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!