Tag Archives: 4D

Weyl Group Orbit Coxeter Sections of the Regular 4D Platonic Polychora

September 23, 2025 jgmoxness Leave a comment

This page presents a comprehensive presentation of the 15 permutations of the six 4D polychora (i.e. 5, 16/8, 24, 600/120 cells). It presents them 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.

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.

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.

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

Platonic, Archimedean, and Catalan Solids

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

Physics

Coxeter Sections for the Swirl Prism 120-Diminished Rectified 600-Cell

September 16, 2025 jgmoxness Leave a comment

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.

The 3D XYZ convex hull projections (with W=0) and the +/- W Coxeter Sections of the rectified 600-Cell with 720 vertices

The Coxeter Sections and XYZ 3D convex hull projections of the Swirl Prism 120-diminished rectified 600-Cell with 600 vertices

The Coxeter Sections for the 120 diminished vertices (i.e. the complement of the Intersection of the Rectified 600-cell and the Swirl 120-Diminished Rectified 600-Cell)

List of vertices for the Rectified 600-cell

List of vertices for the Complement of the Intersection of the Rectified 600-cell and the Swirl 120-Diminished Rectified 600-Cell

Physics

The Isomorphism of H4 and E8

October 28, 2023 jgmoxness Leave a comment

Click here for a PDF of my recent paper. This has been accepted by the arXiv as 2311.01486 (math.GR & hep-th) with an ancillary Mathematica Notebook here. These links have (and will continue to have) minor descriptive improvements/corrections that may not yet be incorporated into arXiv, so the interested reader should check back here for those. Another related follow-on paper on THE ISOMORPHISM OF 3-QUBIT HADAMARDS AND E8 is referenced in the blog post here.

The abstract: This paper gives an explicit isomorphic mapping from the 240 real R^8 roots of the E8 Gossett 4_{21} 8-polytope to two golden ratio scaled copies of the 120 root H4 600-cell quaternion 4-polytope using a traceless 8×8 rotation matrix U with palindromic characteristic polynomial coefficients and a unitary form e^{iU}. It also shows the inverse map from a single H4 600-cell to E8 using a 4D<->8D chiral L<->R mapping function, phi scaling, and U^{-1}. This approach shows that there are actually four copies of each 600-cell living within E8 in the form of chiral H4L+phi H4L+H4R+phi H4R roots. In addition, it demonstrates a quaternion Weyl orbit construction of H4-based 4-polytopes that provides an explicit mapping between E8 and four copies of the tri-rectified Coxeter-Dynkin diagram of H4, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E8-based Grand Unified Theories or GUTs.

Concentric hulls of4_21 in Platonic 3D projection with numeric and symbolic norm distances

2D to the E8 Petrie projection using basis vectors X and Y from (6) with 8-polytope radius 4\sqrt{2} and 483,840 edges of length \sqrt{2} (with 53% of inner edges culled for display clarity

An alternative set of structure constant triads, octonion Fano plane mnemonic, and multiplication table, with decorations showing the palindromic multiplication.

Visualization of the 144 root vertices of S’+T+T’ now identified as the dual snub 24-cell

Breakdown of E8 maximal embeddings at height 248 of content SO(16)=D8 (120,128′)

Archimedean and dual Catalan solids, including their irregular and chiral forms. These were created using quaternion Weyl orbits directly from the A3, B3, and H3 group symmetries listed in the first column

A3 in A4 embeddings of SU(5)=SU(4)xU1
These include the specified 3D quaternion Weyl orbit hulls for each subgroup identified

The Eigenvalues, Eigenvector matrix, and characteristic polynomial coefficients of the unitary form of U as e^{IU} showing a Tr@Re@e^{IU}~4 and a traceless imaginary part

Personal, Sculpture and Art

Info about my creative work: The Universe, Math, Physics, & Art

August 4, 2023 jgmoxness Leave a comment

I think the most well-known (2D) image I’ve created so far (above) is found on Wikipedia’s (WP) math/geometry pages. It is what has been called the Public Relations (PR) or publicity photo for E8, which is an 8D (or 248 dimensional as the math guys count it) geometric object. It has been used in books, papers, math/science conference promotional materials, etc. That WP page also has what many used to consider “uninteresting” – my 3D versions of the same object. E8 is used in various theories to understand how the Universe operates way below the atomic level (i.e. the quantum stuff). I think it is responsible for what I call the shape of the Universe. As others have said, “who knew the Universe had a shape?”

In (very) layman’s terms, that 2D image of E8 looks sort-of-like how the lowly 4D Tesseract cube would look if it grew up into an 8D object, but a bit more interesting. The Tesseract was made (more) popular in the modern Avengers movies. In 2015 before I had ever seen any of those movies, I created a laser etched 3D projection of E8 within a jewel cut optical crystal cube and put a photo of that object lit from below by a blue LED on my website’s main homepage. Someone mentioned the resemblance and I was surprised at the likeness, motivating me to go see the movies.

Another of my more notable math/physics images has to do with the discovery of QuasiCrystals made by a guy, Dan Shechtman, who was ostracized from academia for the “impossible” idea of crystals having rotational symmetries beyond 2,3,4 and 6. He is now a Nobel Laureate. The WP image above is on that QuasiCrystal page as an overlay of part of E8 projected over a picture made from shooting a beam of x-rays at an icosahedral Ho-Mg-Zn quasicrystal. I didn’t actually do that experiment with the x-ray beams, but I did similar ones in my college days on a particle accelerator called the Cockroft-Walton Kevatron – see below for a picture of me (the one with the beard ;–) in the physics lab assembling a moon dust experiment. One of the professors had gotten the accelerator out of Germany after having worked on the Manhattan project.

