Cell-Vertex-Face-Edge First Orientations for D4 24-cell and H4 600-cell 4-Polytopes

This post is a work in process, so check back for the latest visualizations and data. The current update is as of 12/14/2025. Drop me an email if you find anything that needs correction.

Cell-vertex-edge-face first orientations of the OmniTruncated 600 cell (same as 120-cell by Coxeter-Dynkin diagram symmetry). Click the image for SVG version.

The Mathematica NB file for all of the combined and overall hull pairs (like the above PNG), but in all 15 Weyl orbit permutations of the 600-cell is here (50 Mb) or the SVG for all 600-cell orientations here (65 Mb).

An example section visualization – the cell-first-cell600-BiRectified-Sections (click to open the SVG version)

Please see several prior posts presenting (with now much improved) sectioning visualizations of Coxeter’s Regular Polytope book Table V here, 120-diminished rectified 120-cell (aka. Swirl-Prism) here, and 90=6×15 matrices of links to the 15 Weyl orbit permutations (with associated Coxeter-Dynkin diagrams) of the 4-polytope 5, 16/8, 24, 600/120 cell sections/animations here.

This post now presents the 60=[4 cell-first, vertex-first, face-first, and edge-first orientations of 15 Weyl orbits] visualizations for both the self-dual 24-cell and 600-cell (which has the same content as its dual the 120-cell by the symmetry of the Coxeter-Dynkin diagram).

These are generated using 3 quaternionic rotations of the main orbit of each of the 15 Weyl orbit permutations. vZome has a very nice (and fun) website curated by Scott Vorthmann with some information on quaternion generation of cell-vertex-face-edge first orientations here.

The main orbits are (or can also be) generated by using quaternions and the A/B-C/D-F/E/H group theoretic relationships. I actually use octonionic multiplication operations with multiplication tables from the 480 possible where the first (of 7 triads) are the quaternion of 0123. This means the last four dimensions of the octonion (i.e. 4567 or {l, li, lj, lk}) are {0,0,0,0}.

You can navigate the matrices of links below or simply download the ZIP files for the 24-cell here (300 Mb) or the Mathematica NB files here (? Mb) and the 600-cell ZIP here (600 Mb) and NB here (? Mb) .