{"id":4846,"date":"2017-10-21T14:31:53","date_gmt":"2017-10-21T21:31:53","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=4846"},"modified":"2017-10-23T14:01:38","modified_gmt":"2017-10-23T21:01:38","slug":"the-concentric-hulls-of-e8-projected-to-h4","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2017\/10\/21\/the-concentric-hulls-of-e8-projected-to-h4\/","title":{"rendered":"E8 Projected to the Concentric Hulls of H4+H4 Phi"},"content":{"rendered":"<p>The 8 concentric projected 3D hulls from the vertices of the 4_21 polytope (using the split real even E8 roots) and a projection basis from 3 of the 4 rows of the E8 to the H4+H4 Phi folding matrix produce from inner to outer (sorted by vertex Norm):<\/p>\n<ul>\n<li>4 Points at the origin<\/li>\n<li>2 Icosahedron<\/li>\n<li>2 Dodecaheron<\/li>\n<li>4 Icosahedron<\/li>\n<li>1 Icosadodecahedron<\/li>\n<li>2 Dodecaheron<\/li>\n<li>2 Icosahedron<\/li>\n<li>1 Icosadodecahedron<\/li>\n<\/ul>\n<p>Projection Basis:<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/09\/projection1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4515\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/09\/projection1.png\" alt=\"\" width=\"137\" height=\"75\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/E8-polys-phi-projection.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4847\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/E8-polys-phi-projection.png\" alt=\"\" width=\"991\" height=\"1681\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/E8-polys-phi-projection.png 991w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/E8-polys-phi-projection-295x500.png 295w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/E8-polys-phi-projection-604x1024.png 604w\" sizes=\"auto, (max-width: 991px) 100vw, 991px\" \/><\/a><\/p>\n<p>Looking at it as an orthonormal 3D projection of 2 600-cells (from the fully folded E8 to H4+H4 Phi), which is the same (as it should be), but rotated.<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4-H4Phi-orthonormal.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4859\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4-H4Phi-orthonormal.png\" alt=\"\" width=\"991\" height=\"1691\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4-H4Phi-orthonormal.png 991w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4-H4Phi-orthonormal-293x500.png 293w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4-H4Phi-orthonormal-600x1024.png 600w\" sizes=\"auto, (max-width: 991px) 100vw, 991px\" \/><\/a><\/p>\n<p>Here is just one 600-cell (interior).<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4only-orthonormal.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4860\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4only-orthonormal.png\" alt=\"\" width=\"991\" height=\"1409\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4only-orthonormal.png 991w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4only-orthonormal-352x500.png 352w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/hulls-of-H4only-orthonormal-720x1024.png 720w\" sizes=\"auto, (max-width: 991px) 100vw, 991px\" \/><\/a><\/p>\n<p>Here is E7 doing the same procedure:<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/3_21-E7-to-3D-H3-symmetry-concentric-hulls.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4864\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/3_21-E7-to-3D-H3-symmetry-concentric-hulls.png\" alt=\"\" width=\"751\" height=\"546\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/3_21-E7-to-3D-H3-symmetry-concentric-hulls.png 751w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/3_21-E7-to-3D-H3-symmetry-concentric-hulls-500x364.png 500w\" sizes=\"auto, (max-width: 751px) 100vw, 751px\" \/><\/a><\/p>\n<p>And again for E6<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/1_22-E6-to-3D-H3-symmetry-concentric-hulls.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4866\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/1_22-E6-to-3D-H3-symmetry-concentric-hulls.png\" alt=\"\" width=\"741\" height=\"591\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/1_22-E6-to-3D-H3-symmetry-concentric-hulls.png 741w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2017\/10\/1_22-E6-to-3D-H3-symmetry-concentric-hulls-500x399.png 500w\" sizes=\"auto, (max-width: 741px) 100vw, 741px\" \/><\/a><\/p>\n<h4><span style=\"color: #ff0000;\">Wow &#8211; that is nice! Remember, you heard it here first!<\/span><\/h4>\n<p><em><strong>BTW &#8211; if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:\u00a0 jgmoxness@theoryofeverthing.org.\u00a0<\/strong><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The 8 concentric projected 3D hulls from the vertices of the 4_21 polytope (using the split real even E8 roots) and a projection basis from 3 of the 4 rows of the E8 to the H4+H4 Phi folding matrix produce from inner to outer (sorted by vertex Norm): 4 Points at the origin 2 Icosahedron &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2017\/10\/21\/the-concentric-hulls-of-e8-projected-to-h4\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">E8 Projected to the Concentric Hulls of H4+H4 Phi<\/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":[4,34],"class_list":["post-4846","post","type-post","status-publish","format-standard","hentry","category-physics","tag-e8","tag-h4"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4846","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=4846"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4846\/revisions"}],"predecessor-version":[{"id":4858,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4846\/revisions\/4858"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=4846"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=4846"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=4846"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}