{"id":2105,"date":"2014-11-01T22:20:47","date_gmt":"2014-11-02T05:20:47","guid":{"rendered":"http:\/\/theoryofeverything.org\/MyToE\/?p=2105"},"modified":"2017-10-01T09:57:42","modified_gmt":"2017-10-01T16:57:42","slug":"e8-to-3d-projection-using-my-h4-folding-matrix","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2014\/11\/01\/e8-to-3d-projection-using-my-h4-folding-matrix\/","title":{"rendered":"E8 to 3D projection using my H4 folding matrix"},"content":{"rendered":"<p>Best viewed in HD mode&#8230;<\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"500\" src=\"\/\/www.youtube.com\/embed\/LzpbDttfh9g\" frameborder=\"0\" allowfullscreen><\/iframe><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?p=1454\">Last year I showed <\/a>that the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Dynkin_diagram\">Dynkin diagram <\/a>of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/E8_(mathematics)\">Lie Algebra, Group and Lattice of E8 <\/a>is related to the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Coxeter%E2%80%93Dynkin_diagram\">Coxeter-Dynkin <\/a> of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/600-cell\"> 600 Cell of H4<\/a> through a folding of their diagrams:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=1523\" rel=\"attachment wp-att-1523\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2013\/11\/E8-H4-Fold.png\" alt=\"E8-H4-Fold\" width=\"300\" class=\"alignnone size-full wp-image-1523\" \/><\/a><\/p>\n<p>I had found the E8 to H4 folding matrix several years ago after reading <a href=\"http:\/\/journals.tubitak.gov.tr\/physics\/issues\/fiz-98-22-5\/fiz-22-5-8-97022.pdf\">several papers by Koca<\/a> and <a href=\"http:\/\/arxiv.org\/abs\/1110.5228v1\">another paper by Dechant<\/a>, on the topic. The matrix I found is:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2116\" rel=\"attachment wp-att-2116\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/C600.png\" alt=\"C600\" width=\"300\" class=\"alignnone size-full wp-image-2116\" \/><\/a><\/p>\n<p>Notice that C600=Transpose[C600] and the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quaternion\">Quaternion<\/a>&#8211;<a href=\"https:\/\/en.wikipedia.org\/wiki\/Octonion\">Octonion<\/a> like structure (ala. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cayley%E2%80%93Dickson_construction\">Cayley-Dickson<\/a>) within the folding matrix. Only the first 4 rows are needed for folding, but I use the 8&#215;8 matrix to rotate 8D vectors easily.<\/p>\n<p>The following x,y,z vectors project <a href=\"https:\/\/en.wikipedia.org\/wiki\/File:E8Petrie.svg\">E8 to its Petrie projection <\/a>on one face (or 2 of 6 cubic faces, which are the same).<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2118\" rel=\"attachment wp-att-2118\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8projection.png\" alt=\"E8projection\" height=\"100\" class=\"alignnone size-full wp-image-2118\" \/><\/a><\/p>\n<p>On another face are the orthonormal <a href=\"https:\/\/en.wikipedia.org\/wiki\/600-cell\">H4 600 cell <\/a>and the concentric H4\/Phi. There are 6720 edges with 8D Norm&#8217;d length of Sqrt[2], but I only show 1220 in order to prevent obscuring the nodes.<\/p>\n<p>These values were calculated using the dot product of the Inverse[4*C600] with the following H4 Petrie projection vertices (aka. <a href=\"https:\/\/en.wikipedia.org\/wiki\/File:Cell600Petrie.svg\">the Van Oss projection<\/a>), where I&#8217;ve added the z vector for the proper 3D projection. Notice I am still using 8D basis vectors with the last 4 zero, as this maintains scaling due to the left-right symmetries in C600).<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2119\" rel=\"attachment wp-att-2119\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/H4projection.png\" alt=\"H4projection\" height=\"100\" class=\"alignnone size-full wp-image-2119\" \/><\/a><\/p>\n<p>The Split Real Even (SRE) E8 vertices are generated from the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Root_system#E6.2C_E7.2C_E8\">root system <\/a> for the E8 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Dynkin_diagram\">Dynkin diagram  <\/a> and its resulting <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cartan_matrix\">Cartan matrix<\/a>:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2125\" rel=\"attachment wp-att-2125\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8Dynkin.png\" alt=\"E8Dynkin\" width=\"500\" class=\"alignnone size-full wp-image-2125\" \/><\/a><\/p>\n<p>This is input from my <a href=\"http:\/\/theoryofeverything.org\/TOE\/JGM\/ToE_Demonstration.cdf\">Mathematica &#8220;VisibLie&#8221; application<\/a>. It is shown with the assigned physics particles that make up the simple roots matrix entries:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2117\" rel=\"attachment wp-att-2117\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/srE8.png\" alt=\"srE8\" width=\"300\" class=\"alignnone size-full wp-image-2117\" \/><\/a>.