| 18896₆ ||class="entry q1 g0"| 1959₈ ||class="entry q1 g0"| 8637₈ ||class="entry q0 g0"| 13688₈ ||class="entry q1 g0"| 31257₈ ||class="entry q1 g0"| 13443₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (21018, 18647, 24879, 11954, 13673, 29713) ||class="entry q0 g0"| 21018₆ ||class="entry q1 g0"| 18647₈ ||class="entry q1 g0"| 24879₈ ||class="entry q0 g0"| 11954₈ ||class="entry q1 g0"| 13673₈ ||class="entry q1 g0"| 29713₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (24774, 18227, 23575, 7278, 14989, 18729) ||class="entry q0 g0"| 24774₆ ||class="entry q1 g0"| 18227₈ ||class="entry q1 g0"| 23575₈ ||class="entry q0 g0"| 7278₈ ||class="entry q1 g0"| 14989₈ ||class="entry q1 g0"| 18729₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (25028, 5019, 2493, 7532, 28197, 7299) ||class="entry q0 g0"| 25028₆ ||class="entry q1 g0"| 5019₈ ||class="entry q1 g0"| 2493₈ ||class="entry q0 g0"| 7532₈ ||class="entry q1 g0"| 28197₈ ||class="entry q1 g0"| 7299₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27142, 9589, 12711, 5806, 22731, 9369) ||class="entry q0 g0"| 27142₆ ||class="entry q1 g0"| 9589₈ ||class="entry q1 g0"| 12711₈ ||class="entry q0 g0"| 5806₈ ||class="entry q1 g0"| 22731₈ ||class="entry q1 g0"| 9369₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27154, 6493, 3483, 5818, 25827, 6309) ||class="entry q0 g0"| 27154₆ ||class="entry q1 g0"| 6493₈ ||class="entry q1 g0"| 3483₈ ||class="entry q0 g0"| 5818₈ ||class="entry q1 g0"| 25827₈ ||class="entry q1 g0"| 6309₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (1508, 7723, 25467, 31052, 25493, 30277) ||class="entry q0 g0"| 1508₆ ||class="entry q1 g0"| 7723₈ ||class="entry q1 g0"| 25467₁₀ ||class="entry q0 g0"| 31052₈ ||class="entry q1 g0"| 25493₈ ||class="entry q1 g0"| 30277₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3634, 5357, 26461, 29338, 26963, 29283) ||class="entry q0 g0"| 3634₆ ||class="entry q1 g0"| 5357₈ ||class="entry q1 g0"| 26461₁₀ ||class="entry q0 g0"| 29338₈ ||class="entry q1 g0"| 26963₈ ||class="entry q1 g0"| 29283₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (4568, 13867, 19311, 28016, 19349, 24145) ||class="entry q0 g0"| 4568₆ ||class="entry q1 g0"| 13867₈ ||class="entry q1 g0"| 19311₁₀ ||class="entry q0 g0"| 28016₈ ||class="entry q1 g0"| 19349₈ ||class="entry q1 g0"| 24145₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (6936, 21613, 9951, 26544, 10707, 13281) ||class="entry q0 g0"| 6936₆ ||class="entry q1 g0"| 21613₈ ||class="entry q1 g0"| 9951₁₀ ||class="entry q0 g0"| 26544₈ ||class="entry q1 g0"| 10707₈ ||class="entry q1 g0"| 13281₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (10020, 21625, 6903, 23436, 10695, 4041) ||class="entry q0 g0"| 10020₆ ||class="entry q1 g0"| 21625₈ ||class="entry q1 g0"| 6903₁₀ ||class="entry q0 g0"| 23436₈ ||class="entry q1 g0"| 10695₈ ||class="entry q1 g0"| 4041₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (10692, 23433, 7143, 21868, 9783, 3801) ||class="entry q0 g0"| 10692₆ ||class="entry q1 g0"| 23433₈ ||class="entry q1 g0"| 7143₁₀ ||class="entry q0 g0"| 21868₈ ||class="entry q1 g0"| 9783₈ ||class="entry q1 g0"| 3801₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (10704, 26529, 10203, 21880, 6687, 13029) ||class="entry q0 g0"| 10704₆ ||class="entry q1 g0"| 26529₈ ||class="entry q1 g0"| 10203₁₀ ||class="entry q0 g0"| 21880₈ ||class="entry q1 g0"| 6687₈ ||class="entry q1 g0"| 13029₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (12814, 5369, 23413, 20134, 26951, 20043) ||class="entry q0 g0"| 12814₆ ||class="entry q1 g0"| 5369₈ ||class="entry q1 g0"| 23413₁₀ ||class="entry q0 g0"| 20134₈ ||class="entry q1 g0"| 26951₈ ||class="entry q1 g0"| 20043₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (17522, 14121, 8135, 14554, 19095, 2809) ||class="entry q0 g0"| 17522₆ ||class="entry q1 g0"| 14121₈ ||class="entry q1 g0"| 8135₁₀ ||class="entry q0 g0"| 14554₈ ||class="entry q1 g0"| 19095₈ ||class="entry q1 g0"| 2809₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (19090, 14553, 7895, 13882, 17767, 3049) ||class="entry q0 g0"| 19090₆ ||class="entry q1 g0"| 14553₈ ||class="entry q1 g0"| 7895₁₀ ||class="entry q0 g0"| 13882₈ ||class="entry q1 g0"| 17767₈ ||class="entry q1 g0"| 3049₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (19344, 27761, 19325, 14136, 4559, 24131) ||class="entry q0 g0"| 19344₆ ||class="entry q1 g0"| 27761₈ ||class="entry q1 g0"| 19325₁₀ ||class="entry q0 g0"| 14136₈ ||class="entry q1 g0"| 4559₈ ||class="entry q1 g0"| 24131₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (20558, 7977, 14291, 11494, 25239, 8941) ||class="entry q0 g0"| 20558₆ ||class="entry q1 g0"| 7977₈ ||class="entry q1 g0"| 14291₁₀ ||class="entry q0 g0"| 11494₈ ||class="entry q1 g0"| 25239₈ ||class="entry q1 g0"| 8941₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (25222, 11493, 14039, 7726, 20827, 9193) ||class="entry q0 g0"| 25222₆ ||class="entry q1 g0"| 11493₈ ||class="entry q1 g0"| 14039₁₀ ||class="entry q0 g0"| 7726₈ ||class="entry q1 g0"| 20827₈ ||class="entry q1 g0"| 9193₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (25476, 30797, 25469, 7980, 1523, 30275) ||class="entry q0 g0"| 25476₆ ||class="entry q1 g0"| 30797₈ ||class="entry q1 g0"| 25469₁₀ ||class="entry q0 g0"| 7980₈ ||class="entry q1 g0"| 1523₈ ||class="entry q1 g0"| 30275₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26694, 20131, 23399, 5358, 13085, 20057) ||class="entry q0 g0"| 26694₆ ||class="entry q1 g0"| 20131₈ ||class="entry q1 g0"| 23399₁₀ ||class="entry q0 g0"| 5358₈ ||class="entry q1 g0"| 13085₈ ||class="entry q1 g0"| 20057₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26706, 29323, 26459, 5370, 3893, 29285) ||class="entry q0 g0"| 26706₆ ||class="entry q1 g0"| 29323₈ ||class="entry q1 g0"| 26459₁₀ ||class="entry q0 g0"| 5370₈ ||class="entry q1 g0"| 3893₈ ||class="entry q1 g0"| 29285₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (1954, 5021, 28637, 31498, 28195, 31459) ||class="entry q0 g0"| 1954₆ ||class="entry q1 g0"| 5021₈ ||class="entry q1 g0"| 28637₁₂ ||class="entry q0 g0"| 31498₈ ||class="entry q1 g0"| 28195₈ ||class="entry q1 g0"| 31459₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (3188, 6491, 27643, 28892, 25829, 32453) ||class="entry q0 g0"| 3188₆ ||class="entry q1 g0"| 6491₈ ||class="entry q1 g0"| 27643₁₂ ||class="entry q0 g0"| 28892₈ ||class="entry q1 g0"| 25829₈ ||class="entry q1 g0"| 32453₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (5002, 1973, 31733, 28450, 31243, 28363) ||class="entry q0 g0"| 5002₆ ||class="entry q1 g0"| 1973₈ ||class="entry q1 g0"| 31733₁₂ ||class="entry q0 g0"| 28450₈ ||class="entry q1 g0"| 31243₈ ||class="entry q1 g0"| 28363₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (6236, 3443, 32723, 25844, 28877, 27373) ||class="entry q0 g0"| 6236₆ ||class="entry q1 g0"| 3443₈ ||class="entry q1 g0"| 32723₁₂ ||class="entry q0 g0"| 25844₈ ||class="entry q1 g0"| 28877₈ ||class="entry q1 g0"| 27373₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (9332, 12623, 32711, 22748, 19697, 27385) ||class="entry q0 g0"| 9332₆ ||class="entry q1 g0"| 12623₈ ||class="entry q1 g0"| 32711₁₂ ||class="entry q0 g0"| 22748₈ ||class="entry q1 g0"| 19697₈ ||class="entry q1 g0"| 27385₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (12380, 9575, 27631, 19700, 22745, 32465) ||class="entry q0 g0"| 12380₆ ||class="entry q1 g0"| 9575₈ ||class="entry q1 g0"| 27631₁₂ ||class="entry q0 g0"| 19700₈ ||class="entry q1 g0"| 22745₈ ||class="entry q1 g0"| 32465₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (18210, 21023, 31479, 15242, 12193, 28617) ||class="entry q0 g0"| 18210₆ ||class="entry q1 g0"| 21023₈ ||class="entry q1 g0"| 31479₁₂ ||class="entry q0 g0"| 15242₈ ||class="entry q1 g0"| 12193₈ ||class="entry q1 g0"| 28617₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 12, 8, 8, 10) ||class="c"| (21258, 17975, 28383, 12194, 15241, 31713) ||class="entry q0 g0"| 21258₆ ||class="entry q1 g0"| 17975₈ ||class="entry q1 g0"| 28383₁₂ ||class="entry q0 g0"| 12194₈ ||class="entry q1 g0"| 15241₈ ||class="entry q1 g0"| 31713₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1714, 7549, 24621, 31258, 24771, 29971) ||class="entry q0 g0"| 1714₆ ||class="entry q1 g0"| 7549₁₀ ||class="entry q1 g0"| 24621₆ ||class="entry q0 g0"| 31258₈ ||class="entry q1 g0"| 24771₆ ||class="entry q1 g0"| 29971₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1716, 31517, 1611, 31260, 1699, 4981) ||class="entry q0 g0"| 1716₆ ||class="entry q1 g0"| 31517₁₀ ||class="entry q1 g0"| 1611₆ ||class="entry q0 g0"| 31260₈ ||class="entry q1 g0"| 1699₆ ||class="entry q1 g0"| 4981₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (3426, 29147, 621, 29130, 3173, 5971) ||class="entry q0 g0"| 3426₆ ||class="entry q1 g0"| 29147₁₀ ||class="entry q1 g0"| 621₆ ||class="entry q0 g0"| 29130₈ ||class="entry q1 g0"| 3173₆ ||class="entry q1 g0"| 5971₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (3428, 6075, 25611, 29132, 27141, 28981) ||class="entry q0 g0"| 3428₆ ||class="entry q1 g0"| 6075₁₀ ||class="entry q1 g0"| 25611₆ ||class="entry q0 g0"| 29132₈ ||class="entry q1 g0"| 27141₆ ||class="entry q1 g0"| 28981₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4750, 13693, 18489, 28198, 18627, 23815) ||class="entry q0 g0"| 4750₆ ||class="entry q1 g0"| 13693₁₀ ||class="entry q1 g0"| 18489₆ ||class="entry q0 g0"| 28198₈ ||class="entry q1 g0"| 18627₆ ||class="entry q1 g0"| 23815₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (4764, 28469, 4707, 28212, 4747, 1885) ||class="entry q0 g0"| 4764₆ ||class="entry q1 g0"| 28469₁₀ ||class="entry q1 g0"| 4707₆ ||class="entry q0 g0"| 28212₈ ||class="entry q1 g0"| 4747₆ ||class="entry q1 g0"| 1885₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6222, 22331, 9609, 25830, 10885, 12471) ||class="entry q0 g0"| 6222₆ ||class="entry q1 g0"| 22331₁₀ ||class="entry q1 g0"| 9609₆ ||class="entry q0 g0"| 25830₈ ||class="entry q1 g0"| 10885₆ ||class="entry q1 g0"| 12471₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6474, 26099, 5701, 26082, 6221, 891) ||class="entry q0 g0"| 6474₆ ||class="entry q1 g0"| 26099₁₀ ||class="entry q1 g0"| 5701₆ ||class="entry q0 g0"| 26082₈ ||class="entry q1 g0"| 6221₆ ||class="entry q1 g0"| 891₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9330, 22319, 6561, 22746, 10897, 3231) ||class="entry q0 g0"| 9330₆ ||class="entry q1 g0"| 22319₁₀ ||class="entry q1 g0"| 6561₆ ||class="entry q0 g0"| 22746₈ ||class="entry q1 g0"| 10897₆ ||class="entry q1 g0"| 3231₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9570, 22991, 5713, 22986, 9329, 879) ||class="entry q0 g0"| 9570₆ ||class="entry q1 g0"| 22991₁₀ ||class="entry q1 g0"| 5713₆ ||class="entry q0 g0"| 22986₈ ||class="entry q1 g0"| 9329₆ ||class="entry q1 g0"| 879₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10886, 25847, 9357, 22062, 6473, 12723) ||class="entry q0 g0"| 10886₆ ||class="entry q1 g0"| 25847₁₀ ||class="entry q1 g0"| 9357₆ ||class="entry q0 g0"| 22062₈ ||class="entry q1 g0"| 6473₆ ||class="entry q1 g0"| 12723₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10898, 22751, 6321, 22074, 9569, 3471) ||class="entry q0 g0"| 10898₆ ||class="entry q1 g0"| 22751₁₀ ||class="entry q1 g0"| 6321₆ ||class="entry q0 g0"| 22074₈ ||class="entry q1 g0"| 9569₆ ||class="entry q1 g0"| 3471₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (11138, 22079, 5953, 22314, 11137, 639) ||class="entry q0 g0"| 11138₆ ||class="entry q1 g0"| 22079₁₀ ||class="entry q1 g0"| 5953₆ ||class="entry q0 g0"| 22314₈ ||class="entry q1 g0"| 11137₆ ||class="entry q1 g0"| 639₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (12618, 19943, 633, 19938, 12377, 5959) ||class="entry q0 g0"| 12618₆ ||class="entry q1 g0"| 19943₁₀ ||class="entry q1 g0"| 633₆ ||class="entry q0 g0"| 19938₈ ||class="entry q1 g0"| 12377₆ ||class="entry q1 g0"| 5959₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (12632, 6063, 22563, 19952, 27153, 19741) ||class="entry q0 g0"| 12632₆ ||class="entry q1 g0"| 6063₁₀ ||class="entry q1 g0"| 22563₆ ||class="entry q0 g0"| 19952₈ ||class="entry q1 g0"| 27153₆ ||class="entry q1 g0"| 19741₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (17972, 15007, 4961, 15004, 18209, 1631) ||class="entry q0 g0"| 17972₆ ||class="entry q1 g0"| 15007₁₀ ||class="entry q1 g0"| 4961₆ ||class="entry q0 g0"| 15004₈ ||class="entry q1 g0"| 18209₆ ||class="entry q1 g0"| 1631₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18212, 13439, 7313, 15244, 18881, 2479) ||class="entry q0 g0"| 18212₆ ||class="entry q1 g0"| 13439₁₀ ||class="entry q1 g0"| 7313₆ ||class="entry q0 g0"| 15244₈ ||class="entry q1 g0"| 18881₆ ||class="entry q1 g0"| 2479₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18630, 28455, 18475, 13422, 4761, 23829) ||class="entry q0 g0"| 18630₆ ||class="entry q1 g0"| 28455₁₀ ||class="entry q1 g0"| 18475₆ ||class="entry q0 g0"| 13422₈ ||class="entry q1 g0"| 4761₆ ||class="entry q1 g0"| 23829₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18644, 13679, 4721, 13436, 18641, 1871) ||class="entry q0 g0"| 18644₆ ||class="entry q1 g0"| 13679₁₀ ||class="entry q1 g0"| 4721₆ ||class="entry q0 g0"| 13436₈ ||class="entry q1 g0"| 18641₆ ||class="entry q1 g0"| 1871₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (18884, 15247, 7553, 13676, 17969, 2239) ||class="entry q0 g0"| 18884₆ ||class="entry q1 g0"| 15247₁₀ ||class="entry q1 g0"| 7553₆ ||class="entry q0 g0"| 13676₈ ||class="entry q1 g0"| 17969₆ ||class="entry q1 g0"| 2239₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (21020, 11959, 1865, 11956, 21257, 4727) ||class="entry q0 g0"| 21020₆ ||class="entry q1 g0"| 11959₁₀ ||class="entry q1 g0"| 1865₆ ||class="entry q0 g0"| 11956₈ ||class="entry q1 g0"| 21257₆ ||class="entry q1 g0"| 4727₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (21272, 7295, 13445, 12208, 25025, 8635) ||class="entry q0 g0"| 21272₆ ||class="entry q1 g0"| 7295₁₀ ||class="entry q1 g0"| 13445₆ ||class="entry q0 g0"| 12208₈ ||class="entry q1 g0"| 25025₆ ||class="entry q1 g0"| 8635₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (24786, 31515, 24619, 7290, 1701, 29973) ||class="entry q0 g0"| 24786₆ ||class="entry q1 g0"| 31515₁₀ ||class="entry q1 g0"| 24619₆ ||class="entry q0 g0"| 7290₈ ||class="entry q1 g0"| 1701₆ ||class="entry q1 g0"| 29973₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (24788, 7547, 1613, 7292, 24773, 4979) ||class="entry q0 g0"| 24788₆ ||class="entry q1 g0"| 7547₁₀ ||class="entry q1 g0"| 1613₆ ||class="entry q0 g0"| 7292₈ ||class="entry q1 g0"| 24773₆ ||class="entry q1 g0"| 4979₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (25040, 12211, 13697, 7544, 21005, 8383) ||class="entry q0 g0"| 25040₆ ||class="entry q1 g0"| 12211₁₀ ||class="entry q1 g0"| 13697₆ ||class="entry q0 g0"| 7544₈ ||class="entry q1 g0"| 21005₆ ||class="entry q1 g0"| 8383₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27394, 6077, 619, 6058, 27139, 5973) ||class="entry q0 g0"| 27394₆ ||class="entry q1 g0"| 6077₁₀ ||class="entry q1 g0"| 619₆ ||class="entry q0 g0"| 6058₈ ||class="entry q1 g0"| 27139₆ ||class="entry q1 g0"| 5973₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27396, 29149, 25613, 6060, 3171, 28979) ||class="entry q0 g0"| 27396₆ ||class="entry q1 g0"| 29149₁₀ ||class="entry q1 g0"| 25613₆ ||class="entry q0 g0"| 6060₈ ||class="entry q1 g0"| 3171₆ ||class="entry q1 g0"| 28979₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (27408, 19957, 22577, 6072, 12363, 19727) ||class="entry q0 g0"| 27408₆ ||class="entry q1 g0"| 19957₁₀ ||class="entry q1 g0"| 22577₆ ||class="entry q0 g0"| 6072₈ ||class="entry q1 g0"| 12363₆ ||class="entry q1 g0"| 19727₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (1266, 30379, 2797, 30810, 2837, 8147) ||class="entry q0 g0"| 1266₆ ||class="entry q1 g0"| 30379₁₀ ||class="entry q1 g0"| 2797₈ ||class="entry q0 g0"| 30810₈ ||class="entry q1 g0"| 2837₆ ||class="entry q1 g0"| 8147₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (3876, 31853, 3787, 29580, 467, 7157) ||class="entry q0 g0"| 3876₆ ||class="entry q1 g0"| 31853₁₀ ||class="entry q1 g0"| 3787₈ ||class="entry q0 g0"| 29580₈ ||class="entry q1 g0"| 467₆ ||class="entry q1 g0"| 7157₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (4302, 24235, 8953, 27750, 8981, 14279) ||class="entry q0 g0"| 4302₆ ||class="entry q1 g0"| 24235₁₀ ||class="entry q1 g0"| 8953₈ ||class="entry q0 g0"| 27750₈ ||class="entry q1 g0"| 8981₆ ||class="entry q1 g0"| 14279₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (6670, 15597, 20297, 26278, 16723, 23159) ||class="entry q0 g0"| 6670₆ ||class="entry q1 g0"| 15597₁₀ ||class="entry q1 g0"| 20297₈ ||class="entry q0 g0"| 26278₈ ||class="entry q1 g0"| 16723₆ ||class="entry q1 g0"| 23159₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (9778, 15609, 29537, 23194, 16711, 26207) ||class="entry q0 g0"| 9778₆ ||class="entry q1 g0"| 15609₁₀ ||class="entry q1 g0"| 29537₈ ||class="entry q0 g0"| 23194₈ ||class="entry q1 g0"| 16711₆ ||class="entry q1 g0"| 26207₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (10690, 15849, 32129, 21866, 16471, 26815) ||class="entry q0 g0"| 10690₆ ||class="entry q1 g0"| 15849₁₀ ||class="entry q1 g0"| 32129₈ ||class="entry q0 g0"| 21866₈ ||class="entry q1 g0"| 16471₆ ||class="entry q1 g0"| 26815₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (13080, 31865, 13027, 20400, 455, 10205) ||class="entry q0 g0"| 13080₆ ||class="entry q1 g0"| 31865₁₀ ||class="entry q1 g0"| 13027₈ ||class="entry q0 g0"| 20400₈ ||class="entry q1 g0"| 455₆ ||class="entry q1 g0"| 10205₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (17764, 24489, 30289, 14796, 8727, 25455) ||class="entry q0 g0"| 17764₆ ||class="entry q1 g0"| 24489₁₀ ||class="entry q1 g0"| 30289₈ ||class="entry q0 g0"| 14796₈ ||class="entry q1 g0"| 8727₆ ||class="entry q1 g0"| 25455₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (19092, 24249, 30897, 13884, 8967, 28047) ||class="entry q0 g0"| 19092₆ ||class="entry q1 g0"| 24249₁₀ ||class="entry q1 g0"| 30897₈ ||class="entry q0 g0"| 13884₈ ||class="entry q1 g0"| 8967₆ ||class="entry q1 g0"| 28047₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (20824, 30633, 24133, 11760, 2583, 19323) ||class="entry q0 g0"| 20824₆ ||class="entry q1 g0"| 30633₁₀ ||class="entry q1 g0"| 24133₈ ||class="entry q0 g0"| 11760₈ ||class="entry q1 g0"| 2583₆ ||class="entry q1 g0"| 19323₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (25236, 30381, 27789, 7740, 2835, 31155) ||class="entry q0 g0"| 25236₆ ||class="entry q1 g0"| 30381₁₀ ||class="entry q1 g0"| 27789₈ ||class="entry q0 g0"| 7740₈ ||class="entry q1 g0"| 2835₆ ||class="entry q1 g0"| 31155₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 8, 8, 6, 10) ||class="c"| (26946, 31851, 26795, 5610, 469, 32149) ||class="entry q0 g0"| 26946₆ ||class="entry q1 g0"| 31851₁₀ ||class="entry q1 g0"| 26795₈ ||class="entry q0 g0"| 5610₈ ||class="entry q1 g0"| 469₆ ||class="entry q1 g0"| 32149₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (1956, 30205, 2491, 31500, 2115, 7301) ||class="entry q0 g0"| 1956₆ ||class="entry q1 g0"| 30205₁₂ ||class="entry q1 g0"| 2491₈ ||class="entry q0 g0"| 31500₈ ||class="entry q1 g0"| 2115₄ ||class="entry q1 g0"| 7301₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (3186, 32571, 3485, 28890, 645, 6307) ||class="entry q0 g0"| 3186₆ ||class="entry q1 g0"| 32571₁₂ ||class="entry q1 g0"| 3485₈ ||class="entry q0 g0"| 28890₈ ||class="entry q1 g0"| 645₄ ||class="entry q1 g0"| 6307₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (5016, 24061, 8623, 28464, 8259, 13457) ||class="entry q0 g0"| 5016₆ ||class="entry q1 g0"| 24061₁₂ ||class="entry q1 g0"| 8623₈ ||class="entry q0 g0"| 28464₈ ||class="entry q1 g0"| 8259₄ ||class="entry q1 g0"| 13457₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (6488, 16315, 19487, 26096, 16901, 22817) ||class="entry q0 g0"| 6488₆ ||class="entry q1 g0"| 16315₁₂ ||class="entry q1 g0"| 19487₈ ||class="entry q0 g0"| 26096₈ ||class="entry q1 g0"| 16901₄ ||class="entry q1 g0"| 22817₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (9572, 16303, 28727, 22988, 16913, 25865) ||class="entry q0 g0"| 9572₆ ||class="entry q1 g0"| 16303₁₂ ||class="entry q1 g0"| 28727₈ ||class="entry q0 g0"| 22988₈ ||class="entry q1 g0"| 16913₄ ||class="entry q1 g0"| 25865₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (12366, 32559, 12725, 19686, 657, 9355) ||class="entry q0 g0"| 12366₆ ||class="entry q1 g0"| 32559₁₂ ||class="entry q1 g0"| 12725₈ ||class="entry q0 g0"| 19686₈ ||class="entry q1 g0"| 657₄ ||class="entry q1 g0"| 9355₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (17970, 23807, 29959, 15002, 8513, 24633) ||class="entry q0 g0"| 17970₆ ||class="entry q1 g0"| 23807₁₂ ||class="entry q1 g0"| 29959₈ ||class="entry q0 g0"| 15002₈ ||class="entry q1 g0"| 8513₄ ||class="entry q1 g0"| 24633₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (21006, 29951, 23827, 11942, 2369, 18477) ||class="entry q0 g0"| 21006₆ ||class="entry q1 g0"| 29951₁₂ ||class="entry q1 g0"| 23827₈ ||class="entry q0 g0"| 11942₈ ||class="entry q1 g0"| 2369₄ ||class="entry q1 g0"| 18477₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (10900, 16063, 32471, 22076, 17153, 27625) ||class="entry q0 g0"| 10900₆ ||class="entry q1 g0"| 16063₁₂ ||class="entry q1 g0"| 32471₁₂ ||class="entry q0 g0"| 22076₈ ||class="entry q1 g0"| 17153₄ ||class="entry q1 g0"| 27625₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (18882, 24047, 31719, 13674, 8273, 28377) ||class="entry q0 g0"| 18882₆ ||class="entry q1 g0"| 24047₁₂ ||class="entry q1 g0"| 31719₁₂ ||class="entry q0 g0"| 13674₈ ||class="entry q1 g0"| 8273₄ ||class="entry q1 g0"| 28377₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (25026, 30203, 28635, 7530, 2117, 31461) ||class="entry q0 g0"| 25026₆ ||class="entry q1 g0"| 30203₁₂ ||class="entry q1 g0"| 28635₁₂ ||class="entry q0 g0"| 7530₈ ||class="entry q1 g0"| 2117₄ ||class="entry q1 g0"| 31461₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 12, 8, 4, 10) ||class="c"| (27156, 32573, 27645, 5820, 643, 32451) ||class="entry q0 g0"| 27156₆ ||class="entry q1 g0"| 32573₁₂ ||class="entry q1 g0"| 27645₁₂ ||class="entry q0 g0"| 5820₈ ||class="entry q1 g0"| 643₄ ||class="entry q1 g0"| 32451₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (10710, 449, 16829, 21886, 31871, 21635) ||class="entry q0 g0"| 10710₈ ||class="entry q1 g0"| 449₄ ||class="entry q1 g0"| 16829₈ ||class="entry q0 g0"| 21886₁₀ ||class="entry q1 g0"| 31871₁₂ ||class="entry q1 g0"| 21635₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (19350, 2577, 11547, 14142, 30639, 14373) ||class="entry q0 g0"| 19350₈ ||class="entry q1 g0"| 2577₄ ||class="entry q1 g0"| 11547₈ ||class="entry q0 g0"| 14142₁₀ ||class="entry q1 g0"| 30639₁₂ ||class="entry q1 g0"| 14373₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (25494, 8709, 14631, 7998, 24507, 11289) ||class="entry q0 g0"| 25494₈ ||class="entry q1 g0"| 8709₄ ||class="entry q1 g0"| 14631₈ ||class="entry q0 g0"| 7998₁₀ ||class="entry q1 g0"| 24507₁₂ ||class="entry q1 g0"| 11289₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 4, 8, 10, 12, 6) ||class="c"| (26966, 16451, 21655, 5630, 15869, 16809) ||class="entry q0 g0"| 26966₈ ||class="entry q1 g0"| 16451₄ ||class="entry q1 g0"| 21655₈ ||class="entry q0 g0"| 5630₁₀ ||class="entry q1 g0"| 15869₁₂ ||class="entry q1 g0"| 16809₆ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (1526, 17507, 14625, 31070, 14813, 11295) ||class="entry q0 g0"| 1526₈ ||class="entry q1 g0"| 17507₆ ||class="entry q1 g0"| 14625₆ ||class="entry q0 g0"| 31070₁₀ ||class="entry q1 g0"| 14813₁₀ ||class="entry q1 g0"| 11295₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (3894, 9765, 21649, 29598, 23451, 16815) ||class="entry q0 g0"| 3894₈ ||class="entry q1 g0"| 9765₆ ||class="entry q1 g0"| 21649₆ ||class="entry q0 g0"| 29598₁₀ ||class="entry q1 g0"| 23451₁₀ ||class="entry q1 g0"| 16815₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (4574, 20555, 11529, 28022, 11765, 14391) ||class="entry q0 g0"| 4574₈ ||class="entry q1 g0"| 20555₆ ||class="entry q1 g0"| 11529₆ ||class="entry q0 g0"| 28022₁₀ ||class="entry q1 g0"| 11765₁₀ ||class="entry q1 g0"| 14391₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (6942, 12813, 16569, 26550, 20403, 21895) ||class="entry q0 g0"| 6942₈ ||class="entry q1 g0"| 12813₆ ||class="entry q1 g0"| 16569₆ ||class="entry q0 g0"| 26550₁₀ ||class="entry q1 g0"| 20403₁₀ ||class="entry q1 g0"| 21895₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (10038, 3633, 16557, 23454, 29583, 21907) ||class="entry q0 g0"| 10038₈ ||class="entry q1 g0"| 3633₆ ||class="entry q1 g0"| 16557₆ ||class="entry q0 g0"| 23454₁₀ ||class="entry q1 g0"| 29583₁₀ ||class="entry q1 g0"| 21907₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (13086, 6681, 21637, 20406, 26535, 16827) ||class="entry q0 g0"| 13086₈ ||class="entry q1 g0"| 6681₆ ||class="entry q1 g0"| 21637₆ ||class="entry q0 g0"| 20406₁₀ ||class="entry q1 g0"| 26535₁₀ ||class="entry q1 g0"| 16827₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (17782, 1505, 11275, 14814, 30815, 14645) ||class="entry q0 g0"| 17782₈ ||class="entry q1 g0"| 1505₆ ||class="entry q1 g0"| 11275₆ ||class="entry q0 g0"| 14814₁₀ ||class="entry q1 g0"| 30815₁₀ ||class="entry q1 g0"| 14645₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (20830, 4553, 14371, 11766, 27767, 11549) ||class="entry q0 g0"| 20830₈ ||class="entry q1 g0"| 4553₆ ||class="entry q1 g0"| 14371₆ ||class="entry q0 g0"| 11766₁₀ ||class="entry q1 g0"| 27767₁₀ ||class="entry q1 g0"| 11549₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (11158, 27159, 11133, 22334, 6057, 15939) ||class="entry q0 g0"| 11158₈ ||class="entry q1 g0"| 27159₈ ||class="entry q1 g0"| 11133₁₀ ||class="entry q0 g0"| 22334₁₀ ||class="entry q1 g0"| 6057₈ ||class="entry q1 g0"| 15939₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (18902, 25031, 18395, 13694, 7289, 21221) ||class="entry q0 g0"| 18902₈ ||class="entry q1 g0"| 25031₈ ||class="entry q1 g0"| 18395₁₀ ||class="entry q0 g0"| 13694₁₀ ||class="entry q1 g0"| 7289₈ ||class="entry q1 g0"| 21221₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (25046, 18899, 21479, 7550, 13421, 18137) ||class="entry q0 g0"| 25046₈ ||class="entry q1 g0"| 18899₈ ||class="entry q1 g0"| 21479₁₀ ||class="entry q0 g0"| 7550₁₀ ||class="entry q1 g0"| 13421₈ ||class="entry q1 g0"| 18137₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (27414, 11157, 15959, 6078, 22059, 11113) ||class="entry q0 g0"| 27414₈ ||class="entry q1 g0"| 11157₈ ||class="entry q1 g0"| 15959₁₀ ||class="entry q0 g0"| 6078₁₀ ||class="entry q1 g0"| 22059₈ ||class="entry q1 g0"| 11113₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (1974, 12213, 21473, 31518, 21003, 18143) ||class="entry q0 g0"| 1974₈ ||class="entry q1 g0"| 12213₁₀ ||class="entry q1 g0"| 21473₈ ||class="entry q0 g0"| 31518₁₀ ||class="entry q1 g0"| 21003₆ ||class="entry q1 g0"| 18143₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (3446, 19955, 15953, 29150, 12365, 11119) ||class="entry q0 g0"| 3446₈ ||class="entry q1 g0"| 19955₁₀ ||class="entry q1 g0"| 15953₈ ||class="entry q0 g0"| 29150₁₀ ||class="entry q1 g0"| 12365₆ ||class="entry q1 g0"| 11119₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (5022, 15261, 18377, 28470, 17955, 21239) ||class="entry q0 g0"| 5022₈ ||class="entry q1 g0"| 15261₁₀ ||class="entry q1 g0"| 18377₈ ||class="entry q0 g0"| 28470₁₀ ||class="entry q1 g0"| 17955₆ ||class="entry q1 g0"| 21239₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6494, 23003, 10873, 26102, 9317, 16199) ||class="entry q0 g0"| 6494₈ ||class="entry q1 g0"| 23003₁₀ ||class="entry q1 g0"| 10873₈ ||class="entry q0 g0"| 26102₁₀ ||class="entry q1 g0"| 9317₆ ||class="entry q1 g0"| 16199₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (9590, 26087, 10861, 23006, 6233, 16211) ||class="entry q0 g0"| 9590₈ ||class="entry q1 g0"| 26087₁₀ ||class="entry q1 g0"| 10861₈ ||class="entry q0 g0"| 23006₁₀ ||class="entry q1 g0"| 6233₆ ||class="entry q1 g0"| 16211₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (12638, 29135, 15941, 19958, 3185, 11131) ||class="entry q0 g0"| 12638₈ ||class="entry q1 g0"| 29135₁₀ ||class="entry q1 g0"| 15941₈ ||class="entry q0 g0"| 19958₁₀ ||class="entry q1 g0"| 3185₆ ||class="entry q1 g0"| 11131₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (18230, 28215, 18123, 15262, 5001, 21493) ||class="entry q0 g0"| 18230₈ ||class="entry q1 g0"| 28215₁₀ ||class="entry q1 g0"| 18123₈ ||class="entry q0 g0"| 15262₁₀ ||class="entry q1 g0"| 5001₆ ||class="entry q1 g0"| 21493₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (21278, 31263, 21219, 12214, 1953, 18397) ||class="entry q0 g0"| 21278₈ ||class="entry q1 g0"| 31263₁₀ ||class="entry q1 g0"| 21219₈ ||class="entry q0 g0"| 12214₁₀ ||class="entry q1 g0"| 1953₆ ||class="entry q1 g0"| 18397₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (11506, 24255, 7889, 20570, 8961, 3055) ||class="entry q0 g0"| 11506₈ ||class="entry q1 g0"| 24255₁₂ ||class="entry q1 g0"| 7889₈ ||class="entry q0 g0"| 20570₆ ||class="entry q1 g1"| 8961₄ ||class="entry q1 g0"| 3055₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (14542, 30399, 14021, 17510, 2817, 9211) ||class="entry q0 g0"| 14542₈ ||class="entry q1 g0"| 30399₁₂ ||class="entry q1 g0"| 14021₈ ||class="entry q0 g0"| 17510₆ ||class="entry q1 g1"| 2817₄ ||class="entry q1 g0"| 9211₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (20388, 15855, 7137, 13068, 16465, 3807) ||class="entry q0 g0"| 20388₈ ||class="entry q1 g0"| 15855₁₂ ||class="entry q1 g0"| 7137₈ ||class="entry q0 g0"| 13068₆ ||class="entry q1 g1"| 16465₄ ||class="entry q1 g0"| 3807₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (23182, 32111, 23139, 9766, 209, 20317) ||class="entry q0 g0"| 23182₈ ||class="entry q1 g0"| 32111₁₂ ||class="entry q1 g0"| 23139₈ ||class="entry q0 g0"| 9766₆ ||class="entry q1 g1"| 209₄ ||class="entry q1 g0"| 20317₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (26290, 32123, 26187, 6682, 197, 29557) ||class="entry q0 g0"| 26290₈ ||class="entry q1 g0"| 32123₁₂ ||class="entry q1 g0"| 26187₈ ||class="entry q0 g0"| 6682₆ ||class="entry q1 g1"| 197₄ ||class="entry q1 g0"| 29557₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (28004, 30653, 25197, 4556, 2563, 30547) ||class="entry q0 g0"| 28004₈ ||class="entry q1 g0"| 30653₁₂ ||class="entry q1 g0"| 25197₈ ||class="entry q0 g0"| 4556₆ ||class="entry q1 g1"| 2563₄ ||class="entry q1 g0"| 30547₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (29592, 15867, 10185, 3888, 16453, 13047) ||class="entry q0 g0"| 29592₈ ||class="entry q1 g0"| 15867₁₂ ||class="entry q1 g0"| 10185₈ ||class="entry q0 g0"| 3888₆ ||class="entry q1 g1"| 16453₄ ||class="entry q1 g0"| 13047₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (31064, 24509, 19065, 1520, 8707, 24391) ||class="entry q0 g0"| 31064₈ ||class="entry q1 g0"| 24509₁₂ ||class="entry q1 g0"| 19065₈ ||class="entry q0 g0"| 1520₆ ||class="entry q1 g1"| 8707₄ ||class="entry q1 g0"| 24391₁₀ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (15852, 20385, 4047, 16708, 12831, 6897) ||class="entry q0 g0"| 15852₁₀ ||class="entry q1 g0"| 20385₈ ||class="entry q1 g0"| 4047₁₀ ||class="entry q0 g1"| 16708₄ ||class="entry q1 g0"| 12831₈ ||class="entry q1 g0"| 6897₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (15864, 29577, 13299, 16720, 3639, 9933) ||class="entry q0 g0"| 15864₁₀ ||class="entry q1 g0"| 29577₈ ||class="entry q1 g0"| 13299₁₀ ||class="entry q0 g1"| 16720₄ ||class="entry q1 g0"| 3639₈ ||class="entry q1 g0"| 9933₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24250, 11505, 2815, 8722, 20815, 8129) ||class="entry q0 g0"| 24250₁₀ ||class="entry q1 g0"| 11505₈ ||class="entry q1 g0"| 2815₁₀ ||class="entry q0 g1"| 8722₄ ||class="entry q1 g0"| 20815₈ ||class="entry q1 g0"| 8129₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24504, 30809, 24405, 8976, 1511, 19051) ||class="entry q0 g0"| 24504₁₀ ||class="entry q1 g0"| 30809₈ ||class="entry q1 g0"| 24405₁₀ ||class="entry q0 g1"| 8976₄ ||class="entry q1 g0"| 1511₈ ||class="entry q1 g0"| 19051₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30382, 14541, 8959, 2566, 17779, 14273) ||class="entry q0 g0"| 30382₁₀ ||class="entry q1 g0"| 14541₈ ||class="entry q1 g0"| 8959₁₀ ||class="entry q0 g1"| 2566₄ ||class="entry q1 g0"| 17779₈ ||class="entry q1 g0"| 14273₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30636, 27749, 30549, 2820, 4571, 25195) ||class="entry q0 g0"| 30636₁₀ ||class="entry q1 g0"| 27749₈ ||class="entry q1 g0"| 30549₁₀ ||class="entry q0 g1"| 2820₄ ||class="entry q1 g0"| 4571₈ ||class="entry q1 g0"| 25195₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31854, 23179, 20303, 198, 10037, 23153) ||class="entry q0 g0"| 31854₁₀ ||class="entry q1 g0"| 23179₈ ||class="entry q1 g0"| 20303₁₀ ||class="entry q0 g1"| 198₄ ||class="entry q1 g0"| 10037₈ ||class="entry q1 g0"| 23153₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31866, 26275, 29555, 210, 6941, 26189) ||class="entry q0 g0"| 31866₁₀ ||class="entry q1 g0"| 26275₈ ||class="entry q1 g0"| 29555₁₀ ||class="entry q0 g1"| 210₄ ||class="entry q1 g0"| 6941₈ ||class="entry q1 g0"| 26189₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (15592, 32105, 15363, 16448, 215, 10557) ||class="entry q0 g0"| 15592₈ ||class="entry q1 g0"| 32105₁₀ ||class="entry q1 g0"| 15363₆ ||class="entry q0 g1"| 16448₂ ||class="entry q1 g1"| 215₆ ||class="entry q1 g0"| 10557₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (24232, 30393, 20645, 8704, 2823, 17819) ||class="entry q0 g0"| 24232₈ ||class="entry q1 g0"| 30393₁₀ ||class="entry q1 g0"| 20645₆ ||class="entry q0 g1"| 8704₂ ||class="entry q1 g1"| 2823₆ ||class="entry q1 g0"| 17819₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (30376, 24237, 17561, 2560, 8979, 20903) ||class="entry q0 g0"| 30376₈ ||class="entry q1 g0"| 24237₁₀ ||class="entry q1 g0"| 17561₆ ||class="entry q0 g1"| 2560₂ ||class="entry q1 g1"| 8979₆ ||class="entry q1 g0"| 20903₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (31848, 15595, 10537, 192, 16725, 15383) ||class="entry q0 g0"| 31848₈ ||class="entry q1 g0"| 15595₁₀ ||class="entry q1 g0"| 10537₆ ||class="entry q0 g1"| 192₂ ||class="entry q1 g1"| 16725₆ ||class="entry q1 g0"| 15383₈ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (15612, 16705, 63, 16468, 15615, 5377) ||class="entry q0 g0"| 15612₁₀ ||class="entry q1 g1"| 16705₄ ||class="entry q1 g0"| 63₆ ||class="entry q0 g1"| 16468₄ ||class="entry q1 g0"| 15615₁₂ ||class="entry q1 g0"| 5377₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (24490, 8721, 1295, 8962, 24495, 4145) ||class="entry q0 g0"| 24490₁₀ ||class="entry q1 g1"| 8721₄ ||class="entry q1 g0"| 1295₆ ||class="entry q0 g1"| 8962₄ ||class="entry q1 g0"| 24495₁₂ ||class="entry q1 g0"| 4145₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (30634, 2565, 4403, 2818, 30651, 1037) ||class="entry q0 g0"| 30634₁₀ ||class="entry q1 g1"| 2565₄ ||class="entry q1 g0"| 4403₆ ||class="entry q0 g1"| 2818₄ ||class="entry q1 g0"| 30651₁₂ ||class="entry q1 g0"| 1037₄ |- |class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (31868, 195, 5397, 212, 32125, 43) ||class="entry q0 g0"| 31868₁₀ ||class="entry q1 g1"| 195₄ ||class="entry q1 g0"| 5397₆ ||class="entry q0 g1"| 212₄ ||class="entry q1 g0"| 32125₁₂ ||class="entry q1 g0"| 43₄ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (1, 7, 9, 11, 5, 11) ||class="c"| (32768, 38504, 59798, 38782, 33046, 65256) ||class="entry q2 g1"| 32768₁ ||class="entry q2 g1"| 38504₇ ||class="entry q2 g1"| 59798₉ ||class="entry q2 g1"| 38782₁₁ ||class="entry q2 g1"| 33046₅ ||class="entry q2 g1"| 65256₁₁ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (32774, 61448, 36848, 38776, 59254, 39054) ||class="entry q2 g1"| 32774₃ ||class="entry q2 g1"| 61448₅ ||class="entry q2 g1"| 36848₉ ||class="entry q2 g1"| 38776₉ ||class="entry q2 g1"| 59254₁₁ ||class="entry q2 g1"| 39054₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (32786, 52256, 46028, 38764, 56158, 42162) ||class="entry q2 g1"| 32786₃ ||class="entry q2 g1"| 52256₅ ||class="entry q2 g1"| 46028₉ ||class="entry q2 g1"| 38764₉ ||class="entry q2 g1"| 56158₁₁ ||class="entry q2 g1"| 42162₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (32788, 43584, 54698, 38762, 48446, 49876) ||class="entry q2 g1"| 32788₃ ||class="entry q2 g1"| 43584₅ ||class="entry q2 g1"| 54698₉ ||class="entry q2 g1"| 38762₉ ||class="entry q2 g1"| 48446₁₁ ||class="entry q2 g1"| 49876₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (33026, 49856, 48188, 38524, 54718, 43842) ||class="entry q2 g1"| 33026₃ ||class="entry q2 g1"| 49856₅ ||class="entry q2 g1"| 48188₉ ||class="entry q2 g1"| 38524₉ ||class="entry q2 g1"| 54718₁₁ ||class="entry q2 g1"| 43842₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (33028, 42144, 55898, 38522, 46046, 52516) ||class="entry q2 g1"| 33028₃ ||class="entry q2 g1"| 42144₅ ||class="entry q2 g1"| 55898₉ ||class="entry q2 g1"| 38522₉ ||class="entry q2 g1"| 46046₁₁ ||class="entry q2 g1"| 52516₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 5, 9, 9, 11, 7) ||class="c"| (33040, 39048, 58982, 38510, 36854, 61720) ||class="entry q2 g1"| 33040₃ ||class="entry q2 g1"| 39048₅ ||class="entry q2 g1"| 58982₉ ||class="entry q2 g1"| 38510₉ ||class="entry q2 g1"| 36854₁₁ ||class="entry q2 g1"| 61720₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (34944, 40952, 61158, 40958, 34950, 63896) ||class="entry q2 g1"| 34944₃ ||class="entry q2 g1"| 40952₁₁ ||class="entry q2 g1"| 61158₁₁ ||class="entry q2 g1"| 40958₁₃ ||class="entry q2 g1"| 34950₅ ||class="entry q2 g1"| 63896₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (41088, 47084, 64218, 47102, 41106, 60836) ||class="entry q2 g1"| 41088₃ ||class="entry q2 g1"| 47084₁₁ ||class="entry q2 g1"| 64218₁₁ ||class="entry q2 g1"| 47102₁₃ ||class="entry q2 g1"| 41106₅ ||class="entry q2 g1"| 60836₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (43008, 48764, 64938, 49022, 43266, 60116) ||class="entry q2 g1"| 43008₃ ||class="entry q2 g1"| 48764₁₁ ||class="entry q2 g1"| 64938₁₁ ||class="entry q2 g1"| 49022₁₃ ||class="entry q2 g1"| 43266₅ ||class="entry q2 g1"| 60116₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (49280, 55274, 64700, 55294, 49300, 60354) ||class="entry q2 g1"| 49280₃ ||class="entry q2 g1"| 55274₁₁ ||class="entry q2 g1"| 64700₁₁ ||class="entry q2 g1"| 55294₁₃ ||class="entry q2 g1"| 49300₅ ||class="entry q2 g1"| 60354₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (51200, 56954, 64460, 57214, 51460, 60594) ||class="entry q2 g1"| 51200₃ ||class="entry q2 g1"| 56954₁₁ ||class="entry q2 g1"| 64460₁₁ ||class="entry q2 g1"| 57214₁₃ ||class="entry q2 g1"| 51460₅ ||class="entry q2 g1"| 60594₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (57344, 63086, 61424, 63358, 57616, 63630) ||class="entry q2 g1"| 57344₃ ||class="entry q2 g1"| 63086₁₁ ||class="entry q2 g1"| 61424₁₁ ||class="entry q2 g1"| 63358₁₃ ||class="entry q2 g1"| 57616₅ ||class="entry q2 g1"| 63630₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 13, 7, 9, 7, 5) ||class="c"| (33344, 64958, 33622, 38206, 60096, 37928) ||class="entry q2 g1"| 33344₃ ||class="entry q2 g1"| 64958₁₃ ||class="entry q2 g1"| 33622₇ ||class="entry q2 g1"| 38206₉ ||class="entry q2 g1"| 60096₇ ||class="entry q2 g1"| 37928₅ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 13, 7, 9, 7, 5) ||class="c"| (33824, 64478, 34102, 37726, 60576, 37448) ||class="entry q2 g1"| 33824₃ ||class="entry q2 g1"| 64478₁₃ ||class="entry q2 g1"| 34102₇ ||class="entry q2 g1"| 37726₉ ||class="entry q2 g1"| 60576₇ ||class="entry q2 g1"| 37448₅ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 13, 7, 9, 7, 5) ||class="c"| (36872, 61430, 37150, 34678, 63624, 34400) ||class="entry q2 g1"| 36872₃ ||class="entry q2 g1"| 61430₁₃ ||class="entry q2 g1"| 37150₇ ||class="entry q2 g1"| 34678₉ ||class="entry q2 g1"| 63624₇ ||class="entry q2 g1"| 34400₅ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (35216, 37144, 57622, 40686, 34406, 63080) ||class="entry q2 g1"| 35216₅ ||class="entry q2 g1"| 37144₅ ||class="entry q2 g1"| 57622₇ ||class="entry q2 g1"| 40686₁₁ ||class="entry q2 g1"| 34406₇ ||class="entry q2 g1"| 63080₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (41348, 34084, 51478, 46842, 37466, 56936) ||class="entry q2 g1"| 41348₅ ||class="entry q2 g1"| 34084₅ ||class="entry q2 g1"| 51478₇ ||class="entry q2 g1"| 46842₁₁ ||class="entry q2 g1"| 37466₇ ||class="entry q2 g1"| 56936₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (43028, 33364, 49558, 49002, 38186, 55016) ||class="entry q2 g1"| 43028₅ ||class="entry q2 g1"| 33364₅ ||class="entry q2 g1"| 49558₇ ||class="entry q2 g1"| 49002₁₁ ||class="entry q2 g1"| 38186₇ ||class="entry q2 g1"| 55016₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (49538, 33602, 43286, 55036, 37948, 48744) ||class="entry q2 g1"| 49538₅ ||class="entry q2 g1"| 33602₅ ||class="entry q2 g1"| 43286₇ ||class="entry q2 g1"| 55036₁₁ ||class="entry q2 g1"| 37948₇ ||class="entry q2 g1"| 48744₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (51218, 33842, 41366, 57196, 37708, 46824) ||class="entry q2 g1"| 51218₅ ||class="entry q2 g1"| 33842₅ ||class="entry q2 g1"| 41366₇ ||class="entry q2 g1"| 57196₁₁ ||class="entry q2 g1"| 37708₇ ||class="entry q2 g1"| 46824₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (57350, 36878, 35222, 63352, 34672, 40680) ||class="entry q2 g1"| 57350₅ ||class="entry q2 g1"| 36878₅ ||class="entry q2 g1"| 35222₇ ||class="entry q2 g1"| 63352₁₁ ||class="entry q2 g1"| 34672₇ ||class="entry q2 g1"| 40680₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (34962, 50608, 46268, 40940, 53966, 41922) ||class="entry q2 g1"| 34962₅ ||class="entry q2 g1"| 50608₇ ||class="entry q2 g1"| 46268₉ ||class="entry q2 g1"| 40940₁₁ ||class="entry q2 g1"| 53966₉ ||class="entry q2 g1"| 41922₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (34964, 41936, 53978, 40938, 46254, 50596) ||class="entry q2 g1"| 34964₅ ||class="entry q2 g1"| 41936₇ ||class="entry q2 g1"| 53978₉ ||class="entry q2 g1"| 40938₁₁ ||class="entry q2 g1"| 46254₉ ||class="entry q2 g1"| 50596₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (35202, 52048, 47948, 40700, 56366, 44082) ||class="entry q2 g1"| 35202₅ ||class="entry q2 g1"| 52048₇ ||class="entry q2 g1"| 47948₉ ||class="entry q2 g1"| 40700₁₁ ||class="entry q2 g1"| 56366₉ ||class="entry q2 g1"| 44082₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (35204, 44336, 56618, 40698, 47694, 51796) ||class="entry q2 g1"| 35204₅ ||class="entry q2 g1"| 44336₇ ||class="entry q2 g1"| 56618₉ ||class="entry q2 g1"| 40698₁₁ ||class="entry q2 g1"| 47694₉ ||class="entry q2 g1"| 51796₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (41094, 53644, 40124, 47096, 50930, 35778) ||class="entry q2 g1"| 41094₅ ||class="entry q2 g1"| 53644₇ ||class="entry q2 g1"| 40124₉ ||class="entry q2 g1"| 47096₁₁ ||class="entry q2 g1"| 50930₉ ||class="entry q2 g1"| 35778₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (41108, 35780, 50918, 47082, 40122, 53656) ||class="entry q2 g1"| 41108₅ ||class="entry q2 g1"| 35780₇ ||class="entry q2 g1"| 50918₉ ||class="entry q2 g1"| 47082₁₁ ||class="entry q2 g1"| 40122₉ ||class="entry q2 g1"| 53656₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (41346, 58180, 44912, 46844, 62522, 47118) ||class="entry q2 g1"| 41346₅ ||class="entry q2 g1"| 58180₇ ||class="entry q2 g1"| 44912₉ ||class="entry q2 g1"| 46844₁₁ ||class="entry q2 g1"| 62522₉ ||class="entry q2 g1"| 47118₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (41360, 47372, 62762, 46830, 44658, 57940) ||class="entry q2 g1"| 41360₅ ||class="entry q2 g1"| 47372₇ ||class="entry q2 g1"| 62762₉ ||class="entry q2 g1"| 46830₁₁ ||class="entry q2 g1"| 44658₉ ||class="entry q2 g1"| 57940₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (43014, 55324, 39884, 49016, 53090, 36018) ||class="entry q2 g1"| 43014₅ ||class="entry q2 g1"| 55324₇ ||class="entry q2 g1"| 39884₉ ||class="entry q2 g1"| 49016₁₁ ||class="entry q2 g1"| 53090₉ ||class="entry q2 g1"| 36018₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (43026, 58420, 42992, 49004, 62282, 45198) ||class="entry q2 g1"| 43026₅ ||class="entry q2 g1"| 58420₇ ||class="entry q2 g1"| 42992₉ ||class="entry q2 g1"| 49004₁₁ ||class="entry q2 g1"| 62282₉ ||class="entry q2 g1"| 45198₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (43268, 36020, 52838, 48762, 39882, 55576) ||class="entry q2 g1"| 43268₅ ||class="entry q2 g1"| 36020₇ ||class="entry q2 g1"| 52838₉ ||class="entry q2 g1"| 48762₁₁ ||class="entry q2 g1"| 39882₉ ||class="entry q2 g1"| 55576₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (43280, 45212, 62042, 48750, 42978, 58660) ||class="entry q2 g1"| 43280₅ ||class="entry q2 g1"| 45212₇ ||class="entry q2 g1"| 62042₉ ||class="entry q2 g1"| 48750₁₁ ||class="entry q2 g1"| 42978₉ ||class="entry q2 g1"| 58660₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (49286, 45450, 39642, 55288, 42740, 36260) ||class="entry q2 g1"| 49286₅ ||class="entry q2 g1"| 45450₇ ||class="entry q2 g1"| 39642₉ ||class="entry q2 g1"| 55288₁₁ ||class="entry q2 g1"| 42740₉ ||class="entry q2 g1"| 36260₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (49298, 36258, 42726, 55276, 39644, 45464) ||class="entry q2 g1"| 49298₅ ||class="entry q2 g1"| 36258₇ ||class="entry q2 g1"| 42726₉ ||class="entry q2 g1"| 55276₁₁ ||class="entry q2 g1"| 39644₉ ||class="entry q2 g1"| 45464₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (49540, 58658, 53104, 55034, 62044, 55310) ||class="entry q2 g1"| 49540₅ ||class="entry q2 g1"| 58658₇ ||class="entry q2 g1"| 53104₉ ||class="entry q2 g1"| 55034₁₁ ||class="entry q2 g1"| 62044₉ ||class="entry q2 g1"| 55310₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (49552, 55562, 62284, 55022, 52852, 58418) ||class="entry q2 g1"| 49552₅ ||class="entry q2 g1"| 55562₇ ||class="entry q2 g1"| 62284₉ ||class="entry q2 g1"| 55022₁₁ ||class="entry q2 g1"| 52852₉ ||class="entry q2 g1"| 58418₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (51206, 47130, 40362, 57208, 44900, 35540) ||class="entry q2 g1"| 51206₅ ||class="entry q2 g1"| 47130₇ ||class="entry q2 g1"| 40362₉ ||class="entry q2 g1"| 57208₁₁ ||class="entry q2 g1"| 44900₉ ||class="entry q2 g1"| 35540₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (51220, 57938, 51184, 57194, 62764, 53390) ||class="entry q2 g1"| 51220₅ ||class="entry q2 g1"| 57938₇ ||class="entry q2 g1"| 51184₉ ||class="entry q2 g1"| 57194₁₁ ||class="entry q2 g1"| 62764₉ ||class="entry q2 g1"| 53390₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (51458, 35538, 44646, 56956, 40364, 47384) ||class="entry q2 g1"| 51458₅ ||class="entry q2 g1"| 35538₇ ||class="entry q2 g1"| 44646₉ ||class="entry q2 g1"| 56956₁₁ ||class="entry q2 g1"| 40364₉ ||class="entry q2 g1"| 47384₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (51472, 53402, 62524, 56942, 51172, 58178) ||class="entry q2 g1"| 51472₅ ||class="entry q2 g1"| 53402₇ ||class="entry q2 g1"| 62524₉ ||class="entry q2 g1"| 56942₁₁ ||class="entry q2 g1"| 51172₉ ||class="entry q2 g1"| 58178₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (57362, 44070, 46506, 63340, 47960, 41684) ||class="entry q2 g1"| 57362₅ ||class="entry q2 g1"| 44070₇ ||class="entry q2 g1"| 46506₉ ||class="entry q2 g1"| 63340₁₁ ||class="entry q2 g1"| 47960₉ ||class="entry q2 g1"| 41684₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (57364, 51782, 54220, 63338, 56632, 50354) ||class="entry q2 g1"| 57364₅ ||class="entry q2 g1"| 51782₇ ||class="entry q2 g1"| 54220₉ ||class="entry q2 g1"| 63338₁₁ ||class="entry q2 g1"| 56632₉ ||class="entry q2 g1"| 50354₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (57602, 41670, 47706, 63100, 46520, 44324) ||class="entry q2 g1"| 57602₅ ||class="entry q2 g1"| 41670₇ ||class="entry q2 g1"| 47706₉ ||class="entry q2 g1"| 63100₁₁ ||class="entry q2 g1"| 46520₉ ||class="entry q2 g1"| 44324₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 11, 9, 7) ||class="c"| (57604, 50342, 56380, 63098, 54232, 52034) ||class="entry q2 g1"| 57604₅ ||class="entry q2 g1"| 50342₇ ||class="entry q2 g1"| 56380₉ ||class="entry q2 g1"| 