{"id":6899,"date":"2025-02-28T09:09:34","date_gmt":"2025-02-28T16:09:34","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=6899"},"modified":"2026-01-05T08:57:05","modified_gmt":"2026-01-05T15:57:05","slug":"comparing-coxeters-regular-polytopes-table-v-to-moxness-convex-hulls","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2025\/02\/28\/comparing-coxeters-regular-polytopes-table-v-to-moxness-convex-hulls\/","title":{"rendered":"Comparing Coxeter’s Regular Polytopes Table V to Moxness’ Convex Hulls"},"content":{"rendered":"\n

This is a post to present a new visualization that algorithmically generates Coxeter’s Regular Polytopes<\/a> Table V vertex & cell first 600-cell {3,3,5} & 120-cell {5,3,3} polytope sections and compares them with my convex hull projections. I have also visualized the face-first and edge-first versions which Coxeter never addressed. The convex hulls of these match those presented by Robert Webb’s Stella 4D<\/a>.<\/p>\n\n\n\n

The base (H4) objects are generated (from E8 actually<\/a>) using a interactive group theoretic hypercomplex (octonion\/quaternion) hyperdimensional (nD with n<9) Geometric Algebra (GA) \/ Space-Time Algebra (STA)<\/a> capable symbolic engine which I created using Mathematica. <\/p>\n\n\n\n

These are 4D rotated into vertex\/cell\/face\/edge and then grouped & sorted palindromically using Coxeter’s numbering and scaling. See here<\/a>, here<\/a> and here<\/a> for the PDF, interactive 3D Mathematica Notebook, and Wolfram Cloud (respectively). For best performance, I suggest using the free Wolfram Player<\/a> if you don’t own a Mathematica license for local manipulation.<\/p>\n\n\n\n

The hyper-dimensional (4D) vertices used in these panels were also used to generate Koca’s referenced dual snub-24-cell<\/a> as visualized on the that Wikipedia page.<\/p>\n\n\n\n

This is unique (AFAIK). Several of the Table V section entries are not identified by name. This is sometimes due to the irregular faces such that they lack a formal name (e.g. Coxeter labels section 15 of the vertex first {5,3,3} 120-cell as a 1+8 octahedron aka. an octahedron and an irregular truncated octahedron).<\/p>\n\n\n

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Other times it seems they were simply not recognized due to lack of visualization (e.g. while section 6 (& palindromically paired section 10) of the cell first {5,3,3} 120-cell are labeled as rhombicosidodecahedra, section 8 (and my hull #8) is also an irregular form of the rhombicosidodecahedron. Also on that object, sections 4, 5 (& 12, 11 respectively) are irregular truncated icosahedrons.<\/p>\n\n\n\n

On the cell first {5,3,3} 600-cell, unidentified sections 5, 6 (& 11, 10) are seen to be truncated tetrahedra, and 4 (& 12) are irregular cuboctahedra.<\/p>\n\n\n\n

As already referenced above, the most irregular is the vertex first {5,3,3} 120-cell with unidentified sections 2, 4 (& 28, 26) now seen as truncated tetrahedra, sections 6, 14 (& 24, 16) as irregular truncated octahedra, and sections 10 (& 20) as irregular cuboctahedra. Section 9 (& 21) is a pair of truncated tetrahedrons (shown below). <\/p>\n\n\n

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Interestingly, Coxeter’s labels for sections 3, 11 (& 27, 19) are “2 icosahedra” (aka. truncated octahedra). Coxeter’s partial label on 12 (& 18) is “Tetrahedron” for the 4 vertices with 24 others unidentified. These 24 are irregular hexagon (8) and rectangular (6) faces forming an irregular truncated octahedron.<\/p>\n\n\n

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Section 7 (& 23) have 24 vertices that are two irregular cuboctahedra each with triangular (8) and rectangle (6) faces.<\/p>\n\n\n

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Another notable section is found with 9-gon or nonagon faces in sections 13 (& 17), which are generated by the chiral left\/right “chamfered tetrahedrons” (3 per corner). <\/p>\n\n\n

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This likely derives from the 8-simplex A8 SU(9) as a subgroup of E8 as shown in the E8 subgroup tree<\/a> (with more detail in this post<\/a>). This folds to the H4 family of polytopes via my E8 <-> H4 folding matrix<\/a>.<\/p>\n\n\n

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Coxeter’s Regular Polytopes Table V-iii – Vertex First {3,3,5} 600-cell<\/figcaption><\/figure>\n<\/div>\n\n
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Coxeter’s Regular Polytopes Table V-iv-a – Cell First {3,3,5} 600-cell<\/figcaption><\/figure>\n<\/div>\n\n
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Coxeter’s Regular Polytopes Table V-iv-b – Cell First {5,3,3} 120-cell<\/figcaption><\/figure>\n<\/div>\n\n
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Coxeter’s Regular Polytopes Table V-v – Vertex First {5,3,3} 120-cell<\/figcaption><\/figure>\n<\/div>\n\n\n

Since Coxeter never presented any {3,3,5} \/ {5,3,3} edge-first or face-first sectioning, here is my attempt at visualizing them. Please note the top section of the SVG output below now includes the XYZW vertex +\/- & cyclic position permutation base values and other data collected in the process. There is also a 4D perspective projection below that uses a distance factor to change perspective. Note the differences between the edge-first (distance=1) vs. the face-first (distance=10) showing how the 4D perspective changes on both the {3,3,5} 600-cell \/ {5,3,3} 120-cell :<\/p>\n\n\n

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Face First {3,3,5} 600-cell<\/figcaption><\/figure>\n<\/div>\n\n
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Edge First {5,3,3} 120-cell<\/figcaption><\/figure>\n<\/div>\n\n
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Face First {5,3,3} 120-cell<\/figcaption><\/figure>\n<\/div>\n\n\n

And just for more fun… here are combinations of the above displayed together producing 720 and 1200 vertex overall hulls that are nice and\/or interesting!<\/p>\n\n\n

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720 vertex combining vertex first 335 600-cell with cell first 533 120-cell<\/figcaption><\/figure>\n<\/div>\n\n
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cell720p (or prime) 720 vertex combining cell first 335 600-cell with vertex first 533 120-cell<\/figcaption><\/figure>\n<\/div>\n\n
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1200 vertex combining both vertex & cell first 335 600-cell and 533 120-cell<\/figcaption><\/figure>\n<\/div>\n\n
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1200 vertex cell1200p by quaternion calculation:
octSimplify \/@Flatten@prq[[DoubleStruckCapitalA], octExp[Alpha]Lsw, TpT]<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

This is a post to present a new visualization that algorithmically generates Coxeter’s Regular Polytopes Table V vertex & cell first 600-cell {3,3,5} & 120-cell {5,3,3} polytope sections and compares them with my convex hull projections. I have also visualized the face-first and edge-first versions which Coxeter never addressed. The convex hulls of these match … Continue reading Comparing Coxeter’s Regular Polytopes Table V to Moxness’ Convex Hulls<\/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-6899","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\/6899","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=6899"}],"version-history":[{"count":58,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6899\/revisions"}],"predecessor-version":[{"id":7672,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6899\/revisions\/7672"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=6899"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=6899"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=6899"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}