Category Archives: Physics

One of my arXiv papers is featured as a Wolfram Community Editor’s Pick

Please take a look here to view a paper I wrote about in a recent blog post here. The paper registered in arXiv is here and titled: The_Isomorphism_of_H4_and_E8.

As I like to provide more than just type-set text and graphics, I include in the arXiv the Mathematica notebooks with code snippets and actual output w/interactive 3D graphics. This gets attached in the ancillary files and is also linked as “Papers with Code” using my GitHub account here.

That caught the eye of the Wolfram editors who graciously offered to help with this post. My hats off to them. Enjoy the paper, with a follow-on paper: The_Isomorphism_of_3-Qubit_Hadamards_and_E8 here, including the notebook 😉

The Isomorphism of 3-Qubit Hadamards and E8

Click here for a PDF of my latest paper. This has been accepted by the arXiv as 2311.11918 (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.

Abstract: This paper presents several notable properties of the matrix U shown to be related to the isomorphism between H4 and E8. The most significant of these properties is that U.U is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, U.U-(U.U)-1 is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.

The Isomorphism of H4 and E8

Click here for a PDF of my latest 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.

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.
1_42 projections of its 17,280 vertices
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

Collections of 3D Surfaces, 3D Chaotic Attractors, 2D/3D Fractals, and 3D Cellular Automaton

Below are SVG images of output from Pane 1 (Chaos) and Pane 17 (AI) of the VisibLie_E8 visualization tool:

3D Parametric Surfaces
Chaotic Attractors
Fractals
Fractal (same as above) in 3D
Fractals
AI Cellular Autonatons 1_66
AI Cellular Automaton 67_132
AI Cellular Automaton 133_198
AI Cellular Automaton 199_253

Greg Moxness, Tucson AZ

X’d (Tweet’d)

Disdyakis Triacontahedron Coordinates

I discovered an error in the WP article on the Disdyakis Triacontahedron. It seemed the scaling factor for determining the coordinates of the Icosidodecahedron hull was added and changed without any citation references. Since the factor seemed off, I analyzed it using the Koca, Mehmet’s paper and Weisstein, Eric W. “Disdyakis Triacontahedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/DisdyakisTriacontahedron.html

Please see this Mathematica Notebook for the results here or the PDF here.

Below is the SVG image of the notebook.

Greg Moxness, Tucson AZ

A4 Group Orbits & Their Polytope Hulls Using Quaternions

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:

Hasse Visualization
Maximal Embeddings

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: