{"id":5366,"date":"2020-05-04T12:25:46","date_gmt":"2020-05-04T19:25:46","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=5366"},"modified":"2020-05-12T10:58:20","modified_gmt":"2020-05-12T17:58:20","slug":"nested-lattices-of-e8","status":"publish","type":"post","link":"http:\/\/theoryofeverything.org\/theToE\/2020\/05\/04\/nested-lattices-of-e8\/","title":{"rendered":"Nested Lattices of E8 in Complex Projective 4-Space"},"content":{"rendered":"\n

I read an interesting article about a pattern discovered by Warren D. Smith (discussed at length here<\/a>):<\/p>\n\n\n\n

“The sum of the first three terms in the Eisenstein E_4(q)  Series Integers of the Theta series of the E8 lattice is a perfect fourth power: 1 + 240 + 2160 = 2401 = 7^4”<\/p>\n\n\n\n

<\/p>\n\n\n\n

So I decided to visualize the 2401=1+240+2160 vertex patterns of E8 using my Mathematica codebased toolset based on some previous work I put on my Wikipedia talk page<\/a>. <\/p>\n\n\n\n

The image below represents various projections showing 6720 edges of the 240 E8 vertices, plus a black vertex at the origin, and the 2160 Witting Polytope E8 2 _ 41 vertices using the same projection basis (listed at the top of each image along with the color coded vertex overlaps). Click these links for a higher resolution PNG<\/a> or the SVG<\/a> version.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Some of the particular projections of the Witting Polytope may need 8D rotations applied to the basis vectors to find better symmetries with the Gosset, but this is a start using my standard set of projections.<\/p>\n\n\n\n

The 240 vertices of the Gosset Polytope are generated using various permutations:<\/p>\n\n\n\n

(* E8 4_21 vertices *)
e8421 = Union@Join[
Eperms8@{1, 1, 1, 1, 1, 1, 1, 1}\/2,
perms8@{1, 1, 0, 0, 0, 0, 0, 0}];<\/p>\n\n\n\n

The 2160 vertices of the Witting Polytope are generated using various permutations: <\/p>\n\n\n\n

(* E8 2_41 vertices *)
e8241=Union@Join[
perms8[{1,0,0,0,0,0,0,0}4],
perms8[{1,1,1,1,0,0,0,0}2],
Eperms8[({2,0,0,0,0,0,0,0}+1)]]\/4;<\/p>\n\n\n\n

<\/p>\n\n\n\n

Another view shows just the 2160 Witting Polytope vertices. Click these links for a higher resolution PNG<\/a> or the SVG<\/a> version.<\/p>\n\n\n\n

\"\"
2160 Witting Polytope (vertices only)<\/figcaption><\/figure>\n\n\n\n

Another great source of visualizations on E8 and this Witting Polytope is here<\/a>.<\/p>\n\n\n\n

Now visualizing in 3D the structure in 3D using rows 2-4 of the E8->H4 folding matrix, we get: <\/p>\n\n\n\n

\"\"
Witting Polytope with 2160 E8 2 _ 41 vertices sorted by Norm distance into 3D Platonic solid related concentric hull structures using rows 2-4 of the E8->H4 folding matrix. The last image shows overlap counts using colored vertices.<\/figcaption><\/figure><\/div>\n\n\n\n
\"\"
Sets of 8 concentric hulls with hull vertex counts above each.<\/figcaption><\/figure>\n\n\n\n
\"\"
Full set of concentric hulls with color coded vertices showing overlap counts. Norm’d vertex groups in red.<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"

I read an interesting article about a pattern discovered by Warren D. Smith (discussed at length here): “The sum of the first three terms in the Eisenstein E_4(q)  Series Integers of the Theta series of the E8 lattice is a perfect fourth power: 1 + 240 + 2160 = 2401 = 7^4” So I decided … Continue reading Nested Lattices of E8 in Complex Projective 4-Space<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[4,14,16],"class_list":["post-5366","post","type-post","status-publish","format-standard","hentry","category-physics","tag-e8","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=5366"}],"version-history":[{"count":42,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5366\/revisions"}],"predecessor-version":[{"id":5464,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/5366\/revisions\/5464"}],"wp:attachment":[{"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=5366"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=5366"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=5366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}