{"id":3461,"date":"2015-08-17T08:28:09","date_gmt":"2015-08-17T15:28:09","guid":{"rendered":"http:\/\/theoryofeverything.org\/MyToE\/?p=3461"},"modified":"2015-12-27T18:43:45","modified_gmt":"2015-12-27T18:43:45","slug":"fun-with-e8","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2015\/08\/17\/fun-with-e8\/","title":{"rendered":"Fun with Tutte-Coxeter, Beordijk-Coxeter, E8 and H4"},"content":{"rendered":"

In reference to a G+ post by Baez (w\/Greg Egan)<\/a>, it’s interesting to note the link to E8’s outer ring of the Petrie projection of a split real even E8, which creates a Beordijk-Coxeter helix.<\/p>\n

Beordijk-Coxeter helix in 2D
\n\"helix2Db\"<\/a>
\nBeordijk-Coxeter helix in 3D
\n\"cells6004b\"<\/a><\/p>\n

The Beordijk-Coxeter helix connects the nearest 6 vertices on the outer ring. The Tutte-Coxeter graph is created in 3 (blk,grn,red) sets of edges by taking the (outer) ring and skipping (6,8,12) or counting (7,9,13) vertices. It shows there are 2 perfect pentagons and 1 pentagram (with different radii due to the difference in distance between the sets of vertices used).<\/p>\n

\"e8ring8-1b\"<\/a><\/p>\n

Of course, the crystallographic E8 is manifestly related to the 5 fold symmetry of the pentagon, with its integral relationship to the non-crystallographic H4 group (and its Coxeter-Dynkin diagram) through E8 to H4 folding using the Golden ratio Phi<\/a>.<\/p>\n

It is interesting to note that the skipping of 5+(1,3,7) vertices is similar to the creation of the 120 (240) vertex positions of H4 (E8) Petrie projection by adding to the 24 vertices of the 8-cell and 16-cell (which make up the self-dual 24-cell) the 96 vertices of the Snub 24-cell. This is done through 4 rotations skipping 5 vertices.<\/p>\n

Also notice the (1,3,7) are the number of the imaginary parts of Complex, Quaternion, and Octonion numbers, also integrally related to E8.<\/p>\n

\"snub-pics\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In reference to a G+ post by Baez (w\/Greg Egan), it’s interesting to note the link to E8’s outer ring of the Petrie projection of a split real even E8, which creates a Beordijk-Coxeter helix. Beordijk-Coxeter helix in 2D Beordijk-Coxeter helix in 3D The Beordijk-Coxeter helix connects the nearest 6 vertices on the outer ring. … Continue reading Fun with Tutte-Coxeter, Beordijk-Coxeter, E8 and H4<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[8,33,4,34,14],"class_list":["post-3461","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-boerdijk-coxeter","tag-e8","tag-h4","tag-math"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3461","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=3461"}],"version-history":[{"count":1,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3461\/revisions"}],"predecessor-version":[{"id":3888,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/3461\/revisions\/3888"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=3461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=3461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=3461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}