63098₁₁ ||class="entry q2 g1"| 54232₉ ||class="entry q2 g1"| 52034₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (35520, 62510, 33830, 40382, 58192, 37720) ||class="entry q2 g1"| 35520₅ ||class="entry q2 g1"| 62510₉ ||class="entry q2 g1"| 33830₅ ||class="entry q2 g1"| 40382₁₁ ||class="entry q2 g1"| 58192₇ ||class="entry q2 g1"| 37720₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (36000, 62030, 33350, 39902, 58672, 38200) ||class="entry q2 g1"| 36000₅ ||class="entry q2 g1"| 62030₉ ||class="entry q2 g1"| 33350₅ ||class="entry q2 g1"| 39902₁₁ ||class="entry q2 g1"| 58672₇ ||class="entry q2 g1"| 38200₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (41664, 56378, 36890, 46526, 52036, 34660) ||class="entry q2 g1"| 41664₅ ||class="entry q2 g1"| 56378₉ ||class="entry q2 g1"| 36890₅ ||class="entry q2 g1"| 46526₁₁ ||class="entry q2 g1"| 52036₇ ||class="entry q2 g1"| 34660₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (44064, 54218, 37130, 47966, 50356, 34420) ||class="entry q2 g1"| 44064₅ ||class="entry q2 g1"| 54218₉ ||class="entry q2 g1"| 37130₅ ||class="entry q2 g1"| 47966₁₁ ||class="entry q2 g1"| 50356₇ ||class="entry q2 g1"| 34420₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (45192, 52850, 33362, 42998, 55564, 38188) ||class="entry q2 g1"| 45192₅ ||class="entry q2 g1"| 52850₉ ||class="entry q2 g1"| 33362₅ ||class="entry q2 g1"| 42998₁₁ ||class="entry q2 g1"| 55564₇ ||class="entry q2 g1"| 38188₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (47112, 51170, 34082, 44918, 53404, 37468) ||class="entry q2 g1"| 47112₅ ||class="entry q2 g1"| 51170₉ ||class="entry q2 g1"| 34082₅ ||class="entry q2 g1"| 44918₁₁ ||class="entry q2 g1"| 53404₇ ||class="entry q2 g1"| 37468₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (50336, 47708, 36892, 54238, 44322, 34658) ||class="entry q2 g1"| 50336₅ ||class="entry q2 g1"| 47708₉ ||class="entry q2 g1"| 36892₅ ||class="entry q2 g1"| 54238₁₁ ||class="entry q2 g1"| 44322₇ ||class="entry q2 g1"| 34658₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (51776, 46508, 37132, 56638, 41682, 34418) ||class="entry q2 g1"| 51776₅ ||class="entry q2 g1"| 46508₉ ||class="entry q2 g1"| 37132₅ ||class="entry q2 g1"| 56638₁₁ ||class="entry q2 g1"| 41682₇ ||class="entry q2 g1"| 34418₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (53384, 44660, 33844, 51190, 47370, 37706) ||class="entry q2 g1"| 53384₅ ||class="entry q2 g1"| 44660₉ ||class="entry q2 g1"| 33844₅ ||class="entry q2 g1"| 51190₁₁ ||class="entry q2 g1"| 47370₇ ||class="entry q2 g1"| 37706₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (55304, 42980, 33604, 53110, 45210, 37946) ||class="entry q2 g1"| 55304₅ ||class="entry q2 g1"| 42980₉ ||class="entry q2 g1"| 33604₅ ||class="entry q2 g1"| 53110₁₁ ||class="entry q2 g1"| 45210₇ ||class="entry q2 g1"| 37946₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (57920, 40376, 34096, 62782, 35526, 37454) ||class="entry q2 g1"| 57920₅ ||class="entry q2 g1"| 40376₉ ||class="entry q2 g1"| 34096₅ ||class="entry q2 g1"| 62782₁₁ ||class="entry q2 g1"| 35526₇ ||class="entry q2 g1"| 37454₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 7, 7) ||class="c"| (58400, 39896, 33616, 62302, 36006, 37934) ||class="entry q2 g1"| 58400₅ ||class="entry q2 g1"| 39896₉ ||class="entry q2 g1"| 33616₅ ||class="entry q2 g1"| 62302₁₁ ||class="entry q2 g1"| 36006₇ ||class="entry q2 g1"| 37934₇ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (36576, 39320, 59526, 39326, 36582, 65528) ||class="entry q2 g1"| 36576₇ ||class="entry q2 g1"| 39320₇ ||class="entry q2 g1"| 59526₇ ||class="entry q2 g1"| 39326₉ ||class="entry q2 g1"| 36582₉ ||class="entry q2 g1"| 65528₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (45768, 42404, 59538, 42422, 45786, 65516) ||class="entry q2 g1"| 45768₇ ||class="entry q2 g1"| 42404₇ ||class="entry q2 g1"| 59538₇ ||class="entry q2 g1"| 42422₉ ||class="entry q2 g1"| 45786₉ ||class="entry q2 g1"| 65516₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (48168, 43604, 59778, 43862, 48426, 65276) ||class="entry q2 g1"| 48168₇ ||class="entry q2 g1"| 43604₇ ||class="entry q2 g1"| 59778₇ ||class="entry q2 g1"| 43862₉ ||class="entry q2 g1"| 48426₉ ||class="entry q2 g1"| 65276₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (54440, 50114, 59540, 50134, 54460, 65514) ||class="entry q2 g1"| 54440₇ ||class="entry q2 g1"| 50114₇ ||class="entry q2 g1"| 59540₇ ||class="entry q2 g1"| 50134₉ ||class="entry q2 g1"| 54460₉ ||class="entry q2 g1"| 65514₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (55880, 52274, 59780, 52534, 56140, 65274) ||class="entry q2 g1"| 55880₇ ||class="entry q2 g1"| 52274₇ ||class="entry q2 g1"| 59780₇ ||class="entry q2 g1"| 52534₉ ||class="entry q2 g1"| 56140₉ ||class="entry q2 g1"| 65274₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (58976, 61454, 59792, 61726, 59248, 65262) ||class="entry q2 g1"| 58976₇ ||class="entry q2 g1"| 61454₇ ||class="entry q2 g1"| 59792₇ ||class="entry q2 g1"| 61726₉ ||class="entry q2 g1"| 59248₉ ||class="entry q2 g1"| 65262₁₃ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (39624, 36272, 64686, 36278, 39630, 60368) ||class="entry q2 g1"| 39624₇ ||class="entry q2 g1"| 36272₇ ||class="entry q2 g1"| 64686₁₁ ||class="entry q2 g1"| 36278₉ ||class="entry q2 g1"| 39630₉ ||class="entry q2 g1"| 60368₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (40104, 35792, 64206, 35798, 40110, 60848) ||class="entry q2 g1"| 40104₇ ||class="entry q2 g1"| 35792₇ ||class="entry q2 g1"| 64206₁₁ ||class="entry q2 g1"| 35798₉ ||class="entry q2 g1"| 40110₉ ||class="entry q2 g1"| 60848₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (42720, 45452, 64698, 45470, 42738, 60356) ||class="entry q2 g1"| 42720₇ ||class="entry q2 g1"| 45452₇ ||class="entry q2 g1"| 64698₁₁ ||class="entry q2 g1"| 45470₉ ||class="entry q2 g1"| 42738₉ ||class="entry q2 g1"| 60356₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (44640, 47132, 64458, 47390, 44898, 60596) ||class="entry q2 g1"| 44640₇ ||class="entry q2 g1"| 47132₇ ||class="entry q2 g1"| 64458₁₁ ||class="entry q2 g1"| 47390₉ ||class="entry q2 g1"| 44898₉ ||class="entry q2 g1"| 60596₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (46248, 41924, 61170, 41942, 46266, 63884) ||class="entry q2 g1"| 46248₇ ||class="entry q2 g1"| 41924₇ ||class="entry q2 g1"| 61170₁₁ ||class="entry q2 g1"| 41942₉ ||class="entry q2 g1"| 46266₉ ||class="entry q2 g1"| 63884₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (47688, 44084, 61410, 44342, 47946, 63644) ||class="entry q2 g1"| 47688₇ ||class="entry q2 g1"| 44084₇ ||class="entry q2 g1"| 61410₁₁ ||class="entry q2 g1"| 44342₉ ||class="entry q2 g1"| 47946₉ ||class="entry q2 g1"| 63644₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (50912, 53642, 64220, 53662, 50932, 60834) ||class="entry q2 g1"| 50912₇ ||class="entry q2 g1"| 53642₇ ||class="entry q2 g1"| 64220₁₁ ||class="entry q2 g1"| 53662₉ ||class="entry q2 g1"| 50932₉ ||class="entry q2 g1"| 60834₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (52832, 55322, 64940, 55582, 53092, 60114) ||class="entry q2 g1"| 52832₇ ||class="entry q2 g1"| 55322₇ ||class="entry q2 g1"| 64940₁₁ ||class="entry q2 g1"| 55582₉ ||class="entry q2 g1"| 53092₉ ||class="entry q2 g1"| 60114₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (53960, 50594, 61172, 50614, 53980, 63882) ||class="entry q2 g1"| 53960₇ ||class="entry q2 g1"| 50594₇ ||class="entry q2 g1"| 61172₁₁ ||class="entry q2 g1"| 50614₉ ||class="entry q2 g1"| 53980₉ ||class="entry q2 g1"| 63882₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (56360, 51794, 61412, 52054, 56620, 63642) ||class="entry q2 g1"| 56360₇ ||class="entry q2 g1"| 51794₇ ||class="entry q2 g1"| 61412₁₁ ||class="entry q2 g1"| 52054₉ ||class="entry q2 g1"| 56620₉ ||class="entry q2 g1"| 63642₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (62024, 58406, 64952, 58678, 62296, 60102) ||class="entry q2 g1"| 62024₇ ||class="entry q2 g1"| 58406₇ ||class="entry q2 g1"| 64952₁₁ ||class="entry q2 g1"| 58678₉ ||class="entry q2 g1"| 62296₉ ||class="entry q2 g1"| 60102₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (62504, 57926, 64472, 58198, 62776, 60582) ||class="entry q2 g1"| 62504₇ ||class="entry q2 g1"| 57926₇ ||class="entry q2 g1"| 64472₁₁ ||class="entry q2 g1"| 58198₉ ||class="entry q2 g1"| 62776₉ ||class="entry q2 g1"| 60582₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39066, 48174, 52276, 36836, 43856, 56138) ||class="entry q2 g1"| 39066₇ ||class="entry q2 g1"| 48174₉ ||class="entry q2 g1"| 52276₇ ||class="entry q2 g1"| 36836₉ ||class="entry q2 g1"| 43856₇ ||class="entry q2 g1"| 56138₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39068, 55886, 43602, 36834, 52528, 48428) ||class="entry q2 g1"| 39068₇ ||class="entry q2 g1"| 55886₉ ||class="entry q2 g1"| 43602₇ ||class="entry q2 g1"| 36834₉ ||class="entry q2 g1"| 52528₇ ||class="entry q2 g1"| 48428₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39306, 45774, 50116, 36596, 42416, 54458) ||class="entry q2 g1"| 39306₇ ||class="entry q2 g1"| 45774₉ ||class="entry q2 g1"| 50116₇ ||class="entry q2 g1"| 36596₉ ||class="entry q2 g1"| 42416₇ ||class="entry q2 g1"| 54458₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39308, 54446, 42402, 36594, 50128, 45788) ||class="entry q2 g1"| 39308₇ ||class="entry q2 g1"| 54446₉ ||class="entry q2 g1"| 42402₇ ||class="entry q2 g1"| 36594₉ ||class="entry q2 g1"| 50128₇ ||class="entry q2 g1"| 45788₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42150, 48186, 61468, 46040, 43844, 59234) ||class="entry q2 g1"| 42150₇ ||class="entry q2 g1"| 48186₉ ||class="entry q2 g1"| 61468₇ ||class="entry q2 g1"| 46040₉ ||class="entry q2 g1"| 43844₇ ||class="entry q2 g1"| 59234₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42164, 58994, 43590, 46026, 61708, 48440) ||class="entry q2 g1"| 42164₇ ||class="entry q2 g1"| 58994₉ ||class="entry q2 g1"| 43590₇ ||class="entry q2 g1"| 46026₉ ||class="entry q2 g1"| 61708₇ ||class="entry q2 g1"| 48440₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (49862, 55900, 61466, 54712, 52514, 59236) ||class="entry q2 g1"| 49862₇ ||class="entry q2 g1"| 55900₉ ||class="entry q2 g1"| 61466₇ ||class="entry q2 g1"| 54712₉ ||class="entry q2 g1"| 52514₇ ||class="entry q2 g1"| 59236₉ |- |class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (49874, 58996, 52262, 54700, 61706, 56152) ||class="entry q2 g1"| 49874₇ ||class="entry q2 g1"| 58996₉ ||class="entry q2 g1"| 52262₇ ||class="entry q2 g1"| 54700₉ ||class="entry q2 g1"| 61706₇ ||class="entry q2 g1"| 56152₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (33448, 38103, 33601, 64854, 59967, 38185) ||class="entry q2 g1"| 33448₅ ||class="entry q3 g1"| 38103₉ ||class="entry q3 g1"| 33601₅ ||class="entry q2 g1"| 64854₁₁ ||class="entry q3 g1"| 59967₁₁ ||class="entry q3 g1"| 38185₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (33992, 37559, 34081, 64310, 60511, 37705) ||class="entry q2 g1"| 33992₅ ||class="entry q3 g1"| 37559₉ ||class="entry q3 g1"| 34081₅ ||class="entry q2 g1"| 64310₁₁ ||class="entry q3 g1"| 60511₁₁ ||class="entry q3 g1"| 37705₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (35368, 40263, 33841, 62934, 58287, 37465) ||class="entry q2 g1"| 35368₅ ||class="entry q3 g1"| 40263₉ ||class="entry q3 g1"| 33841₅ ||class="entry q2 g1"| 62934₁₁ ||class="entry q3 g1"| 58287₁₁ ||class="entry q3 g1"| 37465₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (35912, 39719, 33361, 62390, 58831, 37945) ||class="entry q2 g1"| 35912₅ ||class="entry q3 g1"| 39719₉ ||class="entry q3 g1"| 33361₅ ||class="entry q2 g1"| 62390₁₁ ||class="entry q3 g1"| 58831₁₁ ||class="entry q3 g1"| 37945₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (37088, 34463, 37129, 61214, 63607, 34657) ||class="entry q2 g1"| 37088₅ ||class="entry q3 g1"| 34463₉ ||class="entry q3 g1"| 37129₅ ||class="entry q2 g1"| 61214₁₁ ||class="entry q3 g1"| 63607₁₁ ||class="entry q3 g1"| 34657₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (40448, 35183, 36889, 57854, 63367, 34417) ||class="entry q2 g1"| 40448₅ ||class="entry q3 g1"| 35183₉ ||class="entry q3 g1"| 36889₅ ||class="entry q2 g1"| 57854₁₁ ||class="entry q3 g1"| 63367₁₁ ||class="entry q3 g1"| 34417₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (41512, 46419, 36877, 56790, 52155, 34405) ||class="entry q2 g1"| 41512₅ ||class="entry q3 g1"| 46419₉ ||class="entry q3 g1"| 36877₅ ||class="entry q2 g1"| 56790₁₁ ||class="entry q3 g1"| 52155₁₁ ||class="entry q3 g1"| 34405₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (45152, 42779, 33349, 53150, 55795, 37933) ||class="entry q2 g1"| 45152₅ ||class="entry q3 g1"| 42779₉ ||class="entry q3 g1"| 33349₅ ||class="entry q2 g1"| 53150₁₁ ||class="entry q3 g1"| 55795₁₁ ||class="entry q3 g1"| 37933₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (46592, 41339, 33829, 51710, 57235, 37453) ||class="entry q2 g1"| 46592₅ ||class="entry q3 g1"| 41339₉ ||class="entry q3 g1"| 33829₅ ||class="entry q2 g1"| 51710₁₁ ||class="entry q3 g1"| 57235₁₁ ||class="entry q3 g1"| 37453₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (50248, 54069, 36875, 48054, 44509, 34403) ||class="entry q2 g1"| 50248₅ ||class="entry q3 g1"| 54069₉ ||class="entry q3 g1"| 36875₅ ||class="entry q2 g1"| 48054₁₁ ||class="entry q3 g1"| 44509₁₁ ||class="entry q3 g1"| 34403₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (53344, 50973, 33827, 44958, 47605, 37451) ||class="entry q2 g1"| 53344₅ ||class="entry q3 g1"| 50973₉ ||class="entry q3 g1"| 33827₅ ||class="entry q2 g1"| 44958₁₁ ||class="entry q3 g1"| 47605₁₁ ||class="entry q3 g1"| 37451₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (54784, 49533, 33347, 43518, 49045, 37931) ||class="entry q2 g1"| 54784₅ ||class="entry q3 g1"| 49533₉ ||class="entry q3 g1"| 33347₅ ||class="entry q2 g1"| 43518₁₁ ||class="entry q3 g1"| 49045₁₁ ||class="entry q3 g1"| 37931₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 13) ||class="c"| (33000, 65281, 59777, 65302, 33257, 65513) ||class="entry q2 g1"| 33000₅ ||class="entry q3 g1"| 65281₉ ||class="entry q3 g1"| 59777₇ ||class="entry q2 g1"| 65302₁₁ ||class="entry q3 g1"| 33257₇ ||class="entry q3 g1"| 65513₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 13) ||class="c"| (36360, 61681, 59537, 61942, 36377, 65273) ||class="entry q2 g1"| 36360₅ ||class="entry q3 g1"| 61681₉ ||class="entry q3 g1"| 59537₇ ||class="entry q2 g1"| 61942₁₁ ||class="entry q3 g1"| 36377₇ ||class="entry q3 g1"| 65273₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 13) ||class="c"| (45600, 52429, 59525, 52702, 45605, 65261) ||class="entry q2 g1"| 45600₅ ||class="entry q3 g1"| 52429₉ ||class="entry q3 g1"| 59525₇ ||class="entry q2 g1"| 52702₁₁ ||class="entry q3 g1"| 45605₇ ||class="entry q3 g1"| 65261₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 13) ||class="c"| (54336, 43691, 59523, 43966, 54339, 65259) ||class="entry q2 g1"| 54336₅ ||class="entry q3 g1"| 43691₉ ||class="entry q3 g1"| 59523₇ ||class="entry q2 g1"| 43966₁₁ ||class="entry q3 g1"| 54339₇ ||class="entry q3 g1"| 65259₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 3) ||class="c"| (38528, 33023, 38761, 59774, 65047, 33025) ||class="entry q2 g1"| 38528₅ ||class="entry q3 g1"| 33023₉ ||class="entry q3 g1"| 38761₉ ||class="entry q2 g1"| 59774₁₁ ||class="entry q3 g1"| 65047₁₁ ||class="entry q3 g1"| 33025₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 3) ||class="c"| (39008, 36623, 38521, 59294, 61927, 32785) ||class="entry q2 g1"| 39008₅ ||class="entry q3 g1"| 36623₉ ||class="entry q3 g1"| 38521₉ ||class="entry q2 g1"| 59294₁₁ ||class="entry q3 g1"| 61927₁₁ ||class="entry q3 g1"| 32785₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 3) ||class="c"| (42056, 45875, 38509, 56246, 52699, 32773) ||class="entry q2 g1"| 42056₅ ||class="entry q3 g1"| 45875₉ ||class="entry q3 g1"| 38509₉ ||class="entry q2 g1"| 56246₁₁ ||class="entry q3 g1"| 52699₁₁ ||class="entry q3 g1"| 32773₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 3) ||class="c"| (49704, 54613, 38507, 48598, 43965, 32771) ||class="entry q2 g1"| 49704₅ ||class="entry q3 g1"| 54613₉ ||class="entry q3 g1"| 38507₉ ||class="entry q2 g1"| 48598₁₁ ||class="entry q3 g1"| 43965₁₁ ||class="entry q3 g1"| 32771₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (34440, 63841, 61409, 63862, 34697, 63881) ||class="entry q2 g1"| 34440₅ ||class="entry q3 g1"| 63841₉ ||class="entry q3 g1"| 61409₁₁ ||class="entry q2 g1"| 63862₁₁ ||class="entry q3 g1"| 34697₇ ||class="entry q3 g1"| 63881₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (34920, 63121, 61169, 63382, 34937, 63641) ||class="entry q2 g1"| 34920₅ ||class="entry q3 g1"| 63121₉ ||class="entry q3 g1"| 61169₁₁ ||class="entry q2 g1"| 63382₁₁ ||class="entry q3 g1"| 34937₇ ||class="entry q3 g1"| 63641₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (37536, 60745, 64457, 60766, 37793, 60833) ||class="entry q2 g1"| 37536₅ ||class="entry q3 g1"| 60745₉ ||class="entry q3 g1"| 64457₁₁ ||class="entry q2 g1"| 60766₁₁ ||class="entry q3 g1"| 37793₇ ||class="entry q3 g1"| 60833₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (38080, 60201, 64937, 60222, 38337, 60353) ||class="entry q2 g1"| 38080₅ ||class="entry q3 g1"| 60201₉ ||class="entry q3 g1"| 64937₁₁ ||class="entry q2 g1"| 60222₁₁ ||class="entry q3 g1"| 38337₇ ||class="entry q3 g1"| 60353₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (39456, 58585, 64697, 58846, 39473, 60113) ||class="entry q2 g1"| 39456₅ ||class="entry q3 g1"| 58585₉ ||class="entry q3 g1"| 64697₁₁ ||class="entry q2 g1"| 58846₁₁ ||class="entry q3 g1"| 39473₇ ||class="entry q3 g1"| 60113₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (40000, 58041, 64217, 58302, 40017, 60593) ||class="entry q2 g1"| 40000₅ ||class="entry q3 g1"| 58041₉ ||class="entry q3 g1"| 64217₁₁ ||class="entry q2 g1"| 58302₁₁ ||class="entry q3 g1"| 40017₇ ||class="entry q3 g1"| 60593₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (41064, 56965, 64205, 57238, 41069, 60581) ||class="entry q2 g1"| 41064₅ ||class="entry q3 g1"| 56965₉ ||class="entry q3 g1"| 64205₁₁ ||class="entry q2 g1"| 57238₁₁ ||class="entry q3 g1"| 41069₇ ||class="entry q3 g1"| 60581₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (42504, 55525, 64685, 55798, 42509, 60101) ||class="entry q2 g1"| 42504₅ ||class="entry q3 g1"| 55525₉ ||class="entry q3 g1"| 64685₁₁ ||class="entry q2 g1"| 55798₁₁ ||class="entry q3 g1"| 42509₇ ||class="entry q3 g1"| 60101₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (46144, 51885, 61157, 52158, 46149, 63629) ||class="entry q2 g1"| 46144₅ ||class="entry q3 g1"| 51885₉ ||class="entry q3 g1"| 61157₁₁ ||class="entry q2 g1"| 52158₁₁ ||class="entry q3 g1"| 46149₇ ||class="entry q3 g1"| 63629₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (49256, 48771, 64683, 49046, 49259, 60099) ||class="entry q2 g1"| 49256₅ ||class="entry q3 g1"| 48771₉ ||class="entry q3 g1"| 64683₁₁ ||class="entry q2 g1"| 49046₁₁ ||class="entry q3 g1"| 49259₇ ||class="entry q3 g1"| 60099₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (50696, 47331, 64203, 47606, 50699, 60579) ||class="entry q2 g1"| 50696₅ ||class="entry q3 g1"| 47331₉ ||class="entry q3 g1"| 64203₁₁ ||class="entry q2 g1"| 47606₁₁ ||class="entry q3 g1"| 50699₇ ||class="entry q3 g1"| 60579₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 11, 7, 9) ||class="c"| (53792, 44235, 61155, 44510, 53795, 63627) ||class="entry q2 g1"| 53792₅ ||class="entry q3 g1"| 44235₉ ||class="entry q3 g1"| 61155₁₁ ||class="entry q2 g1"| 44510₁₁ ||class="entry q3 g1"| 53795₇ ||class="entry q3 g1"| 63627₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (34446, 40705, 35207, 63856, 57833, 40943) ||class="entry q2 g1"| 34446₇ ||class="entry q3 g1"| 40705₇ ||class="entry q3 g1"| 35207₇ ||class="entry q2 g1"| 63856₉ ||class="entry q3 g1"| 57833₉ ||class="entry q3 g1"| 40943₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (34926, 37105, 34967, 63376, 60953, 40703) ||class="entry q2 g1"| 34926₇ ||class="entry q3 g1"| 37105₇ ||class="entry q3 g1"| 34967₇ ||class="entry q2 g1"| 63376₉ ||class="entry q3 g1"| 60953₉ ||class="entry q3 g1"| 40703₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (37554, 46849, 41363, 60748, 51689, 47099) ||class="entry q2 g1"| 37554₇ ||class="entry q3 g1"| 46849₇ ||class="entry q3 g1"| 41363₇ ||class="entry q2 g1"| 60748₉ ||class="entry q3 g1"| 51689₉ ||class="entry q3 g1"| 47099₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (38100, 55041, 49557, 60202, 43497, 55293) ||class="entry q2 g1"| 38100₇ ||class="entry q3 g1"| 55041₇ ||class="entry q3 g1"| 49557₇ ||class="entry q2 g1"| 60202₉ ||class="entry q3 g1"| 43497₉ ||class="entry q3 g1"| 55293₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (39714, 45169, 43283, 58588, 52889, 49019) ||class="entry q2 g1"| 39714₇ ||class="entry q3 g1"| 45169₇ ||class="entry q3 g1"| 43283₇ ||class="entry q2 g1"| 58588₉ ||class="entry q3 g1"| 52889₉ ||class="entry q3 g1"| 49019₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (40260, 53361, 51477, 58042, 44697, 57213) ||class="entry q2 g1"| 40260₇ ||class="entry q3 g1"| 53361₇ ||class="entry q3 g1"| 51477₇ ||class="entry q2 g1"| 58042₉ ||class="entry q3 g1"| 44697₉ ||class="entry q3 g1"| 57213₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (41082, 33997, 41111, 57220, 64037, 46847) ||class="entry q2 g1"| 41082₇ ||class="entry q3 g1"| 33997₇ ||class="entry q3 g1"| 41111₇ ||class="entry q2 g1"| 57220₉ ||class="entry q3 g1"| 64037₉ ||class="entry q3 g1"| 46847₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (42762, 35917, 43271, 55540, 62117, 49007) ||class="entry q2 g1"| 42762₇ ||class="entry q3 g1"| 35917₇ ||class="entry q3 g1"| 43271₇ ||class="entry q2 g1"| 55540₉ ||class="entry q3 g1"| 62117₉ ||class="entry q3 g1"| 49007₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (46416, 50253, 57621, 51886, 47781, 63357) ||class="entry q2 g1"| 46416₇ ||class="entry q3 g1"| 50253₇ ||class="entry q3 g1"| 57621₇ ||class="entry q2 g1"| 51886₉ ||class="entry q3 g1"| 47781₉ ||class="entry q3 g1"| 63357₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (49276, 33451, 49303, 49026, 64579, 55039) ||class="entry q2 g1"| 49276₇ ||class="entry q3 g1"| 33451₇ ||class="entry q3 g1"| 49303₇ ||class="entry q2 g1"| 49026₉ ||class="entry q3 g1"| 64579₉ ||class="entry q3 g1"| 55039₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (50956, 35371, 51463, 47346, 62659, 57199) ||class="entry q2 g1"| 50956₇ ||class="entry q3 g1"| 35371₇ ||class="entry q3 g1"| 51463₇ ||class="entry q2 g1"| 47346₉ ||class="entry q3 g1"| 62659₉ ||class="entry q3 g1"| 57199₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 13) ||class="c"| (54064, 41515, 57619, 44238, 56515, 63355) ||class="entry q2 g1"| 54064₇ ||class="entry q3 g1"| 41515₇ ||class="entry q3 g1"| 57619₇ ||class="entry q2 g1"| 44238₉ ||class="entry q3 g1"| 56515₉ ||class="entry q3 g1"| 63355₁₃ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (33006, 39265, 36839, 65296, 59273, 39311) ||class="entry q2 g1"| 33006₇ ||class="entry q3 g1"| 39265₇ ||class="entry q3 g1"| 36839₁₁ ||class="entry q2 g1"| 65296₉ ||class="entry q3 g1"| 59273₉ ||class="entry q3 g1"| 39311₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (33018, 42313, 46043, 65284, 56225, 42419) ||class="entry q2 g1"| 33018₇ ||class="entry q3 g1"| 42313₇ ||class="entry q3 g1"| 46043₁₁ ||class="entry q2 g1"| 65284₉ ||class="entry q3 g1"| 56225₉ ||class="entry q3 g1"| 42419₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (33020, 49961, 54717, 65282, 48577, 50133) ||class="entry q2 g1"| 33020₇ ||class="entry q3 g1"| 49961₇ ||class="entry q3 g1"| 54717₁₁ ||class="entry q2 g1"| 65282₉ ||class="entry q3 g1"| 48577₉ ||class="entry q3 g1"| 50133₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (34458, 41769, 46523, 63844, 56769, 41939) ||class="entry q2 g1"| 34458₇ ||class="entry q3 g1"| 41769₇ ||class="entry q3 g1"| 46523₁₁ ||class="entry q2 g1"| 63844₉ ||class="entry q3 g1"| 56769₉ ||class="entry q3 g1"| 41939₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (34460, 50505, 54237, 63842, 48033, 50613) ||class="entry q2 g1"| 34460₇ ||class="entry q3 g1"| 50505₇ ||class="entry q3 g1"| 54237₁₁ ||class="entry q2 g1"| 63842₉ ||class="entry q3 g1"| 48033₉ ||class="entry q3 g1"| 50613₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (35178, 41529, 47963, 63124, 56529, 44339) ||class="entry q2 g1"| 35178₇ ||class="entry q3 g1"| 41529₇ ||class="entry q3 g1"| 47963₁₁ ||class="entry q2 g1"| 63124₉ ||class="entry q3 g1"| 56529₉ ||class="entry q3 g1"| 44339₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (35180, 50265, 56637, 63122, 47793, 52053) ||class="entry q2 g1"| 35180₇ ||class="entry q3 g1"| 50265₇ ||class="entry q3 g1"| 56637₁₁ ||class="entry q2 g1"| 63122₉ ||class="entry q3 g1"| 47793₉ ||class="entry q3 g1"| 52053₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (36366, 38545, 36599, 61936, 59513, 39071) ||class="entry q2 g1"| 36366₇ ||class="entry q3 g1"| 38545₇ ||class="entry q3 g1"| 36599₁₁ ||class="entry q2 g1"| 61936₉ ||class="entry q3 g1"| 59513₉ ||class="entry q3 g1"| 39071₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (36618, 42073, 48443, 61684, 55985, 43859) ||class="entry q2 g1"| 36618₇ ||class="entry q3 g1"| 42073₇ ||class="entry q3 g1"| 48443₁₁ ||class="entry q2 g1"| 61684₉ ||class="entry q3 g1"| 55985₉ ||class="entry q3 g1"| 43859₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (36620, 49721, 56157, 61682, 48337, 52533) ||class="entry q2 g1"| 36620₇ ||class="entry q3 g1"| 49721₇ ||class="entry q3 g1"| 56157₁₁ ||class="entry q2 g1"| 61682₉ ||class="entry q3 g1"| 48337₉ ||class="entry q3 g1"| 52533₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (37542, 35625, 40367, 60760, 62913, 35783) ||class="entry q2 g1"| 37542₇ ||class="entry q3 g1"| 35625₇ ||class="entry q3 g1"| 40367₁₁ ||class="entry q2 g1"| 60760₉ ||class="entry q3 g1"| 62913₉ ||class="entry q3 g1"| 35783₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (37556, 53601, 51189, 60746, 44937, 53661) ||class="entry q2 g1"| 37556₇ ||class="entry q3 g1"| 53601₇ ||class="entry q3 g1"| 51189₁₁ ||class="entry q2 g1"| 60746₉ ||class="entry q3 g1"| 44937₉ ||class="entry q3 g1"| 53661₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (38086, 36169, 39887, 60216, 62369, 36263) ||class="entry q2 g1"| 38086₇ ||class="entry q3 g1"| 36169₇ ||class="entry q3 g1"| 39887₁₁ ||class="entry q2 g1"| 60216₉ ||class="entry q3 g1"| 62369₉ ||class="entry q3 g1"| 36263₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (38098, 45409, 42995, 60204, 53129, 45467) ||class="entry q2 g1"| 38098₇ ||class="entry q3 g1"| 45409₇ ||class="entry q3 g1"| 42995₁₁ ||class="entry q2 g1"| 60204₉ ||class="entry q3 g1"| 53129₉ ||class="entry q3 g1"| 45467₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (39462, 33465, 39647, 58840, 64593, 36023) ||class="entry q2 g1"| 39462₇ ||class="entry q3 g1"| 33465₇ ||class="entry q3 g1"| 39647₁₁ ||class="entry q2 g1"| 58840₉ ||class="entry q3 g1"| 64593₉ ||class="entry q3 g1"| 36023₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (39716, 54801, 53109, 58586, 43257, 55581) ||class="entry q2 g1"| 39716₇ ||class="entry q3 g1"| 54801₇ ||class="entry q3 g1"| 53109₁₁ ||class="entry q2 g1"| 58586₉ ||class="entry q3 g1"| 43257₉ ||class="entry q3 g1"| 55581₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (40006, 34009, 40127, 58296, 64049, 35543) ||class="entry q2 g1"| 40006₇ ||class="entry q3 g1"| 34009₇ ||class="entry q3 g1"| 40127₁₁ ||class="entry q2 g1"| 58296₉ ||class="entry q3 g1"| 64049₉ ||class="entry q3 g1"| 35543₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (40258, 46609, 44915, 58044, 51449, 47387) ||class="entry q2 g1"| 40258₇ ||class="entry q3 g1"| 46609₇ ||class="entry q3 g1"| 44915₁₁ ||class="entry q2 g1"| 58044₉ ||class="entry q3 g1"| 51449₉ ||class="entry q3 g1"| 47387₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (41322, 35373, 44903, 56980, 62661, 47375) ||class="entry q2 g1"| 41322₇ ||class="entry q3 g1"| 35373₇ ||class="entry q3 g1"| 44903₁₁ ||class="entry q2 g1"| 56980₉ ||class="entry q3 g1"| 62661₉ ||class="entry q3 g1"| 47375₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (41336, 53349, 62781, 56966, 44685, 58197) ||class="entry q2 g1"| 41336₇ ||class="entry q3 g1"| 53349₇ ||class="entry q3 g1"| 62781₁₁ ||class="entry q2 g1"| 56966₉ ||class="entry q3 g1"| 44685₉ ||class="entry q3 g1"| 58197₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (42522, 33453, 42743, 55780, 64581, 45215) ||class="entry q2 g1"| 42522₇ ||class="entry q3 g1"| 33453₇ ||class="entry q3 g1"| 42743₁₁ ||class="entry q2 g1"| 55780₉ ||class="entry q3 g1"| 64581₉ ||class="entry q3 g1"| 45215₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (42776, 54789, 62301, 55526, 43245, 58677) ||class="entry q2 g1"| 42776₇ ||class="entry q3 g1"| 54789₇ ||class="entry q3 g1"| 62301₁₁ ||class="entry q2 g1"| 55526₉ ||class="entry q3 g1"| 43245₉ ||class="entry q3 g1"| 58677₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (45618, 38533, 45791, 52684, 59501, 42167) ||class="entry q2 g1"| 45618₇ ||class="entry q3 g1"| 38533₇ ||class="entry q3 g1"| 45791₁₁ ||class="entry q2 g1"| 52684₉ ||class="entry q3 g1"| 59501₉ ||class="entry q3 g1"| 42167₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (45858, 39013, 48431, 52444, 59021, 43847) ||class="entry q2 g1"| 45858₇ ||class="entry q3 g1"| 39013₇ ||class="entry q3 g1"| 48431₁₁ ||class="entry q2 g1"| 52444₉ ||class="entry q3 g1"| 59021₉ ||class="entry q3 g1"| 43847₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (45872, 49709, 59253, 52430, 48325, 61725) ||class="entry q2 g1"| 45872₇ ||class="entry q3 g1"| 49709₇ ||class="entry q3 g1"| 59253₁₁ ||class="entry q2 g1"| 52430₉ ||class="entry q3 g1"| 48325₉ ||class="entry q3 g1"| 61725₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (46162, 37093, 46271, 52140, 60941, 41687) ||class="entry q2 g1"| 46162₇ ||class="entry q3 g1"| 37093₇ ||class="entry q3 g1"| 46271₁₁ ||class="entry q2 g1"| 52140₉ ||class="entry q3 g1"| 60941₉ ||class="entry q3 g1"| 41687₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (46402, 40453, 47951, 51900, 57581, 44327) ||class="entry q2 g1"| 46402₇ ||class="entry q3 g1"| 40453₇ ||class="entry q3 g1"| 47951₁₁ ||class="entry q2 g1"| 51900₉ ||class="entry q3 g1"| 57581₉ ||class="entry q3 g1"| 44327₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (49516, 35915, 53095, 48786, 62115, 55567) ||class="entry q2 g1"| 49516₇ ||class="entry q3 g1"| 35915₇ ||class="entry q3 g1"| 53095₁₁ ||class="entry q2 g1"| 48786₉ ||class="entry q3 g1"| 62115₉ ||class="entry q3 g1"| 55567₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (49528, 45155, 62299, 48774, 52875, 58675) ||class="entry q2 g1"| 49528₇ ||class="entry q3 g1"| 45155₇ ||class="entry q3 g1"| 62299₁₁ ||class="entry q2 g1"| 48774₉ ||class="entry q3 g1"| 52875₉ ||class="entry q3 g1"| 58675₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (50716, 33995, 50935, 47586, 64035, 53407) ||class="entry q2 g1"| 50716₇ ||class="entry q3 g1"| 33995₇ ||class="entry q3 g1"| 50935₁₁ ||class="entry q2 g1"| 47586₉ ||class="entry q3 g1"| 64035₉ ||class="entry q3 g1"| 53407₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (50968, 46595, 62779, 47334, 51435, 58195) ||class="entry q2 g1"| 50968₇ ||class="entry q3 g1"| 46595₇ ||class="entry q3 g1"| 62779₁₁ ||class="entry q2 g1"| 47334₉ ||class="entry q3 g1"| 51435₉ ||class="entry q3 g1"| 58195₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (53812, 37091, 53983, 44490, 60939, 50359) ||class="entry q2 g1"| 53812₇ ||class="entry q3 g1"| 37091₇ ||class="entry q3 g1"| 53983₁₁ ||class="entry q2 g1"| 44490₉ ||class="entry q3 g1"| 60939₉ ||class="entry q3 g1"| 50359₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (54052, 40451, 56623, 44250, 57579, 52039) ||class="entry q2 g1"| 54052₇ ||class="entry q3 g1"| 40451₇ ||class="entry q3 g1"| 56623₁₁ ||class="entry q2 g1"| 44250₉ ||class="entry q3 g1"| 57579₉ ||class="entry q3 g1"| 52039₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (54356, 38531, 54463, 43946, 59499, 49879) ||class="entry q2 g1"| 54356₇ ||class="entry q3 g1"| 38531₇ ||class="entry q3 g1"| 54463₁₁ ||class="entry q2 g1"| 43946₉ ||class="entry q3 g1"| 59499₉ ||class="entry q3 g1"| 49879₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (54596, 39011, 56143, 43706, 59019, 52519) ||class="entry q2 g1"| 54596₇ ||class="entry q3 g1"| 39011₇ ||class="entry q3 g1"| 56143₁₁ ||class="entry q2 g1"| 43706₉ ||class="entry q3 g1"| 59019₉ ||class="entry q3 g1"| 52519₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (54608, 42059, 59251, 43694, 55971, 61723) ||class="entry q2 g1"| 54608₇ ||class="entry q3 g1"| 42059₇ ||class="entry q3 g1"| 59251₁₁ ||class="entry q2 g1"| 43694₉ ||class="entry q3 g1"| 55971₉ ||class="entry q3 g1"| 61723₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (34712, 63361, 57361, 63590, 35177, 63097) ||class="entry q2 g1"| 34712₇ ||class="entry q3 g1"| 63361₉ ||class="entry q3 g1"| 57361₅ ||class="entry q2 g1"| 63590₉ ||class="entry q3 g1"| 35177₇ ||class="entry q3 g1"| 63097₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (35192, 63601, 57601, 63110, 34457, 63337) ||class="entry q2 g1"| 35192₇ ||class="entry q3 g1"| 63601₉ ||class="entry q3 g1"| 57601₅ ||class="entry q2 g1"| 63110₉ ||class="entry q3 g1"| 34457₇ ||class="entry q3 g1"| 63337₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (37796, 57217, 51205, 60506, 41321, 56941) ||class="entry q2 g1"| 37796₇ ||class="entry q3 g1"| 57217₉ ||class="entry q3 g1"| 51205₅ ||class="entry q2 g1"| 60506₉ ||class="entry q3 g1"| 41321₇ ||class="entry q3 g1"| 56941₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (38338, 49025, 43011, 59964, 49513, 48747) ||class="entry q2 g1"| 38338₇ ||class="entry q3 g1"| 49025₉ ||class="entry q3 g1"| 43011₅ ||class="entry q2 g1"| 59964₉ ||class="entry q3 g1"| 49513₇ ||class="entry q3 g1"| 48747₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (39476, 55537, 49285, 58826, 42521, 55021) ||class="entry q2 g1"| 39476₇ ||class="entry q3 g1"| 55537₉ ||class="entry q3 g1"| 49285₅ ||class="entry q2 g1"| 58826₉ ||class="entry q3 g1"| 42521₇ ||class="entry q3 g1"| 55021₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (40018, 47345, 41091, 58284, 50713, 46827) ||class="entry q2 g1"| 40018₇ ||class="entry q3 g1"| 47345₉ ||class="entry q3 g1"| 41091₅ ||class="entry q2 g1"| 58284₉ ||class="entry q3 g1"| 50713₇ ||class="entry q3 g1"| 46827₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (41324, 60493, 51457, 56978, 37541, 57193) ||class="entry q2 g1"| 41324₇ ||class="entry q3 g1"| 60493₉ ||class="entry q3 g1"| 51457₅ ||class="entry q2 g1"| 56978₉ ||class="entry q3 g1"| 37541₇ ||class="entry q3 g1"| 57193₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (42524, 58573, 49297, 55778, 39461, 55033) ||class="entry q2 g1"| 42524₇ ||class="entry q3 g1"| 58573₉ ||class="entry q3 g1"| 49297₅ ||class="entry q2 g1"| 55778₉ ||class="entry q3 g1"| 39461₇ ||class="entry q3 g1"| 55033₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (46150, 44237, 34947, 52152, 53797, 40683) ||class="entry q2 g1"| 46150₇ ||class="entry q3 g1"| 44237₉ ||class="entry q3 g1"| 34947₅ ||class="entry q2 g1"| 52152₉ ||class="entry q3 g1"| 53797₇ ||class="entry q3 g1"| 40683₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (49514, 59947, 43265, 48788, 38083, 49001) ||class="entry q2 g1"| 49514₇ ||class="entry q3 g1"| 59947₉ ||class="entry q3 g1"| 43265₅ ||class="entry q2 g1"| 48788₉ ||class="entry q3 g1"| 38083₇ ||class="entry q3 g1"| 49001₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (50714, 58027, 41105, 47588, 40003, 46841) ||class="entry q2 g1"| 50714₇ ||class="entry q3 g1"| 58027₉ ||class="entry q3 g1"| 41105₅ ||class="entry q2 g1"| 47588₉ ||class="entry q3 g1"| 40003₇ ||class="entry q3 g1"| 46841₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (53798, 51883, 34949, 44504, 46147, 40685) ||class="entry q2 g1"| 53798₇ ||class="entry q3 g1"| 51883₉ ||class="entry q3 g1"| 34949₅ ||class="entry q2 g1"| 44504₉ ||class="entry q3 g1"| 46147₇ ||class="entry q3 g1"| 40685₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (44232, 47779, 37149, 54070, 50251, 34677) ||class="entry q2 g1"| 44232₇ ||class="entry q3 g1"| 47779₉ ||class="entry q3 g1"| 37149₇ ||class="entry q2 g1"| 54070₉ ||class="entry q3 g1"| 50251₇ ||class="entry q3 g1"| 34677₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (47328, 44683, 34101, 50974, 53347, 37725) ||class="entry q2 g1"| 47328₇ ||class="entry q3 g1"| 44683₉ ||class="entry q3 g1"| 34101₇ ||class="entry q2 g1"| 50974₉ ||class="entry q3 g1"| 53347₇ ||class="entry q3 g1"| 37725₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (48768, 43243, 33621, 49534, 54787, 38205) ||class="entry q2 g1"| 48768₇ ||class="entry q3 g1"| 43243₉ ||class="entry q3 g1"| 33621₇ ||class="entry q2 g1"| 49534₉ ||class="entry q3 g1"| 54787₇ ||class="entry q3 g1"| 38205₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (51880, 56517, 37147, 46422, 41517, 34675) ||class="entry q2 g1"| 51880₇ ||class="entry q3 g1"| 56517₉ ||class="entry q3 g1"| 37147₇ ||class="entry q2 g1"| 46422₉ ||class="entry q3 g1"| 41517₇ ||class="entry q3 g1"| 34675₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (55520, 52877, 33619, 42782, 45157, 38203) ||class="entry q2 g1"| 55520₇ ||class="entry q3 g1"| 52877₉ ||class="entry q3 g1"| 33619₇ ||class="entry q2 g1"| 42782₉ ||class="entry q3 g1"| 45157₇ ||class="entry q3 g1"| 38203₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (56960, 51437, 34099, 41342, 46597, 37723) ||class="entry q2 g1"| 56960₇ ||class="entry q3 g1"| 51437₉ ||class="entry q3 g1"| 34099₇ ||class="entry q2 g1"| 41342₉ ||class="entry q3 g1"| 46597₇ ||class="entry q3 g1"| 37723₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (58024, 62673, 34087, 40278, 35385, 37711) ||class="entry q2 g1"| 58024₇ ||class="entry q3 g1"| 62673₉ ||class="entry q3 g1"| 34087₇ ||class="entry q2 g1"| 40278₉ ||class="entry q3 g1"| 35385₇ ||class="entry q3 g1"| 37711₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (58568, 62129, 33607, 39734, 35929, 38191) ||class="entry q2 g1"| 58568₇ ||class="entry q3 g1"| 62129₉ ||class="entry q3 g1"| 33607₇ ||class="entry q2 g1"| 39734₉ ||class="entry q3 g1"| 35929₇ ||class="entry q3 g1"| 38191₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (59944, 64833, 33367, 38358, 33705, 37951) ||class="entry q2 g1"| 59944₇ ||class="entry q3 g1"| 64833₉ ||class="entry q3 g1"| 33367₇ ||class="entry q2 g1"| 38358₉ ||class="entry q3 g1"| 33705₇ ||class="entry q3 g1"| 37951₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (60488, 64289, 33847, 37814, 34249, 37471) ||class="entry q2 g1"| 60488₇ ||class="entry q3 g1"| 64289₉ ||class="entry q3 g1"| 33847₇ ||class="entry q2 g1"| 37814₉ ||class="entry q3 g1"| 34249₇ ||class="entry q3 g1"| 37471₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (63104, 57593, 37135, 35198, 40465, 34663) ||class="entry q2 g1"| 63104₇ ||class="entry q3 g1"| 57593₉ ||class="entry q3 g1"| 37135₇ ||class="entry q2 g1"| 35198₉ ||class="entry q3 g1"| 40465₇ ||class="entry q3 g1"| 34663₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (63584, 61193, 36895, 34718, 37345, 34423) ||class="entry q2 g1"| 63584₇ ||class="entry q3 g1"| 61193₉ ||class="entry q3 g1"| 36895₇ ||class="entry q2 g1"| 34718₉ ||class="entry q3 g1"| 37345₇ ||class="entry q3 g1"| 34423₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (33708, 42527, 45197, 64594, 55543, 42725) ||class="entry q2 g1"| 33708₇ ||class="entry q3 g1"| 42527₉ ||class="entry q3 g1"| 45197₇ ||class="entry q2 g1"| 64594₉ ||class="entry q3 g1"| 55543₁₁ ||class="entry q3 g1"| 42725₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (33720, 39479, 36017, 64582, 58591, 39641) ||class="entry q2 g1"| 33720₇ ||class="entry q3 g1"| 39479₉ ||class="entry q3 g1"| 36017₇ ||class="entry q2 g1"| 64582₉ ||class="entry q3 g1"| 58591₁₁ ||class="entry q3 g1"| 39641₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (34250, 50719, 53387, 64052, 47351, 50915) ||class="entry q2 g1"| 34250₇ ||class="entry q3 g1"| 50719₉ ||class="entry q3 g1"| 53387₇ ||class="entry q2 g1"| 64052₉ ||class="entry q3 g1"| 47351₁₁ ||class="entry q3 g1"| 50915₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (34264, 40023, 35537, 64038, 58047, 40121) ||class="entry q2 g1"| 34264₇ ||class="entry q3 g1"| 40023₉ ||class="entry q3 g1"| 35537₇ ||class="entry q2 g1"| 64038₉ ||class="entry q3 g1"| 58047₁₁ ||class="entry q3 g1"| 40121₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35388, 41327, 47117, 62914, 57223, 44645) ||class="entry q2 g1"| 35388₇ ||class="entry q3 g1"| 41327₉ ||class="entry q3 g1"| 47117₇ ||class="entry q2 g1"| 62914₉ ||class="entry q3 g1"| 57223₁₁ ||class="entry q3 g1"| 44645₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35640, 37799, 35777, 62662, 60751, 40361) ||class="entry q2 g1"| 35640₇ ||class="entry q3 g1"| 37799₉ ||class="entry q3 g1"| 35777₇ ||class="entry q2 g1"| 62662₉ ||class="entry q3 g1"| 60751₁₁ ||class="entry q3 g1"| 40361₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35930, 49519, 55307, 62372, 49031, 52835) ||class="entry q2 g1"| 35930₇ ||class="entry q3 g1"| 49519₉ ||class="entry q3 g1"| 55307₇ ||class="entry q2 g1"| 62372₉ ||class="entry q3 g1"| 49031₁₁ ||class="entry q3 g1"| 52835₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (36184, 38343, 36257, 62118, 60207, 39881) ||class="entry q2 g1"| 36184₇ ||class="entry q3 g1"| 38343₉ ||class="entry q3 g1"| 36257₇ ||class="entry q2 g1"| 62118₉ ||class="entry q3 g1"| 60207₁₁ ||class="entry q3 g1"| 39881₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (37346, 53815, 50339, 60956, 44255, 53963) ||class="entry q2 g1"| 37346₇ ||class="entry q3 g1"| 53815₉ ||class="entry q3 g1"| 50339₇ ||class="entry q2 g1"| 60956₉ ||class="entry q3 g1"| 44255₁₁ ||class="entry q3 g1"| 53963₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (37348, 46167, 41669, 60954, 51903, 46253) ||class="entry q2 g1"| 37348₇ ||class="entry q3 g1"| 46167₉ ||class="entry q3 g1"| 41669₇ ||class="entry q2 g1"| 60954₉ ||class="entry q3 g1"| 51903₁₁ ||class="entry q3 g1"| 46253₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (38786, 54359, 49859, 59516, 43711, 54443) ||class="entry q2 g1"| 38786₇ ||class="entry q3 g1"| 54359₉ ||class="entry q3 g1"| 49859₇ ||class="entry q2 g1"| 59516₉ ||class="entry q3 g1"| 43711₁₁ ||class="entry q3 g1"| 54443₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (38788, 45623, 42149, 59514, 52447, 45773) ||class="entry q2 g1"| 38788₇ ||class="entry q3 g1"| 45623₉ ||class="entry q3 g1"| 42149₇ ||class="entry q2 g1"| 59514₉ ||class="entry q3 g1"| 52447₁₁ ||class="entry q3 g1"| 45773₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (38800, 36383, 39065, 59502, 61687, 36593) ||class="entry q2 g1"| 38800₇ ||class="entry q3 g1"| 36383₉ ||class="entry q3 g1"| 39065₇ ||class="entry q2 g1"| 59502₉ ||class="entry q3 g1"| 61687₁₁ ||class="entry q3 g1"| 36593₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39026, 54599, 52259, 59276, 43951, 55883) ||class="entry q2 g1"| 39026₇ ||class="entry q3 g1"| 54599₉ ||class="entry q3 g1"| 52259₇ ||class="entry q2 g1"| 59276₉ ||class="entry q3 g1"| 43951₁₁ ||class="entry q3 g1"| 55883₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39028, 45863, 43589, 59274, 52687, 48173) ||class="entry q2 g1"| 39028₇ ||class="entry q3 g1"| 45863₉ ||class="entry q3 g1"| 43589₇ ||class="entry q2 g1"| 59274₉ ||class="entry q3 g1"| 52687₁₁ ||class="entry q3 g1"| 48173₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39280, 33263, 39305, 59022, 65287, 36833) ||class="entry q2 g1"| 39280₇ ||class="entry q3 g1"| 33263₉ ||class="entry q3 g1"| 39305₇ ||class="entry q2 g1"| 59022₉ ||class="entry q3 g1"| 65287₁₁ ||class="entry q3 g1"| 36833₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (40466, 54055, 51779, 57836, 44495, 56363) ||class="entry q2 g1"| 40466₇ ||class="entry q3 g1"| 54055₉ ||class="entry q3 g1"| 51779₇ ||class="entry q2 g1"| 57836₉ ||class="entry q3 g1"| 44495₁₁ ||class="entry q3 g1"| 56363₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (40468, 46407, 44069, 57834, 52143, 47693) ||class="entry q2 g1"| 40468₇ ||class="entry q3 g1"| 46407₉ ||class="entry q3 g1"| 44069₇ ||class="entry q2 g1"| 57834₉ ||class="entry q3 g1"| 52143₁₁ ||class="entry q3 g1"| 47693₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41532, 35195, 44081, 56770, 63379, 47705) ||class="entry q2 g1"| 41532₇ ||class="entry q3 g1"| 35195₉ ||class="entry q3 g1"| 44081₇ ||class="entry q2 g1"| 56770₉ ||class="entry q3 g1"| 63379₁₁ ||class="entry q3 g1"| 47705₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41772, 34715, 41921, 56530, 63859, 46505) ||class="entry q2 g1"| 41772₇ ||class="entry q3 g1"| 34715₉ ||class="entry q3 g1"| 41921₇ ||class="entry q2 g1"| 56530₉ ||class="entry q3 g1"| 63859₁₁ ||class="entry q3 g1"| 46505₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42062, 54611, 61451, 56240, 43963, 58979) ||class="entry q2 g1"| 42062₇ ||class="entry q3 g1"| 54611₉ ||class="entry q3 g1"| 61451₇ ||class="entry q2 g1"| 56240₉ ||class="entry q3 g1"| 43963₁₁ ||class="entry q3 g1"| 58979₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42076, 36635, 43601, 56226, 61939, 48185) ||class="entry q2 g1"| 42076₇ ||class="entry q3 g1"| 36635₉ ||class="entry q3 g1"| 43601₇ ||class="entry q2 g1"| 56226₉ ||class="entry q3 g1"| 61939₁₁ ||class="entry q3 g1"| 48185₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42316, 33275, 42401, 55986, 65299, 46025) ||class="entry q2 g1"| 42316₇ ||class="entry q3 g1"| 33275₉ ||class="entry q3 g1"| 42401₇ ||class="entry q2 g1"| 55986₉ ||class="entry q3 g1"| 65299₁₁ ||class="entry q3 g1"| 46025₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (45158, 49531, 58403, 53144, 49043, 62027) ||class="entry q2 g1"| 45158₇ ||class="entry q3 g1"| 49531₉ ||class="entry q3 g1"| 58403₇ ||class="entry q2 g1"| 53144₉ ||class="entry q3 g1"| 49043₁₁ ||class="entry q3 g1"| 62027₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (45412, 38355, 45449, 52890, 60219, 42977) ||class="entry q2 g1"| 45412₇ ||class="entry q3 g1"| 38355₉ ||class="entry q3 g1"| 45449₇ ||class="entry q2 g1"| 52890₉ ||class="entry q3 g1"| 60219₁₁ ||class="entry q3 g1"| 42977₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (46598, 50971, 57923, 51704, 47603, 62507) ||class="entry q2 g1"| 46598₇ ||class="entry q3 g1"| 50971₉ ||class="entry q3 g1"| 57923₇ ||class="entry q2 g1"| 51704₉ ||class="entry q3 g1"| 47603₁₁ ||class="entry q3 g1"| 62507₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (46612, 40275, 47129, 51690, 58299, 44657) ||class="entry q2 g1"| 46612₇ ||class="entry q3 g1"| 40275₉ ||class="entry q3 g1"| 47129₇ ||class="entry q2 g1"| 51690₉ ||class="entry q3 g1"| 58299₁₁ ||class="entry q3 g1"| 44657₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (49710, 45877, 61453, 48592, 52701, 58981) ||class="entry q2 g1"| 49710₇ ||class="entry q3 g1"| 45877₉ ||class="entry q3 g1"| 61453₇ ||class="entry q2 g1"| 48592₉ ||class="entry q3 g1"| 52701₁₁ ||class="entry q3 g1"| 58981₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (49722, 36637, 52273, 48580, 61941, 55897) ||class="entry q2 g1"| 49722₇ ||class="entry q3 g1"| 36637₉ ||class="entry q3 g1"| 52273₇ ||class="entry q2 g1"| 48580₉ ||class="entry q3 g1"| 61941₁₁ ||class="entry q3 g1"| 55897₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (49962, 33277, 50113, 48340, 65301, 54697) ||class="entry q2 g1"| 49962₇ ||class="entry q3 g1"| 33277₉ ||class="entry q3 g1"| 50113₇ ||class="entry q2 g1"| 48340₉ ||class="entry q3 g1"| 65301₁₁ ||class="entry q3 g1"| 54697₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50266, 35197, 51793, 48036, 63381, 56377) ||class="entry q2 g1"| 50266₇ ||class="entry q3 g1"| 35197₉ ||class="entry q3 g1"| 51793₇ ||class="entry q2 g1"| 48036₉ ||class="entry q3 g1"| 63381₁₁ ||class="entry q3 g1"| 56377₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (50506, 34717, 50593, 47796, 63861, 54217) ||class="entry q2 g1"| 50506₇ ||class="entry q3 g1"| 34717₉ ||class="entry q3 g1"| 50593₇ ||class="entry