{"id":5991,"date":"2023-02-17T09:11:06","date_gmt":"2023-02-17T16:11:06","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=5991"},"modified":"2023-10-08T10:11:18","modified_gmt":"2023-10-08T17:11:18","slug":"quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/","title":{"rendered":"3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\u00a0\u200bGeometric Group Theory"},"content":{"rendered":"\n<p class=\"has-text-align-center\">Click <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/Quaternion Coxeter-Dynkin Geometric Group Theory-2b.pptx\" target=\"_blank\" rel=\"noopener\" title=\"\">here<\/a><a href=\"https:\/\/1drv.ms\/p\/s!Ah5BIzPGaCWlgaUt3ULb0rNSlX-6Tg?e=eh3QFj\" target=\"_blank\" rel=\"noopener\" title=\"\"> <\/a>to view the Powerpoint presentation.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/Quaternion Coxeter-Dynkin Geometric Group Theory-2b.pptx\" target=\"_blank\" rel=\"noreferrer noopener\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2023\/02\/title-slide-4-1024x576.png\" alt=\"\" class=\"wp-image-6020\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2023\/02\/title-slide-4-1024x576.png 1024w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2023\/02\/title-slide-4-500x281.png 500w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2023\/02\/title-slide-4-1536x864.png 1536w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2023\/02\/title-slide-4.png 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<p>It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. <\/p>\n\n\n\n<p>If you don&#8217;t want to use the interactive Powerpoint, <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/Quaternion Coxeter-Dynkin Geometric Group Theory-2b.pdf\" target=\"_blank\" rel=\"noopener\" title=\"\">here <\/a>is the PDF version.<\/p>\n\n\n\n<p>See these Mathematica notebooks <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/A3-B3-H3-output-1b.nb\" target=\"_blank\" rel=\"noopener\" title=\"\">here<\/a> and <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/A3-B3-H3-output-2b.nb\" target=\"_blank\" rel=\"noopener\" title=\"\">here<\/a> for a more detailed look at the analysis of the Koca papers used to produce the results : <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>Koca, Mehmet;&nbsp;Ozdes&nbsp;Koca, Nazife; Koc, Ramazon (2010). &#8220;Catalan Solids Derived From 3D-Root Systems and&nbsp;Quaternions&#8221;.&nbsp;Journal of Mathematical Physics.&nbsp;<\/em><strong><em>51<\/em><\/strong><em>&nbsp;(4).&nbsp;<\/em><a href=\"https:\/\/en.wikipedia.org\/wiki\/ArXiv_(identifier)\" target=\"_blank\" rel=\"noreferrer noopener\"><em>arXiv<\/em><\/a><em>:<\/em><a href=\"https:\/\/arxiv.org\/abs\/0908.3272\" target=\"_blank\" rel=\"noreferrer noopener\"><em>0908.3272<\/em><\/a><em>.&nbsp;<\/em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Doi_(identifier)\" target=\"_blank\" rel=\"noreferrer noopener\"><em>doi<\/em><\/a><em>:<\/em><a href=\"https:\/\/doi.org\/10.1063%2F1.3356985\" target=\"_blank\" rel=\"noreferrer noopener\"><em>10.1063\/1.3356985<\/em><\/a><em>.<\/em>\u200b<\/li>\n\n\n\n<li><em>Koca, Mehmet; Al-Ajmi,&nbsp;<\/em><em>Mudhahir<\/em><em>;&nbsp;<\/em><em>Ozdes<\/em><em>&nbsp;Koca, Nazife (2011).&nbsp;<\/em><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0024379510005173\" target=\"_blank\" rel=\"noreferrer noopener\"><em>&#8220;Quaternionic representation of snub 24-cell and its&nbsp;<\/em><\/a><a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0024379510005173\" target=\"_blank\" rel=\"noreferrer noopener\"><em>dual polytope derived from E8 root system&#8221;<\/em><\/a><em>.&nbsp;Linear Algebra and Its Applications.&nbsp;<\/em><strong><em>434<\/em><\/strong><em>&nbsp;(4): 977\u2013<\/em><em>989.&nbsp;<\/em><a href=\"https:\/\/en.wikipedia.org\/wiki\/Doi_(identifier)\" target=\"_blank\" rel=\"noreferrer noopener\"><em>doi<\/em><\/a><em>:<\/em><a href=\"https:\/\/doi.org\/10.1016%2Fj.laa.2010.10.005\" target=\"_blank\" rel=\"noreferrer noopener\"><em>10.1016\/j.laa.2010.10.005<\/em><\/a><em>.&nbsp;<\/em><a href=\"https:\/\/en.wikipedia.org\/wiki\/ISSN_(identifier)\" target=\"_blank\" rel=\"noreferrer noopener\"><em>ISSN<\/em><\/a><em>&nbsp;<\/em><a href=\"https:\/\/www.worldcat.org\/issn\/0024-3795\" target=\"_blank\" rel=\"noreferrer noopener\"><em>0024-3795<\/em><\/a><em>.&nbsp;<\/em><a href=\"https:\/\/en.wikipedia.org\/wiki\/S2CID_(identifier)\" target=\"_blank\" rel=\"noreferrer noopener\"><em>S2CID<\/em><\/a><em>&nbsp;<\/em><a href=\"https:\/\/api.semanticscholar.org\/CorpusID:18278359\" target=\"_blank\" rel=\"noreferrer noopener\"><em>18278359<\/em><\/a><em>.<\/em><\/li>\n<\/ul>\n\n\n\n<p>Or scroll down to see the .svg images <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/A3-B3-H3-output-1.svg\" target=\"_blank\" rel=\"noopener\" title=\"\">here <\/a>and <a href=\"https:\/\/theoryofeverything.org\/TOE\/JGM\/A3-B3-H3-output-2.svg\" target=\"_blank\" rel=\"noopener\" title=\"\">here<\/a>.<\/p>\n\n\n\n<p>Greg Moxness, Tucson AZ<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don&#8217;t want to use the interactive Powerpoint, &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\u00a0\u200bGeometric Group Theory<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[8,13,4,34,20],"class_list":["post-5991","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-4d","tag-e8","tag-h4","tag-octonion"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don&#039;t want to use the interactive Powerpoint,\" \/>\n\t<meta name=\"robots\" content=\"max-image-preview:large\" \/>\n\t<meta name=\"author\" content=\"jgmoxness\"\/>\n\t<link rel=\"canonical\" href=\"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/\" \/>\n\t<meta name=\"generator\" content=\"All in One SEO (AIOSEO) 4.9.8\" \/>\n\t\t<meta property=\"og:locale\" content=\"en_US\" \/>\n\t\t<meta property=\"og:site_name\" content=\"Visualizing a Theory of Everything! - Dedicated to the pursuit of beauty and Truth in Nature!\" \/>\n\t\t<meta property=\"og:type\" content=\"article\" \/>\n\t\t<meta property=\"og:title\" content=\"3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \u200bGeometric Group Theory - Visualizing a Theory of Everything!