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 those presented by Robert Webb’s Stella 4D.

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

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

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

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).

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.

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.

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).

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.

Section 7 (& 23) have 24 vertices that are two irregular cuboctahedra each with triangular (8) and rectangle (6) faces.

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).

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

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 :

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!

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 & cell first “sections” in Table V (respectively) 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 and the center section 5 = hull 5).

For completeness, here are the J’ (alternate) & J form of the 120-cells based on quaternion construction from the T & T’ 24-cells. These are also documented in Coxeter’s book “Regular Polytopes” as {5,3,3} vertex first & cell first “sections” in Table V (respectively) 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).

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

For a paper with much of the content below, please see: theoryofeverything.org/TOE/JGM/E8_and_H4_in_QM_and_QC.pdf, and associated Mathematica notebook here.

The wreath product of the Monster group with Z2 (M ≀ Z2) is the BiMonster, a quotient of the Y₅₅₅ Dynkin diagram which contains six E8(Y₁₂₄) Dynkin diagrams and the 23rd Niemeier lattice E8³ root system. This can be reduced to Y₄₄₄ by applying the spider relation to E8³ .

Happy Family genarations:
Red=Mathieu groups (1st)
Green=Leech lattice groups (2nd)
Blue=other Monster subquotients (3rd)
Black=Pariahs

Each color coded node represents one of the 24 Niemeier lattices, and the lines joining them represent the 24-dimensional odd unimodular lattices with no norm 1 vectors. The Coxeter number of the Niemeier lattice is to the left. The red node index number indicates the row in the table above.

”The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that ”the exceptional Lie groups all exist because of the octonions: G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space, see prehomogeneous vector space.”

This E8SubgroupTree.nb Mathematica Notebook (140Mb), viewable using the free viewable Player, which has interactive Tooltips on group nodes (with group data and Hasse diagram) and Tooltips on the subgroup edges (with its individual maximal embedding chart). Also in the notebook is one cell with a table of all of the above data plus the Dynkin diagram based maximal embedding charts and an example 2D or 3D polytope projection.

These models use a tetrahedral mesh generator package << TetGenLink` to analyze the cells created by the edges selected.

Note: The Mathematica auto-generated geometry statistics of 120 vertices, 720 golden-ratio length edges, 1200 faces, and 600 color-coded cells, with a surface area of 11.4 volume of 127.

The transparent shaded blue is the outer hull that forms the Pentakis Icosidodecahedron with a vertex hull having 30 vertices of Norm=1 from the center. All 120 vertices form concentric hulls in groups:

1) two points at the origin
2) two icosahedra (24)
3) two dodecahedra (40)
4) two larger icosahedra (24)
5) one icosidodecahedron (30)

H4 600-Cell with 3D vertex shapes representing Standard Model physics particle assignments and color-coded cells

H4 600-Cell with 1200 unit length edges, 2400 faces and 600-cells. 3D vertex shapes representing Standard Model physics particle assignments and color-coded cells are also shown

The H4 120-cell (J), dual of the 600-cell, showing the hull of the chamfered dodecahedron. It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated. It is shown with physics particle assignments.

Same as above with interior color-coded cells displayed

The Dual H4 120-cell (J’) shown above (and more below).

An animated alternate form 120-cell (J’), which seems to be a new (near-miss Johnson solid?) constructed using D4′ (or T’) vs. D4 (T) above.

The H4 dual 120-cell (J’) build of outer edges and color-coded cells

The H4 dual 120-cell (J’) build of outer edges with physics particle assignments