q2 g1"| 47796₉ ||class="entry q3 g1"| 63861₁₁ ||class="entry q3 g1"| 54217₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (53350, 41341, 57925, 44952, 57237, 62509) ||class="entry q2 g1"| 53350₇ ||class="entry q3 g1"| 41341₉ ||class="entry q3 g1"| 57925₇ ||class="entry q2 g1"| 44952₉ ||class="entry q3 g1"| 57237₁₁ ||class="entry q3 g1"| 62509₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (53602, 37813, 53641, 44700, 60765, 51169) ||class="entry q2 g1"| 53602₇ ||class="entry q3 g1"| 37813₉ ||class="entry q3 g1"| 53641₇ ||class="entry q2 g1"| 44700₉ ||class="entry q3 g1"| 60765₁₁ ||class="entry q3 g1"| 51169₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (54790, 42781, 58405, 43512, 55797, 62029) ||class="entry q2 g1"| 54790₇ ||class="entry q3 g1"| 42781₉ ||class="entry q3 g1"| 58405₇ ||class="entry q2 g1"| 43512₉ ||class="entry q3 g1"| 55797₁₁ ||class="entry q3 g1"| 62029₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (54802, 39733, 55321, 43500, 58845, 52849) ||class="entry q2 g1"| 54802₇ ||class="entry q3 g1"| 39733₉ ||class="entry q3 g1"| 55321₇ ||class="entry q2 g1"| 43500₉ ||class="entry q3 g1"| 58845₁₁ ||class="entry q3 g1"| 52849₉ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (33258, 43945, 48171, 65044, 54593, 43587) ||class="entry q2 g1"| 33258₇ ||class="entry q3 g1"| 43945₉ ||class="entry q3 g1"| 48171₉ ||class="entry q2 g1"| 65044₉ ||class="entry q3 g1"| 54593₇ ||class="entry q3 g1"| 43587₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (33260, 52681, 55885, 65042, 45857, 52261) ||class="entry q2 g1"| 33260₇ ||class="entry q3 g1"| 52681₉ ||class="entry q3 g1"| 55885₉ ||class="entry q2 g1"| 65042₉ ||class="entry q3 g1"| 45857₇ ||class="entry q3 g1"| 52261₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (33272, 61921, 58993, 65030, 36617, 61465) ||class="entry q2 g1"| 33272₇ ||class="entry q3 g1"| 61921₉ ||class="entry q3 g1"| 58993₉ ||class="entry q2 g1"| 65030₉ ||class="entry q3 g1"| 36617₇ ||class="entry q3 g1"| 61465₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (34698, 44489, 47691, 63604, 54049, 44067) ||class="entry q2 g1"| 34698₇ ||class="entry q3 g1"| 44489₉ ||class="entry q3 g1"| 47691₉ ||class="entry q2 g1"| 63604₉ ||class="entry q3 g1"| 54049₇ ||class="entry q3 g1"| 44067₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (34700, 52137, 56365, 63602, 46401, 51781) ||class="entry q2 g1"| 34700₇ ||class="entry q3 g1"| 52137₉ ||class="entry q3 g1"| 56365₉ ||class="entry q2 g1"| 63602₉ ||class="entry q3 g1"| 46401₇ ||class="entry q3 g1"| 51781₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (34938, 44249, 46251, 63364, 53809, 41667) ||class="entry q2 g1"| 34938₇ ||class="entry q3 g1"| 44249₉ ||class="entry q3 g1"| 46251₉ ||class="entry q2 g1"| 63364₉ ||class="entry q3 g1"| 53809₇ ||class="entry q3 g1"| 41667₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (34940, 51897, 53965, 63362, 46161, 50341) ||class="entry q2 g1"| 34940₇ ||class="entry q3 g1"| 51897₉ ||class="entry q3 g1"| 53965₉ ||class="entry q2 g1"| 63362₉ ||class="entry q3 g1"| 46161₇ ||class="entry q3 g1"| 50341₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (36378, 43705, 45771, 61924, 54353, 42147) ||class="entry q2 g1"| 36378₇ ||class="entry q3 g1"| 43705₉ ||class="entry q3 g1"| 45771₉ ||class="entry q2 g1"| 61924₉ ||class="entry q3 g1"| 54353₇ ||class="entry q3 g1"| 42147₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (36380, 52441, 54445, 61922, 45617, 49861) ||class="entry q2 g1"| 36380₇ ||class="entry q3 g1"| 52441₉ ||class="entry q3 g1"| 54445₉ ||class="entry q2 g1"| 61922₉ ||class="entry q3 g1"| 45617₇ ||class="entry q3 g1"| 49861₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) 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57937₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (38340, 55777, 52837, 59962, 42761, 55309) ||class="entry q2 g1"| 38340₇ ||class="entry q3 g1"| 55777₉ ||class="entry q3 g1"| 52837₉ ||class="entry q2 g1"| 59962₉ ||class="entry q3 g1"| 42761₇ ||class="entry q3 g1"| 55309₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (38352, 58825, 62041, 59950, 39713, 58417) ||class="entry q2 g1"| 38352₇ ||class="entry q3 g1"| 58825₉ ||class="entry q3 g1"| 62041₉ ||class="entry q2 g1"| 59950₉ ||class="entry q3 g1"| 39713₇ ||class="entry q3 g1"| 58417₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (39474, 48785, 42723, 58828, 49273, 45195) ||class="entry q2 g1"| 39474₇ ||class="entry q3 g1"| 48785₉ ||class="entry q3 g1"| 42723₉ ||class="entry q2 g1"| 58828₉ ||class="entry q3 g1"| 49273₇ ||class="entry q3 g1"| 45195₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (39728, 59961, 62281, 58574, 38097, 58657) ||class="entry q2 g1"| 39728₇ ||class="entry q3 g1"| 59961₉ ||class="entry q3 g1"| 62281₉ ||class="entry q2 g1"| 58574₉ ||class="entry q3 g1"| 38097₇ ||class="entry q3 g1"| 58657₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (40020, 56977, 50917, 58282, 41081, 53389) ||class="entry q2 g1"| 40020₇ ||class="entry q3 g1"| 56977₉ ||class="entry q3 g1"| 50917₉ ||class="entry q2 g1"| 58282₉ ||class="entry q3 g1"| 41081₇ ||class="entry q3 g1"| 53389₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (40272, 60505, 62761, 58030, 37553, 58177) ||class="entry q2 g1"| 40272₇ ||class="entry q3 g1"| 60505₉ ||class="entry q3 g1"| 62761₉ ||class="entry q2 g1"| 58030₉ ||class="entry q3 g1"| 37553₇ ||class="entry q3 g1"| 58177₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (41070, 47333, 40107, 57232, 50701, 35523) ||class="entry q2 g1"| 41070₇ ||class="entry q3 g1"| 47333₉ ||class="entry q3 g1"| 40107₉ ||class="entry q2 g1"| 57232₉ ||class="entry q3 g1"| 50701₇ ||class="entry q3 g1"| 35523₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (41084, 58029, 50929, 57218, 40005, 53401) ||class="entry q2 g1"| 41084₇ ||class="entry q3 g1"| 58029₉ ||class="entry q3 g1"| 50929₉ ||class="entry q2 g1"| 57218₉ ||class="entry q3 g1"| 40005₇ ||class="entry q3 g1"| 53401₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (42510, 48773, 39627, 55792, 49261, 36003) ||class="entry q2 g1"| 42510₇ ||class="entry q3 g1"| 48773₉ ||class="entry q3 g1"| 39627₉ ||class="entry q2 g1"| 55792₉ ||class="entry q3 g1"| 49261₇ ||class="entry q3 g1"| 36003₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (42764, 59949, 53089, 55538, 38085, 55561) ||class="entry q2 g1"| 42764₇ ||class="entry q3 g1"| 59949₉ ||class="entry q3 g1"| 53089₉ ||class="entry q2 g1"| 55538₉ ||class="entry q3 g1"| 38085₇ ||class="entry q3 g1"| 55561₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (45606, 43693, 36579, 52696, 54341, 39051) ||class="entry q2 g1"| 45606₇ ||class="entry q3 g1"| 43693₉ ||class="entry q3 g1"| 36579₉ ||class="entry q2 g1"| 52696₉ ||class="entry q3 g1"| 54341₇ ||class="entry q3 g1"| 39051₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (45620, 61669, 54457, 52682, 36365, 49873) ||class="entry q2 g1"| 45620₇ ||class="entry q3 g1"| 61669₉ ||class="entry q3 g1"| 54457₉ ||class="entry q2 g1"| 52682₉ ||class="entry q3 g1"| 36365₇ ||class="entry q3 g1"| 49873₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (45860, 65029, 56137, 52442, 33005, 52513) ||class="entry q2 g1"| 45860₇ ||class="entry q3 g1"| 65029₉ ||class="entry q3 g1"| 56137₉ ||class="entry q2 g1"| 52442₉ ||class="entry q3 g1"| 33005₇ ||class="entry q3 g1"| 52513₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (46164, 63109, 53977, 52138, 34925, 50353) ||class="entry q2 g1"| 46164₇ ||class="entry q3 g1"| 63109₉ ||class="entry q3 g1"| 53977₉ ||class="entry q2 g1"| 52138₉ ||class="entry q3 g1"| 34925₇ ||class="entry q3 g1"| 50353₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) 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61667₉ ||class="entry q3 g1"| 45785₉ ||class="entry q2 g1"| 43948₉ ||class="entry q3 g1"| 36363₇ ||class="entry q3 g1"| 42161₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 7) ||class="c"| (54594, 65027, 48425, 43708, 33003, 43841) ||class="entry q2 g1"| 54594₇ ||class="entry q3 g1"| 65027₉ ||class="entry q3 g1"| 48425₉ ||class="entry q2 g1"| 43708₉ ||class="entry q3 g1"| 33003₇ ||class="entry q3 g1"| 43841₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 15) ||class="c"| (48320, 49981, 59797, 49982, 48597, 65533) ||class="entry q2 g1"| 48320₇ ||class="entry q3 g1"| 49981₉ ||class="entry q3 g1"| 59797₉ ||class="entry q2 g1"| 49982₉ ||class="entry q3 g1"| 48597₁₁ ||class="entry q3 g1"| 65533₁₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 15) ||class="c"| (55968, 42331, 59795, 42334, 56243, 65531) ||class="entry q2 g1"| 55968₇ ||class="entry q3 g1"| 42331₉ ||class="entry q3 g1"| 59795₉ ||class="entry q2 g1"| 42334₉ ||class="entry q3 g1"| 56243₁₁ ||class="entry q3 g1"| 65531₁₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 15) ||class="c"| (59016, 39271, 59783, 39286, 59279, 65519) ||class="entry q2 g1"| 59016₇ ||class="entry q3 g1"| 39271₉ ||class="entry q3 g1"| 59783₉ ||class="entry q2 g1"| 39286₉ ||class="entry q3 g1"| 59279₁₁ ||class="entry q3 g1"| 65519₁₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 15) ||class="c"| (59496, 38551, 59543, 38806, 59519, 65279) ||class="entry q2 g1"| 59496₇ ||class="entry q3 g1"| 38551₉ ||class="entry q3 g1"| 59543₉ ||class="entry q2 g1"| 38806₉ ||class="entry q3 g1"| 59519₁₁ ||class="entry q3 g1"| 65279₁₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 5) ||class="c"| (43688, 48323, 38781, 54614, 49707, 33045) ||class="entry q2 g1"| 43688₇ ||class="entry q3 g1"| 48323₉ ||class="entry q3 g1"| 38781₁₁ ||class="entry q2 g1"| 54614₉ ||class="entry q3 g1"| 49707₇ ||class="entry q3 g1"| 33045₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 5) ||class="c"| (52424, 55973, 38779, 45878, 42061, 33043) ||class="entry q2 g1"| 52424₇ ||class="entry q3 g1"| 55973₉ ||class="entry q3 g1"| 38779₁₁ ||class="entry q2 g1"| 45878₉ ||class="entry q3 g1"| 42061₇ ||class="entry q3 g1"| 33043₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 5) ||class="c"| (61664, 59033, 38767, 36638, 39025, 33031) ||class="entry q2 g1"| 61664₇ ||class="entry q3 g1"| 59033₉ ||class="entry q3 g1"| 38767₁₁ ||class="entry q2 g1"| 36638₉ ||class="entry q3 g1"| 39025₇ ||class="entry q3 g1"| 33031₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 5) ||class="c"| (65024, 59753, 38527, 33278, 38785, 32791) ||class="entry q2 g1"| 65024₇ ||class="entry q3 g1"| 59753₉ ||class="entry q3 g1"| 38527₁₁ ||class="entry q2 g1"| 33278₉ ||class="entry q3 g1"| 38785₇ ||class="entry q3 g1"| 32791₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (33706, 49279, 55019, 64596, 48791, 49283) ||class="entry q2 g1"| 33706₇ ||class="entry q3 g1"| 49279₉ ||class="entry q3 g1"| 55019₁₁ ||class="entry q2 g1"| 64596₉ ||class="entry q3 g1"| 48791₁₁ ||class="entry q3 g1"| 49283₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (34252, 41087, 46829, 64050, 56983, 41093) ||class="entry q2 g1"| 34252₇ ||class="entry q3 g1"| 41087₉ ||class="entry q3 g1"| 46829₁₁ ||class="entry q2 g1"| 64050₉ ||class="entry q3 g1"| 56983₁₁ ||class="entry q3 g1"| 41093₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (35386, 50959, 56939, 62916, 47591, 51203) ||class="entry q2 g1"| 35386₇ ||class="entry q3 g1"| 50959₉ ||class="entry q3 g1"| 56939₁₁ ||class="entry q2 g1"| 62916₉ ||class="entry q3 g1"| 47591₁₁ ||class="entry q3 g1"| 51203₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (35932, 42767, 48749, 62370, 55783, 43013) ||class="entry q2 g1"| 35932₇ ||class="entry q3 g1"| 42767₉ ||class="entry q3 g1"| 48749₁₁ ||class="entry q2 g1"| 62370₉ ||class="entry q3 g1"| 55783₁₁ ||class="entry q3 g1"| 43013₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (37360, 34943, 40697, 60942, 63127, 34961) ||class="entry q2 g1"| 37360₇ ||class="entry q3 g1"| 34943₉ ||class="entry q3 g1"| 40697₁₁ ||class="entry q2 g1"| 60942₉ ||class="entry q3 g1"| 63127₁₁ ||class="entry q3 g1"| 34961₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (40720, 34703, 40937, 57582, 63847, 35201) ||class="entry q2 g1"| 40720₇ ||class="entry q3 g1"| 34703₉ ||class="entry q3 g1"| 40937₁₁ ||class="entry q2 g1"| 57582₉ ||class="entry q3 g1"| 63847₁₁ ||class="entry q3 g1"| 35201₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (41518, 54067, 63083, 56784, 44507, 57347) ||class="entry q2 g1"| 41518₇ ||class="entry q3 g1"| 54067₉ ||class="entry q3 g1"| 63083₁₁ ||class="entry q2 g1"| 56784₉ ||class="entry q3 g1"| 44507₁₁ ||class="entry q3 g1"| 57347₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (45172, 39731, 48761, 53130, 58843, 43025) ||class="entry q2 g1"| 45172₇ ||class="entry q3 g1"| 39731₉ ||class="entry q3 g1"| 48761₁₁ ||class="entry q2 g1"| 53130₉ ||class="entry q3 g1"| 58843₁₁ ||class="entry q3 g1"| 43025₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (46852, 37811, 47081, 51450, 60763, 41345) ||class="entry q2 g1"| 46852₇ ||class="entry q3 g1"| 37811₉ ||class="entry q3 g1"| 47081₁₁ ||class="entry q2 g1"| 51450₉ ||class="entry q3 g1"| 60763₁₁ ||class="entry q3 g1"| 41345₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (50254, 46421, 63085, 48048, 52157, 57349) ||class="entry q2 g1"| 50254₇ ||class="entry q3 g1"| 46421₉ ||class="entry q3 g1"| 63085₁₁ ||class="entry q2 g1"| 48048₉ ||class="entry q3 g1"| 52157₁₁ ||class="entry q3 g1"| 57349₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (53362, 40277, 56953, 44940, 58301, 51217) ||class="entry q2 g1"| 53362₇ ||class="entry q3 g1"| 40277₉ ||class="entry q3 g1"| 56953₁₁ ||class="entry q2 g1"| 44940₉ ||class="entry q3 g1"| 58301₁₁ ||class="entry q3 g1"| 51217₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 11, 5) ||class="c"| (55042, 38357, 55273, 43260, 60221, 49537) ||class="entry q2 g1"| 55042₇ ||class="entry q3 g1"| 38357₉ ||class="entry q3 g1"| 55273₁₁ ||class="entry q2 g1"| 43260₉ ||class="entry q3 g1"| 60221₁₁ ||class="entry q3 g1"| 49537₅ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (43240, 55061, 64957, 55062, 43517, 60373) ||class="entry q2 g1"| 43240₇ ||class="entry q3 g1"| 55061₉ ||class="entry q3 g1"| 64957₁₃ ||class="entry q2 g1"| 55062₉ ||class="entry q3 g1"| 43517₁₁ ||class="entry q3 g1"| 60373₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (44680, 53621, 64477, 53622, 44957, 60853) ||class="entry q2 g1"| 44680₇ ||class="entry q3 g1"| 53621₉ ||class="entry q3 g1"| 64477₁₃ ||class="entry q2 g1"| 53622₉ ||class="entry q3 g1"| 44957₁₁ ||class="entry q3 g1"| 60853₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (47776, 50525, 61429, 50526, 48053, 63901) ||class="entry q2 g1"| 47776₇ ||class="entry q3 g1"| 50525₉ ||class="entry q3 g1"| 61429₁₃ ||class="entry q2 g1"| 50526₉ ||class="entry q3 g1"| 48053₁₁ ||class="entry q3 g1"| 63901₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (51432, 46867, 64475, 46870, 51707, 60851) ||class="entry q2 g1"| 51432₇ ||class="entry q3 g1"| 46867₉ ||class="entry q3 g1"| 64475₁₃ ||class="entry q2 g1"| 46870₉ ||class="entry q3 g1"| 51707₁₁ ||class="entry q3 g1"| 60851₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (52872, 45427, 64955, 45430, 53147, 60371) ||class="entry q2 g1"| 52872₇ ||class="entry q3 g1"| 45427₉ ||class="entry q3 g1"| 64955₁₃ ||class="entry q2 g1"| 45430₉ ||class="entry q3 g1"| 53147₁₁ ||class="entry q3 g1"| 60371₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (56512, 41787, 61427, 41790, 56787, 63899) ||class="entry q2 g1"| 56512₇ ||class="entry q3 g1"| 41787₉ ||class="entry q3 g1"| 61427₁₃ ||class="entry q2 g1"| 41790₉ ||class="entry q3 g1"| 56787₁₁ ||class="entry q3 g1"| 63899₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (57576, 40711, 61415, 40726, 57839, 63887) ||class="entry q2 g1"| 57576₇ ||class="entry q3 g1"| 40711₉ ||class="entry q3 g1"| 61415₁₃ ||class="entry q2 g1"| 40726₉ ||class="entry q3 g1"| 57839₁₁ ||class="entry q3 g1"| 63887₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (60936, 37111, 61175, 37366, 60959, 63647) ||class="entry q2 g1"| 60936₇ ||class="entry q3 g1"| 37111₉ ||class="entry q3 g1"| 61175₁₃ ||class="entry q2 g1"| 37366₉ ||class="entry q3 g1"| 60959₁₁ ||class="entry q3 g1"| 63647₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (62112, 36175, 64943, 36190, 62375, 60359) ||class="entry q2 g1"| 62112₇ ||class="entry q3 g1"| 36175₉ ||class="entry q3 g1"| 64943₁₃ ||class="entry q2 g1"| 36190₉ ||class="entry q3 g1"| 62375₁₁ ||class="entry q3 g1"| 60359₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (62656, 35631, 64463, 35646, 62919, 60839) ||class="entry q2 g1"| 62656₇ ||class="entry q3 g1"| 35631₉ ||class="entry q3 g1"| 64463₁₃ ||class="entry q2 g1"| 35646₉ ||class="entry q3 g1"| 62919₁₁ ||class="entry q3 g1"| 60839₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (64032, 34015, 64223, 34270, 64055, 60599) ||class="entry q2 g1"| 64032₇ ||class="entry q3 g1"| 34015₉ ||class="entry q3 g1"| 64223₁₃ ||class="entry q2 g1"| 34270₉ ||class="entry q3 g1"| 64055₁₁ ||class="entry q3 g1"| 60599₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (64576, 33471, 64703, 33726, 64599, 60119) ||class="entry q2 g1"| 64576₇ ||class="entry q3 g1"| 33471₉ ||class="entry q3 g1"| 64703₁₃ ||class="entry q2 g1"| 33726₉ ||class="entry q3 g1"| 64599₁₁ ||class="entry q3 g1"| 60119₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33454, 62135, 58663, 64848, 35935, 62287) ||class="entry q2 g1"| 33454₇ ||class="entry q3 g1"| 62135₁₁ ||class="entry q3 g1"| 58663₉ ||class="entry q2 g1"| 64848₉ ||class="entry q3 g1"| 35935₉ ||class="entry q3 g1"| 62287₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33466, 52895, 55579, 64836, 45175, 53107) ||class="entry q2 g1"| 33466₇ ||class="entry q3 g1"| 52895₁₁ ||class="entry q3 g1"| 55579₉ ||class="entry q2 g1"| 64836₉ ||class="entry q3 g1"| 45175₉ ||class="entry q3 g1"| 53107₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33998, 62679, 58183, 64304, 35391, 62767) ||class="entry q2 g1"| 33998₇ ||class="entry q3 g1"| 62679₁₁ ||class="entry q3 g1"| 58183₉ ||class="entry q2 g1"| 64304₉ ||class="entry q3 g1"| 35391₉ ||class="entry q3 g1"| 62767₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (34012, 44703, 47389, 64290, 53367, 44917) ||class="entry q2 g1"| 34012₇ ||class="entry q3 g1"| 44703₁₁ ||class="entry q3 g1"| 47389₉ ||class="entry q2 g1"| 64290₉ ||class="entry q3 g1"| 53367₉ ||class="entry q3 g1"| 44917₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35374, 64295, 57943, 62928, 34255, 62527) ||class="entry q2 g1"| 35374₇ ||class="entry q3 g1"| 64295₁₁ ||class="entry q3 g1"| 57943₉ ||class="entry q2 g1"| 62928₉ ||class="entry q3 g1"| 34255₉ ||class="entry q3 g1"| 62527₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35626, 51695, 53659, 62676, 46855, 51187) ||class="entry q2 g1"| 35626₇ ||class="entry q3 g1"| 51695₁₁ ||class="entry q3 g1"| 53659₉ ||class="entry q2 g1"| 62676₉ ||class="entry q3 g1"| 46855₉ ||class="entry q3 g1"| 51187₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35918, 64839, 58423, 62384, 33711, 62047) ||class="entry q2 g1"| 35918₇ ||class="entry q3 g1"| 64839₁₁ ||class="entry q3 g1"| 58423₉ ||class="entry q2 g1"| 62384₉ ||class="entry q3 g1"| 33711₉ ||class="entry q3 g1"| 62047₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (36172, 43503, 45469, 62130, 55047, 42997) ||class="entry q2 g1"| 36172₇ ||class="entry q3 g1"| 43503₁₁ ||class="entry q3 g1"| 45469₉ ||class="entry q2 g1"| 62130₉ ||class="entry q3 g1"| 55047₉ ||class="entry q3 g1"| 42997₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (37106, 56535, 52051, 61196, 41535, 56635) ||class="entry q2 g1"| 37106₇ ||class="entry q3 g1"| 56535₁₁ ||class="entry q3 g1"| 52051₉ ||class="entry q2 g1"| 61196₉ ||class="entry q3 g1"| 41535₉ ||class="entry q3 g1"| 56635₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (37108, 47799, 44341, 61194, 50271, 47965) ||class="entry q2 g1"| 37108₇ ||class="entry q3 g1"| 47799₁₁ ||class="entry q3 g1"| 44341₉ ||class="entry q2 g1"| 61194₉ ||class="entry q3 g1"| 50271₉ ||class="entry q3 g1"| 47965₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38534, 59039, 61711, 59768, 39031, 59239) ||class="entry q2 g1"| 38534₇ ||class="entry q3 g1"| 59039₁₁ ||class="entry q3 g1"| 61711₉ ||class="entry q2 g1"| 59768₉ ||class="entry q3 g1"| 39031₉ ||class="entry q3 g1"| 59239₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38546, 55991, 52531, 59756, 42079, 56155) ||class="entry q2 g1"| 38546₇ ||class="entry q3 g1"| 55991₁₁ ||class="entry q3 g1"| 52531₉ ||class="entry q2 g1"| 59756₉ ||class="entry q3 g1"| 42079₉ ||class="entry q3 g1"| 56155₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38548, 48343, 43861, 59754, 49727, 48445) ||class="entry q2 g1"| 38548₇ ||class="entry q3 g1"| 48343₁₁ ||class="entry q3 g1"| 43861₉ ||class="entry q2 g1"| 59754₉ ||class="entry q3 g1"| 49727₉ ||class="entry q3 g1"| 48445₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39014, 59759, 61471, 59288, 38791, 58999) ||class="entry q2 g1"| 39014₇ ||class="entry q3 g1"| 59759₁₁ ||class="entry q3 g1"| 61471₉ ||class="entry q2 g1"| 59288₉ ||class="entry q3 g1"| 38791₉ ||class="entry q3 g1"| 58999₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39266, 56231, 50131, 59036, 42319, 54715) ||class="entry q2 g1"| 39266₇ ||class="entry q3 g1"| 56231₁₁ ||class="entry q3 g1"| 50131₉ ||class="entry q2 g1"| 59036₉ ||class="entry q3 g1"| 42319₉ ||class="entry q3 g1"| 54715₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39268, 48583, 42421, 59034, 49967, 46045) ||class="entry q2 g1"| 39268₇ ||class="entry q3 g1"| 48583₁₁ ||class="entry q3 g1"| 42421₉ ||class="entry q2 g1"| 59034₉ ||class="entry q3 g1"| 49967₉ ||class="entry q3 g1"| 46045₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (40706, 56775, 50611, 57596, 41775, 54235) ||class="entry q2 g1"| 40706₇ ||class="entry q3 g1"| 56775₁₁ ||class="entry q3 g1"| 50611₉ ||class="entry q2 g1"| 57596₉ ||class="entry q3 g1"| 41775₉ ||class="entry q3 g1"| 54235₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (40708, 48039, 41941, 57594, 50511, 46525) ||class="entry q2 g1"| 40708₇ ||class="entry q3 g1"| 48039₁₁ ||class="entry q3 g1"| 41941₉ ||class="entry q2 g1"| 57594₉ ||class="entry q3 g1"| 50511₉ ||class="entry q3 g1"| 46525₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (41530, 61211, 51799, 56772, 37363, 56383) ||class="entry q2 g1"| 41530₇ ||class="entry q3 g1"| 61211₁₁ ||class="entry q3 g1"| 51799₉ ||class="entry q2 g1"| 56772₉ ||class="entry q3 g1"| 37363₉ ||class="entry q3 g1"| 56383₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (41770, 57851, 50599, 56532, 40723, 54223) ||class="entry q2 g1"| 41770₇ ||class="entry q3 g1"| 57851₁₁ ||class="entry q3 g1"| 50599₉ ||class="entry q2 g1"| 56532₉ ||class="entry q3 g1"| 40723₉ ||class="entry q3 g1"| 54223₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42074, 59771, 52279, 56228, 38803, 55903) ||class="entry q2 g1"| 42074₇ ||class="entry q3 g1"| 59771₁₁ ||class="entry q3 g1"| 52279₉ ||class="entry q2 g1"| 56228₉ ||class="entry q3 g1"| 38803₉ ||class="entry q3 g1"| 55903₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42314, 59291, 50119, 55988, 39283, 54703) ||class="entry q2 g1"| 42314₇ ||class="entry q3 g1"| 59291₁₁ ||class="entry q3 g1"| 50119₉ ||class="entry q2 g1"| 55988₉ ||class="entry q3 g1"| 39283₉ ||class="entry q3 g1"| 54703₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42328, 48595, 39325, 55974, 49979, 36853) ||class="entry q2 g1"| 42328₇ ||class="entry q3 g1"| 48595₁₁ ||class="entry q3 g1"| 39325₉ ||class="entry q2 g1"| 55974₉ ||class="entry q3 g1"| 49979₉ ||class="entry q3 g1"| 36853₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (45170, 64851, 55327, 53132, 33723, 52855) ||class="entry q2 g1"| 45170₇ ||class="entry q3 g1"| 64851₁₁ ||class="entry q3 g1"| 55327₉ ||class="entry q2 g1"| 53132₉ ||class="entry q3 g1"| 33723₉ ||class="entry q3 g1"| 52855₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (45424, 43515, 36277, 52878, 55059, 39901) ||class="entry q2 g1"| 45424₇ ||class="entry q3 g1"| 43515₁₁ ||class="entry q3 g1"| 36277₉ ||class="entry q2 g1"| 52878₉ ||class="entry q3 g1"| 55059₉ ||class="entry q3 g1"| 39901₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (46850, 62931, 53647, 51452, 35643, 51175) ||class="entry q2 g1"| 46850₇ ||class="entry q3 g1"| 62931₁₁ ||class="entry q3 g1"| 53647₉ ||class="entry q2 g1"| 51452₉ ||class="entry q3 g1"| 35643₉ ||class="entry q3 g1"| 51175₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (46864, 44955, 35797, 51438, 53619, 40381) ||class="entry q2 g1"| 46864₇ ||class="entry q3 g1"| 44955₁₁ ||class="entry q3 g1"| 35797₉ ||class="entry q2 g1"| 51438₉ ||class="entry q3 g1"| 53619₉ ||class="entry q3 g1"| 40381₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (49724, 59773, 43607, 48578, 38805, 48191) ||class="entry q2 g1"| 49724₇ ||class="entry q3 g1"| 59773₁₁ ||class="entry q3 g1"| 43607₉ ||class="entry q2 g1"| 48578₉ ||class="entry q3 g1"| 38805₉ ||class="entry q3 g1"| 48191₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (49964, 59293, 42407, 48338, 39285, 46031) ||class="entry q2 g1"| 49964₇ ||class="entry q3 g1"| 59293₁₁ ||class="entry q3 g1"| 42407₉ ||class="entry q2 g1"| 48338₉ ||class="entry q3 g1"| 39285₉ ||class="entry q3 g1"| 46031₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (49976, 56245, 39323, 48326, 42333, 36851) ||class="entry q2 g1"| 49976₇ ||class="entry q3 g1"| 56245₁₁ ||class="entry q3 g1"| 39323₉ ||class="entry q2 g1"| 48326₉ ||class="entry q3 g1"| 42333₉ ||class="entry q3 g1"| 36851₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (50268, 61213, 44087, 48034, 37365, 47711) ||class="entry q2 g1"| 50268₇ ||class="entry q3 g1"| 61213₁₁ ||class="entry q3 g1"| 44087₉ ||class="entry q2 g1"| 48034₉ ||class="entry q3 g1"| 37365₉ ||class="entry q3 g1"| 47711₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (50508, 57853, 41927, 47794, 40725, 46511) ||class="entry q2 g1"| 50508₇ ||class="entry q3 g1"| 57853₁₁ ||class="entry q3 g1"| 41927₉ ||class="entry q2 g1"| 47794₉ ||class="entry q3 g1"| 40725₉ ||class="entry q3 g1"| 46511₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (53364, 64309, 47135, 44938, 34269, 44663) ||class="entry q2 g1"| 53364₇ ||class="entry q3 g1"| 64309₁₁ ||class="entry q3 g1"| 47135₉ ||class="entry q2 g1"| 44938₉ ||class="entry q3 g1"| 34269₉ ||class="entry q3 g1"| 44663₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (53616, 51709, 35795, 44686, 46869, 40379) ||class="entry q2 g1"| 53616₇ ||class="entry q3 g1"| 51709₁₁ ||class="entry q3 g1"| 35795₉ ||class="entry q2 g1"| 44686₉ ||class="entry q3 g1"| 46869₉ ||class="entry q3 g1"| 40379₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (55044, 62389, 45455, 43258, 36189, 42983) ||class="entry q2 g1"| 55044₇ ||class="entry q3 g1"| 62389₁₁ ||class="entry q3 g1"| 45455₉ ||class="entry q2 g1"| 43258₉ ||class="entry q3 g1"| 36189₉ ||class="entry q3 g1"| 42983₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (55056, 53149, 36275, 43246, 45429, 39899) ||class="entry q2 g1"| 55056₇ ||class="entry q3 g1"| 53149₁₁ ||class="entry q3 g1"| 36275₉ ||class="entry q2 g1"| 43246₉ ||class="entry q3 g1"| 45429₉ ||class="entry q3 g1"| 39899₁₁ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (33468, 43263, 49021, 64834, 54807, 43285) ||class="entry q2 g1"| 33468₇ ||class="entry q3 g1"| 43263₁₁ ||class="entry q3 g1"| 49021₁₃ ||class="entry q2 g1"| 64834₉ ||class="entry q3 g1"| 54807₉ ||class="entry q3 g1"| 43285₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (34010, 51455, 57211, 64292, 46615, 51475) ||class="entry q2 g1"| 34010₇ ||class="entry q3 g1"| 51455₁₁ ||class="entry q3 g1"| 57211₁₃ ||class="entry q2 g1"| 64292₉ ||class="entry q3 g1"| 46615₉ ||class="entry q3 g1"| 51475₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (35628, 44943, 47101, 62674, 53607, 41365) ||class="entry q2 g1"| 35628₇ ||class="entry q3 g1"| 44943₁₁ ||class="entry q3 g1"| 47101₁₃ ||class="entry q2 g1"| 62674₉ ||class="entry q3 g1"| 53607₉ ||class="entry q3 g1"| 41365₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (36170, 53135, 55291, 62132, 45415, 49555) ||class="entry q2 g1"| 36170₇ ||class="entry q3 g1"| 53135₁₁ ||class="entry q3 g1"| 55291₁₃ ||class="entry q2 g1"| 62132₉ ||class="entry q3 g1"| 45415₉ ||class="entry q3 g1"| 49555₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (37094, 57599, 63343, 61208, 40471, 57607) ||class="entry q2 g1"| 37094₇ ||class="entry q3 g1"| 57599₁₁ ||class="entry q3 g1"| 63343₁₃ ||class="entry q2 g1"| 61208₉ ||class="entry q3 g1"| 40471₉ ||class="entry q3 g1"| 57607₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (40454, 61199, 63103, 57848, 37351, 57367) ||class="entry q2 g1"| 40454₇ ||class="entry q3 g1"| 61199₁₁ ||class="entry q3 g1"| 63103₁₃ ||class="entry q2 g1"| 57848₉ ||class="entry q3 g1"| 37351₉ ||class="entry q3 g1"| 57367₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (41784, 48051, 40957, 56518, 50523, 35221) ||class="entry q2 g1"| 41784₇ ||class="entry q3 g1"| 48051₁₁ ||class="entry q3 g1"| 40957₁₃ ||class="entry q2 g1"| 56518₉ ||class="entry q3 g1"| 50523₉ ||class="entry q3 g1"| 35221₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (45410, 62387, 55279, 52892, 36187, 49543) ||class="entry q2 g1"| 45410₇ ||class="entry q3 g1"| 62387₁₁ ||class="entry q3 g1"| 55279₁₃ ||class="entry q2 g1"| 52892₉ ||class="entry q3 g1"| 36187₉ ||class="entry q3 g1"| 49543₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (46610, 64307, 56959, 51692, 34267, 51223) ||class="entry q2 g1"| 46610₇ ||class="entry q3 g1"| 64307₁₁ ||class="entry q3 g1"| 56959₁₃ ||class="entry q2 g1"| 51692₉ ||class="entry q3 g1"| 34267₉ ||class="entry q3 g1"| 51223₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (50520, 56789, 40955, 47782, 41789, 35219) ||class="entry q2 g1"| 50520₇ ||class="entry q3 g1"| 56789₁₁ ||class="entry q3 g1"| 40955₁₃ ||class="entry q2 g1"| 47782₉ ||class="entry q3 g1"| 41789₉ ||class="entry q3 g1"| 35219₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (53604, 62933, 47087, 44698, 35645, 41351) ||class="entry q2 g1"| 53604₇ ||class="entry q3 g1"| 62933₁₁ ||class="entry q3 g1"| 47087₁₃ ||class="entry q2 g1"| 44698₉ ||class="entry q3 g1"| 35645₉ ||class="entry q3 g1"| 41351₇ |- |class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 7) ||class="c"| (54804, 64853, 48767, 43498, 33725, 43031) ||class="entry q2 g1"| 54804₇ ||class="entry q3 g1"| 64853₁₁ ||class="entry q3 g1"| 48767₁₃ ||class="entry q2 g1"| 43498₉ ||class="entry q3 g1"| 33725₉ ||class="entry q3 g1"| 43031₇ |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> adcbsvad6d32orpk4f387ja4qewgf1g 10 313741 2693777 2679235 2024-12-29T18:07:12Z Watchduck 137431 2693777 wikitext text/x-wiki {| class="wikitable" |style="border-right: 3px solid #a2a9b1;"| [[File:4-ary noble with quadrant 0; zhe 0.svg|200px]] | [[File:4-ary noble with quadrant 1; zhe 854.svg|200px]] | [[File:4-ary noble with quadrant 1; zhe 1334.svg|200px]] | [[File:4-ary noble with quadrant 1; zhe 4382.svg|200px]] |-style="border-top: 3px solid #a2a9b1;"| |style="border-right: 3px solid #a2a9b1;"| [[File:4-ary noble with quadrant 1; zhe 6014.svg|200px]] | [[File:4-ary noble with quadrant 0; zhe 5160.svg|200px]] | [[File:4-ary noble with quadrant 0; zhe 4680.svg|200px]] | [[File:4-ary noble with quadrant 0; zhe 1632.svg|200px]] |}<noinclude> [[Category:Mentors of Boolean functions; chains]] </noinclude> 0bbxqg8viassicqcgs8175wire1a0no 2 317234 2693768 2693681 2024-12-29T16:41:49Z Atcovi 276019 +ch. 14 2693768 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]] * [[User:Atcovi/Health Psychology/Chapter 13: Cancer]] * [[User:Atcovi/Health Psychology/Chapter 14: Cardiovascular Disease]] [[Category:Psychology]] [[Category:Atcovi's Work]] 0w5e1hztee3idky1oirzzvhn38xtovq 0 317254 2693761 2693616 2024-12-29T13:44:41Z Watchduck 137431 2693761 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 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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/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 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"| 10 |class="intpart"| <span class="sortkey">[16, 4, 32, 6]</span><span class="formula"><span class="count">4</span>⋅<span class="size">16</span> + <span class="count">6</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunky chain|chunky chain]]</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 class="size">12</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">3</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great principality|great principality]]</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 class="size">12</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">3</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great dominion|great dominion]]</span> |- |class="number-of-blocks"| 12 |class="intpart"| <span class="sortkey">[16, 8, 32, 4]</span><span class="formula"><span 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 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></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunky burden|chunky burden]]</span> |- |class="number-of-blocks"| 14 |class="intpart"| <span class="sortkey">[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">6</span> + <span class="count">2</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super clan|super clan]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[1, 8, 2, 6, 4, 1, 232, 1]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">232</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry neg|noble symmetry neg]]</span> |- |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 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"| 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/chunky seminar|chunky seminar]]</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 main">[[Boolf prop/3-ary/ultra family|ultra family]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[4, 4, 12, 12, 24, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">4</span> + <span class="count">12</span>⋅<span class="size">12</span> + <span class="count">4</span>⋅<span class="size">24</span></span> |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"| 40 |class="intpart"| <span class="sortkey">[1, 4, 2, 6, 5, 12, 10, 18]</span><span class="formula"><span class="count">4</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">5</span> + <span class="count">18</span>⋅<span class="size">10</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain|chain]]</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"| 96 |class="intpart"| <span class="sortkey">[2, 64, 4, 32]</span><span class="formula"><span class="count">64</span>⋅<span class="size">2</span> + <span class="count">32</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/seminar|seminar]]</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| ]] t5on7bz8dru9bw1lul1cyx3ej5jve0l 2693793 2693761 2024-12-29T19:57:17Z Watchduck 137431 2693793 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"| 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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">[2, 2, 56, 2, 140, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">140</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/changes|changes]]</span> |- |class="number-of-blocks"| 5 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|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 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 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|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 class="size">12</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span class="size">24</span> + <span class="count">3</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/great principality|great principality]]</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 class="size">12</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">2</span>⋅<span 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|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 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 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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"| 16 |class="intpart"| <span class="sortkey">[16, 16]</span><span 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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 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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 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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"| 96 |class="intpart"| <span class="sortkey">[2, 64, 4, 32]</span><span class="formula"><span class="count">64</span>⋅<span class="size">2</span> + <span class="count">32</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/seminar|seminar]]</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| ]] dhx0jlv8951qo12osibqmt8nrkst08b 2693794 2693793 2024-12-29T20:56:51Z Watchduck 137431 2693794 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">[4, 1, 12, 1, 60, 1, 180, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">60</span> + <span class="count">1</span>⋅<span class="size">180</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain length|chain length]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[16, 1, 80, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/anchor|anchor]]</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">[2, 2, 56, 2, 140, 1]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">56</span> + <span class="count">1</span>⋅<span class="size">140</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/changes|changes]]</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> |- |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/sub-prefect weight|sub-prefect 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/zhegalkin deviation weight|zhegalkin deviation 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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"| 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/chunky seminar|chunky seminar]]</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 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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 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|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"| 40 |class="intpart"| <span class="sortkey">[1, 4, 2, 6, 5, 12, 10, 18]</span><span class="formula"><span class="count">4</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">5</span> + <span class="count">18</span>⋅<span class="size">10</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain|chain]]</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"| 96 |class="intpart"| <span class="sortkey">[2, 64, 4, 32]</span><span class="formula"><span class="count">64</span>⋅<span class="size">2</span> + <span class="count">32</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/seminar|seminar]]</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| ]] 1kwmm4sr6efpuxwzdc1kx9qkhhcqpxb 2693799 2693794 2024-12-29T22:03:11Z Watchduck 137431 2693799 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">[4, 1, 12, 1, 60, 1, 180, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">60</span> + <span class="count">1</span>⋅<span class="size">180</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain length|chain length]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[16, 1, 80, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/anchor|anchor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[16, 1, 80, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reduced chain quadrants|reduced chain quadrants]]</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 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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"| 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/chunky seminar|chunky seminar]]</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 main">[[Boolf prop/3-ary/ultra family|ultra family]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[4, 4, 12, 12, 24, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">4</span> + <span class="count">12</span>⋅<span class="size">12</span> + <span class="count">4</span>⋅<span class="size">24</span></span> |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"| 40 |class="intpart"| <span class="sortkey">[1, 4, 2, 6, 5, 12, 10, 18]</span><span class="formula"><span class="count">4</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">5</span> + <span class="count">18</span>⋅<span class="size">10</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain|chain]]</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"| 96 |class="intpart"| <span class="sortkey">[2, 64, 4, 32]</span><span class="formula"><span class="count">64</span>⋅<span class="size">2</span> + <span class="count">32</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/seminar|seminar]]</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| ]] 0x0zxsldegerc74qlqb4z7sujqqvs4v 2693832 2693799 2024-12-30T00:01:42Z Watchduck 137431 2693832 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">[4, 1, 12, 1, 60, 1, 180, 1]</span><span class="formula"><span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">60</span> + <span class="count">1</span>⋅<span class="size">180</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain length|chain length]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[16, 1, 80, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/anchor|anchor]]</span> |- |class="number-of-blocks"| 4 |class="intpart"| <span class="sortkey">[16, 1, 80, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/reduced chain quadrants|reduced chain quadrants]]</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> + 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|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 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|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 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|class="number-of-blocks"| 7 |class="intpart"| <span class="sortkey">[16, 2, 32, 3, 64, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">32</span> + <span class="count">2</span>⋅<span class="size">64</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/village|village]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[2, 4, 10, 3, 218, 1]</span><span class="formula"><span class="count">4</span>⋅<span class="size">2</span> + <span class="count">3</span>⋅<span class="size">10</span> + <span class="count">1</span>⋅<span class="size">218</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/atomvals|atomvals]]</span> |- |class="number-of-blocks"| 8 |class="intpart"| <span class="sortkey">[4, 1, 12, 1, 20, 3, 60, 3]</span><span class="formula"><span class="count">1</span>⋅<span class="size">4</span> + <span 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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/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 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 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weight]]</span> |- |class="number-of-blocks"| 10 |class="intpart"| <span class="sortkey">[16, 4, 32, 6]</span><span class="formula"><span class="count">4</span>⋅<span class="size">16</span> + <span class="count">6</span>⋅<span class="size">32</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunky chain|chunky chain]]</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"| 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dominion|great dominion]]</span> |- |class="number-of-blocks"| 12 |class="intpart"| <span class="sortkey">[16, 8, 32, 4]</span><span class="formula"><span 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 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></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chunky burden|chunky burden]]</span> |- |class="number-of-blocks"| 14 |class="intpart"| <span class="sortkey">[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]</span><span class="formula"><span class="count">2</span>⋅<span class="size">2</span> + <span class="count">2</span>⋅<span class="size">6</span> + <span class="count">2</span>⋅<span class="size">8</span> + <span class="count">2</span>⋅<span class="size">16</span> + <span class="count">4</span>⋅<span class="size">24</span> + <span class="count">2</span>⋅<span class="size">48</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/super clan|super clan]]</span> |- |class="number-of-blocks"| 16 |class="intpart"| <span class="sortkey">[1, 8, 2, 6, 4, 1, 232, 1]</span><span class="formula"><span class="count">8</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">232</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/noble symmetry neg|noble symmetry neg]]</span> |- |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 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"| 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/chunky seminar|chunky seminar]]</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 main">[[Boolf prop/3-ary/ultra family|ultra family]]</span> |- |class="number-of-blocks"| 20 |class="intpart"| <span class="sortkey">[4, 4, 12, 12, 24, 4]</span><span class="formula"><span class="count">4</span>⋅<span class="size">4</span> + <span class="count">12</span>⋅<span class="size">12</span> + <span class="count">4</span>⋅<span class="size">24</span></span> |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"| 40 |class="intpart"| <span class="sortkey">[1, 4, 2, 6, 5, 12, 10, 18]</span><span class="formula"><span class="count">4</span>⋅<span class="size">1</span> + <span class="count">6</span>⋅<span class="size">2</span> + <span class="count">12</span>⋅<span class="size">5</span> + <span class="count">18</span>⋅<span class="size">10</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/chain|chain]]</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"| 96 |class="intpart"| <span class="sortkey">[2, 64, 4, 32]</span><span class="formula"><span class="count">64</span>⋅<span class="size">2</span> + <span class="count">32</span>⋅<span class="size">4</span></span> |class="props"| <span class="prop main">[[Boolf prop/3-ary/seminar|seminar]]</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| ]] 4ieyiukvjwmpoihz3gfhgnm5tfr04yi 0 317256 2693755 2693060 2024-12-29T12:20:33Z Watchduck 137431 2693755 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> [[File:Set of 3-ary Boolean functions 87183322370842425795587566660030343945541236792690890543826997033971843334915.svg|thumb|500px|The values 0, 1, 8, 9... form the block of [[Boolf prop/3-ary/great patron|great patron]] 0.]] <source lang="python"> val = (boolf.twin(3) ^ boolf.mentor(3)).zhe </source> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">64</span></span> Integer partition: &nbsp; <span class="count">64</span>⋅<span class="size">4</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| twin mentor !class="block"| block |- |class="size"| 4 |class="prop"| 0 |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 4 |class="prop"| 232 |class="block"| <span class="block-list">[1, 41, 73, 97]</span>[[File:Set_of_3-ary_Boolean_functions_158456334473261643125401583618.svg|420px]] |- |class="size"| 4 |class="prop"| 191 |class="block"| <span class="block-list">[2, 42, 74, 98]</span>[[File:Set_of_3-ary_Boolean_functions_316912668946523286250803167236.svg|420px]] |- |class="size"| 4 |class="prop"| 87 |class="block"| <span class="block-list">[3, 43, 75, 99]</span>[[File:Set_of_3-ary_Boolean_functions_633825337893046572501606334472.svg|420px]] |- |class="size"| 4 |class="prop"| 223 |class="block"| <span class="block-list">[4, 44, 76, 100]</span>[[File:Set_of_3-ary_Boolean_functions_1267650675786093145003212668944.svg|420px]] |- |class="size"| 4 |class="prop"| 55 |class="block"| <span class="block-list">[5, 45, 77, 101]</span>[[File:Set_of_3-ary_Boolean_functions_2535301351572186290006425337888.svg|420px]] |- |class="size"| 4 |class="prop"| 96 |class="block"| <span class="block-list">[6, 46, 78, 102]</span>[[File:Set_of_3-ary_Boolean_functions_5070602703144372580012850675776.svg|420px]] |- |class="size"| 