<br \/>\nPlease note, the Cartan matrix can also be generated by srE8.Transpose[srE8]<\/p>\n<p>It takes Transpose[srE8] and applies the dot product against the 120 positive and 120 negative algebra roots generated by the <a href=\"http:\/\/www.equaonline.com\/math\/SuperLie\/\">Mathematica &#8220;SuperLie&#8221; package <\/a>(shown along with its Hasse diagram, which I generate in the full version of my Mathematica notebook):<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2122\" rel=\"attachment wp-att-2122\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8_Hasse.png\" alt=\"E8_Hasse\" width=\"500\" class=\"alignnone size-full wp-image-2122\" \/><\/a><\/p>\n<p>Interestingly, E8 in addition to containing the 8D structures D8 and BC8 and the <a href=\"https:\/\/commons.wikimedia.org\/wiki\/Category:Polychora\">4D Polychora<\/a>, contains the 7D E7, as well as the 6D structures of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/6-cube\">Hexeract or 6 Cube<\/a>. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rhombic_triacontahedron#Related_polyhedra\">I also showed several years ago that this projects down to the 3D as the Rhombic Triacontahedron<\/a>. The Rhombic Triacontahedron (of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quasicrystal\">Quasicrystal <\/a>fame) contains the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Platonic_solid\">Platonic Solids <\/a> including the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Icosahedron\">Icosahedron <\/a>and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Dodecahedron\">Dodecahedron<\/a>!   This is shown to be done through folding of the 6-Cube using rows 2 through 4 of the E8 to H4 folding matrix!!<\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2160\" rel=\"attachment wp-att-2160\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/D6_H3fold.png\" alt=\"D6_H3fold\" width=\"300\" class=\"alignnone size-full wp-image-2160\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2155\" rel=\"attachment wp-att-2155\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8-8D-Polytopes-2.jpg\" alt=\"E8-8D-Polytopes-2\" width=\"500\" class=\"alignnone size-full wp-image-2155\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2156\" rel=\"attachment wp-att-2156\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8-4D-Polychora-2.jpg\" alt=\"E8-4D-Polychora-2\" width=\"500\" class=\"alignnone size-full wp-image-2156\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2157\" rel=\"attachment wp-att-2157\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/E8-3D-Platonic-2.png\" alt=\"E8-3D-Platonic-2\" width=\"500\" class=\"alignnone size-full wp-image-2157\" \/><\/a><\/p>\n<p><a href=\"http:\/\/theoryofeverything.org\/MyToE\/?attachment_id=2139\" rel=\"attachment wp-att-2139\"><img decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/MyToE\/wp-content\/uploads\/2014\/11\/outE821b.png\" alt=\"outE821b\" width=\"500\" class=\"alignnone size-full wp-image-2139\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Best viewed in HD mode&#8230; Last year I showed that the Dynkin diagram of the Lie Algebra, Group and Lattice of E8 is related to the Coxeter-Dynkin of the 600 Cell of H4 through a folding of their diagrams: I had found the E8 to H4 folding matrix several years ago after reading several papers &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2014\/11\/01\/e8-to-3d-projection-using-my-h4-folding-matrix\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">E8 to 3D projection using my H4 folding matrix<\/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":[13,4,16],"class_list":["post-2105","post","type-post","status-publish","format-standard","hentry","category-physics","tag-4d","tag-e8","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2105","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=2105"}],"version-history":[{"count":2,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2105\/revisions"}],"predecessor-version":[{"id":4593,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2105\/revisions\/4593"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=2105"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=2105"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=2105"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}