E8 and its 4D children, the 600-cell and 120-cell (pages on which I have some work, amongst others) and its grandkids (2 of the 3D 5 Platonic Solids, one of which is the 3D version of the 2D Pentagon) are all related to the Fibonacci numbers and the Golden Ratio. So that kind of explains why most of my 2D art, 3D objects and sculptures (e.g. furniture like the dodecahedron table below), and 4D youtube animations all use the Golden Ratio theme.

Greg Moxness, Tucson AZ

Physics

A4 Group Orbits & Their Polytope Hulls Using Quaternions

May 10, 2023 jgmoxness Leave a comment

Updated: 05/11/2023

This post is a Mathematica evaluation of A4 Group Orbits & Their Polytope Hulls Using Quaternions and nD Weyl Orbits. This is a work-in-process, but the current results are encouraging for evaluating ToE’s.

Please see the this PDF or this Mathematica Notebook (.nb) for the details.

The paper being referenced in this analysis is here.

Of course, as in most cases, this capability has been incorporated into the VisibLie_E8 Demonstration codebase. See below for example screen shots:

This is an example .svg file output from the same interactive demonstration:

SVG file output from the VisibLie_E8 Coxeter-Dynkin Interactive Demonstration Pane showing the OmniTruncated A4 Group Vertex Hulls

More code and output images below:

Physics

Comparing two different 5-Cell Cartesian Coordinate sets

April 8, 2023 jgmoxness Leave a comment

This is to support an analysis of the 4D polytope called the 5-Cell. For more of the context, there is discussion on the Wikipedia Talk page for the 5-Cell here. I created a visualization of the these. Click here for the Mathematica Notebook or see the image below:

Greg Moxness

Physics

The (diminished) 120-Cell and it’s relationships to the 5-Cell (A4), the 600-Cell (H4), the two 24-Cells (D4), the Dual Snub 24-Cell, and of course E8!

April 1, 2023 jgmoxness Leave a comment

Updated: 05/03/2023

This post is a Mathematica evaluation of important H4 polytopes involved in the Quaternion construction of nD Weyl Orbits.

Please see the this PDF or this Mathematica Notebook (.nb) for the details.

The paper being referenced in this analysis is here.

I’ve added this PDF or this Mathematica Notebook (.nb) which is having a bit of fun visualizing various (3D) orbits of diminished 120-cell convex hulls. The #5 subset of 408 vertices (diminishing 192) is completely internal to the normal 3D projection of the 120-cell to the chamfered dodecahedron. The #3 has 12 sets of 2 (out of 3) dual snub 24-cell kite cells (that is 24 out of the 96 total cells).

Physics

3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin Geometric Group Theory

February 17, 2023 jgmoxness Leave a comment

Click here to view the Powerpoint presentation.

It shows code and output from my VisibLie-E8 tool generating all Platonic, Archimedean and Catalan 3D solids (including known and a few new 4D polychora from my discovery of the E8->H4 folding matrix) from quaternions given their Weyl Orbits.

If you don’t want to use the interactive Powerpoint, here is the PDF version.

See these Mathematica notebooks here and here for a more detailed look at the analysis of the Koca papers used to produce the results :

Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010). “Catalan Solids Derived From 3D-Root Systems and Quaternions”. Journal of Mathematical Physics. 51 (4). arXiv:0908.3272. doi:10.1063/1.3356985.
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). “Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system”. Linear Algebra and Its Applications. 434 (4): 977–989. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.

Or scroll down to see the .svg images here and here.

Greg Moxness, Tucson AZ

Physics

Visualizing the Quaternion Generated Dual to the Snub 24 Cell

August 10, 2022 jgmoxness Leave a comment

I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here.

3D Visualization of the outer hull of the 144 vertex Dual Snub 24 Cell, with vertices colored by overlap count:
* The (42) yellow have no overlaps.
* The (51) orange have 2 overlaps.
* The (18) tetrahedral hull surfaces are uniquely colored.

What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T(24).

3D Visualization of the outer hull of the alternate 96 vertex Snub 24 Cell (S’)

Visualization of the concentric hulls of the Alternate Snub 24 Cell

Various 2D Coxeter Plane Projections with vertex overlap color coding.

3D Visualization of the outer hull of M(192) as one of the W(D4) C3 orbits of the 120-Cell (600)

3D Visualization of the outer hull of N(288) that are the 120-Cell (600) Complement of
the W(D4) C3 orbits T'(24)+S'(96)+M (192)

3D Visualization of the outer hull of the 120-Cell (600) generated using T’

3D Animation of the 5 quaternion generated 24-cell outer hulls consecutively adding to make the 600-Cell.

3D Animation of the 5 quaternion generated 600-cell outer hulls consecutively adding to make the 120-Cell.

Physics

Interactive Cloud VisibLie-E8 4D Periodic Table

April 15, 2022 jgmoxness Leave a comment

This is a link to the free cloud Mathematica demonstration. (Note: You need to enable “Dynamic Behavior” aka. interactivity in the upper left corner).

Please bear in mind that this demonstration is written for a full Mathematica licensed viewer. The cloud deployments are limited in interactivity, especially those that involve 3D and significant computation. Also, be patient – it takes a minute to load and more than a few seconds to respond to any mouse click interactions.

The utility of the cloud demo of this 4D (3D+color) Periodic Table is in visualizing it in 2D or 3D (from the left side menu) and building up n=1 to 8. Select the Stowe vs. Scerri display for different 3D models. The explode view slider helps distribute the lattices in the model.

The 2D/3D electron density representations for each atom’s orbitals are too slow for the cloud, so they don’t show anything. The isotope and list-picker of internet curated element data also does not function.

For an explanation of this pane #10 in the suite of 18 VisibLie-E8 demonstrations, please see this link.