This post is a work in process, so check back for the latest visualizations and data. Today (Dec. 31 2025) I finally finished a major update to the section SVG files from the first update of 12/16/2025. Have a blessed New Year for 2026 and beyond!

These were improved a bit on 01/04/2026 and on 01/06/2026 I added more informative Thumbnail icons in the matrix links that include the vertex-edge-face-cell OverallHull images. On 01/12/2026 I added the Snub 24-cell orientations. On 01/20/2026 I improved the visualizations. On 01/26/2026 I increased the thumbnail font and corrected a 24-cell Wythoffian permutation definition.

So if you downloaded previous files, you may want to take another look, as some of the output had anomalous data that didn’t render, along with other errors (due to the way I computed approximate equality between norms). In order to get this done, I realized I was thrashing my Solid State Drive (SSD) due to memory swapping. It died due to that (SSDs have limited writes!), so I had to rebuild the machine with 64 Gb RAM to keep the computations in RAM.

Drop me an email if you find anything that needs correction.

For fully interactive Mathematica NB (and higher quality SVG, PDF, as well as PNG files for all of the combined and overall hull pairs as shown above) in all 15 Weyl orbit permutations, download them here:

H4 600-cell: NB (50 Mb) SVG (50 Mb) PDF (25 Mb)

F4 288-cell: NB (10 Mb) SVG (25 Mb) PDF (10 Mb)

D4 24-cell: NB (5 Mb) SVG (15 Mb) PDF (10 Mb)

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 the self-dual 24-cell (D4), the disphenoidal 288-cell (F4), and H4 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 4 quaternionic rotations. The one off a main orbit orientation (i.e. using the e₀=1 identity quaternion rotation) is either the vertex or cell first orientation for each of the 15 Weyl orbit permutations, with the other rotation being e₂+e₃. Then for edge and face first orientation rotations, depending on the 24, 288, or 600 cell type, they will be one of either e₁+e₂+e₃ & e₁+2e₂+e₃ (for 24 & 288 cells) or e₀+φe₁ & e₀+e₁+e₂ (for 600/120 cells).

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 SVG sectioning for the 600-cell here (900 Mb), 288-cell here (200 Mb), and 24-cell here (150 Mb).

SVG section visualizations:

Orientations for H4 600-cell SVG section file links:

              Cell                          Vertex                             Edge                                 Face

Orientations for F4 Disphenoidal 288-cell SVG section file links:

              Cell                          Vertex                             Edge                                 Face

Orientations for D4 24-cell SVG section file links:

              Cell                          Vertex                             Edge                                 Face

MP3 section animations:

Orientations for H4 600-cell MP3 section animation file links:

              Cell                          Vertex                             Edge                                 Face

Orientations for F4 Disphenoidal 288-cell MP3 section animation file links:

              Cell                          Vertex                             Edge                                 Face

Orientations for D4 24-cell MP3 section animation file links:

              Cell                          Vertex                             Edge                                 Face

An interesting set of prisms are in the F4 disphenoidal 288-cell’s omni-truncated sections (below):

If you have access to Wolfram’s Mathematica, see my notebook with code and data that may be helpful here (135Mb). A PDF is here (65Mb). This has detail on the octonion (E8² and E8³) based constructions for both the 24D Leech lattices as well as the 16D Barnes-Wall lattice shown at the bottom of this post.

The Leech lattice is a sub-quotient of the largest of the sporadic finite simple groups, namely the Monster group. It is related to (and can be constructed by) E₈ and the 23rd Niemeier lattice of E₈³. The 24D Leech lattice is interesting to the physics of sphere packing, error-correcting codes, and possibly unifying General Relativity (GR) with Quantum Mechanics (QM). String theory was founded on the ideas related to its relationship to the Monster group and how 24 dimensions relate to the bosonic energy levels of the partition function on a torus. For more on the Monster and Group Theory, see my post here.

But taking things down a 8D (Bott periodic) notch, roughly speaking the 16D Barnes-Wall lattice is related to E₈ in the same way by BW₁₆=E₈².

I found some very nice recent visualizations on the BW₁₆ WP article by Misaki Ohta with related arXiv papers. He provided links to related code and data here. Below are my renderings using that provided information.

