{"id":543,"date":"2011-12-10T13:32:03","date_gmt":"2011-12-10T20:32:03","guid":{"rendered":"http:\/\/theoryofeverything.org\/wordpress\/?p=543"},"modified":"2017-10-01T11:26:45","modified_gmt":"2017-10-01T18:26:45","slug":"ho-mg-zn-quasicrystal-electron-diffraction-from-e8-5cube-projection","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2011\/12\/10\/ho-mg-zn-quasicrystal-electron-diffraction-from-e8-5cube-projection\/","title":{"rendered":"Ho-Mg-Zn QuasiCrystal Electron Diffraction with E8 5Cube Projection"},"content":{"rendered":"<figure id=\"attachment_544\" aria-describedby=\"caption-attachment-544\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/theoryofeverything.org\/wordpress\/wp-content\/uploads\/2011\/12\/Ho-Mg-Zn_E8-5Cube.png\"><img loading=\"lazy\" decoding=\"async\" title=\"Ho-Mg-Zn_E8-5Cube\" src=\"http:\/\/theoryofeverything.org\/wordpress\/wp-content\/uploads\/2011\/12\/Ho-Mg-Zn_E8-5Cube-300x300.png\" alt=\"\" width=\"500\" height=\"500\" \/><\/a><figcaption id=\"caption-attachment-544\" class=\"wp-caption-text\">Ho-Mg-Zn QuasiCrystal Electron Diffraction from E8 5Cube Projection<\/figcaption><\/figure>\n<p>This is an image of <a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Zn-Mg-HoDiffraction.JPG\">the electron diffraction pattern of an icosahedral Zn-Mg-Ho quasicrystal <\/a>with an overlay of a 5-Cube projection from the 240 vertices of the split real even E8 Lie Group.<\/p>\n<p>The basis vectors for the E8 projection are shown (1:1 with the ring of gray vertices with the last 3 of 8 dimensions 0).<\/p>\n<p>There are 2480 overlapping edge lines from the 240 E8 vertices. They have norm\u2019d unit length calculated from the 5 non-zero projected dimensions of E8. Of these, 32 inner vertices and 80 edges belong to the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Penteract\">5D 5-Cube (Penteract)<\/a> proper. Edges are shown with colors assigned based on origin vertex distance from the outer perimeter.<\/p>\n<p>The vertex colors of the 5-Cube projection represent E8 vertex overlaps. These are:<br \/>\nInView vertices={color{overlap,count},&#8230;}Total<br \/>\n{LightGreen{1,20},Pink{5,20},Gray{10,10},Orange{20,1}}51<br \/>\nAll vertices={color{overlap,count},&#8230;}Total<br \/>\n{LightGreen{1,20},Pink{5,100},Gray{10,100},Orange{20,20}}240<\/p>\n<p>A related projection of a <a href=\"http:\/\/en.wikipedia.org\/wiki\/Hexeract\">6D 6-Cube (Hexeract)<\/a> into a perspective 3D object using the Golden Ratio [Phi]. This particular projection is used to understand the structure of QuasiCrystals. The specific basis vectors are:<br \/>\nx = {1, [Phi], 0, -1, [Phi], 0}<br \/>\ny = {[Phi], 0, 1, [Phi], 0, -1}<br \/>\nz = {0, 1, [Phi], 0, -1, [Phi]}<br \/>\nThere are 64 vertices and 192 unit length edges forming pentagonal symmetry along specific axis (as well as hexagonal symmetries on other axis).<\/p>\n<figure id=\"attachment_583\" aria-describedby=\"caption-attachment-583\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/theoryofeverything.org\/wordpress\/wp-content\/uploads\/2011\/12\/6Cube-H4.jpg\"><img decoding=\"async\" class=\"size-large wp-image-583\" title=\"6Cube-H4\" src=\"http:\/\/theoryofeverything.org\/wordpress\/wp-content\/uploads\/2011\/12\/6Cube-H4-1024x1024.jpg\" alt=\"\" width=\"500\" \/><\/a><figcaption id=\"caption-attachment-583\" class=\"wp-caption-text\">Hexeract orthographically projected to 3D using Golden Ratio<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>This is an image of the electron diffraction pattern of an icosahedral Zn-Mg-Ho quasicrystal with an overlay of a 5-Cube projection from the 240 vertices of the split real even E8 Lie Group. The basis vectors for the E8 projection are shown (1:1 with the ring of gray vertices with the last 3 of 8 &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2011\/12\/10\/ho-mg-zn-quasicrystal-electron-diffraction-from-e8-5cube-projection\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Ho-Mg-Zn QuasiCrystal Electron Diffraction with E8 5Cube Projection<\/span> <span class=\"meta-nav\">&rarr;<\/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":[4,14,16],"class_list":["post-543","post","type-post","status-publish","format-standard","hentry","category-physics","tag-e8","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/543","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=543"}],"version-history":[{"count":1,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/543\/revisions"}],"predecessor-version":[{"id":4622,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/543\/revisions\/4622"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=543"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=543"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=543"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}