{"id":6802,"date":"2025-01-23T12:36:53","date_gmt":"2025-01-23T19:36:53","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=6802"},"modified":"2025-04-06T18:00:43","modified_gmt":"2025-04-07T01:00:43","slug":"a-visual-overview-of-how-the-h4-600-cells-embed-into-e8","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2025\/01\/23\/a-visual-overview-of-how-the-h4-600-cells-embed-into-e8\/","title":{"rendered":"A Visual Overview of how the H4 600-cell(s) embed into E8"},"content":{"rendered":"\n

Please see this Powerpoint<\/a> for an interactive presentation (160 Mb).<\/a><\/p>\n\n\n

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2D projection of 8,16,24, snub-24 and 600 cells as part of E8 vertex Petrie projection<\/figcaption><\/figure>\n<\/div>\n\n\n

Below (and here<\/a> and here<\/a> in PDF and Mathematica Notebook form or here<\/a> on the Cloud) are the I & I’ 600-cells based on quaternion construction from the T & T’ 24-cells. These are also documented in Coxeter’s book “Regular Polytopes” as {3,3,5} vertex first & cell first “sections” in Table V (respectively)<\/strong> vs. the projections shown here with one of the four dimensions projected to 0 such that the hulls are palindromic pairs of sections (e.g. for 9 sections, section 1-4 pairs with 9-6 in vertex counts equal to hull vertex counts 1-4 <\/strong><\/strong>and the center section 5 = hull 5).<\/strong><\/p>\n\n\n

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The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows pairs of:

1) points at the origin

2) icosahedrons

3) dodecahedrons

4) icosahedrons

5) and a single icosadodecahedron

for a total of 120 vertices and an overall outer 3D hull of the Pentakis Icosidodecahedron.<\/figcaption><\/figure>\n<\/div>\n\n
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Alternate 600-Cell Convex Hulls. The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:

1) a pair of tetrahedrons (or single cube)

2) a pair of tetrahedrons (or single cube)

3) a pair of icosahedrons

4) truncated cube

5) rhombicuboctahedron

6) rhombicuboctahedron

7) a pair of tetrahedrons (or single cube)

8) cuboctahedron

for a total of 120 vertices and an overall outer 3D hull of a near-miss of the Pentakis Icosidodecahedron.<\/figcaption><\/figure>\n<\/div>\n\n\n

For completeness, here are the J’ (alternate) <\/strong> & J form of the 120-cells based on quaternion construction from the T & T’ 24-cells<\/strong>. These are also documented in Coxeter’s book “Regular Polytopes” as {5,3,3} vertex first & cell first “sections”<\/strong> in Table V (respectively)<\/strong> vs. the projections shown here with one of the four dimensions projected to 0 such that the hulls are palindromic pairs of sections (e.g. for 31 sections, section 0-14 pairs with 30-16 in vertex counts equal to hull vertex counts 1-15 and the center section 15 = hull 16).<\/strong><\/strong><\/p>\n\n\n

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Hull 1 has 2 points at the origin.
Hulls 2 & 6 are cubes.
Hulls 3 & 5 are rhombicuboctahedrons.
Hulls 4 & 12 are each pairs of truncated octahedrons.
Hulls 7 & 15 are truncated cuboctahedrons.
Hull 11 is a truncated cube.
Hulls 8, 9, 10, 13, 14, and 16 are unnamed solids.<\/figcaption><\/figure>\n<\/div>\n\n
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Hulls 1, 2, & 7 are each overlapping pairs of dodecahedrons.
Hull 3 is a pair of icosidodecahedrons.
Hulls 4 & 5 are each pairs of truncated icosahedrons.
Hulls 6 & 8 are each rhombicosidodecahedrons.<\/figcaption><\/figure>\n<\/div>\n\n\n

Below is an interactive Mathematica 4D visualization of the 8,16, 24, 120, and 600 cells (including alternate forms 24p, 120p, 600p). Click here<\/a> or on the image below to open in a new tab.<\/p>\n\n\n

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\"\"<\/a><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

Please see this Powerpoint for an interactive presentation (160 Mb). Below (and here and here in PDF and Mathematica Notebook form or here on the Cloud) are the I & I’ 600-cells based on quaternion construction from the T & T’ 24-cells. These are also documented in Coxeter’s book “Regular Polytopes” as {3,3,5} vertex first … Continue reading A Visual Overview of how the H4 600-cell(s) embed into E8<\/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":[8,4,34,14,16],"class_list":["post-6802","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-e8","tag-h4","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6802","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=6802"}],"version-history":[{"count":21,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6802\/revisions"}],"predecessor-version":[{"id":7080,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6802\/revisions\/7080"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=6802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=6802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=6802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}