It is interesting to note that if one projects the 61440 2nd shortest 16D vectors of BW₁₆  using orthogonal projection mapping ℝ¹⁶ → ℝ² (i.e. to B₈ basis vectors [x=Cos, y=Sin]@(0-15)π/8 used for E₈ below vs. the proper (0-15)π/16 B₁₆ as done above), we get a similar result as projecting the 2nd shortest 8D vectors of E₈ using the same basis shown as grey vertices in Fig 2 of arXiv:2506.11725:

Projecting the 2160 2nd shortest 8D vectors of E₈ using the B₈ basis vectors including 69120 edges of 8D norm=√8 with colors assigned by projected edge length (notice the similarity in the vertex locations of the figure above):

Projecting the 4320 shortest 16D vectors of BW₁₆  using orthogonal projection mapping ℝ¹⁶ → ℝ² using the B₈ basis vectors [x=Cos, y=Sin]@(0-15)π/8 gives the results shown in Fig. 2 as red and blue vertices:

This time with edges colored by projected edge lengths:

Now, getting creative with the 16D projections of the 4230 BW₁₆, if we double the E8 Petrie basis vectors we get a similar result with 480+1 visible vertices with one at the origin w/240 overlapping vertices. There are 240 yellow with no overlaps and 240 blue with 16 overlaps each. There are 604800 norm=4 edges using roughly the same color pallet as my WP E₈ Petrie image. This makes it easier to see the difference between the 4230 BW₁₆ and the normal E₈ Petrie projection, which seem to share the grid-like pattern of the 24-cells within the folded E₈ = H₄ + φ H₄ :

This is the same as above, but using the 61440 vertex BW₁₆ with no edges (given there are millions of them). Please note that the number of visible vertices is the same as E8₂₄₁ at 2160!

Octonion based Barnes-Wall and Leech Lattice visualizations based on my implementation of work from Geoffrey Dixon and Robert Wilson:

This page presents the comprehensive 15 permutations of the six 4D polychora (i.e. 5, 16/8, 24, 600/120 cells) in full SVG format for convenient use in high quality academic papers on the topic (e.g. simply save the SVG and edit in Inkscape or other tool to produce PDF or PNG, etc.). You are free to use these under Creative Commons Attribution-ShareAlike 4.0 International CC BY-SA 4.0 with appropriate attribution. Drop me an email if you find anything that needs correction.

My cell-first and vertex-first {5,3,3} 120-cell section visualizations with subgroup sections (when the inscribed solid has more than one permutation in its orbit) are available in Wikipedia Commons here and here. Let me know if you need customized hyper-complex and/or hyper-dimensional group-theoretic projection/section visualizations, as my extensive Mathematica code-base may be able to generate it.

The content below uses a 6×15 matrix of 4D to 3D projections of the convex hulls, each with a link to a page with that objects Coxeter section decomposition. I want to give a shout-out (cite) qfbox.info  and polytope.miraheze.org for providing vertex coordinates used in the creation of the objects. The Vertex-Edge-Face-Cell first sections are in another one of my posts here.

There is also another 6×15 table with links to video animations of the sections in the 4D to 3D flatlander from Left/Right (or minus to plus). This is an analog of 3D objects passing through the 2D planar world of flatlanders.

Also available is a Mathematica notebook (360Mb) with code to produce the 4D vertex data as well as the visualized interactive 3D objects. See also an overview of these lists explained in a this Powerpoint presentation, which covers the isomorphisms between them due to the symmetries in the A4, BC4, and H4 Coxeter-Dynkin diagrams that represent them.

Below are some screenshots from a PPT I created to explain the organization. Some of this content is now on Wikipedia from my updates to those pages. The PPT is available here (in PDF here) or you can click on the PNG images (below) to get the SVG versions:

The chart below is here on Wikipedia (WP):

I took screenshots of a few qfbox.info pages to annotate the differences in layout below. Please note the palindromic binary Weyl group orbits and operation equivalences between them:

The Coxeter-Dynkin diagram relationships between the BC4, D4, and F4 are as follows (my apologies to those who might be irritated by my use of filled nodes instead of the “ringed” nodes):

The Coxeter-Dynkin diagram relationship between the D4 (dual 24-cell parent) + D4 Snub 24-cell = H4 are:

But let’s not forget the larger symmetry of these 4-polytopes, as they are related to the largest exceptional algebra and group of E8. Indeed, E8 is isomorphic to H4+H4φ through a simple rotation (folding) matrix.

Links to the SVG Section Files:

Links to Sectioning MP4 Video Animations (i.e. 4D to 3D Flatlander):

For completeness, I am also including a table of 3D polyhedra, namely the Platonic, Archimedean, and Catalan Solids including
their irregular and chiral forms. These were created using
quaternion Weyl orbits directly from the A3, B3, and H3 group
symmetries

.

Another post with more detail on the 5-cell (A4) SU(5)->SU(4) maximal subgroup content generated via quaternions is here.

This is an analysis of the Coxeter Sections for the Swirl Prism 120-Diminished Rectified 600-Cell. The source data, which can be found on either of two polychora sites (qfbox.info and polytope.miraheze.org) seems to have two missing vertices 598 as generated by their permutation lists (vs. the intended 600). The data below, with an added two vertices of {-1,-1,φ³, φ³} and {1,-1,φ³, φ³} in the suspected error sections #3 & #15, is shown in data and 3D convex hull and Coxeter sectioning forms.