4 |class="prop"| 136 |class="block"| <span class="block-list">[7, 47, 79, 103]</span>[[File:Set_of_3-ary_Boolean_functions_10141205406288745160025701351552.svg|420px]] |- |class="size"| 4 |class="prop"| 150 |class="block"| <span class="block-list">[8, 32, 64, 104]</span>[[File:Set_of_3-ary_Boolean_functions_20282409603670117168025255805184.svg|420px]] |- |class="size"| 4 |class="prop"| 126 |class="block"| <span class="block-list">[9, 33, 65, 105]</span>[[File:Set_of_3-ary_Boolean_functions_40564819207340234336050511610368.svg|420px]] |- |class="size"| 4 |class="prop"| 41 |class="block"| <span class="block-list">[10, 34, 66, 106]</span>[[File:Set_of_3-ary_Boolean_functions_81129638414680468672101023220736.svg|420px]] |- |class="size"| 4 |class="prop"| 193 |class="block"| <span class="block-list">[11, 35, 67, 107]</span>[[File:Set_of_3-ary_Boolean_functions_162259276829360937344202046441472.svg|420px]] |- |class="size"| 4 |class="prop"| 73 |class="block"| <span class="block-list">[12, 36, 68, 108]</span>[[File:Set_of_3-ary_Boolean_functions_324518553658721874688404092882944.svg|420px]] |- |class="size"| 4 |class="prop"| 161 |class="block"| <span class="block-list">[13, 37, 69, 109]</span>[[File:Set_of_3-ary_Boolean_functions_649037107317443749376808185765888.svg|420px]] |- |class="size"| 4 |class="prop"| 246 |class="block"| <span class="block-list">[14, 38, 70, 110]</span>[[File:Set_of_3-ary_Boolean_functions_1298074214634887498753616371531776.svg|420px]] |- |class="size"| 4 |class="prop"| 30 |class="block"| <span class="block-list">[15, 39, 71, 111]</span>[[File:Set_of_3-ary_Boolean_functions_2596148429269774997507232743063552.svg|420px]] |- |class="size"| 4 |class="prop"| 247 |class="block"| <span class="block-list">[16, 56, 88, 112]</span>[[File:Set_of_3-ary_Boolean_functions_5192297168019837521933159091994624.svg|420px]] |- |class="size"| 4 |class="prop"| 31 |class="block"| <span class="block-list">[17, 57, 89, 113]</span>[[File:Set_of_3-ary_Boolean_functions_10384594336039675043866318183989248.svg|420px]] |- |class="size"| 4 |class="prop"| 72 |class="block"| <span class="block-list">[18, 58, 90, 114]</span>[[File:Set_of_3-ary_Boolean_functions_20769188672079350087732636367978496.svg|420px]] |- |class="size"| 4 |class="prop"| 160 |class="block"| <span class="block-list">[19, 59, 91, 115]</span>[[File:Set_of_3-ary_Boolean_functions_41538377344158700175465272735956992.svg|420px]] |- |class="size"| 4 |class="prop"| 40 |class="block"| <span class="block-list">[20, 60, 92, 116]</span>[[File:Set_of_3-ary_Boolean_functions_83076754688317400350930545471913984.svg|420px]] |- |class="size"| 4 |class="prop"| 192 |class="block"| <span class="block-list">[21, 61, 93, 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122]</span>[[File:Set_of_3-ary_Boolean_functions_5316911983144499194894812657794154496.svg|420px]] |- |class="size"| 4 |class="prop"| 54 |class="block"| <span class="block-list">[27, 51, 83, 123]</span>[[File:Set_of_3-ary_Boolean_functions_10633823966288998389789625315588308992.svg|420px]] |- |class="size"| 4 |class="prop"| 190 |class="block"| <span class="block-list">[28, 52, 84, 124]</span>[[File:Set_of_3-ary_Boolean_functions_21267647932577996779579250631176617984.svg|420px]] |- |class="size"| 4 |class="prop"| 86 |class="block"| <span class="block-list">[29, 53, 85, 125]</span>[[File:Set_of_3-ary_Boolean_functions_42535295865155993559158501262353235968.svg|420px]] |- |class="size"| 4 |class="prop"| 1 |class="block"| <span class="block-list">[30, 54, 86, 126]</span>[[File:Set_of_3-ary_Boolean_functions_85070591730311987118317002524706471936.svg|420px]] |- |class="size"| 4 |class="prop"| 233 |class="block"| <span class="block-list">[31, 55, 87, 127]</span>[[File:Set_of_3-ary_Boolean_functions_170141183460623974236634005049412943872.svg|420px]] |- |class="size"| 4 |class="prop"| 23 |class="block"| <span class="block-list">[128, 168, 200, 224]</span>[[File:Set_of_3-ary_Boolean_functions_26959948274088684427801709786033152441864002583575393903246944763904.svg|420px]] |- |class="size"| 4 |class="prop"| 255 |class="block"| <span class="block-list">[129, 169, 201, 225]</span>[[File:Set_of_3-ary_Boolean_functions_53919896548177368855603419572066304883728005167150787806493889527808.svg|420px]] |- |class="size"| 4 |class="prop"| 168 |class="block"| <span class="block-list">[130, 170, 202, 226]</span>[[File:Set_of_3-ary_Boolean_functions_107839793096354737711206839144132609767456010334301575612987779055616.svg|420px]] |- |class="size"| 4 |class="prop"| 64 |class="block"| <span class="block-list">[131, 171, 203, 227]</span>[[File:Set_of_3-ary_Boolean_functions_215679586192709475422413678288265219534912020668603151225975558111232.svg|420px]] |- |class="size"| 4 |class="prop"| 200 |class="block"| <span class="block-list">[132, 172, 204, 228]</span>[[File:Set_of_3-ary_Boolean_functions_431359172385418950844827356576530439069824041337206302451951116222464.svg|420px]] |- |class="size"| 4 |class="prop"| 32 |class="block"| <span class="block-list">[133, 173, 205, 229]</span>[[File:Set_of_3-ary_Boolean_functions_862718344770837901689654713153060878139648082674412604903902232444928.svg|420px]] |- |class="size"| 4 |class="prop"| 119 |class="block"| <span class="block-list">[134, 174, 206, 230]</span>[[File:Set_of_3-ary_Boolean_functions_1725436689541675803379309426306121756279296165348825209807804464889856.svg|420px]] |- |class="size"| 4 |class="prop"| 159 |class="block"| <span class="block-list">[135, 175, 207, 231]</span>[[File:Set_of_3-ary_Boolean_functions_3450873379083351606758618852612243512558592330697650419615608929779712.svg|420px]] |- |class="size"| 4 |class="prop"| 129 |class="block"| <span class="block-list">[136, 160, 192, 232]</span>[[File:Set_of_3-ary_Boolean_functions_6901746346796840889171604044765304430724280570065829093129176852987904.svg|420px]] |- |class="size"| 4 |class="prop"| 105 |class="block"| <span class="block-list">[137, 161, 193, 233]</span>[[File:Set_of_3-ary_Boolean_functions_13803492693593681778343208089530608861448561140131658186258353705975808.svg|420px]] |- |class="size"| 4 |class="prop"| 62 |class="block"| <span class="block-list">[138, 162, 194, 234]</span>[[File:Set_of_3-ary_Boolean_functions_27606985387187363556686416179061217722897122280263316372516707411951616.svg|420px]] |- |class="size"| 4 |class="prop"| 214 |class="block"| <span class="block-list">[139, 163, 195, 235]</span>[[File:Set_of_3-ary_Boolean_functions_55213970774374727113372832358122435445794244560526632745033414823903232.svg|420px]] |- |class="size"| 4 |class="prop"| 94 |class="block"| <span class="block-list">[140, 164, 196, 236]</span>[[File:Set_of_3-ary_Boolean_functions_110427941548749454226745664716244870891588489121053265490066829647806464.svg|420px]] |- |class="size"| 4 |class="prop"| 182 |class="block"| <span class="block-list">[141, 165, 197, 237]</span>[[File:Set_of_3-ary_Boolean_functions_220855883097498908453491329432489741783176978242106530980133659295612928.svg|420px]] |- |class="size"| 4 |class="prop"| 225 |class="block"| <span class="block-list">[142, 166, 198, 238]</span>[[File:Set_of_3-ary_Boolean_functions_441711766194997816906982658864979483566353956484213061960267318591225856.svg|420px]] |- |class="size"| 4 |class="prop"| 9 |class="block"| <span class="block-list">[143, 167, 199, 239]</span>[[File:Set_of_3-ary_Boolean_functions_883423532389995633813965317729958967132707912968426123920534637182451712.svg|420px]] |- |class="size"| 4 |class="prop"| 224 |class="block"| <span class="block-list">[144, 184, 216, 240]</span>[[File:Set_of_3-ary_Boolean_functions_1766847170090676022660412852537468678429999273317197014843191772047212544.svg|420px]] |- |class="size"| 4 |class="prop"| 8 |class="block"| <span class="block-list">[145, 185, 217, 241]</span>[[File:Set_of_3-ary_Boolean_functions_3533694340181352045320825705074937356859998546634394029686383544094425088.svg|420px]] |- |class="size"| 4 |class="prop"| 95 |class="block"| <span class="block-list">[146, 186, 218, 242]</span>[[File:Set_of_3-ary_Boolean_functions_7067388680362704090641651410149874713719997093268788059372767088188850176.svg|420px]] |- |class="size"| 4 |class="prop"| 183 |class="block"| <span class="block-list">[147, 187, 219, 243]</span>[[File:Set_of_3-ary_Boolean_functions_14134777360725408181283302820299749427439994186537576118745534176377700352.svg|420px]] |- |class="size"| 4 |class="prop"| 63 |class="block"| <span class="block-list">[148, 188, 220, 244]</span>[[File:Set_of_3-ary_Boolean_functions_28269554721450816362566605640599498854879988373075152237491068352755400704.svg|420px]] |- |class="size"| 4 |class="prop"| 215 |class="block"| <span class="block-list">[149, 189, 221, 245]</span>[[File:Set_of_3-ary_Boolean_functions_56539109442901632725133211281198997709759976746150304474982136705510801408.svg|420px]] |- |class="size"| 4 |class="prop"| 128 |class="block"| <span class="block-list">[150, 190, 222, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078218885803265450266422562397995419519953492300608949964273411021602816.svg|420px]] |- |class="size"| 4 |class="prop"| 104 |class="block"| <span class="block-list">[151, 191, 223, 247]</span>[[File:Set_of_3-ary_Boolean_functions_226156437771606530900532845124795990839039906984601217899928546822043205632.svg|420px]] |- |class="size"| 4 |class="prop"| 118 |class="block"| <span class="block-list">[152, 176, 208, 248]</span>[[File:Set_of_3-ary_Boolean_functions_452312848583677764512750242677738991171946451439834175447313734237415276544.svg|420px]] |- |class="size"| 4 |class="prop"| 158 |class="block"| <span class="block-list">[153, 177, 209, 249]</span>[[File:Set_of_3-ary_Boolean_functions_904625697167355529025500485355477982343892902879668350894627468474830553088.svg|420px]] |- |class="size"| 4 |class="prop"| 201 |class="block"| <span class="block-list">[154, 178, 210, 250]</span>[[File:Set_of_3-ary_Boolean_functions_1809251394334711058051000970710955964687785805759336701789254936949661106176.svg|420px]] |- |class="size"| 4 |class="prop"| 33 |class="block"| <span class="block-list">[155, 179, 211, 251]</span>[[File:Set_of_3-ary_Boolean_functions_3618502788669422116102001941421911929375571611518673403578509873899322212352.svg|420px]] |- |class="size"| 4 |class="prop"| 169 |class="block"| <span class="block-list">[156, 180, 212, 252]</span>[[File:Set_of_3-ary_Boolean_functions_7237005577338844232204003882843823858751143223037346807157019747798644424704.svg|420px]] |- |class="size"| 4 |class="prop"| 65 |class="block"| <span class="block-list">[157, 181, 213, 253]</span>[[File:Set_of_3-ary_Boolean_functions_14474011154677688464408007765687647717502286446074693614314039495597288849408.svg|420px]] |- |class="size"| 4 |class="prop"| 22 |class="block"| <span class="block-list">[158, 182, 214, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948022309355376928816015531375295435004572892149387228628078991194577698816.svg|420px]] |- |class="size"| 4 |class="prop"| 254 |class="block"| <span class="block-list">[159, 183, 215, 255]</span>[[File:Set_of_3-ary_Boolean_functions_57896044618710753857632031062750590870009145784298774457256157982389155397632.svg|420px]] |} [[Category:Boolf prop/3-ary|nameless 2]] ibrwoja2s2mbz5dt4eknrfuodwqja2d 0 317611 2693754 2024-12-29T12:01:25Z Watchduck 137431 New resource with "<templatestyles src="Boolf prop/blocks.css" /> [[File:Set of 3-ary Boolean functions 1569276182135508504381241476088783068524472619773441605632.svg|thumb|500px|{{Boolf prop 3-ary|chain}} [62, 169, 86, 87, 190] of length 5]] [[File:Set of 3-ary Boolean functions 5070602703144372580012850675776.svg|thumb|500px|{{Boolf prop 3-ary|twin mentor}} 96]] File:Set of 3-ary Boolean functions 28608766866473769627983418348657563907728747872949673495630586973941286305020.svg|thumb|..." 2693754 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> [[File:Set of 3-ary Boolean functions 1569276182135508504381241476088783068524472619773441605632.svg|thumb|500px|{{Boolf prop 3-ary|chain}} [62, 169, 86, 87, 190] of length 5]] [[File:Set of 3-ary Boolean functions 5070602703144372580012850675776.svg|thumb|500px|{{Boolf prop 3-ary|twin mentor}} 96]] [[File:Set of 3-ary Boolean functions 28608766866473769627983418348657563907728747872949673495630586973941286305020.svg|thumb|500px|{{Boolf prop 3-ary binary|great quaestor sword dominion }} 2]] <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">4</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">1</span>⋅<span class="size">60</span> + <span class="count">1</span>⋅<span class="size">180</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain length !class="block"| block |- |class="size"| 4 |class="prop"| 1 |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 60 |class="prop"| 5 |class="block"| <span class="block-list small">[1, 8, 9, 22, 23, 30, 31, 32, 33, 41, 54, 55, 62, 63, 64, 65, 73, 86, 87, 94, 95, 97, 104, 105, 118, 119, 126, 127, 128, 129, 136, 137, 150, 151, 158, 159, 160, 161, 168, 169, 182, 183, 190, 191, 192, 193, 200, 201, 214, 215, 222, 223, 224, 225, 232, 233, 246, 247, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_87183322370842425795587566660030343945541236792611662376590366212409142543106.svg|420px]] |- |class="size"| 180 |class="prop"| 10 |class="block"| [[File:Set_of_3-ary_Boolean_functions_28608766866473769627983418348657563907728644021935710395735775730746595736764.svg|420px]] |- |class="size"| 12 |class="prop"| 2 |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |} [[Category:Boolf prop/3-ary|chain length]] cckdoflw0xyg2pib47b61fcyqgopkyg 0 317612 2693757 2024-12-29T12:38:04Z Watchduck 137431 New resource with "<templatestyles src="Boolf prop/blocks.css" /> <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">10</span></span> Integer partition: &nbsp; <span class="count">4</span>⋅<span class="size">16</span> + <span class="count">6</span>⋅<span class="size">32</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky chain !class="block"| block |- |class="siz..." 2693757 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">10</span></span> Integer partition: &nbsp; <span class="count">4</span>⋅<span class="size">16</span> + <span class="count">6</span>⋅<span class="size">32</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky chain !class="block"| block |- |class="size"| 16 |class="prop"| 0 |class="block"| <span class="block-list">[0, 1, 22, 23, 104, 105, 126, 127, 128, 129, 150, 151, 232, 233, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844087633226186939369601060787799465383022669001262447603868028627564101635.svg|420px]] |- |class="size"| 32 |class="prop"| 2 |class="block"| <span class="block-list small">[2, 3, 20, 21, 42, 43, 60, 61, 66, 67, 84, 85, 106, 107, 124, 125, 130, 131, 148, 149, 170, 171, 188, 189, 194, 195, 212, 213, 234, 235, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_21711099552972694259875045268464461806683409121387009693514626718151614660620.svg|420px]] |- |class="size"| 32 |class="prop"| 4 |class="block"| <span class="block-list small">[4, 5, 18, 19, 36, 37, 50, 51, 76, 77, 90, 91, 108, 109, 122, 123, 132, 133, 146, 147, 164, 165, 178, 179, 204, 205, 218, 219, 236, 237, 250, 251]</span>[[File:Set_of_3-ary_Boolean_functions_5428085468087667240591646904279872958047222271303001218131541626195815432240.svg|420px]] |- |class="size"| 32 |class="prop"| 6 |class="block"| <span class="block-list small">[6, 7, 14, 15, 16, 17, 24, 25, 102, 103, 110, 111, 112, 113, 120, 121, 134, 135, 142, 143, 144, 145, 152, 153, 230, 231, 238, 239, 240, 241, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1363569398552478199278750422282454737804137663287534690226178186135048798400.svg|420px]] |- |class="size"| 16 |class="prop"| 8 |class="block"| <span class="block-list">[8, 9, 30, 31, 96, 97, 118, 119, 136, 137, 158, 159, 224, 225, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234717317289792731912505239812105433800834279578846437447474378012885760.svg|420px]] |- |class="size"| 32 |class="prop"| 10 |class="block"| <span class="block-list small">[10, 11, 28, 29, 34, 35, 52, 53, 74, 75, 92, 93, 98, 99, 116, 117, 138, 139, 156, 157, 162, 163, 180, 181, 202, 203, 220, 221, 226, 227, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_84808987683731736979549389857862882249710924980580170943516537973582597120.svg|420px]] |- |class="size"| 32 |class="prop"| 12 |class="block"| <span class="block-list small">[12, 13, 26, 27, 44, 45, 58, 59, 68, 69, 82, 83, 100, 101, 114, 115, 140, 141, 154, 155, 172, 173, 186, 187, 196, 197, 210, 211, 228, 229, 242, 243]</span>[[File:Set_of_3-ary_Boolean_functions_21203458859717450452813733559663024377301133068365161988570178364429578240.svg|420px]] |- |class="size"| 16 |class="prop"| 32 |class="block"| <span class="block-list">[32, 33, 54, 55, 72, 73, 94, 95, 160, 161, 182, 183, 200, 201, 222, 223]</span>[[File:Set_of_3-ary_Boolean_functions_20219964821195502573845983519444008947711945227919536382652721397760.svg|420px]] |- |class="size"| 32 |class="prop"| 38 |class="block"| <span class="block-list small">[38, 39, 46, 47, 48, 49, 56, 57, 70, 71, 78, 79, 80, 81, 88, 89, 166, 167, 174, 175, 176, 177, 184, 185, 198, 199, 206, 207, 208, 209, 216, 217]</span>[[File:Set_of_3-ary_Boolean_functions_317480740805612630213248498566966758923182560826153727120795238400.svg|420px]] |- |class="size"| 16 |class="prop"| 40 |class="block"| <span class="block-list">[40, 41, 62, 63, 64, 65, 86, 87, 168, 169, 190, 191, 192, 193, 214, 215]</span>[[File:Set_of_3-ary_Boolean_functions_78984242290550520156748855280404341698104021866145148313544949760.svg|420px]] |} [[Category:Boolf prop/3-ary|chunky chain]] j5tulwb67yvy886n7wyyndlfz9s7mw3 2693758 2693757 2024-12-29T12:57:43Z Watchduck 137431 2693758 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 5305717504095245911590468586269161187957798664794375308018027854244659200.svg | caption1 = chunky '''seminar''' 6 | image2 = Set of 3-ary Boolean functions 1358263681048382953367159953696185576616179864622740314918160158280804139200.svg | caption2 = chunky '''seminar''' 14 | image3 = Set of 3-ary Boolean functions 1363569398552478199278750422282454737804137663287534690226178186135048798400.svg | caption3 = chunky '''chain''' 6 | footer = Chunky chains of size 32 correspond to two {{Boolf prop 3-ary|chunky seminar}}s. }} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">10</span></span> Integer partition: &nbsp; <span class="count">4</span>⋅<span class="size">16</span> + <span class="count">6</span>⋅<span class="size">32</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chunky chain !class="block"| block |- |class="size"| 16 |class="prop"| 0 |class="block"| <span class="block-list">[0, 1, 22, 23, 104, 105, 126, 127, 128, 129, 150, 151, 232, 233, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844087633226186939369601060787799465383022669001262447603868028627564101635.svg|420px]] |- |class="size"| 32 |class="prop"| 2 |class="block"| <span class="block-list small">[2, 3, 20, 21, 42, 43, 60, 61, 66, 67, 84, 85, 106, 107, 124, 125, 130, 131, 148, 149, 170, 171, 188, 189, 194, 195, 212, 213, 234, 235, 252, 253]</span>[[File:Set_of_3-ary_Boolean_functions_21711099552972694259875045268464461806683409121387009693514626718151614660620.svg|420px]] |- |class="size"| 32 |class="prop"| 4 |class="block"| <span class="block-list small">[4, 5, 18, 19, 36, 37, 50, 51, 76, 77, 90, 91, 108, 109, 122, 123, 132, 133, 146, 147, 164, 165, 178, 179, 204, 205, 218, 219, 236, 237, 250, 251]</span>[[File:Set_of_3-ary_Boolean_functions_5428085468087667240591646904279872958047222271303001218131541626195815432240.svg|420px]] |- |class="size"| 32 |class="prop"| 6 |class="block"| <span class="block-list small">[6, 7, 14, 15, 16, 17, 24, 25, 102, 103, 110, 111, 112, 113, 120, 121, 134, 135, 142, 143, 144, 145, 152, 153, 230, 231, 238, 239, 240, 241, 248, 249]</span>[[File:Set_of_3-ary_Boolean_functions_1363569398552478199278750422282454737804137663287534690226178186135048798400.svg|420px]] |- |class="size"| 16 |class="prop"| 8 |class="block"| <span class="block-list">[8, 9, 30, 31, 96, 97, 118, 119, 136, 137, 158, 159, 224, 225, 246, 247]</span>[[File:Set_of_3-ary_Boolean_functions_339234717317289792731912505239812105433800834279578846437447474378012885760.svg|420px]] |- |class="size"| 32 |class="prop"| 10 |class="block"| <span class="block-list small">[10, 11, 28, 29, 34, 35, 52, 53, 74, 75, 92, 93, 98, 99, 116, 117, 138, 139, 156, 157, 162, 163, 180, 181, 202, 203, 220, 221, 226, 227, 244, 245]</span>[[File:Set_of_3-ary_Boolean_functions_84808987683731736979549389857862882249710924980580170943516537973582597120.svg|420px]] |- |class="size"| 32 |class="prop"| 12 |class="block"| <span class="block-list small">[12, 13, 26, 27, 44, 45, 58, 59, 68, 69, 82, 83, 100, 101, 114, 115, 140, 141, 154, 155, 172, 173, 186, 187, 196, 197, 210, 211, 228, 229, 242, 243]</span>[[File:Set_of_3-ary_Boolean_functions_21203458859717450452813733559663024377301133068365161988570178364429578240.svg|420px]] |- |class="size"| 16 |class="prop"| 32 |class="block"| <span class="block-list">[32, 33, 54, 55, 72, 73, 94, 95, 160, 161, 182, 183, 200, 201, 222, 223]</span>[[File:Set_of_3-ary_Boolean_functions_20219964821195502573845983519444008947711945227919536382652721397760.svg|420px]] |- |class="size"| 32 |class="prop"| 38 |class="block"| <span class="block-list small">[38, 39, 46, 47, 48, 49, 56, 57, 70, 71, 78, 79, 80, 81, 88, 89, 166, 167, 174, 175, 176, 177, 184, 185, 198, 199, 206, 207, 208, 209, 216, 217]</span>[[File:Set_of_3-ary_Boolean_functions_317480740805612630213248498566966758923182560826153727120795238400.svg|420px]] |- |class="size"| 16 |class="prop"| 40 |class="block"| <span class="block-list">[40, 41, 62, 63, 64, 65, 86, 87, 168, 169, 190, 191, 192, 193, 214, 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Allez étudier, vous ne pouvez pas y échapper ? 🫶🏻 [[Special:Contributions/102.96.35.202|102.96.35.202]] ([[User talk:102.96.35.202|discuss]]) 15:48, 29 December 2024 (UTC) 8wgpzn8hvied79pz8pirrzppkxjumcq 2693766 2693765 2024-12-29T16:40:25Z Atcovi 276019 {{Talk header}} 2693766 wikitext text/x-wiki {{Talk header}} 6ujz0t3lkt6jsf7d1r360l6l7wj3njb 2 317615 2693769 2024-12-29T16:43:59Z Atcovi 276019 +14.1 2693769 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == l61lbix7cl5shl9oznx8c83uoho4715 2693770 2693769 2024-12-29T16:57:04Z Atcovi 276019 /* 14.1 - Cardiovascular Disease: Definitions and Prevalence */ 2693770 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. 3kfwt04o7hxywuvg5o1nqr60m0brfj9 2693771 2693770 2024-12-29T16:58:49Z Atcovi 276019 /* 14.1 - Cardiovascular Disease: Definitions and Prevalence */ 2693771 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. b5vvazszvdkqkd4t4n73q5k8xy3wbpa 2693784 2693771 2024-12-29T19:26:45Z Atcovi 276019 2693784 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). s2n12id1kvd81fpdoeoq0t00eooc04c 2693785 2693784 2024-12-29T19:27:33Z Atcovi 276019 /* 14.2 - Cultural Variations and Developmental Issues */ 2693785 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). 6xhevbnqmsdangryr6qchyhtiim0d13 2693786 2693785 2024-12-29T19:32:02Z Atcovi 276019 2693786 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). 5fa8rlanuvb2tegzs3jz1favto87rod 2693787 2693786 2024-12-29T19:33:36Z Atcovi 276019 /* 14.3 - Correlates of Cardiovascular Disease */ 2693787 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Physiological Correlates === # '''Atherosclerosis and Arteriosclerosis:''' #* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks. #* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages. # '''Risk Factors:''' #* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history. #* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking. #* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity. # '''Diabetes and CVD:''' #* Both type 1 and type 2 diabetes significantly increase CVD risk. #* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis. #* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes. # '''Discrimination and CVD:''' #* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation. ---- === Psychological Correlates === # '''Emotions and Personality Traits:''' #* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk. #* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks. # '''Social Support:''' #* Social networks and emotional support play crucial roles in preventing and managing CVD. #* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack. #* Marital status, social participation, and perceived support significantly affect outcomes. # '''Socioeconomic Status (SES):''' #* In high-income countries, low SES correlates with increased CVD risk. #* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles. ---- === Key Findings === * Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks. * Psychological and social factors are as critical as physiological factors in both prevention and recovery. * Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management. === Stress and Cardiovascular Health === City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow. Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems. Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk. Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take. Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks. In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). pm7fls4qfnqvpxwcx7y5ozs2wiey3ar 2693788 2693787 2024-12-29T19:38:15Z Atcovi 276019 2693788 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Physiological Correlates === # '''Atherosclerosis and Arteriosclerosis:''' #* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks. #* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages. # '''Risk Factors:''' #* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history. #* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking. #* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity. # '''Diabetes and CVD:''' #* Both type 1 and type 2 diabetes significantly increase CVD risk. #* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis. #* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes. # '''Discrimination and CVD:''' #* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation. ---- === Psychological Correlates === # '''Emotions and Personality Traits:''' #* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk. #* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks. # '''Social Support:''' #* Social networks and emotional support play crucial roles in preventing and managing CVD. #* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack. #* Marital status, social participation, and perceived support significantly affect outcomes. # '''Socioeconomic Status (SES):''' #* In high-income countries, low SES correlates with increased CVD risk. #* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles. ---- === Key Findings === * Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks. * Psychological and social factors are as critical as physiological factors in both prevention and recovery. * Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management. === Stress and Cardiovascular Health === City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow. Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems. Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk. Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take. Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks. In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). == 14.4 - Health Behaviors and Cardiovascular Disease == === Raw Textbook Page === Tobacco Use The number one behavior to avoid if you want to minimize your risk for CVDs is smoking (Mermelstein, 2019). You can see why we have spent so much time on this topic (see Chapter 8). Tobacco use poses a significant challenge to global health, with more than 1 billion active tobacco users (Roth et al., 2020), and together with being an important risk factor for other chronic diseases such as cancer (see Chapter 13), cigarette smoking has also been identified as an important factor in the development of CVDs (Bishop, 2019). In 2019, tobacco use (primary smoking, secondhand smoke, chewing tobacco), accounted for 8.71 million deaths globally, of which 36.7% were due to CVDs (Roth et al., 2020). Also, in 2019, global rates of smoking remained higher among males than females (33.5% and 6.8%, respectively), resulting in 75.4% of smoking-related deaths occurring among males (Roth et al., 2020). However, through environmental exposure, tobacco also poses a risk to those who do not smoke. For example, Khoramdad et al. (2020), in a meta-analysis of case-control studies, containing 10,672 participants, found that passive smoking increased the risk of CVD by 28%, with people exposed to secondhand smoke at home and at work at highest risk. Similarly, in a meta-analysis of cohort studies, containing 2,313,935 participants, passive smoking increased the risk of CVD by 12%, with people exposed to secondhand smoke at home, work and public places at highest risk (Khoramdad et al., 2020). In one of the clearest demonstrations of the impact of tobacco use, Doll et al. (2004) followed approximately 35,000 British doctors from 1951 to 2001 and found that the dangers of smoking varied with cohorts. Males born in 1900 through 1930 who smoked only cigarettes and continued smoking died on average about 10 years sooner than lifelong nonsmokers. Quitting was beneficial regardless of the age at which it was attempted. Males who quit when 60, 50, 40, or 30 years of age gained, respectively, about 3, 6, 9, or 10 years of life expectancy (Doll et al., 2004). Smoking also influences CVD though its impact on physiological systems. For example, smoking has been associated with changes in blood pressure, arterial stiffening, increased inflammation, insulin resistance, and increased HDL cholesterol levels (Gallucci et al., 2020). What effects do smoking bans or governmental regulations to curb smoking have? Consider the following case. In 1993, the state of Massachusetts introduced the Massachusetts Tobacco Control Program (MTCP). This reduced how many people smoked in the state by 29%. That’s not all. There was a 31% decline in death rates due to CVD (from 1993 to 2003, 425 fewer CVD deaths; Kabir et al., 2008). Additionally, smoke-free laws have been associated with lower odds of commencing smoking, of being a current smoker and fewer days smoking among American adolescents and young adults (Song et al., 2015). Furthermore, despite some people’s concerns, in their review, Monson and Arsenault (2017) found that increased restrictions on smoking in public locations, has not been associated with increased smoking at home. Also, since the introduction of public smoking bans, children have been found to be exposed to less secondhand smoke at home (Nanninga et al., 2018). Benefits have been found when people reduce or cease smoking. For example, people who reduce their daily cigarette consumption from “heavy” (15–20 cigarettes per day or more) to “light” (less than 10 cigarettes per day) have been found to have a significantly lower risk of CVD (CHD and stroke) than those who continue heavy smoking (Chang et al., 2021). Also, people who reduce from heavy to moderate (10–19 cigarettes per day), and heavy to light smoking, have shown significantly reduced risk for CHD (ischemic heart disease and heart attack), compared with continuing heavy smokers (Chang et al., 2021). Additionally, ceasing smoking confers benefits; people who cease smoking show a rapid decline in CVD risk compared to those who continue to smoke, resulting in significantly lower risk of CVD within 5 years of cessation (Duncan et al., 2019). However, compared to never smoking, it takes 10 to 15 years for past heavy smoking to cease being associated with higher risk of CVD (Duncan et al., 2019). Over time, smoking cessation also results in reduced CRP levels (predictors of CVD as discussed earlier) (Gallus et al., 2018). A warning message from Health Canada. Smoking and Cardiovascular Disease. A warning on a Canadian cigarette packet. Those who can quit smoking significantly reduce their risk for contracting a cardiovascular disease. Long Description People around the world smoke. Smoking is more common in some cultures than in others, and the link between CVD and smoking is not the same across cultures. For example, smoking rates in China are among the highest in the world, with rates declining, but approximately 50% of males ages over 15 smoke (Ma et al., 2020). The level of smoking in China is such that, 30% of the world’s current smokers reside there and Chinese people now account for nearly one third of all tobacco-attributable deaths (Roth et al., 2020). In other countries, high levels of other unhealthy behaviors such as poor diet also accompany high smoking rates. Recent attention has turned to the presence of risk factors that accentuate the effect of smoking, and some of the most important are dietary factors and cholesterol. Diet Your food choices play a large role in your overall health and well-being. What you eat determines the levels of nutrients available for your cells and ensures the smooth and healthy functioning of your bodily symptoms (Greger, 2015). There are many dietary factors that influence the incidence and progression of CVD (Figure 14.7), and diet is a factor that affects the interaction between culture, psychology, and behavior (Van Horn et al., 2008). Your diet can influence your cholesterol level, your blood pressure, your tolerance for glucose (and consequently your risk for diabetes), your likelihood to be overweight, and even how your blood coagulates. Each of these factors is associated with the development of CVD (Carson et al., 2020; Fung et al., 2008). Cholesterol is an important risk factor for CVD. It is found in most animal products and is an important component of cell walls and membranes. Cholesterol is also a main component of plaque. As discussed in Chapter 8, we have high-density lipoprotein (HDL) and low-density lipoprotein (LDL) cholesterol. As the level of cholesterol in the blood increases, the risk of CVD increases as well (Jeong et al., 2018). LDLs seem to be the primary factor, and a number of treatments (e.g., statins) aim to reduce the LDL levels in the bloodstream (Rached & Santos, 2020). A high cholesterol level is even more likely to cause CVD in people with other health issues such as diabetes, where reducing cholesterol may reduce CV risk by 20% to 50% (Vesa et al., 2020). Recommendations are no longer provided about specific levels of cholesterol required to prevent heart disease; rather, the focus is on healthy dietary patterns. Additionally, while previously it was recommended to reduce saturated fats, views about this have changed more recently, with suggestions that there is increasing evidence to the contrary about reducing saturated fatty acids (SFA; Astrup et al., 2020). It has been argued that meta-analyses show that reducing SFA confers no benefits for CVD or total mortality and that the health effects of foods cannot be predicted by their nutrient group alone; overall macronutrient distribution or dietary pattern must be considered (Astrup et al., 2020). Furthermore, not all foods that contain SFA are problematic. For example, whole-fat dairy, dark chocolate, and unprocessed meat, are SFA-rich foods but are not associated with increased CVD risk (Astrup et al., 2020). Therefore, Astrup et al. (2020) concluded that no strong evidence exists that having population level limits on saturated fat consumption in the United States will prevent CVD or reduce mortality. Similarly, Krauss and Kris-Etherton (2020) argued that the evidence used to support recommendations to reduce saturated fat is often based on low-quality research designs and incorrect beliefs and assumptions and that dietary recommendations to limit SFA send the wrong messages to the public and health professionals. They argued that it is best to have dietary guidelines that encourage eating whole, natural foods and following specific dietary patterns. Others have similarly suggested that rather than target a specific nutrient such as saturated fat intake, CVD risk and overall health would likely see greater improvement by focusing on overall diet quality and eliminating processed foods such as simple carbohydrates (Gershuni, 2018). Diets including higher consumption of fish have been shown to be associated with lower risk of CHD and CHD mortality (Zhang et al., 2020). However, while this is valuable, as noted above, dietary patterns appear more important than specific nutrient groups. Several dietary patterns have been explored for their potential to reduce CVD risk, with the American Heart Association recommending Dietary Approaches to Stop Hypertension (DASH)-stylediets or Mediterranean-style diets (Carson et al., 2020). The DASH diet was first developed and examined by Sacks and colleagues in 1995 with the aim to reduce blood pressure (Sacks, 1995). In this approach, researchers calculate a DASH score based on eight food and nutrient components (fruits, vegetables, whole grains, nuts and legumes, low-fat dairy, red and processed meats, sweetened beverages, and sodium; Fung et al., 2008). In perhaps the most extensive study of the approach, Fung et al. (2008) examined the diet of 88,517 female nurses seven times between 1980 and 2004. There was a direct negative correlation between DASH scores and heart disease; the better the score the less likely a heart attack. The DASH score was also significantly associated with lower risk of stroke and lower levels of CRP. In subsequent meta-analytic research, the DASH diet has been associated with reduced risk of CAD (Yang et al., 2019) and reduced risk of all-cause and CVD and stroke mortality (Soltani et al., 2020). Additionally, in a recent umbrella review, the DASH diet was associated with decreased incidence of CVD, CHD, stroke, diabetes, and lower blood pressure, total cholesterol, LDL cholesterol, and body weight (Chiavaroli et al., 2019). Different food items including a variety of fruits and vegetables, whole grain products, peas, dairy products, salt, sugar, knives, and spoons spread on a table. Fresh Vegetables. A dietary pattern with a variety of fruits and vegetables, whole grains, healthy proteins and minimal processed foods, salt, sugar and alcohol is recommended for cardiovascular health. Another dietary pattern with demonstrated benefit for CVD is the Mediterranean diet. While this dietary pattern may vary slightly according to location, it typically involves high consumption of fruits, vegetables, legumes, grains and cereals (mainly whole grains), nuts, and fish, moderate consumption of milk and dairy products, low consumption of meat and meat products, a low to moderate wine consumption and olive oil (Rosato et al., 2019). As can be seen, the Mediterranean diet shares some similarities to the DASH diet. In a meta-analytic study, it has been associated with lower risk for CVD, CHD/heart attack, stroke and ischaemic stroke but not hemorrhagic stroke (Rosato et al., 2019). Similarly, in an umbrella review, containing 12,800,000 people, higher adherence to a Mediterranean diet was associated with reduced CVD incidence and mortality, CHD and heart attack incidence and overall mortality (Dinu et al., 2018). Finally, researchers have explored an adaption of the Mediterranean diet, (referred to as an Indo-Mediterranean diet). In an illustrative intervention, Singh et al. (2002) recruited 1,000 patients with angina pectoris and myocardial infarctions and had half of them eat the Indo-Mediterranean diet (a diet rich in whole grains, fruits, vegetables, walnuts, and almonds). The control group ate a diet suggested by the National Cholesterol Education Program. Interestingly, the intervention group had fewer heart problems, including heart attacks (both fatal and nonfatal), and showed lower cholesterol levels than the control group. Similarly, over 2 years, people eating the Indo-Mediterranean diet have shown significantly lower rates of pre-heart failure, heart failure, and cardiac arrythmia than a control group (Singh et al., 2020). While the DASH and Mediterranean diets confer cardiovascular and other health benefits, more recently it has been suggested that the Indo-Mediterranean diet has wider food diversity and may produce better outcomes as it includes foods and spices that have enhanced anti-inflammatory and cardioprotective effects (i.e., more whole grains such as millets, porridge, beans, brown rice, and spices such as turmeric, cumin, cloves, black pepper, cardamom, cinnamon, and coriander; Singh et al., 2022). The Indo-Mediterranean diet also excludes animal products, except fish and satisfies all 10 qualities of high-quality foods (i.e., low glycemic index, high nutrient density, food diversity, no trans fat, no or low refined sugar, low salt, moderate healthy fat, high fiber, beneficial effects on gut microbiota, and no preoxidation of foods (Singh et al., 2022). A comparison of the diets is shown in Table 14.2. As can be seen, culture plays an important role in CVD via its influence on diet. A related cultural diet component is alcohol. For example, the French have relatively low levels of CVD even though French food is known to be rich in saturated fats. This French paradox (see Chapter 8) has been linked to moderate consumption of alcohol. One or two glasses of wine per day seem to reduce the incidence of CVD (Castaldo et al., 2019; Snopek et al., 2018). Too much alcohol (e.g., binge drinking) does not lessen your CVD risk (Day & Rudd, 2019; Degerud et al., 2018). Research suggests a J-shaped association between alcohol consumption and CVD. Low levels of drinking relate to lower rates of CVD than abstaining (Degerud et al., 2021; Zhang et al., 2021). Do not be mistaken: This is not a reason for underage drinking, and binge drinking is still not a good thing. Physical Activity In terms of physical activity, not only is exercising useful in reducing the risks of CVD but being physically inactive actually increases the risks (Duran et al., 2022; Varghese et al., 2016). In this respect, standing still actually makes you slide backward on the continuum of health. In a global look at physical activity, the INTERHEART study, leisure time as well as mild to moderate occupational physical activity resulted in lower rates of CVD. Owners of cars and televisions tended to exhibit more sedentary behavior and showed an increased risk of CVD (Held et al., 2012). A key component of treatment and many behavioral interventions to reduce CVD hence includes some form of physical activity. Dibben et al. (2021) presented a review of the effectiveness of exercise-based cardiac rehabilitation in people with CHD. Their study included 85 trials with a total of 23,430 people and found that short-term (6–12 months) exercise-based cardiac rehabilitation was associated with fewer hospitalizations for all causes, fewer deaths from all causes, and fewer deaths from heart attacks. At medium- and long-term follow-up, exercise-based cardiac rehabilitation was associated with a large reduction in deaths from CVD. Additionally, a recent worldwide study involving more than 15,000 participants examined the relationship between self-reported physical activity and both cardiovascular and all-cause mortality. In this study individuals reporting the highest levels of physical activity showed significantly reduced all-cause mortality as well as significantly reduced cardiovascular mortality (Steward et al., 2017). The role of physical activity in the primary and secondary prevention of CVD has also been investigated. Jeong et al. (2019) examined the link between physical activity and CVD mortality among 131,558 people with CVD and 310,240 without CVD. For both groups, physical activity was associated with reduced risk of all-cause death, with stronger results for people with CVD. Furthermore, people with CVD who engaged in high levels of physical activity had a similar or lower risk of mortality than people without CVD (Jeong et al., 2019). People with CVD are often given an exercise prescription (a specific exercise plan) tailored to their needs with variations in the type, frequency, duration, and intensity of exercise. Exercise plans for people with CVD often begin with moderate intensity aerobic exercise followed by resistance training, with a gradual increase in intensity (Tucker et al., 2022). 03cphoplwdzbd9fy90vempubvvbkyi2 2693789 2693788 2024-12-29T19:39:12Z Atcovi 276019 /* 14.4 - Health Behaviors and Cardiovascular Disease */ 2693789 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Physiological Correlates === # '''Atherosclerosis and Arteriosclerosis:''' #* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks. #* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages. # '''Risk Factors:''' #* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history. #* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking. #* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity. # '''Diabetes and CVD:''' #* Both type 1 and type 2 diabetes significantly increase CVD risk. #* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis. #* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes. # '''Discrimination and CVD:''' #* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation. ---- === Psychological Correlates === # '''Emotions and Personality Traits:''' #* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk. #* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks. # '''Social Support:''' #* Social networks and emotional support play crucial roles in preventing and managing CVD. #* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack. #* Marital status, social participation, and perceived support significantly affect outcomes. # '''Socioeconomic Status (SES):''' #* In high-income countries, low SES correlates with increased CVD risk. #* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles. ---- === Key Findings === * Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks. * Psychological and social factors are as critical as physiological factors in both prevention and recovery. * Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management. === Stress and Cardiovascular Health === City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow. Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems. Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk. Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take. Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks. In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). == 14.4 - Health Behaviors and Cardiovascular Disease == ==== Impact of Smoking on Global Health and CVD ==== * '''Prevalence:''' Over 1 billion active tobacco users globally (Roth et al., 2020). * '''Mortality:''' Tobacco use caused 8.71 million deaths in 2019; 36.7% linked to cardiovascular diseases (CVDs). * '''Gender Disparity:''' Smoking rates are higher in males (33.5%) compared to females (6.8%), with 75.4% of smoking-related deaths among men. * '''Passive Smoking:''' Increased CVD risk by 28% at home/work (case-control studies) and by 12% (cohort studies). ==== Physiological Effects of Smoking ==== * Smoking impacts physiological systems, contributing to high blood pressure, arterial stiffening, inflammation, insulin resistance, and cholesterol changes. * Historical data from Doll et al. (2004) revealed a 10-year reduction in life expectancy for lifelong smokers born between 1900–1930, with life expectancy improvements for those who quit at any age. ==== Benefits of Smoking Reduction and Cessation ==== * Reducing daily cigarette consumption significantly lowers risks of coronary heart disease (CHD) and stroke (Chang et al., 2021). * Smoking cessation shows rapid risk reduction within 5 years, with near-normalized CVD risk after 10–15 years (Duncan et al., 2019). ==== Policy Interventions and Effectiveness ==== * Massachusetts Tobacco Control Program (MTCP, 1993): ** Smoking prevalence decreased by 29%. ** CVD-related deaths reduced by 31% over a decade. * Smoke-free laws lower smoking initiation rates and reduce exposure to secondhand smoke, including among children. ---- === Diet and CVD: Role and Patterns === ==== Cholesterol and Risk ==== * High LDL cholesterol is a primary risk factor, particularly in conjunction with diabetes (20–50% reduced CV risk when cholesterol is controlled). ==== Dietary Patterns and Recommendations ==== * '''Saturated Fat Debate:''' ** Earlier guidelines on reducing saturated fat intake have been questioned; focus is now on overall dietary patterns and quality (Astrup et al., 2020). * '''Beneficial Diets:''' ** DASH Diet: Emphasizes fruits, vegetables, whole grains, and low sodium; reduces risks for CHD, stroke, and high CRP levels. ** Mediterranean Diet: Similar to DASH, includes olive oil, nuts, and moderate wine consumption; linked to lower CHD/stroke incidence and mortality. ** Indo-Mediterranean Diet: Enhances DASH/Mediterranean diets with anti-inflammatory ingredients like turmeric, brown rice, and spices, offering greater diversity and benefits. ==== Alcohol's Role ==== * Moderate wine consumption is associated with lower CVD risk (French Paradox). However, binge drinking or excess consumption negates these benefits. ---- === Physical Activity: Prevention and Recovery === ==== Global Trends and Impact ==== * Inactivity increases CVD risks (INTERHEART study). Ownership of cars/TVs correlates with sedentary behavior and CVD prevalence. ==== Rehabilitation and Outcomes ==== * Exercise-based cardiac rehabilitation (6–12 months): Fewer deaths and hospitalizations for CVD patients (Dibben et al., 2021). * Physical activity significantly reduces all-cause and CVD-specific mortality, even in patients with existing CVD (Jeong et al., 2019). ==== Prescriptive Exercise Plans ==== * Designed for intensity and progression: Moderate aerobic activity transitioning to resistance training for tailored recovery (Tucker et al., 2022). This structured summary highlights the multifaceted approach to CVD prevention and management, focusing on lifestyle changes, policy interventions, and cultural considerations. === Raw Textbook Page === Tobacco Use The number one behavior to avoid if you want to minimize your risk for CVDs is smoking (Mermelstein, 2019). You can see why we have spent so much time on this topic (see Chapter 8). Tobacco use poses a significant challenge to global health, with more than 1 billion active tobacco users (Roth et al., 2020), and together with being an important risk factor for other chronic diseases such as cancer (see Chapter 13), cigarette smoking has also been identified as an important factor in the development of CVDs (Bishop, 2019). In 2019, tobacco use (primary smoking, secondhand smoke, chewing tobacco), accounted for 8.71 million deaths globally, of which 36.7% were due to CVDs (Roth et al., 2020). Also, in 2019, global rates of smoking remained higher among males than females (33.5% and 6.8%, respectively), resulting in 75.4% of smoking-related deaths occurring among males (Roth et al., 2020). However, through environmental exposure, tobacco also poses a risk to those who do not smoke. For example, Khoramdad et al. (2020), in a meta-analysis of case-control studies, containing 10,672 participants, found that passive smoking increased the risk of CVD by 28%, with people exposed to secondhand smoke at home and at work at highest risk. Similarly, in a meta-analysis of cohort studies, containing 2,313,935 participants, passive smoking increased the risk of CVD by 12%, with people exposed to secondhand smoke at home, work and public places at highest risk (Khoramdad et al., 2020). In one of the clearest demonstrations of the impact of tobacco use, Doll et al. (2004) followed approximately 35,000 British doctors from 1951 to 2001 and found that the dangers of smoking varied with cohorts. Males born in 1900 through 1930 who smoked only cigarettes and continued smoking died on average about 10 years sooner than lifelong nonsmokers. Quitting was beneficial regardless of the age at which it was attempted. Males who quit when 60, 50, 40, or 30 years of age gained, respectively, about 3, 6, 9, or 10 years of life expectancy (Doll et al., 2004). Smoking also influences CVD though its impact on physiological systems. For example, smoking has been associated with changes in blood pressure, arterial stiffening, increased inflammation, insulin resistance, and increased HDL cholesterol levels (Gallucci et al., 2020). What effects do smoking bans or governmental regulations to curb smoking have? Consider the following case. In 1993, the state of Massachusetts introduced the Massachusetts Tobacco Control Program (MTCP). This reduced how many people smoked in the state by 29%. That’s not all. There was a 31% decline in death rates due to CVD (from 1993 to 2003, 425 fewer CVD deaths; Kabir et al., 2008). Additionally, smoke-free laws have been associated with lower odds of commencing smoking, of being a current smoker and fewer days smoking among American adolescents and young adults (Song et al., 2015). Furthermore, despite some people’s concerns, in their review, Monson and Arsenault (2017) found that increased restrictions on smoking in public locations, has not been associated with increased smoking at home. Also, since the introduction of public smoking bans, children have been found to be exposed to less secondhand smoke at home (Nanninga et al., 2018). Benefits have been found when people reduce or cease smoking. For example, people who reduce their daily cigarette consumption from “heavy” (15–20 cigarettes per day or more) to “light” (less than 10 cigarettes per day) have been found to have a significantly lower risk of CVD (CHD and stroke) than those who continue heavy smoking (Chang et al., 2021). Also, people who reduce from heavy to moderate (10–19 cigarettes per day), and heavy to light smoking, have shown significantly reduced risk for CHD (ischemic heart disease and heart attack), compared with continuing heavy smokers (Chang et al., 2021). Additionally, ceasing smoking confers benefits; people who cease smoking show a rapid decline in CVD risk compared to those who continue to smoke, resulting in significantly lower risk of CVD within 5 years of cessation (Duncan et al., 2019). However, compared to never smoking, it takes 10 to 15 years for past heavy smoking to cease being associated with higher risk of CVD (Duncan et al., 2019). Over time, smoking cessation also results in reduced CRP levels (predictors of CVD as discussed earlier) (Gallus et al., 2018). A warning message from Health Canada. Smoking and Cardiovascular Disease. A warning on a Canadian cigarette packet. Those who can quit smoking significantly reduce their risk for contracting a cardiovascular disease. Long Description People around the world smoke. Smoking is more common in some cultures than in others, and the link between CVD and smoking is not the same across cultures. For example, smoking rates in China are among the highest in the world, with rates declining, but approximately 50% of males ages over 15 smoke (Ma et al., 2020). The level of smoking in China is such that, 30% of the world’s current smokers reside there and Chinese people now account for nearly one third of all tobacco-attributable deaths (Roth et al., 2020). In other countries, high levels of other unhealthy behaviors such as poor diet also accompany high smoking rates. Recent attention has turned to the presence of risk factors that accentuate the effect of smoking, and some of the most important are dietary factors and cholesterol. Diet Your food choices play a large role in your overall health and well-being. What you eat determines the levels of nutrients available for your cells and ensures the smooth and healthy functioning of your bodily symptoms (Greger, 2015). There are many dietary factors that influence the incidence and progression of CVD (Figure 14.7), and diet is a factor that affects the interaction between culture, psychology, and behavior (Van Horn et al., 2008). Your diet can influence your cholesterol level, your blood pressure, your tolerance for glucose (and consequently your risk for diabetes), your likelihood to be overweight, and even how your blood coagulates. Each of these factors is associated with the development of CVD (Carson et al., 2020; Fung et al., 2008). Cholesterol is an important risk factor for CVD. It is found in most animal products and is an important component of cell walls and membranes. Cholesterol is also a main component of plaque. As discussed in Chapter 8, we have high-density lipoprotein (HDL) and low-density lipoprotein (LDL) cholesterol. As the level of cholesterol in the blood increases, the risk of CVD increases as well (Jeong et al., 2018). LDLs seem to be the primary factor, and a number of treatments (e.g., statins) aim to reduce the LDL levels in the bloodstream (Rached & Santos, 2020). A high cholesterol level is even more likely to cause CVD in people with other health issues such as diabetes, where reducing cholesterol may reduce CV risk by 20% to 50% (Vesa et al., 2020). Recommendations are no longer provided about specific levels of cholesterol required to prevent heart disease; rather, the focus is on healthy dietary patterns. Additionally, while previously it was recommended to reduce saturated fats, views about this have changed more recently, with suggestions that there is increasing evidence to the contrary about reducing saturated fatty acids (SFA; Astrup et al., 2020). It has been argued that meta-analyses show that reducing SFA confers no benefits for CVD or total mortality and that the health effects of foods cannot be predicted by their nutrient group alone; overall macronutrient distribution or dietary pattern must be considered (Astrup et al., 2020). Furthermore, not all foods that contain SFA are problematic. For example, whole-fat dairy, dark chocolate, and unprocessed meat, are SFA-rich foods but are not associated with increased CVD risk (Astrup et al., 2020). Therefore, Astrup et al. (2020) concluded that no strong evidence exists that having population level limits on saturated fat consumption in the United States will prevent CVD or reduce mortality. Similarly, Krauss and Kris-Etherton (2020) argued that the evidence used to support recommendations to reduce saturated fat is often based on low-quality research designs and incorrect beliefs and assumptions and that dietary recommendations to limit SFA send the wrong messages to the public and health professionals. They argued that it is best to have dietary guidelines that encourage eating whole, natural foods and following specific dietary patterns. Others have similarly suggested that rather than target a specific nutrient such as saturated fat intake, CVD risk and overall health would likely see greater improvement by focusing on overall diet quality and eliminating processed foods such as simple carbohydrates (Gershuni, 2018). Diets including higher consumption of fish have been shown to be associated with lower risk of CHD and CHD mortality (Zhang et al., 2020). However, while this is valuable, as noted above, dietary patterns appear more important than specific nutrient groups. Several dietary patterns have been explored for their potential to reduce CVD risk, with the American Heart Association recommending Dietary Approaches to Stop Hypertension (DASH)-stylediets or Mediterranean-style diets (Carson et al., 2020). The DASH diet was first developed and examined by Sacks and colleagues in 1995 with the aim to reduce blood pressure (Sacks, 1995). In this approach, researchers calculate a DASH score based on eight food and nutrient components (fruits, vegetables, whole grains, nuts and legumes, low-fat dairy, red and processed meats, sweetened beverages, and sodium; Fung et al., 2008). In perhaps the most extensive study of the approach, Fung et al. (2008) examined the diet of 88,517 female nurses seven times between 1980 and 2004. There was a direct negative correlation between DASH scores and heart disease; the better the score the less likely a heart attack. The DASH score was also significantly associated with lower risk of stroke and lower levels of CRP. In subsequent meta-analytic research, the DASH diet has been associated with reduced risk of CAD (Yang et al., 2019) and reduced risk of all-cause and CVD and stroke mortality (Soltani et al., 2020). Additionally, in a recent umbrella review, the DASH diet was associated with decreased incidence of CVD, CHD, stroke, diabetes, and lower blood pressure, total cholesterol, LDL cholesterol, and body weight (Chiavaroli et al., 2019). Different food items including a variety of fruits and vegetables, whole grain products, peas, dairy products, salt, sugar, knives, and spoons spread on a table. Fresh Vegetables. A dietary pattern with a variety of fruits and vegetables, whole grains, healthy proteins and minimal processed foods, salt, sugar and alcohol is recommended for cardiovascular health. Another dietary pattern with demonstrated benefit for CVD is the Mediterranean diet. While this dietary pattern may vary slightly according to location, it typically involves high consumption of fruits, vegetables, legumes, grains and cereals (mainly whole grains), nuts, and fish, moderate consumption of milk and dairy products, low consumption of meat and meat products, a low to moderate wine consumption and olive oil (Rosato et al., 2019). As can be seen, the Mediterranean diet shares some similarities to the DASH diet. In a meta-analytic study, it has been associated with lower risk for CVD, CHD/heart attack, stroke and ischaemic stroke but not hemorrhagic stroke (Rosato et al., 2019). Similarly, in an umbrella review, containing 12,800,000 people, higher adherence to a Mediterranean diet was associated with reduced CVD incidence and mortality, CHD and heart attack incidence and overall mortality (Dinu et al., 2018). Finally, researchers have explored an adaption of the Mediterranean diet, (referred to as an Indo-Mediterranean diet). In an illustrative intervention, Singh et al. (2002) recruited 1,000 patients with angina pectoris and myocardial infarctions and had half of them eat the Indo-Mediterranean diet (a diet rich in whole grains, fruits, vegetables, walnuts, and almonds). The control group ate a diet suggested by the National Cholesterol Education Program. Interestingly, the intervention group had fewer heart problems, including heart attacks (both fatal and nonfatal), and showed lower cholesterol levels than the control group. Similarly, over 2 years, people eating the Indo-Mediterranean diet have shown significantly lower rates of pre-heart failure, heart failure, and cardiac arrythmia than a control group (Singh et al., 2020). While the DASH and Mediterranean diets confer cardiovascular and other health benefits, more recently it has been suggested that the Indo-Mediterranean diet has wider food diversity and may produce better outcomes as it includes foods and spices that have enhanced anti-inflammatory and cardioprotective effects (i.e., more whole grains such as millets, porridge, beans, brown rice, and spices such as turmeric, cumin, cloves, black pepper, cardamom, cinnamon, and coriander; Singh et al., 2022). The Indo-Mediterranean diet also excludes animal products, except fish and satisfies all 10 qualities of high-quality foods (i.e., low glycemic index, high nutrient density, food diversity, no trans fat, no or low refined sugar, low salt, moderate healthy fat, high fiber, beneficial effects on gut microbiota, and no preoxidation of foods (Singh et al., 2022). A comparison of the diets is shown in Table 14.2. As can be seen, culture plays an important role in CVD via its influence on diet. A related cultural diet component is alcohol. For example, the French have relatively low levels of CVD even though French food is known to be rich in saturated fats. This French paradox (see Chapter 8) has been linked to moderate consumption of alcohol. One or two glasses of wine per day seem to reduce the incidence of CVD (Castaldo et al., 2019; Snopek et al., 2018). Too much alcohol (e.g., binge drinking) does not lessen your CVD risk (Day & Rudd, 2019; Degerud et al., 2018). Research suggests a J-shaped association between alcohol consumption and CVD. Low levels of drinking relate to lower rates of CVD than abstaining (Degerud et al., 2021; Zhang et al., 2021). Do not be mistaken: This is not a reason for underage drinking, and binge drinking is still not a good thing. Physical Activity In terms of physical activity, not only is exercising useful in reducing the risks of CVD but being physically inactive actually increases the risks (Duran et al., 2022; Varghese et al., 2016). In this respect, standing still actually makes you slide backward on the continuum of health. In a global look at physical activity, the INTERHEART study, leisure time as well as mild to moderate occupational physical activity resulted in lower rates of CVD. Owners of cars and televisions tended to exhibit more sedentary behavior and showed an increased risk of CVD (Held et al., 2012). A key component of treatment and many behavioral interventions to reduce CVD hence includes some form of physical activity. Dibben et al. (2021) presented a review of the effectiveness of exercise-based cardiac rehabilitation in people with CHD. Their study included 85 trials with a total of 23,430 people and found that short-term (6–12 months) exercise-based cardiac rehabilitation was associated with fewer hospitalizations for all causes, fewer deaths from all causes, and fewer deaths from heart attacks. At medium- and long-term follow-up, exercise-based cardiac rehabilitation was associated with a large reduction in deaths from CVD. Additionally, a recent worldwide study involving more than 15,000 participants examined the relationship between self-reported physical activity and both cardiovascular and all-cause mortality. In this study individuals reporting the highest levels of physical activity showed significantly reduced all-cause mortality as well as significantly reduced cardiovascular mortality (Steward et al., 2017). The role of physical activity in the primary and secondary prevention of CVD has also been investigated. Jeong et al. (2019) examined the link between physical activity and CVD mortality among 131,558 people with CVD and 310,240 without CVD. For both groups, physical activity was associated with reduced risk of all-cause death, with stronger results for people with CVD. Furthermore, people with CVD who engaged in high levels of physical activity had a similar or lower risk of mortality than people without CVD (Jeong et al., 2019). People with CVD are often given an exercise prescription (a specific exercise plan) tailored to their needs with variations in the type, frequency, duration, and intensity of exercise. Exercise plans for people with CVD often begin with moderate intensity aerobic exercise followed by resistance training, with a gradual increase in intensity (Tucker et al., 2022). 43ld5ur2sy3b6pnwgwx3jatnuhusqfs 2693790 2693789 2024-12-29T19:43:04Z Atcovi 276019 2693790 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Physiological Correlates === # '''Atherosclerosis and Arteriosclerosis:''' #* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks. #* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages. # '''Risk Factors:''' #* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history. #* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking. #* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity. # '''Diabetes and CVD:''' #* Both type 1 and type 2 diabetes significantly increase CVD risk. #* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis. #* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes. # '''Discrimination and CVD:''' #* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation. ---- === Psychological Correlates === # '''Emotions and Personality Traits:''' #* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk. #* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks. # '''Social Support:''' #* Social networks and emotional support play crucial roles in preventing and managing CVD. #* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack. #* Marital status, social participation, and perceived support significantly affect outcomes. # '''Socioeconomic Status (SES):''' #* In high-income countries, low SES correlates with increased CVD risk. #* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles. ---- === Key Findings === * Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks. * Psychological and social factors are as critical as physiological factors in both prevention and recovery. * Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management. === Stress and Cardiovascular Health === City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow. Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems. Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk. Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take. Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks. In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). == 14.4 - Health Behaviors and Cardiovascular Disease == ==== Impact of Smoking on Global Health and CVD ==== * '''Prevalence:''' Over 1 billion active tobacco users globally (Roth et al., 2020). * '''Mortality:''' Tobacco use caused 8.71 million deaths in 2019; 36.7% linked to cardiovascular diseases (CVDs). * '''Gender Disparity:''' Smoking rates are higher in males (33.5%) compared to females (6.8%), with 75.4% of smoking-related deaths among men. * '''Passive Smoking:''' Increased CVD risk by 28% at home/work (case-control studies) and by 12% (cohort studies). ==== Physiological Effects of Smoking ==== * Smoking impacts physiological systems, contributing to high blood pressure, arterial stiffening, inflammation, insulin resistance, and cholesterol changes. * Historical data from Doll et al. (2004) revealed a 10-year reduction in life expectancy for lifelong smokers born between 1900–1930, with life expectancy improvements for those who quit at any age. ==== Benefits of Smoking Reduction and Cessation ==== * Reducing daily cigarette consumption significantly lowers risks of coronary heart disease (CHD) and stroke (Chang et al., 2021). * Smoking cessation shows rapid risk reduction within 5 years, with near-normalized CVD risk after 10–15 years (Duncan et al., 2019). ==== Policy Interventions and Effectiveness ==== * Massachusetts Tobacco Control Program (MTCP, 1993): ** Smoking prevalence decreased by 29%. ** CVD-related deaths reduced by 31% over a decade. * Smoke-free laws lower smoking initiation rates and reduce exposure to secondhand smoke, including among children. ---- === Diet and CVD: Role and Patterns === ==== Cholesterol and Risk ==== * High LDL cholesterol is a primary risk factor, particularly in conjunction with diabetes (20–50% reduced CV risk when cholesterol is controlled). ==== Dietary Patterns and Recommendations ==== * '''Saturated Fat Debate:''' ** Earlier guidelines on reducing saturated fat intake have been questioned; focus is now on overall dietary patterns and quality (Astrup et al., 2020). * '''Beneficial Diets:''' ** DASH Diet: Emphasizes fruits, vegetables, whole grains, and low sodium; reduces risks for CHD, stroke, and high CRP levels. ** Mediterranean Diet: Similar to DASH, includes olive oil, nuts, and moderate wine consumption; linked to lower CHD/stroke incidence and mortality. ** Indo-Mediterranean Diet: Enhances DASH/Mediterranean diets with anti-inflammatory ingredients like turmeric, brown rice, and spices, offering greater diversity and benefits. ==== Alcohol's Role ==== * Moderate wine consumption is associated with lower CVD risk (French Paradox). However, binge drinking or excess consumption negates these benefits. ---- === Physical Activity: Prevention and Recovery === ==== Global Trends and Impact ==== * Inactivity increases CVD risks (INTERHEART study). Ownership of cars/TVs correlates with sedentary behavior and CVD prevalence. ==== Rehabilitation and Outcomes ==== * Exercise-based cardiac rehabilitation (6–12 months): Fewer deaths and hospitalizations for CVD patients (Dibben et al., 2021). * Physical activity significantly reduces all-cause and CVD-specific mortality, even in patients with existing CVD (Jeong et al., 2019). ==== Prescriptive Exercise Plans ==== * Designed for intensity and progression: Moderate aerobic activity transitioning to resistance training for tailored recovery (Tucker et al., 2022). This structured summary highlights the multifaceted approach to CVD prevention and management, focusing on lifestyle changes, policy interventions, and cultural considerations. === Raw Textbook Page === Tobacco Use The number one behavior to avoid if you want to minimize your risk for CVDs is smoking (Mermelstein, 2019). You can see why we have spent so much time on this topic (see Chapter 8). Tobacco use poses a significant challenge to global health, with more than 1 billion active tobacco users (Roth et al., 2020), and together with being an important risk factor for other chronic diseases such as cancer (see Chapter 13), cigarette smoking has also been identified as an important factor in the development of CVDs (Bishop, 2019). In 2019, tobacco use (primary smoking, secondhand smoke, chewing tobacco), accounted for 8.71 million deaths globally, of which 36.7% were due to CVDs (Roth et al., 2020). Also, in 2019, global rates of smoking remained higher among males than females (33.5% and 6.8%, respectively), resulting in 75.4% of smoking-related deaths occurring among males (Roth et al., 2020). However, through environmental exposure, tobacco also poses a risk to those who do not smoke. For example, Khoramdad et al. (2020), in a meta-analysis of case-control studies, containing 10,672 participants, found that passive smoking increased the risk of CVD by 28%, with people exposed to secondhand smoke at home and at work at highest risk. Similarly, in a meta-analysis of cohort studies, containing 2,313,935 participants, passive smoking increased the risk of CVD by 12%, with people exposed to secondhand smoke at home, work and public places at highest risk (Khoramdad et al., 2020). In one of the clearest demonstrations of the impact of tobacco use, Doll et al. (2004) followed approximately 35,000 British doctors from 1951 to 2001 and found that the dangers of smoking varied with cohorts. Males born in 1900 through 1930 who smoked only cigarettes and continued smoking died on average about 10 years sooner than lifelong nonsmokers. Quitting was beneficial regardless of the age at which it was attempted. Males who quit when 60, 50, 40, or 30 years of age gained, respectively, about 3, 6, 9, or 10 years of life expectancy (Doll et al., 2004). Smoking also influences CVD though its impact on physiological systems. For example, smoking has been associated with changes in blood pressure, arterial stiffening, increased inflammation, insulin resistance, and increased HDL cholesterol levels (Gallucci et al., 2020). What effects do smoking bans or governmental regulations to curb smoking have? Consider the following case. In 1993, the state of Massachusetts introduced the Massachusetts Tobacco Control Program (MTCP). This reduced how many people smoked in the state by 29%. That’s not all. There was a 31% decline in death rates due to CVD (from 1993 to 2003, 425 fewer CVD deaths; Kabir et al., 2008). Additionally, smoke-free laws have been associated with lower odds of commencing smoking, of being a current smoker and fewer days smoking among American adolescents and young adults (Song et al., 2015). Furthermore, despite some people’s concerns, in their review, Monson and Arsenault (2017) found that increased restrictions on smoking in public locations, has not been associated with increased smoking at home. Also, since the introduction of public smoking bans, children have been found to be exposed to less secondhand smoke at home (Nanninga et al., 2018). Benefits have been found when people reduce or cease smoking. For example, people who reduce their daily cigarette consumption from “heavy” (15–20 cigarettes per day or more) to “light” (less than 10 cigarettes per day) have been found to have a significantly lower risk of CVD (CHD and stroke) than those who continue heavy smoking (Chang et al., 2021). Also, people who reduce from heavy to moderate (10–19 cigarettes per day), and heavy to light smoking, have shown significantly reduced risk for CHD (ischemic heart disease and heart attack), compared with continuing heavy smokers (Chang et al., 2021). Additionally, ceasing smoking confers benefits; people who cease smoking show a rapid decline in CVD risk compared to those who continue to smoke, resulting in significantly lower risk of CVD within 5 years of cessation (Duncan et al., 2019). However, compared to never smoking, it takes 10 to 15 years for past heavy smoking to cease being associated with higher risk of CVD (Duncan et al., 2019). Over time, smoking cessation also results in reduced CRP levels (predictors of CVD as discussed earlier) (Gallus et al., 2018). A warning message from Health Canada. Smoking and Cardiovascular Disease. A warning on a Canadian cigarette packet. Those who can quit smoking significantly reduce their risk for contracting a cardiovascular disease. Long Description People around the world smoke. Smoking is more common in some cultures than in others, and the link between CVD and smoking is not the same across cultures. For example, smoking rates in China are among the highest in the world, with rates declining, but approximately 50% of males ages over 15 smoke (Ma et al., 2020). The level of smoking in China is such that, 30% of the world’s current smokers reside there and Chinese people now account for nearly one third of all tobacco-attributable deaths (Roth et al., 2020). In other countries, high levels of other unhealthy behaviors such as poor diet also accompany high smoking rates. Recent attention has turned to the presence of risk factors that accentuate the effect of smoking, and some of the most important are dietary factors and cholesterol. Diet Your food choices play a large role in your overall health and well-being. What you eat determines the levels of nutrients available for your cells and ensures the smooth and healthy functioning of your bodily symptoms (Greger, 2015). There are many dietary factors that influence the incidence and progression of CVD (Figure 14.7), and diet is a factor that affects the interaction between culture, psychology, and behavior (Van Horn et al., 2008). Your diet can influence your cholesterol level, your blood pressure, your tolerance for glucose (and consequently your risk for diabetes), your likelihood to be overweight, and even how your blood coagulates. Each of these factors is associated with the development of CVD (Carson et al., 2020; Fung et al., 2008). Cholesterol is an important risk factor for CVD. It is found in most animal products and is an important component of cell walls and membranes. Cholesterol is also a main component of plaque. As discussed in Chapter 8, we have high-density lipoprotein (HDL) and low-density lipoprotein (LDL) cholesterol. As the level of cholesterol in the blood increases, the risk of CVD increases as well (Jeong et al., 2018). LDLs seem to be the primary factor, and a number of treatments (e.g., statins) aim to reduce the LDL levels in the bloodstream (Rached & Santos, 2020). A high cholesterol level is even more likely to cause CVD in people with other health issues such as diabetes, where reducing cholesterol may reduce CV risk by 20% to 50% (Vesa et al., 2020). Recommendations are no longer provided about specific levels of cholesterol required to prevent heart disease; rather, the focus is on healthy dietary patterns. Additionally, while previously it was recommended to reduce saturated fats, views about this have changed more recently, with suggestions that there is increasing evidence to the contrary about reducing saturated fatty acids (SFA; Astrup et al., 2020). It has been argued that meta-analyses show that reducing SFA confers no benefits for CVD or total mortality and that the health effects of foods cannot be predicted by their nutrient group alone; overall macronutrient distribution or dietary pattern must be considered (Astrup et al., 2020). Furthermore, not all foods that contain SFA are problematic. For example, whole-fat dairy, dark chocolate, and unprocessed meat, are SFA-rich foods but are not associated with increased CVD risk (Astrup et al., 2020). Therefore, Astrup et al. (2020) concluded that no strong evidence exists that having population level limits on saturated fat consumption in the United States will prevent CVD or reduce mortality. Similarly, Krauss and Kris-Etherton (2020) argued that the evidence used to support recommendations to reduce saturated fat is often based on low-quality research designs and incorrect beliefs and assumptions and that dietary recommendations to limit SFA send the wrong messages to the public and health professionals. They argued that it is best to have dietary guidelines that encourage eating whole, natural foods and following specific dietary patterns. Others have similarly suggested that rather than target a specific nutrient such as saturated fat intake, CVD risk and overall health would likely see greater improvement by focusing on overall diet quality and eliminating processed foods such as simple carbohydrates (Gershuni, 2018). Diets including higher consumption of fish have been shown to be associated with lower risk of CHD and CHD mortality (Zhang et al., 2020). However, while this is valuable, as noted above, dietary patterns appear more important than specific nutrient groups. Several dietary patterns have been explored for their potential to reduce CVD risk, with the American Heart Association recommending Dietary Approaches to Stop Hypertension (DASH)-stylediets or Mediterranean-style diets (Carson et al., 2020). The DASH diet was first developed and examined by Sacks and colleagues in 1995 with the aim to reduce blood pressure (Sacks, 1995). In this approach, researchers calculate a DASH score based on eight food and nutrient components (fruits, vegetables, whole grains, nuts and legumes, low-fat dairy, red and processed meats, sweetened beverages, and sodium; Fung et al., 2008). In perhaps the most extensive study of the approach, Fung et al. (2008) examined the diet of 88,517 female nurses seven times between 1980 and 2004. There was a direct negative correlation between DASH scores and heart disease; the better the score the less likely a heart attack. The DASH score was also significantly associated with lower risk of stroke and lower levels of CRP. In subsequent meta-analytic research, the DASH diet has been associated with reduced risk of CAD (Yang et al., 2019) and reduced risk of all-cause and CVD and stroke mortality (Soltani et al., 2020). Additionally, in a recent umbrella review, the DASH diet was associated with decreased incidence of CVD, CHD, stroke, diabetes, and lower blood pressure, total cholesterol, LDL cholesterol, and body weight (Chiavaroli et al., 2019). Different food items including a variety of fruits and vegetables, whole grain products, peas, dairy products, salt, sugar, knives, and spoons spread on a table. Fresh Vegetables. A dietary pattern with a variety of fruits and vegetables, whole grains, healthy proteins and minimal processed foods, salt, sugar and alcohol is recommended for cardiovascular health. Another dietary pattern with demonstrated benefit for CVD is the Mediterranean diet. While this dietary pattern may vary slightly according to location, it typically involves high consumption of fruits, vegetables, legumes, grains and cereals (mainly whole grains), nuts, and fish, moderate consumption of milk and dairy products, low consumption of meat and meat products, a low to moderate wine consumption and olive oil (Rosato et al., 2019). As can be seen, the Mediterranean diet shares some similarities to the DASH diet. In a meta-analytic study, it has been associated with lower risk for CVD, CHD/heart attack, stroke and ischaemic stroke but not hemorrhagic stroke (Rosato et al., 2019). Similarly, in an umbrella review, containing 12,800,000 people, higher adherence to a Mediterranean diet was associated with reduced CVD incidence and mortality, CHD and heart attack incidence and overall mortality (Dinu et al., 2018). Finally, researchers have explored an adaption of the Mediterranean diet, (referred to as an Indo-Mediterranean diet). In an illustrative intervention, Singh et al. (2002) recruited 1,000 patients with angina pectoris and myocardial infarctions and had half of them eat the Indo-Mediterranean diet (a diet rich in whole grains, fruits, vegetables, walnuts, and almonds). The control group ate a diet suggested by the National Cholesterol Education Program. Interestingly, the intervention group had fewer heart problems, including heart attacks (both fatal and nonfatal), and showed lower cholesterol levels than the control group. Similarly, over 2 years, people eating the Indo-Mediterranean diet have shown significantly lower rates of pre-heart failure, heart failure, and cardiac arrythmia than a control group (Singh et al., 2020). While the DASH and Mediterranean diets confer cardiovascular and other health benefits, more recently it has been suggested that the Indo-Mediterranean diet has wider food diversity and may produce better outcomes as it includes foods and spices that have enhanced anti-inflammatory and cardioprotective effects (i.e., more whole grains such as millets, porridge, beans, brown rice, and spices such as turmeric, cumin, cloves, black pepper, cardamom, cinnamon, and coriander; Singh et al., 2022). The Indo-Mediterranean diet also excludes animal products, except fish and satisfies all 10 qualities of high-quality foods (i.e., low glycemic index, high nutrient density, food diversity, no trans fat, no or low refined sugar, low salt, moderate healthy fat, high fiber, beneficial effects on gut microbiota, and no preoxidation of foods (Singh et al., 2022). A comparison of the diets is shown in Table 14.2. As can be seen, culture plays an important role in CVD via its influence on diet. A related cultural diet component is alcohol. For example, the French have relatively low levels of CVD even though French food is known to be rich in saturated fats. This French paradox (see Chapter 8) has been linked to moderate consumption of alcohol. One or two glasses of wine per day seem to reduce the incidence of CVD (Castaldo et al., 2019; Snopek et al., 2018). Too much alcohol (e.g., binge drinking) does not lessen your CVD risk (Day & Rudd, 2019; Degerud et al., 2018). Research suggests a J-shaped association between alcohol consumption and CVD. Low levels of drinking relate to lower rates of CVD than abstaining (Degerud et al., 2021; Zhang et al., 2021). Do not be mistaken: This is not a reason for underage drinking, and binge drinking is still not a good thing. Physical Activity In terms of physical activity, not only is exercising useful in reducing the risks of CVD but being physically inactive actually increases the risks (Duran et al., 2022; Varghese et al., 2016). In this respect, standing still actually makes you slide backward on the continuum of health. In a global look at physical activity, the INTERHEART study, leisure time as well as mild to moderate occupational physical activity resulted in lower rates of CVD. Owners of cars and televisions tended to exhibit more sedentary behavior and showed an increased risk of CVD (Held et al., 2012). A key component of treatment and many behavioral interventions to reduce CVD hence includes some form of physical activity. Dibben et al. (2021) presented a review of the effectiveness of exercise-based cardiac rehabilitation in people with CHD. Their study included 85 trials with a total of 23,430 people and found that short-term (6–12 months) exercise-based cardiac rehabilitation was associated with fewer hospitalizations for all causes, fewer deaths from all causes, and fewer deaths from heart attacks. At medium- and long-term follow-up, exercise-based cardiac rehabilitation was associated with a large reduction in deaths from CVD. Additionally, a recent worldwide study involving more than 15,000 participants examined the relationship between self-reported physical activity and both cardiovascular and all-cause mortality. In this study individuals reporting the highest levels of physical activity showed significantly reduced all-cause mortality as well as significantly reduced cardiovascular mortality (Steward et al., 2017). The role of physical activity in the primary and secondary prevention of CVD has also been investigated. Jeong et al. (2019) examined the link between physical activity and CVD mortality among 131,558 people with CVD and 310,240 without CVD. For both groups, physical activity was associated with reduced risk of all-cause death, with stronger results for people with CVD. Furthermore, people with CVD who engaged in high levels of physical activity had a similar or lower risk of mortality than people without CVD (Jeong et al., 2019). People with CVD are often given an exercise prescription (a specific exercise plan) tailored to their needs with variations in the type, frequency, duration, and intensity of exercise. Exercise plans for people with CVD often begin with moderate intensity aerobic exercise followed by resistance training, with a gradual increase in intensity (Tucker et al., 2022). == 14.5 - Treatment Options == === Raw Textbook Page === The specific treatment for CVD is determined by the severity of the symptoms, the size and quantity of areas with ischemias (reduced blood flow), how well the left ventricle of the heart is pumping, and other medical factors such as severity of chest pain. As you can tell from the previous sections, unhealthy lifestyles (e.g., bad eating habits and not getting enough physical activity) are key determinants of whether you will develop a CVD. Correspondingly, changing health behaviors is one of the most critical treatment options to prevent the development of symptoms, relieve the symptoms, and lower the risk of heart attack and death. The primary goal will be to change unhealthy behaviors. The patient is normally admitted to a cardiac rehabilitation program. Rehabilitation programs educate patients on the best way to change their lifestyles and use a combination of physical activity, health and nutrition, and psychological and social support, to improve their overall functioning, physical and psychological quality of life and prevent death (Chindhy et al., 2020). If the person smokes, a smoking cessation program will be prescribed. They will also receive consultations on how to change diet, reduce salt intake, and eat more nutritionally balanced meals. If excessive drinking or not enough physical activity is the issue, it is critical to tackle each of these problems. People may even be told to start taking aspirin. Aspirin, you say? That’s right, not just any headache or pain killer, but aspirin. Many studies have shown that aspirin is beneficial for people with known CHD and probably beneficial for people at high risk of CHD (Bartolucci et al., 2011). In a meta-analysis containing approximately 90,000 people, aspirin reduced the risk of CVD events (i.e., heart attack, stroke, and cardiovascular death combined) and nonfatal heart attack (Bartoloucci et al., 2011). However, there are some risks associated with taking aspirin, including bleeding inside the brain or gastrointestinal tract (García Rodríguez et al., 2016). While the benefits of aspirin therapy outweigh the risks among people who have had a cardiac event (i.e., heart attack or stroke) (Weisman & Brunton, 2022), this treatment is not recommended for individuals with migraine (Kurth et al., 2011) or healthy older people (over 65 years) without CVD (McNeil et al., 2018). Debate remains about the use of aspirin in the primary prevention of CVD among people who have not had a cardiovascular event, as low-dose use is associated with reduced risk for CVD but also increased risk for bleeding (Weisman & Brunton, 2022). Aspirin may be necessary to help restore the blood flow to the affected parts of the heart if the ischemias are serious or the disease continues to worsen despite measures to slow it down (sometimes because the person continues the unhealthy behaviors). In people with critical conditions, surgery is often needed. Surgery There are two main forms of invasive surgery to deal with blockages: angioplasty and cardiac bypass surgery (American Heart Association, 2020). Angioplasty (also known as balloon angioplasty, coronary artery balloon dilation, and percutaneous coronary interventions) is a procedure done to open a partially blocked blood vessel so that blood can flow through it more easily (Figure 14.8). It is most often done on arteries that deliver blood to the heart (coronary arteries) when they are narrowed or blocked by atherosclerosis. The procedure involves the insertion of a thin, flexible tube (catheter) through an artery in the groin or arm, which is carefully guided into the artery that is narrowed. This is not a comfortable procedure. Once the tube reaches the narrowed artery, a small balloon at the end of the tube is inflated. The balloon may remain inflated from 20 seconds to 3 minutes. The pressure from the inflated balloon pushes the plaque against the wall of the artery opening up the passageway to improve blood flow. Once the fat and calcium build-up is compressed, a small, expandable wire tube called a stent is sometimes inserted into the artery to keep it open and reduce the likelihood of further blockages. The procedure make take anywhere from 30 minutes to several hours. Similar alternatives to angioplasty exist. One alternative is laser angioplasty, where the procedure largely mirrors that for angioplasty, however the catheter has a laser at the end rather than a balloon and pulsating beams of light are used to vaporize the blockage. Individuals may also be offered atherectomy, again like angioplasty, however the catheter has a rotating shaver at the end to cut away the blockage. Sometimes rather than removing the blockage, it may be best to just go around it. If you are driving to work, and you hear that there is a traffic jam ahead, you may take an alternative route. In the circulatory system, there are few alternative routes for blood to take, so medical personnel create one. Cardiac bypass surgery (also known as coronary artery bypass graft surgery) involves taking a blood vessel (vein or artery, referred to as a graft) from elsewhere in the body (usually the chest or leg) and using it to redirect blood flow around a severely blocked artery. Typically, the heart is stopped temporarily with a cardiopulmonary bypass machine (known as a heart-lung machine) used to provide oxygen and pump blood during the surgery; in some cases, the surgery can be done without stopping the heart, instead off-pump cardiac bypass uses a stabilizing device to enable work on the heart while it is still beating. Once complete, blood is redirected through the new blood vessel, bypassing the blocked artery and restoring blood flow to the affected portion of the heart muscle. One or more grafts may be needed to restore blood flow, with the surgery taking several hours. Traditionally, cardiac bypass surgery required open heart surgery, whereby access to the heart was gained by cutting through and separating the sternum (chest bone). Less