{"id":4015,"date":"2016-03-16T17:14:19","date_gmt":"2016-03-16T17:14:19","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=4015"},"modified":"2016-03-16T18:56:24","modified_gmt":"2016-03-16T18:56:24","slug":"e8-in-e6-petrie-projection","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2016\/03\/16\/e8-in-e6-petrie-projection\/","title":{"rendered":"E8 in E6 Petrie Projection"},"content":{"rendered":"

An article<\/a> (interview) with John Baez<\/a> used an E8 projection which I introduced to Wikipedia in Feb of 2010 here<\/a>. Technically, it is E8 projected to the E6 Coxeter plane. <\/p>\n

\"E8<\/a><\/p>\n

The projection uses X Y basis vectors of:
\nX = {-Sqrt[3] + 1, 0, 1, 1, 0, 0, 0, 0};
\nY = {0, Sqrt[3] – 1, -1, 1, 0, 0, 0, 0};<\/p>\n

Resulting in vertex overlaps of:
\n24 Yellow with 1 overlap
\n24 Dark Blue each with 8 overlaps (192 vertices)
\n1 Light Blue with 24 overlaps (24 vertices)<\/p>\n

After doing this for a few example symmetries<\/a>, Tom took my idea of projecting higher dimensional objects to the 2D (and 3D) symmetries of lower dimensional subgroups – and ran with it in 2D – producing a ton of visualizations across WP. \ud83d\ude42<\/p>\n

It was one of those that was subsequently used that article from the 4_21 E8 WP page<\/a>. <\/p>\n

Here is a representation of E6 in the E6 Coxeter plane:
\n\"E6inE6\"<\/a><\/p>\n

Resulting in vertex overlaps of:
\n24 Yellow with 1 overlap
\n24 Orange each with 2 overlaps (48 vertices)<\/p>\n","protected":false},"excerpt":{"rendered":"

An article (interview) with John Baez used an E8 projection which I introduced to Wikipedia in Feb of 2010 here. Technically, it is E8 projected to the E6 Coxeter plane. The projection uses X Y basis vectors of: X = {-Sqrt[3] + 1, 0, 1, 1, 0, 0, 0, 0}; Y = {0, Sqrt[3] – … Continue reading E8 in E6 Petrie Projection<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"image","meta":{"footnotes":""},"categories":[3],"tags":[11,4,14,16],"class_list":["post-4015","post","type-post","status-publish","format-image","hentry","category-physics","tag-2d","tag-e8","tag-math","tag-physics","post_format-post-format-image"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4015","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=4015"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4015\/revisions"}],"predecessor-version":[{"id":4027,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4015\/revisions\/4027"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=4015"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=4015"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=4015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}