{"id":2308,"date":"2014-12-01T10:56:53","date_gmt":"2014-12-01T17:56:53","guid":{"rendered":"http:\/\/theoryofeverything.org\/MyToE\/?p=2308"},"modified":"2017-10-01T12:14:57","modified_gmt":"2017-10-01T19:14:57","slug":"hopf-fibration-and-chaotic-attractors-etc","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2014\/12\/01\/hopf-fibration-and-chaotic-attractors-etc\/","title":{"rendered":"Hopf Fibration and Chaotic Attractors, etc."},"content":{"rendered":"

I’ve added some new features to my VisibLie_E8 ToE Demonstration<\/a>. Some of it comes from Richard Hennigan’s Rotating The Hopf Fibration<\/a> and Enrique Zeleny’s A Collection Of Chaotic Attractors<\/a> . These are excellent demonstrations that I’ve now included with the features of my integrated ToE demonstration, since they are not only great visualizations, but relate to the high-dimensional physics of E8, octonions and their projections. This gives the opportunity to change the background and color schemes, as well as output 3D models or stereoscopic L\/R and red-cyan anaglyph images.<\/p>\n

\"outChaos3\"<\/a><\/p>\n

\"Hopf3Dstereo\"<\/a><\/p>\n

\"outChaos2\"<\/a><\/p>\n

\"outChaos1\"<\/a><\/p>\n

I’ve also used David Madore’s<\/a> help to calculate the symbolic value of the E7 18-gon and 20-gon symmetries of E8. It uses the nth roots of unity (18 and 20, in this case) and applies a recursive dot product matrix based on the Weyl group centralizer elements of a given conjugacy class of E8. I ended up using a combination of Mathematica Group Theory built-in functions, SuperLie<\/a> and also LieART<\/a> packages. These symbolic projection values are:<\/p>\n

\"18-20-gonProjection\"<\/a><\/p>\n

\"outE802\"<\/a><\/p>\n

\"20-gon\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve added some new features to my VisibLie_E8 ToE Demonstration. Some of it comes from Richard Hennigan’s Rotating The Hopf Fibration and Enrique Zeleny’s A Collection Of Chaotic Attractors . These are excellent demonstrations that I’ve now included with the features of my integrated ToE demonstration, since they are not only great visualizations, but relate … Continue reading Hopf Fibration and Chaotic Attractors, etc.<\/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":[8,12,30,14],"class_list":["post-2308","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-cdf","tag-chaos","tag-math"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2308","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=2308"}],"version-history":[{"count":1,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2308\/revisions"}],"predecessor-version":[{"id":4653,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/2308\/revisions\/4653"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=2308"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=2308"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=2308"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}