invasive options, referred to as minimally invasive heart surgery, are now sometimes used. In these procedures, small cuts, known as “ports” are made in the chest. Medical instruments are then passed through the ports to attach chest arteries or leg veins to redirect blood flow around a blocked artery. The surgeon uses a camera and video monitor to guide their work, rather than direct observation; sometimes robotic machinery is also used to guide the procedure. Even with minimally invasive heart surgery, the exact procedure may vary. In port-access coronary artery bypass (PACAB), the heart is stopped and a heart-lung machine is used, while minimally invasive direct coronary artery bypass (MIDCAB) often avoids the heart-lung machine, operating while the heart is still beating. Cardiac bypass surgeries typically require hospital stays of 4 to 7 days. Behavioral Interventions Previous sections of this chapter highlight the central role of health behaviors (i.e., tobacco use, diet, physical activity) in the development of CVD and, in turn, cardiac events such as heart attack. Given this role, behavioral interventions to change health behaviors are an essential part of CVD treatment. Two forms of behavioral intervention often utilized for people with CVD are cardiac rehabilitation and stress management interventions. Cardiac Rehabilitation Most health psychological behavioral interventions for CVD take the form of cardiac rehabilitation programs. These programs may take different forms and comprise many different components. Most programs are center-based programs, meaning cardiac rehabilitation is undertaken within medically supervised facilities. However, more recently cardiac rehabilitation has evolved to include home-based cardiac rehabilitation, where the program is provided predominantly or entirely outside of medical settings, using indirect physical activity supervision and remote coaching. Additionally, there has been an increase in technology-assisted cardiac rehabilitation which can be used alone or in conjunction with center-based or home-based cardiac rehabilitation. The effectiveness of cardiac rehabilitation programs is hard to assess because they have many different components and can be delivered in varying formats. If there is a change in risk, a decrease in mortality, or an increase in quality of life, any one of the components could have caused it (or even an interaction of several components). That said, meta-analytical studies have shown that rehabilitation programs in general have accounted for reductions in mortality compared with that for control groups (Bishop, 2019). Also, programs with higher levels of physical activity have produced better results (Dibben et al., 2018). International reviews of cardiac rehabilitation practices worldwide found varying data regarding program length and delivery. For example, Pesah et al. (2017), in a review of cardiac rehabilitation in 50 countries, found that programs lasted for a median of 20 sessions and were generally delivered by physicians, nurses, and physiotherapists, while in Europe, Benzer et al. (2017) found the average number of sessions was 43.5 (3 to 96 sessions). A clinical trial of a community-based cardiac rehabilitation program in the Netherlands showed significant improvement in risk factors such as lack of exercise, smoking, and being overweight (Minneboo et al., 2017). Rehabilitation not only leads to increases in healthy behaviors and decreases in unhealthy behaviors, but it also saves money. Despite the proven benefits of cardiac rehabilitation, referral, participation, and adherence rates are relatively low (Chindhy et al., 2020; Chong et al., 2021). In fact, more than 80% of eligible people do not participate (Thomas et al., 2019). Some barriers have the potential to affect all people, however others demonstrate cultural variation. Barriers to cardiac rehabilitation include low referrals and endorsement by health professionals, low SES, cost, travel, age, medical comorbidities, being a smoker, and poor psychological well-being (Chindhy et al., 2020; Resurrección et al., 2019). Sex is also a significant barrier, with females found to be 11% less likely to be referred to cardiac rehabilitation than males (Li et al., 2018). Females are also more likely not to attend cardiac rehabilitation or to not complete the program (Chindhy et al., 2020; Resurrección et al., 2019). Many reasons have been proposed, including cardiac rehabilitation programs being tailored to males and male dominated, females lacking finances, being single, of older age, and prioritizing the needs of their families over their own (Chindhy et al., 2020; Clark et al., 2012; Galati et al., 2018). Furthermore, race is also a barrier to cardiac rehabilitation participation. Compared to White Americans, Black Americans, Latinx Americans, and Asian Americans have been found to be 20%, 36%, and 50% less likely, respectively, to be referred to cardiac rehabilitation (Li et al., 2018). While home-based and technology-assisted cardiac rehabilitation programs may not address sex and racial disparities, as they offer comparable results to center-based rehabilitation (i.e., Chong et al., 2021; Thomas et al., 2019), they may offer valuable alternatives to people who face barriers in attending center-based rehabilitation. Mobile-health approaches to cardiac rehabilitation have also been suggested as an alternative to traditional center-based cardiac rehabilitation (Bostrom et al., 2020). Recently, yoga-based cardiac rehabilitation has also been explored as a way to provide affordable access to services that may improve cardiovascular health for people in low- and middle-income countries and subgroups within high-income countries (females, older people, and people of low SES) (Prabhakaran et al., 2020). A review of six randomized controlled trials examining yoga-based cardiac rehabilitation (where yoga was added to standard care) found that this approach appears to confer some benefits. From the studies conducted to date, included in Bruce et al.’s review (2021), yoga-based cardiac rehabilitation was associated with improvements in some physiological and biochemical markers including reduced body mass index; a significant reduction in BMI, heart rate, SBP and DBP; and a trend toward better glycemic control, but findings have been mixed. For example, one study showed no significant differences in a range of cardiovascular physiological risk factors, and two studies found no significant change in triglycerides, total cholesterol, HDL, and LDL cholesterol at three months (Bruce et al., 2021). In contrast, one study found significant reductions in total triglycerides, HDL, and VLDL at 1 year. Improvements in patient perceptions of cardiac function and self-reported return to pre-heart attack activities, as well as improved parasympathetic activity and cardiac autonomic tone were noted in some studies, and there was a trend toward improved left ventricular systolic function (Bruce et al., 2021). Only one study examined major adverse cardiac events (composite of any cause death, nonfatal MI, nonfatal CVA, or emergency cardiovascular hospitalization) and found that while people in the yoga-based cardiac rehabilitation group experienced less total major adverse cardiac events, the difference was not significant. With regard to psychological health, people who participated in yoga-based cardiac rehabilitation reported significant improvements in depression, anxiety, stress and quality of life (Bruce et al., 2021). While the findings from yoga-based cardiac rehabilitation appear promising, due to the small number of studies to date with diverse populations of differing inclusion criteria, varying rehabilitation programs, and relatively small sample sizes, the findings should be viewed with some caution. Further research with larger samples and longer follow-up would be beneficial in better understanding any possible benefits from yoga-based cardiac rehabilitation. Interventions to Reduce Stress and Negative Emotions Given the number of psychological factors involved in CVD, a number of interventions to reduce stress and negative emotions have also been tried, and the news is good (Blumenthal et al., 2016; Sherwood et al., 2017). Richards et al. (2018) reviewed 35 studies that collectively evaluated 10,703 patients who received psychological interventions for CHD and found that while psychological intervention did not reduce the risk of total mortality at a median follow-up of 13 months, it reduced the risk of cardiac mortality at a median follow-up of 57 months. Psychological interventions also showed no evidence of reducing the risk of a subsequent nonfatal heart attack or need for revascularization (treatment to restore blood supply). Reductions in depression symptoms, anxiety, and stress were found for patients who received a psychological intervention compared to a control group. The psychological interventions varied but targeted depression, anxiety, stress, and Type A behavior including anger and hostility, and used approaches such as relaxation techniques, awareness and self-monitoring, cognitive challenging/restructuring techniques, and emotional support and client-led discussion. Many interventions also included attempts to increase awareness about cardiac risk factors and behavior change techniques for cardiac risk factors (i.e., diet, smoking). One of the earliest and most ambitious interventions was conducted by Friedman et al. (1986). They observed 1,013 heart patients for 4.5 years to determine whether their Type A behaviors could be altered. There were three experimental groups: A control section of 270 patients received group cardiac counseling, an experimental section of 592 patients received both group cardiac counseling and Type A behavioral counseling, and 151 patients served as a comparison group. The results were startling. At the end of the study, 35.1% of participants given cardiac and Type A behavior counseling reduced their Type A behavior compared with 9.8% of participants given only cardiac counseling, and the heart attack recurrence rate was only 12.9%. The recurrence rate in the control group was 21.2%, and the comparison group fared worse (recurrence rate of 28.2%). There was also a significant difference in the number of cardiac deaths between the experimental and control participants, clearly showing that altering Type A behavior reduces cardiac morbidity and mortality (Friedman et al., 1986). Therefore, as hostility is a critical component of Type A behavior, interventions designed specifically to reduce hostility appeared worthy of further exploration. However, Hamieh et al. (2020) in an examination of data from the GAZEL cohort study, a study where middle-age French electricity and gas workers were surveyed annually commencing in 1989, found that from 1994 to 2014, among 10,304 participants followed prospectively for cardiac events, irritability was the only baseline hostility trait associated with cardiac events. Furthermore, this association was no longer significant when considered with depression, but depression remained significantly associated with cardiac events, even when considered with all hostility traits. Hamieh et al. (2020) concluded that interventions that aim to reduce the risk of hostility-related cardiac events should target depression as a modifiable factor. Interventions to improve stress management and increase social support also reduce the effects of CVD (Blumenthal et al., 2016; Mondesir et al., 2018; Suchy-Dicey et al., 2022; Sherwood et al., 2017). Blumenthal et al. (2016) examined the effects of standard cardiac rehabilitation compared to cardiac rehabilitation with stress management training over a 5-year follow-up period in 151 patients aged 36 to 84 with established CHD. Patients in the standard cardiac rehabilitation group received education about CHD, nutrition counseling, two classes on the role of stress in CHD, and exercised 35 minutes (three times per week). In contrast, those in the cardiac rehabilitation with stress management training group received the same information and completed the same exercise as the standard group but also completed 12 weekly 1.5-hour sessions of stress management training involving education, group support, and cognitive behavior therapy. Blumenthal et al. (2016) found that patients in the cardiac rehabilitation and stress management group demonstrated greater reduction in overall stress and lower rates of cardiac events (18% compared to 33%). Both groups showed improvements in CHD biomarkers, lipids, physical activity and exercise capacity and experienced less cardiac events than patients who did not receive either form of cardiac rehabilitation. Given the positive role played by a healthy diet (discussed previously), it is not surprising that many interventions focus on changing diets. The OmniHeart Trial compared three diets designed to reduce CVD risk (Kovell et al., 2020). One diet was high in carbohydrates. Two diets replaced carbohydrates with either unsaturated fat or protein. The lower carbohydrate diets improved the CVD risk factors. Recently, it was shown that a healthy diet, regardless of what type of macronutrients it contains (fat vs. carbohydrate patterns) reduced inflammation markers associated with the development of CVD and, therefore, it was suggested that diet recommendations should be simplified to emphasize healthy foods rather than one specific dietary pattern (Kovell et al., 2020). 00zrmntlasyk2j06qj4ejm558xnw64f 2693791 2693790 2024-12-29T19:44:03Z Atcovi 276019 /* 14.5 - Treatment Options */ 2693791 wikitext text/x-wiki == 14.1 - Cardiovascular Disease: Definitions and Prevalence == * '''Definition:''' Diseases affecting the heart and circulatory system, including: ** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension. ** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases. * '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity. ---- ==== Coronary Heart Disease (CHD): ==== * '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD). * '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms. * '''Key Statistics:''' ** Leading cause of death in the U.S. (1 in 5 deaths in 2020). ** Affects 20.5 million Americans aged 20+. ** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females). ** An American has a heart attack every 40 seconds. ---- ==== Hypertension (High Blood Pressure): ==== * '''Definition:''' Chronic elevation of blood pressure. ** Normal: <120/80 mm Hg. ** Elevated: 120–129/<80 mm Hg. ** Stage 1: 130–139/80–89 mm Hg. ** Stage 2: >140/90 mm Hg. ** Hypertensive Crisis: >180/120 mm Hg. * '''Prevalence:''' ** Affects 116.4 million Americans (58.7M males, 57.7M females). ** Higher rates in Black Americans (57.6% males, 53.2% females). * '''Risk Factors:''' Obesity, poor diet, lack of physical activity. * '''Awareness Issue:''' Over 35% of cases go undiagnosed. ---- ==== Stroke: ==== * '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation. * '''Types:''' ** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases). ** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%). ** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes. * '''Impact:''' ** 7 million Americans have had a stroke. ** Leading cause of disability and 4th (females) and 5th (males) leading cause of death. ** Higher mortality in Southeastern U.S. ("Stroke Belt"). * '''Risk Disparities:''' ** Males have higher incidence; females experience more strokes due to longer life expectancy. ---- ==== Global and Cultural Trends: ==== * '''CVD Mortality Rates:''' ** Highest in Eastern Europe (e.g., Lithuania, Russia). ** Declining rates in the U.S., U.K., and Brazil. ** High disparities in low-income countries. * '''Framingham Heart Study:''' ** Longitudinal study initiated in 1948 in Framingham, Massachusetts. ** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors. ** Continues to provide insights with third-generation participants. ---- ==== Key Terms and Concepts: ==== * '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension. * '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity. * '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath. * '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness. ---- ==== Trends and Recommendations: ==== * Efforts to reduce mortality through public health education and lifestyle interventions. * Emphasis on addressing disparities in diagnosis and treatment across different demographic groups. ===Raw Textbook Page=== Diseases resulting from problems with the heart and the circulatory system are all gathered under the general heading of cardiovascular disease (CVD). The most common are coronary heart disease (CHD) and heart failure (both commonly referred to as heart attacks), strokes, and hypertension or high blood pressure (medically referred to as essential hypertension). Others include abnormal heart rhythms, congenital heart disease (abnormalities present at birth that could relate to the heart muscle, valves, or blood vessels), heart valve failure, electrical conduction disorder, heart muscle disease (cardiomyopathy), rheumatic heart disease (damage to the heart muscle and valves from rheumatic fever), pulmonary heart disease (enlargement of the right ventricle of the heart), peripheral artery disease (narrowing of the arteries that carry blood from the heart to other parts of the body such as the arms and legs), cerebrovascular disease (a condition affecting the blood vessels in the brain), and diseases of the veins, arteries, and lymph nodes (the last three collectively are called vascular diseases). Many other medical conditions put you at risk for CVD. Major risk factors include high blood pressure, prediabetes/diabetes, kidney disease, and obesity (Bays et al., 2022). To get a better feel for how CVDs develop, this is a good time to refresh yourself on the circulatory system (described in Chapter 4). Coronary heart disease (CHD), the leading cause of death in America, is a condition in which the small blood vessels that supply blood and oxygen to the heart narrow due the accumulation of fat or scar tissue. It is also called coronary artery disease (CAD). As the coronary arteries narrow, blood flow to the heart can slow down or even stop, causing chest pain, shortness of breath, or a heart attack. Warning signs of heart attack include chest discomfort (i.e., fullness, pressure, squeezing, pain) that lasts more than a few minutes and which may come and go; pain or discomfort in the back, neck, jaw, stomach or one or both arms; shortness of breath; cold sweat; nausea or vomiting; and light-headedness (American Heart Association, 2022). Importantly, symptoms may vary by sex, with females somewhat more likely to experience shortness of breath, nausea or vomiting, and jaw or back pain compared with males (American Heart Association, 2022). Evidence suggests that, overall, increasing numbers of Americans can state the five most common signs of heart attack (i.e., jaw/neck/back discomfort, weakness or light-headedness, chest discomfort, arm/shoulder discomfort, and shortness of breath), increasing from 39.6% in 2008 to 50.2% in 2017 (Fang et al., 2019). However, while CHD is the leading cause of death for females in the United States, between 2009 and 2019, knowledge of this fact declined among females of all races/ethnicities and ages (except those age 65 years or older) (Cushman et al., 2021). Concerningly, knowledge about heart attack warning signs also decreased; there was a significant difference over the 10-year period for White American females on 7 out of 13 warning signs, 3 out of 13 warning signs for Black American females and Latino American females, with no significant differences for Asian American females (Cushman et al., 2021). Estimates suggest that 20.5 million Americans age 20 years or older have CHD, with a total prevalence of 7.1% (males 8.7% and females 5.8%; Tsao et al., 2023). Additionally, CHD prevalence varies by race/ethnicity, with rates of 4.4% for Asian Americans, 5.4% for Black Americans, 5.7% for White Americans, and 8.6% for Native Americans age 18 years or older (Tsao et al., 2023). CHD is the leading cause of death for American males and females, with heart disease responsible for 1 in 5 deaths in 2020, and an American having a heart attack every 40 seconds (CDC, 2022). Americans experience 605,000 new heart attacks and 200,000 recurrent heart attacks annually, with the average age of first heart attack 65.6 years for males and 72 years for females (Tsao et al., 2023). Rates of CHD and CHD outcomes vary according to race/ethnicity and sex (see Table 14.1 and Figure 14.2). For a further exploration of cultural factors, refer to the section below. Hypertension or high blood pressure is a condition in which the blood pressure (the force with which blood flows through the blood vessels) remains chronically elevated. Blood pressure that is below 120/80 mm Hg is considered normal, while blood pressure that is 120–129/80 mm Hg is considered elevated (American Heart Association, 2017). Hypertension is diagnosed when blood pressure is above these levels. Hypertension has two stages—Stage 1: blood pressure is 130–139/80–89 mm Hg, and Stage 2: blood pressure is above 140/90 mm Hg (American Heart Association, 2017). Any blood pressure above this level (above 180/120 mm Hg) is considered a hypertensive crisis (American Heart Association, 2017). It has been estimated that among Americans 20 years or older, 116.4 million have hypertension (58.7 million males and 57.7 million females; Benjamin et al., 2019). Hypertension varies by age, sex and race. Between 2013 and 2016, the prevalence of hypertension was 26.1% for Americans ages 20 to 44 years, 59.2% for those 45 to 64 years, and 78.2% for people aged 65+ (Benjamin et al., 2019). Males have higher prevalence of hypertension in younger age ranges (before age 65); after 65, hypertension is more prevalent in females (Benjamin et al., 2019). Black Americans have some of the highest rates of hypertension in the world; 57.6% and 53.2% among males and females, respectively, with Black Americans more likely to develop hypertension than White Americans (Benjamin et al., 2019). In a 10-year longitudinal cohort study of 10,801 American adults, Howard et al. (2017) found that Black American males had a 24% higher incidence of hypertension than White American males, and that while rates increased for White American females with age, Black American females ages 45 to 54 had a 93% higher risk of hypertension than White American females (this reduced to 18% higher risk at age 75+). Similarly, in a study of 4,060,585 adults with overweight or obesity, Young et al. (2018) found that the odds of hypertension were significantly higher for Americans of most races/ethnicities when compared with White Americans. That is, the odds of hypertension compared to White Americans were as follows: Native Hawaiians/other Pacific Islanders (85% higher), Black Americans (double), Asian Americans (42% higher), Native Americans (17% higher) and Latino Americans (4% lower). Regarding prevalence, hypertension was most prevalent among Black Americans (47.3%) and least prevalent among Latinx Americans (27.7%; Young et al., 2018). Many factors may contribute to age, sex and racial disparities. While research continues to explore these factors, it is also pertinent to note that typically, there are no obvious symptoms of hypertension, with more than 35% of people unaware that they have it (Benjamin et al., 2019). This lack of awareness is problematic as untreated hypertension increases the risk for heart attack and stroke. Blood Pressure. High blood pressure is one of the easiest ways to detect a risk for heart problems and is associated with obesity, bad diets, and not enough physical activity. Stroke is a type of CVD that affects the arteries leading to and within the brain. A stroke occurs when a blood vessel to the brain is either blocked by a clot or bursts. When that happens, the part of the brain affected cannot get the blood and oxygen it needs and begins to die (American Stroke Association, 2023a). Ischemic stroke (when a blood vessel that supplies blood to the brain is obstructed), is the most common, accounting for 87% of all strokes (American Stroke Association, 2023b). Stroke may also take a number of other forms including hemorrhagic stroke (when a blood vessel ruptures, bleeding into the brain, compressing surrounding brain tissue; 13% of strokes; American Stroke Association, 2023c), transient ischemic attack or mini-stroke (where blood flow to the brain is temporarily blocked), brain stem stroke (when the stroke occurs in the brain stem), and cryptogenic stroke (strokes without a known cause) (American Stroke Association, 2023d). Stroke is the fourth-leading cause of death in females and fifth-leading cause of death in males in the United States, and is the leading cause of disability among adults of both sexes (Bushnell et al., 2018). Additionally, rehabilitation costs and lost productivity result in a financial burden that costs the United States more than $75 billion per year (Bushnell et al., 2018). Approximately 7 million Americans ages over 20 years report having had a stroke (Benjamin et al., 2019), with predictions that by 2030, compared to 2012, an additional 3.4 million Americans ages over 18 years will have experienced a stroke (Ovbiagele et al., 2013). It is suggested that 795,000 Americans, one every 40 seconds, experience a stroke (Benjamin et al., 2019). In 2016, stroke accounted for approximately one in every 19 American deaths, with a person dying on average every 3 minutes and 42 seconds. Geographical disparities exist, with people in the American Southeast (i.e., North Carolina, South Carolina, Georgia, Tennessee, Mississippi, Alabama, Louisiana, and Arkansas), known as the “stroke belt,” experiencing higher rates of stroke mortality (Benjamin et al., 2019). Prevalence of stroke varies according to age, sex and race. The relationship between sex and stroke is complex with males and females both at increased risk as they age (Benjamin et al., 2019). Males have higher incidence of stroke than females, but due in part to differences in life expectancy, females experience approximately 55,000 more strokes per year than males in the United States (Bushnell et al., 2018). We will discuss more of the physiological and psychological aspects of CVDs later in the chapter. Prevalence of Cardiovascular Disease By 2030, almost 23.6 million people will die from CVD worldwide (World Health Organization [WHO], 2017). Although that is still a lot of people, the good news is that heart disease death rates have been dropping. In the United States, the annual death rate attributed to CHD dropped 19.2% between 2010 and 2020 (Tsao et al., 2023). Also, between 2010 and 2020, age-adjusted stroke deaths dropped from 39.1 to 38.8 deaths per 100,000 people; however, while age-adjusted stroke death rates decreased 2.3% for females, they increased 1.3% for males (Tsao et al., 2023). Deaths from CVD have dropped also in some European countries (WHO, 2017). Yet heart problems and those related to the circulatory system are still the leading causes of death globally, disproportionately affecting low- and middle-income countries (Thomas et al., 2018; Figure 14.3) and problems due to heart disease vary from country to country. Figure 14.3 Description Fatality due to heart disease is affected also by differences in health behaviors among countries. Some of these varying major risk factors are blood pressure, blood cholesterol, smoking, physical activity, and diet. Heart disease is responsible for high mortality in countries of varying geographic location and income, with the highest mortality observed in Lithuania, the Republic of Moldova, The Russian Federation, Hungary, Romania, and the Czech Republic (Nowbar et al., 2019). Heart disease mortality trends are similar in the United States, United Kingdom, and Brazil (high- and middle-income countries), while although trending down, mortality rates are “strikingly high” in Ukraine (a lower middle-income country) compared to other countries (Nowbar et al., 2019). The number of deaths from varying forms of CVD in the United States during 2020 is shown in Figure 14.4. Figure 14.4 Description Much of our understanding of heart disease comes from a large-scope longitudinal study begun more than 50 years ago. In the late 1940s, the U.S. Public Health Service selected the town of Framingham, Massachusetts, to be the site of a large-scale study to understand why heart disease had become North America’s number one killer. A total of more than 5,000 healthy male and female residents between ages 30 and 60 were enrolled as the first cohort of participants. Every 2 to 4 years, study participants are given extensive medical examinations including a medical history, blood tests, and other tests of current health status. The Framingham Heart Study was the first to establish a relationship between levels of cholesterol and high blood pressure and their effect on heart disease risk (Apel et al., 1997). The researchers found that a lifestyle with a bad diet, sedentary living, smoking, and unrestrained weight gain accelerated the occurrence of cardiovascular problems. Even today, new information about heart disease is based on the latest assessments made with Framingham participants (Juul et al., 2021; Pollevick et al., 2021). The children and grandchildren of the original cohort participate today and are referred to as the offspring or third-generation cohorts (Park et al., 2021; Sawicki et al., 2021). Let’s take a look at cultural variations in these often-fatal diseases. == 14.2 - Cultural Variations and Developmental Issues == Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023). Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018). Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018). Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD. '''Key Points:''' # '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities. # '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity. # '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks. # '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes. # '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications. # '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks === Raw Textbook Page === CVDs are a common global problem, with prevalence increasing from 271 million in 1990 to 530 million in 2019, and mortality increasing from 12.1 million in 1990 to 18.6 million in 2019 (Roth et al., 2020). Moreover, CVDs are likely to increase further due to population growth and aging, particularly in Northern Africa, Western, Central and Southern Asia, Latin America, the Caribbean and Eastern and Southeast Asia (Roth et al., 2020). The highest CVD mortality rates are currently found in Uzbekistan, the Solomon Islands, and Tajikistan, while CVD mortality is at its lowest in France, Peru, and Japan (Roth et al., 2020). As with other chronic illnesses, there are some significant cultural differences in the incidence of CVDs (Chand et al., 2017). Variation in CVDs occur not only between but also within countries. As shown in Figure 14.5, Hispanic Americans show the lowest numbers of deaths due to heart disease followed by White Americans, Black Americans, and Asian or Pacific Islander Americans who do not show large differences. In the case of stroke, American females experience more strokes than American males (Tsao et al., 2023). Among females, Black American females experience the highest numbers of strokes, followed by White American females and Hispanic American females; Asian American females have the lowest numbers of stroke (Tsao et al., 2023). The incidence variations are often due to differences in health literacy, knowledge levels about the disease, differences in health behaviors, and risk factors among different cultural groups (Bell et al., 2018; Magnani et al., 2018). For example, Hamner and Wilder (2008) used the Coronary Heart Disease Knowledge Test to measure knowledge of CVD in rural Alabama females. The average on the test was 8.50 (out of 20). The participants were at significant risk for CVD. They recognized that smoking and obesity were issues, but were less aware of factors such as personality, oral contraceptive use, hypertension, diabetes, and family history. How you would fare on the test? You should do pretty well once you get to the end of this chapter. There are many cultural differences that could account for higher CVD incidence. The higher risk for heart attack shown by South Asians is attributed, in part, to a higher prevalence of diabetes (Gholap et al., 2011; Volgman et al., 2018) and other risk factors (e.g., Bathula et al., 2008) in some South Asian populations. Higher rates can also be due to the psychological experiences of different groups that relate to CVD (Baker et al., 2001). For example, hostility, anger, and social support, each of which play a key role in the development of CVD, also vary across cultures and map onto larger cultural dimensions (see Chapter 3). Hypertension also shows strong cultural differences and is found at a higher rate in Black Americans (Tsao et al., 2023). Some evidence suggests an intriguing sociobiological reason. High blood pressure is positively correlated to dark skin color, which could induce more discrimination (i.e., because of being darker skinned) (Laidley et al., 2019). Anger and other components of the Type A personality such as competitiveness and time urgency are closely tied to the individualistic or collectivistic dimension of culture. People in individualistic cultures are more competitive, which is viewed as a desirable trait. Those in collectivistic cultures are more cooperative, and competition against members of one’s group is often discouraged. Social support, similarly, is seen more in collectivistic cultures (Shavitt et al., 2016; Wang & Lau, 2015). Time orientation varies with another cultural dimension: fluid time versus fixed time. In fixed-time cultures, such as most cultures in North America, time is fixed: When you say you will meet someone at 10:00 a.m., you mean exactly that. In fluid-time cultures, such as those in India and among certain groups such as the Maori in New Zealand and Native Americans, meeting someone at 10:00 a.m. really means you will show up anywhere between 10:15 a.m. and 10:30 a.m. This is understood and expected, and no one is frustrated when someone is late. In fact, in many East Indian American communities, people set appointments and specify whether they mean an exact time or Indian Style Time. Accelerated blood pressure due to time constraints is generally less common in such cultures, and the stress of being late accordingly is different as well. There is some indication that increased risk of CHD in some ethnic populations may be due to basic differences in the epidemiology of atherosclerosis, the accumulation of fatty substances in the blood vessels. Some racial and ethnic populations are also inadequately prescribed antiplatelet therapy—daily aspirin doses—despite their higher risk (Johansen et al., 2015). As discussed in Chapter 9, there are cultural differences in health care–seeking behavior and patient–practitioner interactions. For example, South Asians do not use ambulances as often when experiencing heart emergencies (reflecting cultural differences or possibly geographical proximity to hospitals; Ben-Shlomo et al., 2008). Additionally, in a recent review, Chinese and South Asian people in Canada and non-White people in America were also shown to delay presenting for medical care when experiencing chest pain (Wechkunanukul et al., 2017). There is also evidence of differences in how doctors manage patients with chest pain according to their cultural backgrounds. Doctors may have a lower threshold for giving thrombolytic therapy (treatment that breaks up blood clots) to South Asian men with chest pain because they are aware of the increased risk of CHD in this population (Ben-Shlomo et al., 2008). Furthermore, sex or gender concordance between doctor and patient (being the same sex or gender) may be important in health outcomes, including for cardiac conditions. For example, Greenwood et al. (2018), examined mortality among people admitted to Florida hospitals for heart attack between 1991 and 2010, finding that males and females experienced similar outcomes when treated by a female doctor, but that female patients treated by male doctors had higher mortality. Lau et al. (2021) and the Cardiovascular Disease in Women Section of the American College of Cardiology, in a recent systematic review examining patient–provider sex/gender concordance in studies from 2009 to 2019 in a range of medical settings, reported similar findings, noting that 6 out of 8 studies found that gender concordance affected clinical outcomes. New areas of research investigate biological markers of CHD that may vary by culture. For example, C-reactive protein (CRP) concentrations are associated with risk of CHD (Peikert et al., 2020; Zhuang et al., 2019). Zahodne et al. (2019) conducted a longitudinal study of 12,382 people enrolled in the Health and Retirement Study to examine relationships between race/ethnicity, social disadvantage and inflammation. The study population was 77.8% non-Latinx White American, 13.1% non-Latinx Black American, and 9.0% Latinx American. At baseline, both non-Latinx Black Americans and Latinx Americans had higher CRP than non-Latinx White Americans. Also, non-Latinx Black Americans demonstrated the greatest increase in CRP over the 4-year study period. A range of psychosocial factors, including educational disadvantage, depression, external locus of control, discrimination, and smoking, were proposed as pathways to explain CRP levels. In another detailed study of the links among culture, biology, and CHD, Nasir et al. (2008) assessed nearly 7,000 individuals in the Multi-Ethnic Study of Atherosclerosis (MESA). The researchers focused on different types of calcifications that correspond to atherosclerosis and CHD. Clear ethnic differences emerged. The highest prevalence of calcifications was observed in White Americans, followed by Latinx Americans and Black Americans, with the lowest levels of calcification among Chinese Americans. Recently, Budoff et al. (2018) examined relationships between calcification and cardiovascular health outcomes over a 10-year period among 6,814 participants from the MESA cohort, finding that cardiovascular event rates ranged from 1.3% to 5.6% for people with a calcification score of 0, compared to 13.1% to 25.6% for people with a calcification score above 300. While calcification rates differed via race/ethnicity, with calcification present among 43.5% of Black Americans, 50.2% of Chinese Americans, 45.2% of Latinx Americans, and 57% of non-Latinx White Americans, the relationship between calcification score and cardiovascular events was independent of race/ethnicity. The exact relationship between calcification and race remains unclear. For example, Lewis et al. (2006) found that everyday discrimination (combined racial and nonracial discrimination), but not racial discrimination alone, was associated with calcification in Black American females. Also, Everage et al. (2012) found surprisingly that calcification was inversely correlated to perceived racism among Black Americans. While Reddy et al. (2022) reported that Black Americans living in low racially segregated neighborhoods in young adulthood were about half as likely to develop calcification as those living in high segregation neighborhoods. Two other cultural dimensions beyond ethnicity can also influence CVD. In the control versus constraint dimension (Trompenaars, 1997), control cultures believe that they have absolute control of their outcomes (similar to having an internal locus of control). Contrastingly, people in constraint cultures believe everything is in the hands of God or fate. Those in control cultures may have higher levels of anxiety and stress and correspondingly have more risk for heart problems (Baker et al., 2001). Similarly, the level of emotionality may make a difference as well. Neutral cultures, such as that of Japan, do not sanction the open display of emotions. In contrast, affect cultures, such as that of Italy, place a premium on the display of emotions. Not expressing and processing emotions could also lead to higher levels of CVD (Chapman et al., 2013). Sex differences, another element of culture, appear in et al. (2008) tested for the not just the incidence of CVD (Bishop, 2019) but also in patient–practitioner interactions. Adams sources of uncertainty and sex bias in doctors’ diagnoses and decision making relating to CVD. They randomly selected male and female doctors in England and the United States and showed them video clips of actors portraying patients with CVD. The video clips included patients of different ages, sexes, ethnicities and socioeconomic status (SES). The doctors were interviewed about their decision making. Adams et al. (2008) found differences in male and female doctors’ responses to different types of patient information. The female doctors remembered information differently than did the males (e.g., more patient cues). All doctors paid more attention to male patients’ age and considered more age-related disease possibilities for males than females. Additionally, Colella et al. (2015), in a meta-analysis of 19 studies containing 241,613 participants, found that females are significantly less likely to be referred to cardiac rehabilitation than males; males are almost 1.5 times more likely to be referred. Developmental Issues in Cardiovascular Disease A number of developmental issues connect to CVD. For example, low birth weight is now known to be associated with increased rates of CVD (Liang et al., 2021). Suboptimal growth in infancy and rapid childhood weight gain exacerbates the effects of impaired prenatal growth. As we age, we have a greater risk for developing CVDs. A large part of this risk is due to wear and tear on our arteries and the accumulation of plaque that increases with time, but this risk also has psychosocial correlates. People go through different stages of physical, social, and cognitive development. Erikson, for example, hypothesized that we progress through eight different life stages, each with its own challenges and milestones. The life changes that accompany social development can serve as stressors that, in turn, can lead to higher risks for CVDs. Some milestones include puberty, graduation from high school or college, a first job, and perhaps losing a first job. Even relationships, dating, and marriage can be important correlates of CVDs (see the upcoming section on stress for more on the impact of transitions). Satisfying relationships can provide social support, and acrimonious relationships or divorces could raise blood pressure and otherwise negatively impact health. Important negative life events corresponding to developmental stages are often present in the lead-up to a heart attack. For example, risk of heart attack is higher in people who have experienced divorce (Dupre et al., 2015), and the incidence of heart attack has been found to increase 21-fold within 24-hours after the death of a significant person in one’s life (Mostofsky et al., 2012). Also, people who have never married, are divorced or widowed, have been found to be at higher risk of death after a heart attack than people who are married (Dupre & Nelson, 2016). Other life events such as natural disasters have also been noted to be related to heart attack. In the wake of Hurricane Sandy, the heart attack incidence increased 22%, with a 31% increase in 30-day mortality (Swerdel et al., 2014). The findings are more mixed with regards to earthquakes, but some studies have shown a relationship between heart attack incidence and earthquakes (and tsunamis) (Bazoukis et al., 2018). Although heart disease has been studied more extensively in males, studies of females also show that heart disease patients experience a significantly larger number of negative life changes, many of them related to family life (Vaccarino & Bremner, 2017). Beyond developmentally related life events, it is also important to factor in development and take a life span approach to CVD because one of the main psychosocial correlates of CVD—social support—changes over time and social isolation and loneliness are associated with higher risk of CVD (Valtorta et al., 2016), with loneliness appearing more important than social isolation (Valtorta et al., 2018). Younger women with poor social networks have higher levels of heart problems (Carroll et al., 2013). Surprisingly, older retired females with more extensive social networks also had a higher incidence of cardiac problems. The additional networks for older females could have come with more mental burdens. To understand conflicting data such as these, it is important to look at how social support changes over the life span (see Chapter 6 and Eslami et al., 2017). == 14.3 - Correlates of Cardiovascular Disease == === Physiological Correlates === # '''Atherosclerosis and Arteriosclerosis:''' #* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks. #* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages. # '''Risk Factors:''' #* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history. #* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking. #* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity. # '''Diabetes and CVD:''' #* Both type 1 and type 2 diabetes significantly increase CVD risk. #* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis. #* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes. # '''Discrimination and CVD:''' #* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation. ---- === Psychological Correlates === # '''Emotions and Personality Traits:''' #* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk. #* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks. # '''Social Support:''' #* Social networks and emotional support play crucial roles in preventing and managing CVD. #* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack. #* Marital status, social participation, and perceived support significantly affect outcomes. # '''Socioeconomic Status (SES):''' #* In high-income countries, low SES correlates with increased CVD risk. #* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles. ---- === Key Findings === * Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks. * Psychological and social factors are as critical as physiological factors in both prevention and recovery. * Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management. === Stress and Cardiovascular Health === City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow. Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems. Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk. Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take. Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks. In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems === Raw Textbook Page === Physiological Correlates of Cardiovascular Disease The primary physiological antecedent for the incidence of CVDs is atherosclerosis (Heart Research Institute, 2022). Microscopic accumulations of fats within artery walls progress to visible fatty streaks as early as childhood (Fernando et al., 2020). This accumulation of fat, often in the form of plaque, reduces and sometimes blocks the arteries supplying blood to the heart. The plaque build-up can get so great as to tear the artery, creating a snag in which a blood clot can form to block the artery. Lesions are sometimes observed in adolescents and become increasingly common with age. This interference in blood flow to the heart is what causes heart attacks. A related condition is the hardening of the arteries, or arteriosclerosis, in which the arteries lose their elasticity and are more susceptible to blockages from clots or plaques (Andrus et al., 2015). Some of the main physiological risk factors cannot be changed: age, sex, and family history. As mentioned, the risk for having a CVD increases as a person gets older, and males younger than 50 are more likely to develop a problem. At around 50 years of age, corresponding to when most females reach menopause, females experience a significant elevation in their risk for CVD (El Khoudary et al., 2020). It is also clear that having a parent or relative with a CVD greatly increases the incidence rates of CVDs (Yamada et al., 2008). Genetic linkage analyses of families and sibling pairs have implicated several loci and candidate genes in predisposition to CVD, and genes that contribute to genetic susceptibility to these conditions are beginning to be identified (Qureshi et al., 2015). Other physiological factors predicting the incidences of CVDs are high blood pressure, diabetes, high cholesterol level, inactivity, and being overweight or obese (Bishop, 2019). For example, people with type 1 diabetes have a two to eight times increased risk of CVD and death (Katsarou et al., 2017). While Einarson et al. (2018), in a review containing 4,549,481 people, found that over a 10 year period among people with type 2 diabetes, the prevalence of CVD was 32.2% (27.6% in males and 27.2% in females); males had higher prevalence for each specific type of CVD (stroke, angina, heart attack, heart failure, CAD) than females. Notably, females with gestational diabetes are more than twice as likely to develop CVD within the first decade after pregnancy and this increased risk is not dependent on whether they develop type 2 diabetes (Kramer et al., 2019). In fact, CVD is the most common cause of death among people with diabetes (Baena-Díez et al., 2016). Einarson et al. (2018) recently found that CVD accounted for 50.3% of all deaths among people with type 2 diabetes, with most deaths from CAD (29.7%). The increased risk of CVD in type 2 diabetes is due, in part, to irregularities with fat storage and metabolism. Diabetic dyslipidemia, for example, is characterized by elevated triglycerides, low high-density lipoprotein cholesterol (HDL) and increased low-density lipoprotein cholesterol (LDL) particles (Jialal & Singh, 2019). As you can see, this second group of physiological factors can all be modified depending on a person’s health behaviors (more on this later). Cross sections of normal artery and artery affected by atherosclerosis, on the left and right respectively. Cross-Sections of Human Arteries. The artery on the left is normal (clear wide opening). The artery on the right has a much smaller opening due to the accumulation of plaque (atherosclerosis). Another important issue in the modifiable physiological risk factors is the fact that different cultural groups vary in their genetic predispositions. Some ethnic groups are more prone to high blood pressure (van Laer et al., 2018). Some groups are more prone to be overweight. These underlying differences in risk factors could explain cultural differences in CVD. Hypertension is significantly higher in Black Americans than White Americans and minority status is significantly associated with diabetes (Bell et al., 2018). Additionally, research suggests that cardiovascular reactivity may vary according to race/ethnicity and sexuality. Salomon and Jagusztyn (2008) looked at the relationship between discrimination and cardiovascular responses to interpersonal incivility among Black Americans, Latinx Americans, and White Americans. Participants completed a measure of past discrimination which was related to higher resting systolic blood pressure (SBP) among Latinx American participants and lower resting SBP among White American participants. Reporting being discriminated against was related to attenuated SBP and heart reactivity among Latinx American participants. Discrimination was not related to resting levels or reactivity among Black American participants. Their findings suggest that the relationship between discrimination and cardiovascular risk differs by ethnicity. Similarly, Goosby et al. (2015) examined the relationship between perceived discrimination and cardiovascular responses in Black Americans ages 10 to 15 years. Participants completed a survey about perceived daily discrimination and measures of CRP, SBP and diastolic blood (DBP) pressure were taken. Being discriminated against was significantly associated with higher CRP, SBP and DBP, even after controlling for age, sex, body mass index, waist circumference and mother’s education. Cardiovascular reactivity may also be related to sexuality. Juster et al. (2019) explored the relationship between cardiovascular reactivity and sexuality and disclosure of sexuality in Canadian adults. Participants were exposed to laboratory stressors, with measures of heart rate and blood pressure collected in visits before and after stressor exposure. Compared to sex and age-matched heterosexual people, gay/bisexual males demonstrated higher heart rates, and lesbian/bisexual females demonstrated marginally higher mean arterial blood pressure. Disclosure of sexuality was unrelated to cardiovascular reactivity. As testified to by this empirical evidence, cardiovascular reactivity—changes in heart rate and blood pressure in response to stress—varies greatly between individuals. This reactivity is a key physiological risk factor for the development of CVD (Wirtz & von Känel, 2017). As discussed in Chapter 8, there are also significant ethnic and geographical differences in obesity, tobacco use, diet, and activity levels—all key health behaviors related to CHD (discussed in detail below). Psychological Correlates of Cardiovascular Disease There is strong evidence for a link between psychological characteristics and the development of CVDs (Levine et al., 2021; Neylon et al., 2013), making this set of diseases a prime candidate for use of the biopsychosocial approach of health psychology. Of all the chronic diseases, CVDs are perhaps most illustrative of the importance of focusing on psychological factors together with biological factors (Levine et al., 2021). Psychological factors, such as personality traits (e.g., hostility), anger, depression, social support, and stress, and health behaviors, such as dietary habits and physical activity, have all been intrinsically tied to the incidence and progression of CVDs (Levine et al., 2021). Perhaps the best-known controversy regarding the psychological causes of heart attacks revolves around a constellation of personality characteristics called the Type A personality (see Chapter 6). Friedman and Rosenman (1974) noted that people with heart conditions who showed a sense of time urgency (always doing more than one thing at the same time), competitiveness, and hostility in their interactions with other people were found to have a higher risk for CHD. In contrast, the Type B personality is relaxed, patient, and easygoing. The original finding that was greeted with enthusiasm did not bear further examination (Shekelle et al., 1985). However, a number of people still believe that having a Type A personality, in general, is not necessarily a positive attribute. What seems more accurate is that being hostile is the problem (Chida & Steptoe, 2009; Suls, 2013). Hostility and anger are negative emotions that can trigger a heart attack and even sudden death among individuals who are at risk, with anger or emotional upset common an hour before a heart attack (Smyth et al., 2016). A study of 1,000 males over a 30-year follow-up period found that high scores on a measure of trait anger were associated with a three- to sixfold increase in CHD (Chang et al., 2002). Additionally, outbursts of anger have been associated with increased risk of heart attack, acute coronary syndrome, ischemic and hemorrhagic stroke, and arrhythmia in the two hours after an outburst (Mostofsky et al., 2014). Hostility can also lessen the benefits of receiving social support (Holt-Lunstad et al., 2008). The role of hostility is so critical that for every dollar spent on anger-management treatments or hostility therapy, there is an approximate savings of two dollars in hospitalization costs in the following 6 months (Davidson et al., 2007). How’s that for a good deal? The relationship between two variables is rarely straightforward. Research has begun to look for the factors that may mediate the relationship between hostility and the increased risk of CHD (see Chapter 6 for more on mediation). One such variable is carotenoid, a substance in most plants that is known to have antioxidant properties. Antioxidants, in turn, may be mediators for atherosclerosis. Ohira et al. (2008) found that high hostility predicted future low levels of some serum carotenoids, which may help to explain the association of hostility and cardiovascular risk observed in epidemiologic studies. Other negative emotions play a role in angina, heart attack, heart failure, stroke (Daskalopoulou et al., 2016), sudden cardiac death (Daskalopoulou et al., 2016; Obrova et al., 2022), recurrent major cardiovascular events (Sverre et al., 2020), and re-admission to hospital among people with heart failure (Sokoreli et al., 2018). Feeling sad and depressed may also increase your likelihood of heart problems and the progression of CVD (Bishop, 2019). Meta-analyses have found that the presence of clinically significant depression can increase the risk of CHD by 30% to 90% among otherwise healthy individuals (Carney & Freedland, 2017). In an early, important study, Anda et al. (1993) studied the relationship of both depressed affect and hopelessness to CHD incidence using data from a large cohort of 2,832 North American adults. The participants had no history of CHD or serious illness at baseline. Anda et al. (1993) found that people who were depressed were significantly more likely to have a fatal heart attack (relative risk 1.5). Depression was also associated with an increased risk of nonfatal heart attacks. Similarly, in a more recent meta-analysis, containing 323,709 participants, Wu and Kling (2016), found that depression is associated with a significantly higher risk of heart attack and coronary death. Also, in other studies, depression after a diagnosis of CAD has been found to confer a two-fold higher risk of death (May et al., 2017). Depression has also been associated with adverse outcomes after heart attack. In an early study of survivors of heart attacks, Blumenthal (2008) found that depression was related to increased mortality in the 6-month period after the first heart attack. In a recent longitudinal exploration of the relationship between depression and post-heart attack outcomes, Worcester et al. (2019) found that depression was a significant predictor of death at 5, 10, and 15 years (but not 20 and 25 years) post-heart attack, independent of age and heart attack severity. Moreover, people with mild depression demonstrated greater mortality than those with low or moderate to severe depression. Feeling hopeless, an emotion often accompanying depression, can independently predict the incidence of CVD as well (Gidron et al., 2007). Indeed, when exploring longitudinal relationships between hopelessness, depression and hypertension, it was found that hopelessness had a direct relationship on SBP but not DBP, whereas, depression did not have a direct relationship with either blood pressure measure (Roane et al., 2017). A variety of different types of social supports also relate to CVD (Uchino et al., 2020). One model explains that stress and depression may both positively and negatively influence social support (perceived and received), which in turn is associated with biological pathways (i.e., health behaviors and treatment adherence) and psychological pathways (i.e., appraisals and quality of life) (Uchino et al., 2020). Both of these pathways interact with each other and are seen to link to biological pathways such as immune, cardiovascular, and neuroendocrine function, which are hypothesized to directly influence CVD development and progression (Uchino et al., 2020). Social support could influence the development of CVD by buffering the person from the effects of stress, consequently safeguarding the person from the deleterious effects that stress has on the circulatory system, including cardiovascular reactivity, although findings in this regard have been mixed (Creaven et al., 2020; Teoh & Hilmert, 2018). For example, Lee and Way (2019) found that social support was only associated with reduced CRP for people with high self-esteem, while Creaven et al. (2020) found that higher social support was associated with increased SBP and DBP reactivity, more suggestive of a dual-effects model of social support. Teoh and Hilmert’s (2018) systematic review of the relationship between social support and cardiovascular reactivity, also explored a dual-effects model. They hypothesized that when people are more engaged during stress, social support provides social comfort, reducing cardiovascular reactivity, and when not engaged, social support acts as social encouragement, increasing cardiovascular reactivity. Teoh and Hilmert (2018) found that, overall, people with social support had less cardiovascular reactivity than those without support. Also, their hypothesis was partially supported; when more engaged, social support reduced SBP and marginally reduced DBP reactivity. Supportive networks also ensure that a person is more likely to get help and to comply with doctor’s orders. If a male is at risk for a heart attack and they are not supposed to eat fatty foods, smoke, or drink too much, a supportive partner and good friends are likely to make sure that they do not. In support of this link, Schultz et al. (2017), followed over 6,000 people in a cohort study for a median of 3.7 years, finding that unmarried people with suspected or confirmed heart conditions had higher risk of all-cause death, cardiovascular death, and cardiovascular death or heart attack than married people. People with CHD or CHD risk factors with large social networks are also more likely to adhere to medication recommendations (Mondesir et al., 2018), while among people attending cardiac rehabilitation, those with larger social networks, have better coping efficacy and health behaviors (Tkatch et al., 2011). Similarly, larger and closer social networks are associated with improved motor functioning in people participating in rehabilitation after stroke (Podury et al., 2021). However, in a meta-analysis, functional support (the function the support serves for a person) and not structural support (marital status or living arrangement) was associated with overall and medication adherence in people with hypertension (Magrin et al., 2015). Four men of different cultures stood outdoors and talked in a happy mood. One of them is holding a few burning incense sticks. Social Support. Social support can aid people in living a healthy lifestyle, following medical advice, and achieving good physical, psychological, and social health. Not having healthy social networks, another measure of social support, is also related to the incidence of CVD, but some studies show this relationship is not always significant when demographic differences such as income level or marital status are accounted for (Morris et al., 2008). In a sizable study on the power of social support, Sundquist et al. (2004) examined whether low social participation predicted incidence rates of CHD. They followed 6,861 Swedish females and males for nearly 10 years and found that persons with low social participation (as measured by an interview) had the highest risk of CHD. They were more than twice as likely (relative risk 2.15) to have another heart attack than those with high social participation. This increased risk remained even after controls were added for education and smoking habits (Sundquist et al., 2004). In another large study, Ikeda et al. (2008) examined prospectively the association between social support and risk of CHD and stroke incidence and mortality within a cohort of 44,152 Japanese males and females. Low social support was associated with higher risk of stroke mortality in males. Similarly, within a cohort study of 7,846 British older people, while small social network was associated with CVD mortality, network size appeared less important than SES trajectories, with a stable low SES associated with nearly twice the risk of CVD mortality and 94% higher risk of CVD death than people with a stable high SES (Stringhini et al., 2018). Sometimes not getting social support can be fatal. People in the Enhancing Recovery in Coronary Heart Disease (ENRICHD) trial with depression and those with lower perceived social support (even without elevated depressive symptoms) were at increased risk for death (Lett et al., 2007). The importance of social support remains over time, with higher levels of social support after heart attack associated with lower risk for death during hospitalization and at 10 to 13 years (Weiss-Faratci et al., 2016). Such findings have prompted a call for standardized screening for depression in patients with CVD (Jha et al., 2019) and a form of special social support screening for CVD patients with depression (Thombs, 2008). Not having enough social support is often related to not having enough resources in general and is strongly linked to SES. The data regarding SES and the risk of CVD is still evolving. In high-income countries, low SES is consistently associated with CVD risk factors and CVD; however, evidence is more limited for low-income and middle-income countries (de Mestral & Stringhini, 2017). Recently, Rosengren et al. (2021), in the Prospective Urban Rural Epidemiologic (PURE) study, examined CVD risk and mortality in adults 35 to 70 years from 367 urban and 302 rural communities in 20 countries. The final sample contained 154,169 people from five low-income, 11 middle-income, and four high-income countries, followed for an average of 7.5 years. They found that people with lower education (the proxy measure of SES) in low-income and middle-income countries had higher CVD incidence and mortality. While they had better risk factor profiles, people with lower education in low-income and middle-income countries also had poorer health care. Further research about the relationship between SES in CVD in low-income and middle-income countries is still required. In an early study exploring the relationship between SES and CVD, Marmot et al. (1991) studied British civil servants in the Whitehall part of London (the Whitehall studies). They showed that higher rates of CVD were seen in males of lower employment grade. At every rung of the bureaucratic ladder, males in the lower positions were worse off. There are other possible mediators. In a study of an ethnically diverse population, participants with more emotional social support showed higher high-density lipoprotein (HDL) cholesterol levels. The mediators? Physical activity and wine intake. These are but two health behaviors influenced by social support that may have the result of reducing CVD risk (Fischer et al., 2008). Stress You may have heard of the television show and movies of the same title, Sex and the City, but have you considered that a show could be called Stress and the City? In fact, city living marks the brain (Abbott, 2011). A range of environmental factors associated with city living have been linked with CVD, such as traffic density, pollution, the amount of green space, walkability, access to fast food restaurants and supermarkets, and living in disadvantaged neighborhoods (Bhatnagar, 2017; Malambo et al., 2016). Living in an urban environment can be associated with increased stress. For example, frequent exposure to noise is associated with increased stress and physiological changes in blood pressure, heart rate, and cardiac output, which can be compounded by exposure to air pollution and contribute to CVD risk (Bhatnagar, 2017). A little stress can go a long way. Not only does being stressed influence your health behaviors (as described in Chapter 8), but it also increases your likelihood of developing a number of diseases. In Chapter 5, we described how our body’s reactions to short-term physical stressors are thought to have evolved to get us out of danger. You should also remember that these same responses, when activated for a long period of time (chronic stress), can begin to break down the body’s systems (e.g., allostatic load). Cardiovascular problems are some of the most common ways that the body breaks down. Superficially, the relationship is pretty intuitive. What happens when we get stressed (Figure 14.6)? At the physiological core of the response, the catecholamines and cortisol pumped into the bloodstream increase blood flow, thus raising blood pressure. The heart is pumping faster, and blood is shunting around the body faster. There are also changes in how we metabolize food for energy (details in Chapter 5). This constellation of factors that accompany the experience and process of stress take a toll on the circulatory system and aid in the incidence of CVD (Kivimäki & Steptoe, 2018). Personal life stress and work stress have been associated with a 1.1 to 1.6-fold increase risk of incident CHD and stroke (Kivimäki & Steptoe, 2018). Stress may impact cardiovascular health risk via effects on cardiac electrical stability, lowered threshold for arrythmia (irregular heart beat), increased blood pressure, increased inflammation, increased coagulation/blood clot formation, plaque disruption or reduced blood flow to the heart (Kivimäki & Steptoe, 2018). A sizeable body of research documents how various stressors, particularly those at work, accentuate CVD. No one mechanism explains the relationship between work stress and CVD (Kivimäki & Kawachi, 2015); possible mechanisms may include impacts on blood pressure, metabolic syndrome, and lifestyle behaviors (Kivimäki & Kawachi, 2015), or increased cardiac electrical instability and hypercoagulability (increased likelihood of developing blood clots) (Virtanen & Kivimäki, 2018). Together with the work front, environmental stress (especially that caused by low SES) and stress from interpersonal relationships at home have also been associated with the incidence of CVD. Even watching sports can be stressful and lead to a heart attack! Lin et al. (2019), in a meta-analysis of 13 studies, examined the relationship between the emotional stress of watching football and the risk of cardiovascular events. They found that watching a football match was associated with increased risk of nonfatal CVD (combined heart attack and stroke), nonfatal heart attack, and fatal CVD (combined heart attack and stroke), for males and females (although risk was slightly higher for male spectators). The risk of fatal CVD was higher when spectators’ teams lost. So, take it easy during March Madness and the Super Bowl! Acute stress (e.g., a person being the victim of an assault or having to deliver a very difficult presentation) can also trigger heart problems if the individual already is at risk because of atherosclerosis (Lagraauw et al., 2015). Early work showed that catastrophic events such as earthquakes and the death of a spouse could also initiate cardiovascular changes (Fagundes et al., 2018). Bazoukis et al. (2018) reviewed the impact of major earthquakes on the incidence of acute coronary syndromes, finding that the Armenia, Athens, Christchurch, Great East Japan, Great Hanshin-Awaji, Niigata-Chuetsu, Northridge, Noto Peninsula, and Sichuan earthquakes were all associated with increased incidence of acute coronary syndromes. The nature and timing of the effects varied. However, not all earthquakes showed such effects, with the Loma Prieta, Newcastle, and Thessaloniki earthquakes not significantly related to acute coronary syndrome admissions or cardiac mortality. Therefore, Bazoukis et al. (2018) stated that definitive conclusions cannot be made regarding the type or strength of relationships between earthquakes and adverse cardiac health outcomes. The work-stress and CVD relationship has garnered the most research attention (Kivimäki & Kawachi, 2015). We can all acknowledge the fact that working can be stressful. Even if you enjoy your job, having to work can still challenge the body. The stress from work can be even more dangerous if you are overworked, have too many roles to fulfill, are not clear what your job role is, are bored with your job, or do not have support at work. No matter what the exact cause, work stress can accentuate the chances of developing CVD. Even job status may be important. For example, Stringhini et al. (2017), in a multicohort meta-analysis containing 1.7 million people, examining occupational class (as a proxy for SES) as a risk factor for premature mortality found that people who worked in low occupational positions (i.e., lower clerical, services and sales workers, unskilled, semiskilled and skilled workers) had higher risk of mortality than those in higher occupational positions (i.e., higher professionals and managers, higher clerical, services, and sales workers); males—hazard ratio of 1.42, females—hazard ratio of 1.34. Similarly, people in lower occupational positions were at higher risk of death from CVD than those in higher occupational positions, even after adjusting for age, sex, race/ethnicity, marital status, SES, and seven health risk factors (i.e., harmful alcohol use, insufficient physical activity, current tobacco use, elevated blood pressure, salt/sodium intake, diabetes, and obesity) (Stringhini et al., 2017). There are also some interesting cultural differences in the work–stress relationship. For example, the pattern of lower status occupations having higher risk may be a “Western” phenomenon. In Japan, people that work in higher status occupations (i.e., professional and managerial roles), are the ones at higher risk for CHD, even after controlling for alcohol and tobacco use (Zaitsu et al., 2019). Indeed, in Japan, there is even a term for death from overwork: karoshi (Eguchi et al., 2016). Like Japan, in Korea the relationship between occupational status and CVD differs from that reported in “Western” countries. In a sample of nearly 23,000 Korean workers, Ahn et al. (2019) found no differences in hypertension, hyperlipidemia, or CVD for males or females employed in five occupational groups—unskilled manual workers, unskilled managers and professionals, clerks, service and sales workers, and skilled manual workers. Recent work on stress and heart disease aims to examine how stress can increase negative health behaviors such as smoking (Byrne & Mazanov, 2016) and a concept called vital exhaustion. Vital exhaustion is marked by feeling low, extreme fatigue, and increased irritability. People with CVD who show vital exhaustion are twice as likely to have recurring heart attacks (Frestad & Prescott, 2017). == 14.4 - Health Behaviors and Cardiovascular Disease == ==== Impact of Smoking on Global Health and CVD ==== * '''Prevalence:''' Over 1 billion active tobacco users globally (Roth et al., 2020). * '''Mortality:''' Tobacco use caused 8.71 million deaths in 2019; 36.7% linked to cardiovascular diseases (CVDs). * '''Gender Disparity:''' Smoking rates are higher in males (33.5%) compared to females (6.8%), with 75.4% of smoking-related deaths among men. * '''Passive Smoking:''' Increased CVD risk by 28% at home/work (case-control studies) and by 12% (cohort studies). ==== Physiological Effects of Smoking ==== * Smoking impacts physiological systems, contributing to high blood pressure, arterial stiffening, inflammation, insulin resistance, and cholesterol changes. * Historical data from Doll et al. (2004) revealed a 10-year reduction in life expectancy for lifelong smokers born between 1900–1930, with life expectancy improvements for those who quit at any age. ==== Benefits of Smoking Reduction and Cessation ==== * Reducing daily cigarette consumption significantly lowers risks of coronary heart disease (CHD) and stroke (Chang et al., 2021). * Smoking cessation shows rapid risk reduction within 5 years, with near-normalized CVD risk after 10–15 years (Duncan et al., 2019). ==== Policy Interventions and Effectiveness ==== * Massachusetts Tobacco Control Program (MTCP, 1993): ** Smoking prevalence decreased by 29%. ** CVD-related deaths reduced by 31% over a decade. * Smoke-free laws lower smoking initiation rates and reduce exposure to secondhand smoke, including among children. ---- === Diet and CVD: Role and Patterns === ==== Cholesterol and Risk ==== * High LDL cholesterol is a primary risk factor, particularly in conjunction with diabetes (20–50% reduced CV risk when cholesterol is controlled). ==== Dietary Patterns and Recommendations ==== * '''Saturated Fat Debate:''' ** Earlier guidelines on reducing saturated fat intake have been questioned; focus is now on overall dietary patterns and quality (Astrup et al., 2020). * '''Beneficial Diets:''' ** DASH Diet: Emphasizes fruits, vegetables, whole grains, and low sodium; reduces risks for CHD, stroke, and high CRP levels. ** Mediterranean Diet: Similar to DASH, includes olive oil, nuts, and moderate wine consumption; linked to lower CHD/stroke incidence and mortality. ** Indo-Mediterranean Diet: Enhances DASH/Mediterranean diets with anti-inflammatory ingredients like turmeric, brown rice, and spices, offering greater diversity and benefits. ==== Alcohol's Role ==== * Moderate wine consumption is associated with lower CVD risk (French Paradox). However, binge drinking or excess consumption negates these benefits. ---- === Physical Activity: Prevention and Recovery === ==== Global Trends and Impact ==== * Inactivity increases CVD risks (INTERHEART study). Ownership of cars/TVs correlates with sedentary behavior and CVD prevalence. ==== Rehabilitation and Outcomes ==== * Exercise-based cardiac rehabilitation (6–12 months): Fewer deaths and hospitalizations for CVD patients (Dibben et al., 2021). * Physical activity significantly reduces all-cause and CVD-specific mortality, even in patients with existing CVD (Jeong et al., 2019). ==== Prescriptive Exercise Plans ==== * Designed for intensity and progression: Moderate aerobic activity transitioning to resistance training for tailored recovery (Tucker et al., 2022). This structured summary highlights the multifaceted approach to CVD prevention and management, focusing on lifestyle changes, policy interventions, and cultural considerations. === Raw Textbook Page === Tobacco Use The number one behavior to avoid if you want to minimize your risk for CVDs is smoking (Mermelstein, 2019). You can see why we have spent so much time on this topic (see Chapter 8). Tobacco use poses a significant challenge to global health, with more than 1 billion active tobacco users (Roth et al., 2020), and together with being an important risk factor for other chronic diseases such as cancer (see Chapter 13), cigarette smoking has also been identified as an important factor in the development of CVDs (Bishop, 2019). In 2019, tobacco use (primary smoking, secondhand smoke, chewing tobacco), accounted for 8.71 million deaths globally, of which 36.7% were due to CVDs (Roth et al., 2020). Also, in 2019, global rates of smoking remained higher among males than females (33.5% and 6.8%, respectively), resulting in 75.4% of smoking-related deaths occurring among males (Roth et al., 2020). However, through environmental exposure, tobacco also poses a risk to those who do not smoke. For example, Khoramdad et al. (2020), in a meta-analysis of case-control studies, containing 10,672 participants, found that passive smoking increased the risk of CVD by 28%, with people exposed to secondhand smoke at home and at work at highest risk. Similarly, in a meta-analysis of cohort studies, containing 2,313,935 participants, passive smoking increased the risk of CVD by 12%, with people exposed to secondhand smoke at home, work and public places at highest risk (Khoramdad et al., 2020). In one of the clearest demonstrations of the impact of tobacco use, Doll et al. (2004) followed approximately 35,000 British doctors from 1951 to 2001 and found that the dangers of smoking varied with cohorts. Males born in 1900 through 1930 who smoked only cigarettes and continued smoking died on average about 10 years sooner than lifelong nonsmokers. Quitting was beneficial regardless of the age at which it was attempted. Males who quit when 60, 50, 40, or 30 years of age gained, respectively, about 3, 6, 9, or 10 years of life expectancy (Doll et al., 2004). Smoking also influences CVD though its impact on physiological systems. For example, smoking has been associated with changes in blood pressure, arterial stiffening, increased inflammation, insulin resistance, and increased HDL cholesterol levels (Gallucci et al., 2020). What effects do smoking bans or governmental regulations to curb smoking have? Consider the following case. In 1993, the state of Massachusetts introduced the Massachusetts Tobacco Control Program (MTCP). This reduced how many people smoked in the state by 29%. That’s not all. There was a 31% decline in death rates due to CVD (from 1993 to 2003, 425 fewer CVD deaths; Kabir et al., 2008). Additionally, smoke-free laws have been associated with lower odds of commencing smoking, of being a current smoker and fewer days smoking among American adolescents and young adults (Song et al., 2015). Furthermore, despite some people’s concerns, in their review, Monson and Arsenault (2017) found that increased restrictions on smoking in public locations, has not been associated with increased smoking at home. Also, since the introduction of public smoking bans, children have been found to be exposed to less secondhand smoke at home (Nanninga et al., 2018). Benefits have been found when people reduce or cease smoking. For example, people who reduce their daily cigarette consumption from “heavy” (15–20 cigarettes per day or more) to “light” (less than 10 cigarettes per day) have been found to have a significantly lower risk of CVD (CHD and stroke) than those who continue heavy smoking (Chang et al., 2021). Also, people who reduce from heavy to moderate (10–19 cigarettes per day), and heavy to light smoking, have shown significantly reduced risk for CHD (ischemic heart disease and heart attack), compared with continuing heavy smokers (Chang et al., 2021). Additionally, ceasing smoking confers benefits; people who cease smoking show a rapid decline in CVD risk compared to those who continue to smoke, resulting in significantly lower risk of CVD within 5 years of cessation (Duncan et al., 2019). However, compared to never smoking, it takes 10 to 15 years for past heavy smoking to cease being associated with higher risk of CVD (Duncan et al., 2019). Over time, smoking cessation also results in reduced CRP levels (predictors of CVD as discussed earlier) (Gallus et al., 2018). A warning message from Health Canada. Smoking and Cardiovascular Disease. A warning on a Canadian cigarette packet. Those who can quit smoking significantly reduce their risk for contracting a cardiovascular disease. Long Description People around the world smoke. Smoking is more common in some cultures than in others, and the link between CVD and smoking is not the same across cultures. For example, smoking rates in China are among the highest in the world, with rates declining, but approximately 50% of males ages over 15 smoke (Ma et al., 2020). The level of smoking in China is such that, 30% of the world’s current smokers reside there and Chinese people now account for nearly one third of all tobacco-attributable deaths (Roth et al., 2020). In other countries, high levels of other unhealthy behaviors such as poor diet also accompany high smoking rates. Recent attention has turned to the presence of risk factors that accentuate the effect of smoking, and some of the most important are dietary factors and cholesterol. Diet Your food choices play a large role in your overall health and well-being. What you eat determines the levels of nutrients available for your cells and ensures the smooth and healthy functioning of your bodily symptoms (Greger, 2015). There are many dietary factors that influence the incidence and progression of CVD (Figure 14.7), and diet is a factor that affects the interaction between culture, psychology, and behavior (Van Horn et al., 2008). Your diet can influence your cholesterol level, your blood pressure, your tolerance for glucose (and consequently your risk for diabetes), your likelihood to be overweight, and even how your blood coagulates. Each of these factors is associated with the development of CVD (Carson et al., 2020; Fung et al., 2008). Cholesterol is an important risk factor for CVD. It is found in most animal products and is an important component of cell walls and membranes. Cholesterol is also a main component of plaque. As discussed in Chapter 8, we have high-density lipoprotein (HDL) and low-density lipoprotein (LDL) cholesterol. As the level of cholesterol in the blood increases, the risk of CVD increases as well (Jeong et al., 2018). LDLs seem to be the primary factor, and a number of treatments (e.g., statins) aim to reduce the LDL levels in the bloodstream (Rached & Santos, 2020). A high cholesterol level is even more likely to cause CVD in people with other health issues such as diabetes, where reducing cholesterol may reduce CV risk by 20% to 50% (Vesa et al., 2020). Recommendations are no longer provided about specific levels of cholesterol required to prevent heart disease; rather, the focus is on healthy dietary patterns. Additionally, while previously it was recommended to reduce saturated fats, views about this have changed more recently, with suggestions that there is increasing evidence to the contrary about reducing saturated fatty acids (SFA; Astrup et al., 2020). It has been argued that meta-analyses show that reducing SFA confers no benefits for CVD or total mortality and that the health effects of foods cannot be predicted by their nutrient group alone; overall macronutrient distribution or dietary pattern must be considered (Astrup et al., 2020). Furthermore, not all foods that contain SFA are problematic. For example, whole-fat dairy, dark chocolate, and unprocessed meat, are SFA-rich foods but are not associated with increased CVD risk (Astrup et al., 2020). Therefore, Astrup et al. (2020) concluded that no strong evidence exists that having population level limits on saturated fat consumption in the United States will prevent CVD or reduce mortality. Similarly, Krauss and Kris-Etherton (2020) argued that the evidence used to support recommendations to reduce saturated fat is often based on low-quality research designs and incorrect beliefs and assumptions and that dietary recommendations to limit SFA send the wrong messages to the public and health professionals. They argued that it is best to have dietary guidelines that encourage eating whole, natural foods and following specific dietary patterns. Others have similarly suggested that rather than target a specific nutrient such as saturated fat intake, CVD risk and overall health would likely see greater improvement by focusing on overall diet quality and eliminating processed foods such as simple carbohydrates (Gershuni, 2018). Diets including higher consumption of fish have been shown to be associated with lower risk of CHD and CHD mortality (Zhang et al., 2020). However, while this is valuable, as noted above, dietary patterns appear more important than specific nutrient groups. Several dietary patterns have been explored for their potential to reduce CVD risk, with the American Heart Association recommending Dietary Approaches to Stop Hypertension (DASH)-stylediets or Mediterranean-style diets (Carson et al., 2020). The DASH diet was first developed and examined by Sacks and colleagues in 1995 with the aim to reduce blood pressure (Sacks, 1995). In this approach, researchers calculate a DASH score based on eight food and nutrient components (fruits, vegetables, whole grains, nuts and legumes, low-fat dairy, red and processed meats, sweetened beverages, and sodium; Fung et al., 2008). In perhaps the most extensive study of the approach, Fung et al. (2008) examined the diet of 88,517 female nurses seven times between 1980 and 2004. There was a direct negative correlation between DASH scores and heart disease; the better the score the less likely a heart attack. The DASH score was also significantly associated with lower risk of stroke and lower levels of CRP. In subsequent meta-analytic research, the DASH diet has been associated with reduced risk of CAD (Yang et al., 2019) and reduced risk of all-cause and CVD and stroke mortality (Soltani et al., 2020). Additionally, in a recent umbrella review, the DASH diet was associated with decreased incidence of CVD, CHD, stroke, diabetes, and lower blood pressure, total cholesterol, LDL cholesterol, and body weight (Chiavaroli et al., 2019). Different food items including a variety of fruits and vegetables, whole grain products, peas, dairy products, salt, sugar, knives, and spoons spread on a table. Fresh Vegetables. A dietary pattern with a variety of fruits and vegetables, whole grains, healthy proteins and minimal processed foods, salt, sugar and alcohol is recommended for cardiovascular health. Another dietary pattern with demonstrated benefit for CVD is the Mediterranean diet. While this dietary pattern may vary slightly according to location, it typically involves high consumption of fruits, vegetables, legumes, grains and cereals (mainly whole grains), nuts, and fish, moderate consumption of milk and dairy products, low consumption of meat and meat products, a low to moderate wine consumption and olive oil (Rosato et al., 2019). As can be seen, the Mediterranean diet shares some similarities to the DASH diet. In a meta-analytic study, it has been associated with lower risk for CVD, CHD/heart attack, stroke and ischaemic stroke but not hemorrhagic stroke (Rosato et al., 2019). Similarly, in an umbrella review, containing 12,800,000 people, higher adherence to a Mediterranean diet was associated with reduced CVD incidence and mortality, CHD and heart attack incidence and overall mortality (Dinu et al., 2018). Finally, researchers have explored an adaption of the Mediterranean diet, (referred to as an Indo-Mediterranean diet). In an illustrative intervention, Singh et al. (2002) recruited 1,000 patients with angina pectoris and myocardial infarctions and had half of them eat the Indo-Mediterranean diet (a diet rich in whole grains, fruits, vegetables, walnuts, and almonds). The control group ate a diet suggested by the National Cholesterol Education Program. Interestingly, the intervention group had fewer heart problems, including heart attacks (both fatal and nonfatal), and showed lower cholesterol levels than the control group. Similarly, over 2 years, people eating the Indo-Mediterranean diet have shown significantly lower rates of pre-heart failure, heart failure, and cardiac arrythmia than a control group (Singh et al., 2020). While the DASH and Mediterranean diets confer cardiovascular and other health benefits, more recently it has been suggested that the Indo-Mediterranean diet has wider food diversity and may produce better outcomes as it includes foods and spices that have enhanced anti-inflammatory and cardioprotective effects (i.e., more whole grains such as millets, porridge, beans, brown rice, and spices such as turmeric, cumin, cloves, black pepper, cardamom, cinnamon, and coriander; Singh et al., 2022). The Indo-Mediterranean diet also excludes animal products, except fish and satisfies all 10 qualities of high-quality foods (i.e., low glycemic index, high nutrient density, food diversity, no trans fat, no or low refined sugar, low salt, moderate healthy fat, high fiber, beneficial effects on gut microbiota, and no preoxidation of foods (Singh et al., 2022). A comparison of the diets is shown in Table 14.2. As can be seen, culture plays an important role in CVD via its influence on diet. A related cultural diet component is alcohol. For example, the French have relatively low levels of CVD even though French food is known to be rich in saturated fats. This French paradox (see Chapter 8) has been linked to moderate consumption of alcohol. One or two glasses of wine per day seem to reduce the incidence of CVD (Castaldo et al., 2019; Snopek et al., 2018). Too much alcohol (e.g., binge drinking) does not lessen your CVD risk (Day & Rudd, 2019; Degerud et al., 2018). Research suggests a J-shaped association between alcohol consumption and CVD. Low levels of drinking relate to lower rates of CVD than abstaining (Degerud et al., 2021; Zhang et al., 2021). Do not be mistaken: This is not a reason for underage drinking, and binge drinking is still not a good thing. Physical Activity In terms of physical activity, not only is exercising useful in reducing the risks of CVD but being physically inactive actually increases the risks (Duran et al., 2022; Varghese et al., 2016). In this respect, standing still actually makes you slide backward on the continuum of health. In a global look at physical activity, the INTERHEART study, leisure time as well as mild to moderate occupational physical activity resulted in lower rates of CVD. Owners of cars and televisions tended to exhibit more sedentary behavior and showed an increased risk of CVD (Held et al., 2012). A key component of treatment and many behavioral interventions to reduce CVD hence includes some form of physical activity. Dibben et al. (2021) presented a review of the effectiveness of exercise-based cardiac rehabilitation in people with CHD. Their study included 85 trials with a total of 23,430 people and found that short-term (6–12 months) exercise-based cardiac rehabilitation was associated with fewer hospitalizations for all causes, fewer deaths from all causes, and fewer deaths from heart attacks. At medium- and long-term follow-up, exercise-based cardiac rehabilitation was associated with a large reduction in deaths from CVD. Additionally, a recent worldwide study involving more than 15,000 participants examined the relationship between self-reported physical activity and both cardiovascular and all-cause mortality. In this study individuals reporting the highest levels of physical activity showed significantly reduced all-cause mortality as well as significantly reduced cardiovascular mortality (Steward et al., 2017). The role of physical activity in the primary and secondary prevention of CVD has also been investigated. Jeong et al. (2019) examined the link between physical activity and CVD mortality among 131,558 people with CVD and 310,240 without CVD. For both groups, physical activity was associated with reduced risk of all-cause death, with stronger results for people with CVD. Furthermore, people with CVD who engaged in high levels of physical activity had a similar or lower risk of mortality than people without CVD (Jeong et al., 2019). People with CVD are often given an exercise prescription (a specific exercise plan) tailored to their needs with variations in the type, frequency, duration, and intensity of exercise. Exercise plans for people with CVD often begin with moderate intensity aerobic exercise followed by resistance training, with a gradual increase in intensity (Tucker et al., 2022). == 14.5 - Treatment Options == === Treatment for Cardiovascular Disease (CVD) === Treatment of CVD varies based on symptom severity, ischemic areas, left ventricle function, and other medical factors. Unhealthy lifestyles are key contributors to CVD development, making behavioral changes a critical aspect of prevention and treatment. ==== Cardiac Rehabilitation Programs ==== * '''Purpose:''' Improve functioning, quality of life, and reduce mortality. * '''Components:''' Physical activity, nutrition counseling, psychological and social support. * '''Types:''' ** '''Center-based programs''' (traditional, supervised facilities). ** '''Home-based and technology-assisted programs''' (remote or hybrid models). ** '''Yoga-based rehabilitation''' (emerging option, especially in low-income areas). * '''Effectiveness:''' Linked to reduced mortality and increased quality of life, though barriers like low participation rates persist. ==== Aspirin Therapy ==== * '''Benefits:''' Reduces risk of CVD events, especially in patients with prior cardiac events (Bartolucci et al., 2011). * '''Risks:''' Bleeding complications; not recommended for migraines or healthy older adults (Weisman & Brunton, 2022; McNeil et al., 2018). * '''Usage:''' Debated for primary prevention but necessary in severe ischemias or worsening disease. ==== Surgical Interventions ==== # '''Angioplasty:''' Opens blocked arteries using a balloon or alternative tools (e.g., lasers or shavers). Often followed by stent placement to maintain artery openness. # '''Cardiac Bypass Surgery:''' #* Creates alternative blood routes using grafts. #* Variants include minimally invasive options (e.g., robotic surgery, port-access procedures). #* Recovery involves 4–7 hospital days. ==== Behavioral and Psychological Interventions ==== # '''Stress Management:''' #* Training combined with cardiac rehabilitation reduces stress and cardiac event rates (Blumenthal et al., 2016). #* Cognitive-behavioral therapy (CBT) and relaxation techniques improve psychological well-being and reduce hostility (Friedman et al., 1986; Hamieh et al., 2020). # '''Dietary Interventions:''' #* Healthy diets reduce inflammation and CVD risks (Kovell et al., 2020). #* Emphasis should be on healthy food choices rather than specific macronutrient patterns === Raw Textbook Page === The specific treatment for CVD is determined by the severity of the symptoms, the size and quantity of areas with ischemias (reduced blood flow), how well the left ventricle of the heart is pumping, and other medical factors such as severity of chest pain. As you can tell from the previous sections, unhealthy lifestyles (e.g., bad eating habits and not getting enough physical activity) are key determinants of whether you will develop a CVD. Correspondingly, changing health behaviors is one of the most critical treatment options to prevent the development of symptoms, relieve the symptoms, and lower the risk of heart attack and death. The primary goal will be to change unhealthy behaviors. The patient is normally admitted to a cardiac rehabilitation program. Rehabilitation programs educate patients on the best way to change their lifestyles and use a combination of physical activity, health and nutrition, and psychological and social support, to improve their overall functioning, physical and psychological quality of life and prevent death (Chindhy et al., 2020). If the person smokes, a smoking cessation program will be prescribed. They will also receive consultations on how to change diet, reduce salt intake, and eat more nutritionally balanced meals. If excessive drinking or not enough physical activity is the issue, it is critical to tackle each of these problems. People may even be told to start taking aspirin. Aspirin, you say? That’s right, not just any headache or pain killer, but aspirin. Many studies have shown that aspirin is beneficial for people with known CHD and probably beneficial for people at high risk of CHD (Bartolucci et al., 2011). In a meta-analysis containing approximately 90,000 people, aspirin reduced the risk of CVD events (i.e., heart attack, stroke, and cardiovascular death combined) and nonfatal heart attack (Bartoloucci et al., 2011). However, there are some risks associated with taking aspirin, including bleeding inside the brain or gastrointestinal tract (García Rodríguez et al., 2016). While the benefits of aspirin therapy outweigh the risks among people who have had a cardiac event (i.e., heart attack or stroke) (Weisman & Brunton, 2022), this treatment is not recommended for individuals with migraine (Kurth et al., 2011) or healthy older people (over 65 years) without CVD (McNeil et al., 2018). Debate remains about the use of aspirin in the primary prevention of CVD among people who have not had a cardiovascular event, as low-dose use is associated with reduced risk for CVD but also increased risk for bleeding (Weisman & Brunton, 2022). Aspirin may be necessary to help restore the blood flow to the affected parts of the heart if the ischemias are serious or the disease continues to worsen despite measures to slow it down (sometimes because the person continues the unhealthy behaviors). In people with critical conditions, surgery is often needed. Surgery There are two main forms of invasive surgery to deal with blockages: angioplasty and cardiac bypass surgery (American Heart Association, 2020). Angioplasty (also known as balloon angioplasty, coronary artery balloon dilation, and percutaneous coronary interventions) is a procedure done to open a partially blocked blood vessel so that blood can flow through it more easily (Figure 14.8). It is most often done on arteries that deliver blood to the heart (coronary arteries) when they are narrowed or blocked by atherosclerosis. The procedure involves the insertion of a thin, flexible tube (catheter) through an artery in the groin or arm, which is carefully guided into the artery that is narrowed. This is not a comfortable procedure. Once the tube reaches the narrowed artery, a small balloon at the end of the tube is inflated. The balloon may remain inflated from 20 seconds to 3 minutes. The pressure from the inflated balloon pushes the plaque against the wall of the artery opening up the passageway to improve blood flow. Once the fat and calcium build-up is compressed, a small, expandable wire tube called a stent is sometimes inserted into the artery to keep it open and reduce the likelihood of further blockages. The procedure make take anywhere from 30 minutes to several hours. Similar alternatives to angioplasty exist. One alternative is laser angioplasty, where the procedure largely mirrors that for angioplasty, however the catheter has a laser at the end rather than a balloon and pulsating beams of light are used to vaporize the blockage. Individuals may also be offered atherectomy, again like angioplasty, however the catheter has a rotating shaver at the end to cut away the blockage. Sometimes rather than removing the blockage, it may be best to just go around it. If you are driving to work, and you hear that there is a traffic jam ahead, you may take an alternative route. In the circulatory system, there are few alternative routes for blood to take, so medical personnel create one. Cardiac bypass surgery (also known as coronary artery bypass graft surgery) involves taking a blood vessel (vein or artery, referred to as a graft) from elsewhere in the body (usually the chest or leg) and using it to redirect blood flow around a severely blocked artery. Typically, the heart is stopped temporarily with a cardiopulmonary bypass machine (known as a heart-lung machine) used to provide oxygen and pump blood during the surgery; in some cases, the surgery can be done without stopping the heart, instead off-pump cardiac bypass uses a stabilizing device to enable work on the heart while it is still beating. Once complete, blood is redirected through the new blood vessel, bypassing the blocked artery and restoring blood flow to the affected portion of the heart muscle. One or more grafts may be needed to restore blood flow, with the surgery taking several hours. Traditionally, cardiac bypass surgery required open heart surgery, whereby access to the heart was gained by cutting through and separating the sternum (chest bone). Less invasive options, referred to as minimally invasive heart surgery, are now sometimes used. In these procedures, small cuts, known as “ports” are made in the chest. Medical instruments are then passed through the ports to attach chest arteries or leg veins to redirect blood flow around a blocked artery. The surgeon uses a camera and video monitor to guide their work, rather than direct observation; sometimes robotic machinery is also used to guide the procedure. Even with minimally invasive heart surgery, the exact procedure may vary. In port-access coronary artery bypass (PACAB), the heart is stopped and a heart-lung machine is used, while minimally invasive direct coronary artery bypass (MIDCAB) often avoids the heart-lung machine, operating while the heart is still beating. Cardiac bypass surgeries typically require hospital stays of 4 to 7 days. Behavioral Interventions Previous sections of this chapter highlight the central role of health behaviors (i.e., tobacco use, diet, physical activity) in the development of CVD and, in turn, cardiac events such as heart attack. Given this role, behavioral interventions to change health behaviors are an essential part of CVD treatment. Two forms of behavioral intervention often utilized for people with CVD are cardiac rehabilitation and stress management interventions. Cardiac Rehabilitation Most health psychological behavioral interventions for CVD take the form of cardiac rehabilitation programs. These programs may take different forms and comprise many different components. Most programs are center-based programs, meaning cardiac rehabilitation is undertaken within medically supervised facilities. However, more recently cardiac rehabilitation has evolved to include home-based cardiac rehabilitation, where the program is provided predominantly or entirely outside of medical settings, using indirect physical activity supervision and remote coaching. Additionally, there has been an increase in technology-assisted cardiac rehabilitation which can be used alone or in conjunction with center-based or home-based cardiac rehabilitation. The effectiveness of cardiac rehabilitation programs is hard to assess because they have many different components and can be delivered in varying formats. If there is a change in risk, a decrease in mortality, or an increase in quality of life, any one of the components could have caused it (or even an interaction of several components). That said, meta-analytical studies have shown that rehabilitation programs in general have accounted for reductions in mortality compared with that for control groups (Bishop, 2019). Also, programs with higher levels of physical activity have produced better results (Dibben et al., 2018). International reviews of cardiac rehabilitation practices worldwide found varying data regarding program length and delivery. For example, Pesah et al. (2017), in a review of cardiac rehabilitation in 50 countries, found that programs lasted for a median of 20 sessions and were generally delivered by physicians, nurses, and physiotherapists, while in Europe, Benzer et al. (2017) found the average number of sessions was 43.5 (3 to 96 sessions). A clinical trial of a community-based cardiac rehabilitation program in the Netherlands showed significant improvement in risk factors such as lack of exercise, smoking, and being overweight (Minneboo et al., 2017). Rehabilitation not only leads to increases in healthy behaviors and decreases in unhealthy behaviors, but it also saves money. Despite the proven benefits of cardiac rehabilitation, referral, participation, and adherence rates are relatively low (Chindhy et al., 2020; Chong et al., 2021). In fact, more than 80% of eligible people do not participate (Thomas et al., 2019). Some barriers have the potential to affect all people, however others demonstrate cultural variation. Barriers to cardiac rehabilitation include low referrals and endorsement by health professionals, low SES, cost, travel, age, medical comorbidities, being a smoker, and poor psychological well-being (Chindhy et al., 2020; Resurrección et al., 2019). Sex is also a significant barrier, with females found to be 11% less likely to be referred to cardiac rehabilitation than males (Li et al., 2018). Females are also more likely not to attend cardiac rehabilitation or to not complete the program (Chindhy et al., 2020; Resurrección et al., 2019). Many reasons have been proposed, including cardiac rehabilitation programs being tailored to males and male dominated, females lacking finances, being single, of older age, and prioritizing the needs of their families over their own (Chindhy et al., 2020; Clark et al., 2012; Galati et al., 2018). Furthermore, race is also a barrier to cardiac rehabilitation participation. Compared to White Americans, Black Americans, Latinx Americans, and Asian Americans have been found to be 20%, 36%, and 50% less likely, respectively, to be referred to cardiac rehabilitation (Li et al., 2018). While home-based and technology-assisted cardiac rehabilitation programs may not address sex and racial disparities, as they offer comparable results to center-based rehabilitation (i.e., Chong et al., 2021; Thomas et al., 2019), they may offer valuable alternatives to people who face barriers in attending center-based rehabilitation. Mobile-health approaches to cardiac rehabilitation have also been suggested as an alternative to traditional center-based cardiac rehabilitation (Bostrom et al., 2020). Recently, yoga-based cardiac rehabilitation has also been explored as a way to provide affordable access to services that may improve cardiovascular health for people in low- and middle-income countries and subgroups within high-income countries (females, older people, and people of low SES) (Prabhakaran et al., 2020). A review of six randomized controlled trials examining yoga-based cardiac rehabilitation (where yoga was added to standard care) found that this approach appears to confer some benefits. From the studies conducted to date, included in Bruce et al.’s review (2021), yoga-based cardiac rehabilitation was associated with improvements in some physiological and biochemical markers including reduced body mass index; a significant reduction in BMI, heart rate, SBP and DBP; and a trend toward better glycemic control, but findings have been mixed. For example, one study showed no significant differences in a range of cardiovascular physiological risk factors, and two studies found no significant change in triglycerides, total cholesterol, HDL, and LDL cholesterol at three months (Bruce et al., 2021). In contrast, one study found significant reductions in total triglycerides, HDL, and VLDL at 1 year. Improvements in patient perceptions of cardiac function and self-reported return to pre-heart attack activities, as well as improved parasympathetic activity and cardiac autonomic tone were noted in some studies, and there was a trend toward improved left ventricular systolic function (Bruce et al., 2021). Only one study examined major adverse cardiac events (composite of any cause death, nonfatal MI, nonfatal CVA, or emergency cardiovascular hospitalization) and found that while people in the yoga-based cardiac rehabilitation group experienced less total major adverse cardiac events, the difference was not significant. With regard to psychological health, people who participated in yoga-based cardiac rehabilitation reported significant improvements in depression, anxiety, stress and quality of life (Bruce et al., 2021). While the findings from yoga-based cardiac rehabilitation appear promising, due to the small number of studies to date with diverse populations of differing inclusion criteria, varying rehabilitation programs, and relatively small sample sizes, the findings should be viewed with some caution. Further research with larger samples and longer follow-up would be beneficial in better understanding any possible benefits from yoga-based cardiac rehabilitation. Interventions to Reduce Stress and Negative Emotions Given the number of psychological factors involved in CVD, a number of interventions to reduce stress and negative emotions have also been tried, and the news is good (Blumenthal et al., 2016; Sherwood et al., 2017). Richards et al. (2018) reviewed 35 studies that collectively evaluated 10,703 patients who received psychological interventions for CHD and found that while psychological intervention did not reduce the risk of total mortality at a median follow-up of 13 months, it reduced the risk of cardiac mortality at a median follow-up of 57 months. Psychological interventions also showed no evidence of reducing the risk of a subsequent nonfatal heart attack or need for revascularization (treatment to restore blood supply). Reductions in depression symptoms, anxiety, and stress were found for patients who received a psychological intervention compared to a control group. The psychological interventions varied but targeted depression, anxiety, stress, and Type A behavior including anger and hostility, and used approaches such as relaxation techniques, awareness and self-monitoring, cognitive challenging/restructuring techniques, and emotional support and client-led discussion. Many interventions also included attempts to increase awareness about cardiac risk factors and behavior change techniques for cardiac risk factors (i.e., diet, smoking). One of the earliest and most ambitious interventions was conducted by Friedman et al. (1986). They observed 1,013 heart patients for 4.5 years to determine whether their Type A behaviors could be altered. There were three experimental groups: A control section of 270 patients received group cardiac counseling, an experimental section of 592 patients received both group cardiac counseling and Type A behavioral counseling, and 151 patients served as a comparison group. The results were startling. At the end of the study, 35.1% of participants given cardiac and Type A behavior counseling reduced their Type A behavior compared with 9.8% of participants given only cardiac counseling, and the heart attack recurrence rate was only 12.9%. The recurrence rate in the control group was 21.2%, and the comparison group fared worse (recurrence rate of 28.2%). There was also a significant difference in the number of cardiac deaths between the experimental and control participants, clearly showing that altering Type A behavior reduces cardiac morbidity and mortality (Friedman et al., 1986). Therefore, as hostility is a critical component of Type A behavior, interventions designed specifically to reduce hostility appeared worthy of further exploration. However, Hamieh et al. (2020) in an examination of data from the GAZEL cohort study, a study where middle-age French electricity and gas workers were surveyed annually commencing in 1989, found that from 1994 to 2014, among 10,304 participants followed prospectively for cardiac events, irritability was the only baseline hostility trait associated with cardiac events. Furthermore, this association was no longer significant when considered with depression, but depression remained significantly associated with cardiac events, even when considered with all hostility traits. Hamieh et al. (2020) concluded that interventions that aim to reduce the risk of hostility-related cardiac events should target depression as a modifiable factor. Interventions to improve stress management and increase social support also reduce the effects of CVD (Blumenthal et al., 2016; Mondesir et al., 2018; Suchy-Dicey et al., 2022; Sherwood et al., 2017). Blumenthal et al. (2016) examined the effects of standard cardiac rehabilitation compared to cardiac rehabilitation with stress management training over a 5-year follow-up period in 151 patients aged 36 to 84 with established CHD. Patients in the standard cardiac rehabilitation group received education about CHD, nutrition counseling, two classes on the role of stress in CHD, and exercised 35 minutes (three times per week). In contrast, those in the cardiac rehabilitation with stress management training group received the same information and completed the same exercise as the standard group but also completed 12 weekly 1.5-hour sessions of stress management training involving education, group support, and cognitive behavior therapy. Blumenthal et al. (2016) found that patients in the cardiac rehabilitation and stress management group demonstrated greater reduction in overall stress and lower rates of cardiac events (18% compared to 33%). Both groups showed improvements in CHD biomarkers, lipids, physical activity and exercise capacity and experienced less cardiac events than patients who did not receive either form of cardiac rehabilitation. Given the positive role played by a healthy diet (discussed previously), it is not surprising that many interventions focus on changing diets. The OmniHeart Trial compared three diets designed to reduce CVD risk (Kovell et al., 2020). One diet was high in carbohydrates. Two diets replaced carbohydrates with either unsaturated fat or protein. The lower carbohydrate diets improved the CVD risk factors. Recently, it was shown that a healthy diet, regardless of what type of macronutrients it contains (fat vs. carbohydrate patterns) reduced inflammation markers associated with the development of CVD and, therefore, it was suggested that diet recommendations should be simplified to emphasize healthy foods rather than one specific dietary pattern (Kovell et al., 2020). 3vteek4u3aabzndduff7og8d6rt7tj9 14 317616 2693774 2024-12-29T18:03:24Z Watchduck 137431 New resource with "[[Category:Mentors of Boolean functions]]" 2693774 wikitext text/x-wiki [[Category:Mentors of Boolean functions]] 4ei6wz63hc5r966dprp6lwap259r3na 2693783 2693774 2024-12-29T18:13:05Z Watchduck 137431 2693783 wikitext text/x-wiki * {{tl|Template:Mentors of Boolean functions/cycles/style.css}} [[Category:Mentors of Boolean functions]] fc7itq1cr4rzvaj6ph74nzpcw4cqdpf 0 317617 2693792 2024-12-29T19:57:04Z Watchduck 137431 New resource with "<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">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| anchor !class="block"| block |- |class="size"| 16..." 2693792 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">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| anchor !class="block"| block |- |class="size"| 16 |class="prop"| 0 |class="block"| <span class="block-list">[0, 1, 22, 23, 104, 105, 126, 127, 128, 129, 150, 151, 232, 233, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844087633226186939369601060787799465383022669001262447603868028627564101635.svg|420px]] |- |class="size"| 80 |class="prop"| 40 |class="block"| [[File:Set_of_3-ary_Boolean_functions_21795908540735410239145145178479073544213524388065693886324288404438742207500.svg|420px]] |- |class="size"| 80 |class="prop"| 72 |class="block"| [[File:Set_of_3-ary_Boolean_functions_5449288947167349512239963211685519501868532352083311608039648187212966408240.svg|420px]] |- |class="size"| 80 |class="prop"| 96 |class="block"| [[File:Set_of_3-ary_Boolean_functions_1702804116187248732816275557735515341804905256490296097489779387633856922560.svg|420px]] |} [[Category:Boolf prop/3-ary|anchor]] 5njqm6apmlyurs854k3fb90jge7jviz 2693804 2693792 2024-12-29T22:36:56Z Watchduck 137431 2693804 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> XOR of all entries in a {{Boolf prop 3-ary|chain}}, a fixed point of the [[Mentors of Boolean functions|mentor]] permutation. <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">4</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| anchor !class="block"| block |- |class="size"| 16 |class="prop"| 0 |class="block"| <span class="block-list">[0, 1, 22, 23, 104, 105, 126, 127, 128, 129, 150, 151, 232, 233, 254, 255]</span>[[File:Set_of_3-ary_Boolean_functions_86844087633226186939369601060787799465383022669001262447603868028627564101635.svg|420px]] |- |class="size"| 80 |class="prop"| 40 |class="block"| [[File:Set_of_3-ary_Boolean_functions_21795908540735410239145145178479073544213524388065693886324288404438742207500.svg|420px]] |- |class="size"| 80 |class="prop"| 72 |class="block"| [[File:Set_of_3-ary_Boolean_functions_5449288947167349512239963211685519501868532352083311608039648187212966408240.svg|420px]] |- |class="size"| 80 |class="prop"| 96 |class="block"| [[File:Set_of_3-ary_Boolean_functions_1702804116187248732816275557735515341804905256490296097489779387633856922560.svg|420px]] |} [[Category:Boolf prop/3-ary|anchor]] oa8mafc2fmlsa0xr5vtpiws5wfvfyuj 0 317618 2693795 2024-12-29T20:57:15Z Watchduck 137431 New resource with "<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">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf..." 2693795 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">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| (0,) |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| (1, 3, 3, 2, 3) |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| (0, 0) |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| (0, 1, 1, 2, 2) |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| (0, 2, 1, 0, 3) |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] ouiedrlb08bq8dluyssl9f0398tx76c 2693796 2693795 2024-12-29T20:59:41Z Watchduck 137431 2693796 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">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] sa3opgqos9plbqpv5qgtaynhgslepv3 2693797 2693796 2024-12-29T21:15:17Z Watchduck 137431 2693797 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] stpcz4nzclff9afhcdbntv5vbio7dek 2693808 2693797 2024-12-29T22:43:26Z Watchduck 137431 2693808 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} See also {{Boolf prop 3-ary|reduced chain quadrants}}, {{Boolf prop 3-ary|chain}}, {{Boolf prop 3-ary|chunky chain}}, {{Boolf prop 3-ary|chain length}}, {{Boolf prop 3-ary|anchor}}. <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] ajofj2j0pacvmxri8bbtpge2i0xznbq 2693811 2693808 2024-12-29T22:46:12Z Watchduck 137431 2693811 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] stpcz4nzclff9afhcdbntv5vbio7dek 2693821 2693811 2024-12-29T23:19:29Z Watchduck 137431 2693821 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="prop"| twin mentors of chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="prop"| {0} |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="prop"| {23, 104, 105, 232, 254} |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="prop"| {41, 73, 97, 137, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 247} |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="prop"| {30, 31, 54, 55, 62, 86, 87, 94, 118, 169, 190, 201, 222, 225, 246} |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="prop"| {40, 72, 96} |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="prop"| {8, 9, 32, 33, 63, 64, 65, 95, 119, 136, 158, 160, 182, 192, 214} |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="prop"| {22, 126, 127, 129, 150} |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="prop"| {1, 128, 151, 233, 255} |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] en57er8mtpsrohkvpuwr2px54v94hgg 2693822 2693821 2024-12-29T23:24:43Z Watchduck 137431 2693822 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{Collapsible START|relationships|collapsed gap-below}} {{multiple image | align = left | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 58122214886935002256303026678885186177490199581866974724862928333937250926594.svg | caption1 = '''chain quadrants''' [1, 3, 3, 2, 3] | image2 = Set of 3-ary Boolean functions 19037172575477343356526600596078200258739634279505979654031961182739418841216.svg | caption2 = '''chain quadrants''' [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] | image3 = Set of 3-ary Boolean functions 77159387462412345612829627274963386436229833861372954378894889516676669767810.svg | caption3 = reduced chain quadrants (0, 1, 1, 3) &nbsp; <small>This means that there is no quadrant 0, and quadrant 3 appears three times more than quadrants 1 and 2.</small> | footer = {{Boolf prop 3-ary|reduced chain quadrants|Reduced chain quadrants}} show only the ratio among the quadrants. }} {{Collapsible END}} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="prop"| twin mentors of chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="prop"| {0} |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="prop"| {23, 104, 105, 232, 254} |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="prop"| {41, 73, 97, 137, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 247} |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="prop"| {30, 31, 54, 55, 62, 86, 87, 94, 118, 169, 190, 201, 222, 225, 246} |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="prop"| {40, 72, 96} |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="prop"| {8, 9, 32, 33, 63, 64, 65, 95, 119, 136, 158, 160, 182, 192, 214} |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="prop"| {22, 126, 127, 129, 150} |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="prop"| {1, 128, 151, 233, 255} |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] a67b8g613ommbvsz9m67zhhywi7kszr 2693829 2693822 2024-12-29T23:58:18Z Watchduck 137431 2693829 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{Collapsible START|relationships|collapsed gap-below}} {{multiple image | align = left | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 53919896548177368855603419572066304883728005167150787806493889527808.svg | caption1 = twin mentor 255 | image2 = Set of 3-ary Boolean functions 113078272805699813627635278165817567741368919429995007057106809660918202368.svg | caption2 = '''chain quadrants''' [0, 2, 1, 0, 3] | image3 = Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg | caption3 = chain [22, 129, 126, 127, 150] | footer = Each matrix is a union of {{Boolf prop 3-ary|twin mentor}}s &ndash; and of course of {{Boolf prop 3-ary|chain}}s. }} {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 58122214886935002256303026678885186177490199581866974724862928333937250926594.svg | caption1 = '''chain quadrants''' [1, 3, 3, 2, 3] | image2 = Set of 3-ary Boolean functions 19037172575477343356526600596078200258739634279505979654031961182739418841216.svg | caption2 = '''chain quadrants''' [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] | image3 = Set of 3-ary Boolean functions 77159387462412345612829627274963386436229833861372954378894889516676669767810.svg | caption3 = reduced chain quadrants (0, 1, 1, 3) &nbsp; <small>This means that there is no quadrant 0, and quadrant 3 appears three times more than quadrants 1 and 2.</small> | footer = {{Boolf prop 3-ary|reduced chain quadrants|Reduced chain quadrants}} show only the ratio among the quadrants. }} {{Collapsible END}} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">8</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="prop"| twin mentors of chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="prop"| {0}<br>[[File:Set of 3-ary Boolean functions 0.svg|420px]] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="prop"| {23, 104, 105, 232, 254}<br>[[File:Set of 3-ary Boolean functions 718.svg|420px]] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="prop"| {41, 73, 97, 137, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 247}<br>[[File:Set of 3-ary Boolean functions 2512.svg|420px]] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="prop"| {30, 31, 54, 55, 62, 86, 87, 94, 118, 169, 190, 201, 222, 225, 246}<br>[[File:Set of 3-ary Boolean functions 1870.svg|420px]] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="prop"| {40, 72, 96}<br>[[File:Set of 3-ary Boolean functions 208.svg|420px]] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="prop"| {8, 9, 32, 33, 63, 64, 65, 95, 119, 136, 158, 160, 182, 192, 214}<br>[[File:Set of 3-ary Boolean functions 1530.svg|420px]] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="prop"| {22, 126, 127, 129, 150}<br>[[File:Set of 3-ary Boolean functions 554.svg|420px]] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="prop"| {1, 128, 151, 233, 255}<br>[[File:Set of 3-ary Boolean functions 768.svg|420px]] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] hjgs3avedshcdq5es6egg3m7it7seml 2693830 2693829 2024-12-29T23:59:16Z Watchduck 137431 2693830 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">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="prop"| twin mentors of chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="prop"| {0}<br>[[File:Set of 3-ary Boolean functions 1.svg|420px]] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="prop"| {23, 104, 105, 232, 254}<br>[[File:Set of 3-ary Boolean functions 28948029211075395646456533686927839240342948678366341991205962436344268455936.svg|420px]] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="prop"| {41, 73, 97, 137, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 247}<br>[[File:Set of 3-ary Boolean functions 226156464784210963389959305026653201078608060145693328958752407227704803328.svg|420px]] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="prop"| {30, 31, 54, 55, 62, 86, 87, 94, 118, 169, 190, 201, 222, 225, 246}<br>[[File:Set of 3-ary Boolean functions 113078272805699813627635278164390319112886450516355946563377144087168679936.svg|420px]] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="prop"| {40, 72, 96}<br>[[File:Set of 3-ary Boolean functions 79228167236630821562700791808.svg|420px]] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="prop"| {8, 9, 32, 33, 63, 64, 65, 95, 119, 136, 158, 160, 182, 192, 214}<br>[[File:Set of 3-ary Boolean functions 26328079200371016051500959966719612948350653464368632466704958208.svg|420px]] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="prop"| {22, 126, 127, 129, 150}<br>[[File:Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg|420px]] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="prop"| {1, 128, 151, 233, 255}<br>[[File:Set of 3-ary Boolean functions 57896058422150791292913067373858532976411591521721281395904175926709546123266.svg|420px]] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] q5x2hw5x5outmxteutdmlouuxsev7xx 2693831 2693830 2024-12-30T00:00:38Z Watchduck 137431 2693831 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">1</span>⋅<span class="size">4</span> + <span class="count">1</span>⋅<span class="size">12</span> + <span class="count">3</span>⋅<span class="size">20</span> + <span class="count">3</span>⋅<span class="size">60</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| chain quadrants !class="prop"| twin mentors of chain quadrants !class="block"| block |- |class="size"| 4 |class="prop"| [0] |class="prop"| {0}<br>[[File:Set of 3-ary Boolean functions 1.svg|300px]] |class="block"| <span class="block-list">[0, 40, 72, 96]</span>[[File:Set_of_3-ary_Boolean_functions_79228167236630821562700791809.svg|420px]] |- |class="size"| 20 |class="prop"| [1, 3, 3, 2, 3] |class="prop"| {23, 104, 105, 232, 254}<br>[[File:Set of 3-ary Boolean functions 28948029211075395646456533686927839240342948678366341991205962436344268455936.svg|300px]] |class="block"| <span class="block-list small">[1, 41, 73, 97, 128, 137, 151, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 233, 247, 255]</span>[[File:Set_of_3-ary_Boolean_functions_58122214886935002256303026678885186177490199581866974724862928333937250926594.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 2, 1, 0, 3, 0, 2, 1, 0, 3] |class="prop"| {41, 73, 97, 137, 159, 161, 168, 183, 191, 193, 200, 215, 223, 224, 247}<br>[[File:Set of 3-ary Boolean functions 226156464784210963389959305026653201078608060145693328958752407227704803328.svg|300px]] |class="block"| <span class="block-list small">[2, 4, 10, 11, 12, 13, 16, 24, 25, 34, 35, 36, 37, 42, 44, 48, 49, 56, 66, 67, 68, 69, 74, 76, 80, 81, 88, 98, 100, 106, 107, 108, 109, 112, 120, 121, 130, 132, 135, 144, 147, 149, 170, 172, 175, 184, 187, 189, 202, 204, 207, 216, 219, 221, 226, 228, 231, 240, 243, 245]</span>[[File:Set_of_3-ary_Boolean_functions_72444724046062282054372241607084548726185460892053405940208098623562529812.svg|420px]] |- |class="size"| 60 |class="prop"| [0, 1, 1, 2, 2, 0, 1, 1, 2, 2] |class="prop"| {30, 31, 54, 55, 62, 86, 87, 94, 118, 169, 190, 201, 222, 225, 246}<br>[[File:Set of 3-ary Boolean functions 113078272805699813627635278164390319112886450516355946563377144087168679936.svg|300px]] |class="block"| <span class="block-list small">[3, 5, 14, 15, 17, 26, 27, 28, 29, 38, 39, 43, 45, 50, 51, 52, 53, 57, 70, 71, 75, 77, 82, 83, 84, 85, 89, 99, 101, 110, 111, 113, 122, 123, 124, 125, 138, 140, 142, 152, 154, 156, 162, 164, 166, 176, 178, 180, 194, 196, 198, 208, 210, 212, 234, 236, 238, 248, 250, 252]</span>[[File:Set_of_3-ary_Boolean_functions_9499149566950363989402445510972279100262824281537677335763606449383614365736.svg|420px]] |- |class="size"| 12 |class="prop"| [0, 0] |class="prop"| {40, 72, 96}<br>[[File:Set of 3-ary Boolean functions 79228167236630821562700791808.svg|300px]] |class="block"| <span class="block-list">[6, 18, 20, 46, 58, 60, 78, 90, 92, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851013963099894811243194690568256.svg|420px]] |- |class="size"| 60 |class="prop"| [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] |class="prop"| {8, 9, 32, 33, 63, 64, 65, 95, 119, 136, 158, 160, 182, 192, 214}<br>[[File:Set of 3-ary Boolean functions 26328079200371016051500959966719612948350653464368632466704958208.svg|300px]] |class="block"| <span class="block-list small">[7, 19, 21, 47, 59, 61, 79, 91, 93, 103, 115, 117, 131, 133, 134, 139, 141, 143, 145, 146, 148, 153, 155, 157, 163, 165, 167, 171, 173, 174, 177, 179, 181, 185, 186, 188, 195, 197, 199, 203, 205, 206, 209, 211, 213, 217, 218, 220, 227, 229, 230, 235, 237, 239, 241, 242, 244, 249, 251, 253]</span>[[File:Set_of_3-ary_Boolean_functions_19037172575477343356526600596078200258739634279505979654031961182739418841216.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 1, 1, 2, 2] |class="prop"| {22, 126, 127, 129, 150}<br>[[File:Set of 3-ary Boolean functions 1427248628482468913639060493729665573749522432.svg|300px]] |class="block"| <span class="block-list small">[8, 9, 23, 32, 33, 63, 64, 65, 95, 104, 105, 119, 136, 158, 160, 182, 192, 214, 232, 254]</span>[[File:Set_of_3-ary_Boolean_functions_28948029211101723725656904702979340200309668291314692644670331068810973414144.svg|420px]] |- |class="size"| 20 |class="prop"| [0, 2, 1, 0, 3] |class="prop"| {1, 128, 151, 233, 255}<br>[[File:Set of 3-ary Boolean functions 57896058422150791292913067373858532976411591521721281395904175926709546123266.svg|300px]] |class="block"| <span class="block-list small">[22, 30, 31, 54, 55, 62, 86, 87, 94, 118, 126, 127, 129, 150, 169, 190, 201, 222, 225, 246]</span>[[File:Set_of_3-ary_Boolean_functions_113078272805699813627635278165817567741368919429995007057106809660918202368.svg|420px]] |} [[Category:Boolf prop/3-ary|chain quadrants]] gnjfyvu77ccbuqu5wnut55negxcayp4 0 317619 2693798 2024-12-29T22:00:51Z 2601:CA:4280:490:546F:C4FA:A5B7:BED3 Created a page for mathematics education. 2693798 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> bo751v2v6tg5zv4rrgiptys3sh8plr3 2693801 2693798 2024-12-29T22:10:39Z Websterwong 2931187 Added a section to host learning resources 2693801 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> == Mathematics Teacher Education Resources == qknn4vy7h1oq3mie6wpuut20pk8gi4f 2693803 2693801 2024-12-29T22:27:54Z Websterwong 2931187 /* Mathematics Teacher Education Resources */ Added one resource. 2693803 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> == Mathematics Teacher Education Resources == * Levels of cognitive demand of tasks<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> i0mndyjxcrade53vzmlz53xuwyv8q8i 2693806 2693803 2024-12-29T22:41:00Z Websterwong 2931187 Red linked a page. 2693806 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> == Mathematics Teacher Education Resources == * [[Levels of cognitive demand of tasks]]<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> 10iurs0b6l4bqvkfmvy3kqlkmwfut3a 2693813 2693806 2024-12-29T22:56:49Z MathXplore 2888076 +[[Category:Education]]; +[[Category:Mathematics]] using [[Help:Gadget-HotCat|HotCat]] 2693813 wikitext text/x-wiki Mathematics education is a discipline that studies the learning and teaching of mathematics.<ref>{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}</ref> == Mathematics Teacher Education Resources == * [[Levels of cognitive demand of tasks]]<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> [[Category:Education]] [[Category:Mathematics|education]] 0ed3vvsz64ads19zotr988hailwrh3d 0 317620 2693800 2024-12-29T22:03:47Z Watchduck 137431 New resource with "<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">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| reduced chain quadrants !class="block"| block |- |..." 2693800 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">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| reduced chain quadrants !class="block"| block |- |class="size"| 16 |class="prop"| (1, 0, 0, 0) |class="block"| <span class="block-list">[0, 6, 18, 20, 40, 46, 58, 60, 72, 78, 90, 92, 96, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851093191267131442064757391360065.svg|420px]] |- |class="size"| 80 |class="prop"| (0, 1, 1, 3) |class="block"| [[File:Set_of_3-ary_Boolean_functions_77159387462412345612829627274963386436229833861372954378894889516676669767810.svg|420px]] |- |class="size"| 80 |class="prop"| (2, 1, 1, 1) |class="block"| [[File:Set_of_3-ary_Boolean_functions_185522996851762095682007519772902116467554380322048412997314908284480732180.svg|420px]] |- |class="size"| 80 |class="prop"| (1, 2, 2, 0) |class="block"| [[File:Set_of_3-ary_Boolean_functions_38447178778052087715059350213951619300572492572852369980433937518194587779880.svg|420px]] |} [[Category:Boolf prop/3-ary|reduced chain quadrants]] slylgutwgu0w5mbff633qm9r4510oqx 2693805 2693800 2024-12-29T22:37:11Z Watchduck 137431 2693805 wikitext text/x-wiki <templatestyles src="Boolf prop/blocks.css" /> {{multiple image | align = right | perrow = 1 | total_width = 500 | image1 = Set of 3-ary Boolean functions 58122214886935002256303026678885186177490199581866974724862928333937250926594.svg | caption1 = chain quadrants [1, 3, 3, 2, 3] | image2 = Set of 3-ary Boolean functions 19037172575477343356526600596078200258739634279505979654031961182739418841216.svg | caption2 = chain quadrants [1, 3, 3, 2, 3, 1, 3, 3, 2, 3] | image3 = Set of 3-ary Boolean functions 77159387462412345612829627274963386436229833861372954378894889516676669767810.svg | caption3 = '''reduced chain quadrants''' (0, 1, 1, 3) &nbsp; <small>This means that there is no quadrant 0, and quadrant 3 appears three times more than quadrants 1 and 2.</small> | footer = ''Reduced'' refers to the ratio of {{Boolf prop 3-ary|chain quadrants}} to each other. }} <div class="intpart"> <span class="number-of-blocks">Number of blocks: &nbsp; <span class="count">4</span></span> Integer partition: &nbsp; <span class="count">1</span>⋅<span class="size">16</span> + <span class="count">3</span>⋅<span class="size">80</span> </div> {| class="wikitable sortable boolf-blocks" !class="size"| <abbr title="block size">#</abbr> !class="prop"| reduced chain quadrants !class="block"| block |- |class="size"| 16 |class="prop"| (1, 0, 0, 0) |class="block"| <span class="block-list">[0, 6, 18, 20, 40, 46, 58, 60, 72, 78, 90, 92, 96, 102, 114, 116]</span>[[File:Set_of_3-ary_Boolean_functions_103851093191267131442064757391360065.svg|420px]] |- |class="size"| 80 |class="prop"| (0, 1, 1, 3) |class="block"| [[File:Set_of_3-ary_Boolean_functions_77159387462412345612829627274963386436229833861372954378894889516676669767810.svg|420px]] |- |class="size"| 80 |class="prop"| (2, 1, 1, 1) |class="block"| [[File:Set_of_3-ary_Boolean_functions_185522996851762095682007519772902116467554380322048412997314908284480732180.svg|420px]] |- |class="size"| 80 |class="prop"| (1, 2, 2, 0) |class="block"| [[File:Set_of_3-ary_Boolean_functions_38447178778052087715059350213951619300572492572852369980433937518194587779880.svg|420px]] |} [[Category:Boolf prop/3-ary|reduced chain quadrants]] fr04ximnxeuttows4ikd96sa2jcrgdl 0 317621 2693807 2024-12-29T22:42:59Z Websterwong 2931187 Created this page. 2693807 wikitext text/x-wiki Smith and Stein<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423?seq=1|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> outline a card sort activity that could be given to mathematics teachers to introduce them to the levels of cognitive demand. 78yq31yc6pebc8c8ws8xb4g6a1bm85b 2693810 2693807 2024-12-29T22:45:34Z Websterwong 2931187 Added a heading. 2693810 wikitext text/x-wiki == Teaching ideas == Smith and Stein<ref>{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423?seq=1|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}</ref> outline a card sort activity that could be given to mathematics teachers to introduce them to the levels of cognitive demand. 1l0343bttaw4ds0t48a5rfsr4pfugsw 1 317622 2693809 2024-12-29T22:44:44Z Websterwong 2931187 /* Creating a lesson based on the article */ new section 2693809 wikitext text/x-wiki == Creating a lesson based on the article == I would like to make a lesson plan here based on the article (http://mathedseminar.pbworks.com/w/file/fetch/92864991/Smith%20and%20Stein%20-%201998%20-%20Selecting%20and%20Creating%20Mathematical%20Tasks%20From%20Re.pdf), but I am not sure how copyright comes into play. I appreciate any input on this. [[User:Websterwong|Websterwong]] ([[User talk:Websterwong|discuss]] • [[Special:Contributions/Websterwong|contribs]]) 22:44, 29 December 2024 (UTC) 9f2fffpxvi8fx55atoqcx5qvv3qavby 3 317623 2693818 2024-12-29T23:12:38Z MathXplore 2888076 New resource with "== December 2024 == {{subst:uw-vandalism1}} ~~~~" 2693818 wikitext text/x-wiki == December 2024 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 23:12, 29 December 2024 (UTC) ha1c6xzbs93iklj9ckq2tnlfwxr199w 6 317624 2693828 2024-12-29T23:54:48Z Young1lim 21186 {{Information |Description=LIB.2A: Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693828 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2A: Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |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}} n5qr6bonxw0t51mknd7grrd4ipr20g6 6 317625 2693839 2024-12-30T02:17:01Z Young1lim 21186 {{Information |Description=LIB.2B: Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693839 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2B: Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |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}} 12er4nst3dhkhh8l036qm8dm75amk5d 6 317626 2693842 2024-12-30T07:22:13Z Young1lim 21186 {{Information |Description=DIR.1A: Shared Library Names (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693842 wikitext text/x-wiki == Summary == {{Information |Description=DIR.1A: Shared Library Names (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |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}} 0zxuivgp97jbsamsxlrv0r3t2gurvar 6 317628 2693845 2024-12-30T08:19:13Z Young1lim 21186 {{Information |Description=DIR.2A: Managing Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693845 wikitext text/x-wiki == Summary == {{Information |Description=DIR.2A: Managing Shared Libraries (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |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}} qcnnfxl278jspx5xsjy14cq3nno0n7f 6 317629 2693848 2024-12-30T08:55:01Z Young1lim 21186 {{Information |Description=API.2A: Functions (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2693848 wikitext text/x-wiki == Summary == {{Information |Description=API.2A: Functions (20241230 - 20241228) |Source={{own|Young1lim}} |Date=2024-12-30 |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}} am9ukp6o8tfei613xzcuem0d3usiy1r