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 distances2D 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 columnA3 in A4 embeddings of SU(5)=SU(4)xU1 These include the specified 3D quaternion Weyl orbit hulls for each subgroup identifiedThe 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
This is constructed from VisibLie_E8 and uses the Mathematica package Mathgroup. A comprehensive PowerPoint/PDF that gives an overview of the underlying theory and VisibLie_E8 tool capability is in this post.
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.
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:
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).
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.
This uses the latest Mathematica version 13.2 with code tweaks to better understand what is being presented, including the E8 Algebra roots, weights, and heights. This is used on the E8 group theory page of WikiPedia.
For a more graphic intensive E8 Hasse visualization, see below with both positive and negative roots and more node information and graphics. See here and here for the Mathematica notebook and PDF respectively.
Double Hasse E8 Visualization
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