\" \/>\n\t\t<meta property=\"og:description\" content=\"Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don&#039;t want to use the interactive Powerpoint,\" \/>\n\t\t<meta property=\"og:url\" content=\"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/\" \/>\n\t\t<meta property=\"article:published_time\" content=\"2023-02-17T16:11:06+00:00\" \/>\n\t\t<meta property=\"article:modified_time\" content=\"2023-10-08T17:11:18+00:00\" \/>\n\t\t<meta name=\"twitter:card\" content=\"summary\" \/>\n\t\t<meta name=\"twitter:title\" content=\"3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \u200bGeometric Group Theory - Visualizing a Theory of Everything!\" \/>\n\t\t<meta name=\"twitter:description\" content=\"Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don&#039;t want to use the interactive Powerpoint,\" \/>\n\t\t<script type=\"application\/ld+json\" class=\"aioseo-schema\">\n\t\t\t{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"BlogPosting\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#blogposting\",\"name\":\"3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \\u200bGeometric Group Theory - Visualizing a Theory of Everything!\",\"headline\":\"3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\\u00a0\\u200bGeometric Group Theory\",\"author\":{\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/#author\"},\"publisher\":{\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/#person\"},\"image\":{\"@type\":\"ImageObject\",\"url\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/wp-content\\\/uploads\\\/2023\\\/02\\\/title-slide-4-1024x576.png\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#articleImage\"},\"datePublished\":\"2023-02-17T09:11:06-07:00\",\"dateModified\":\"2023-10-08T10:11:18-07:00\",\"inLanguage\":\"en-US\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#webpage\"},\"isPartOf\":{\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#webpage\"},\"articleSection\":\"Physics, 3D, 4D, E8, H4, Octonion\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#breadcrumblist\",\"itemListElement\":[{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE#listItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"name\":\"Physics\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"position\":2,\"name\":\"Physics\",\"item\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/\",\"nextItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#listItem\",\"name\":\"3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\\u00a0\\u200bGeometric Group Theory\"},\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE#listItem\",\"name\":\"Home\"}},{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#listItem\",\"position\":3,\"name\":\"3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\\u00a0\\u200bGeometric Group Theory\",\"previousItem\":{\"@type\":\"ListItem\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/topics\\\/physics\\\/#listItem\",\"name\":\"Physics\"}}]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/#person\",\"name\":\"jgmoxness\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/#author\",\"url\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/author\\\/jgmoxness\\\/\",\"name\":\"jgmoxness\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/#webpage\",\"url\":\"https:\\\/\\\/theoryofeverything.org\\\/theToE\\\/2023\\\/02\\\/17\\\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\\\/\",\"name\":\"3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \\u200bGeometric Group Theory - Visualizing a Theory of Everything!\",\"description\":\"Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8->H4 folding matrix) from quaternions given their Weyl Orbits. 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It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don't want to use the interactive Powerpoint,","og:url":"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/","article:published_time":"2023-02-17T16:11:06+00:00","article:modified_time":"2023-10-08T17:11:18+00:00","twitter:card":"summary","twitter:title":"3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \u200bGeometric Group Theory - Visualizing a Theory of Everything!","twitter:description":"Click here to view the Powerpoint presentation. It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8-&gt;H4 folding matrix) from quaternions given their Weyl Orbits. If you don't want to use the interactive Powerpoint,"},"aioseo_meta_data":{"post_id":"5991","title":null,"description":null,"keywords":[],"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":[],"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"BlogPosting","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","local_seo":null,"limit_modified_date":false,"created":"2023-02-17 16:05:01","updated":"2025-06-10 15:54:51"},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/\" title=\"Physics\">Physics<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t3D &amp; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin \u200bGeometric Group Theory\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/theoryofeverything.org\/theToE"},{"label":"Physics","link":"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/"},{"label":"3D &#038; 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin\u00a0\u200bGeometric Group Theory","link":"https:\/\/theoryofeverything.org\/theToE\/2023\/02\/17\/quaternion-weyl-orbits-from-coxeter-dynkin-geometric-group-theory\/"}],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5991","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=5991"}],"version-history":[{"count":31,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5991\/revisions"}],"predecessor-version":[{"id":6311,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5991\/revisions\/6311"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=5991"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=5991"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=5991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}