In several papers on BKS proofs, Arthur Ruuge’s “Exceptional and Non-Crystallograpic Root Systems and the Kochen-Specker Theorem” https://arxiv.org/abs/0906.2696v1 and Mordecai Waegell & P.K. Aravind’s “Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8” https://arxiv.org/abs/1502.04350v2, in addition to E8, E6 and E7 is studied. Using the visualization developed for my recent paper and prior papers, I present here the related visualizations for E6 and E7 as discussed in those papers.
The 72 E6 vertices derived from E8 and projected to 3D using the “E8->H4” basis vectors. The 15*72=1080 edges are shown in the upper left, the 36 anti-podal rays are shown in the upper right along with the hull group vertex counts and norm distance. The bottom image shows the 4 hulls – yellow Icosahedron (12), cyan dodecahedron (20) and the orange/pink pentagonal prisms (40)).The 126 E7 vertices derived from E8 and projected to 3D using the “E8->H4” basis vectors. The 16*126=2016 edges are shown in the upper left, the 63 anti-podal rays are shown in the upper right along with the hull group vertex counts and norm distance. The bottom image shows the 4 hulls – orange 2 overlapping Icosahedrons (24), pink2 overlapping dodecahedron (40), and cyan & gray icosidodecahedrons(60) with 2 vertices at the origin.
Using rows 2 through 4 of a unimodular 8x8 rotation matrix, the vertices of E8 421, 241, and 142 are projected to 3D and then gathered & tallied into groups by the norm of their projected locations. The resulting Platonic and Archimedean solid 3D structures are then used to study E8’s relationship to other research areas, such as sphere packings in Grassmannian spaces, using E8 Eisenstein Theta Series in recent proofs for optimal 8D and 24D sphere packings, nested lattices, and quantum basis critical parity proofs of the Bell-Kochen-Specker (BKS) theorem.
A few new Figures from the paper.
FIG. 6: Pair of overlapping rhombicosidodecahedrons from 3rd largest hull of the 74 hulls in 142FIG. 13: 421 & Polytope projected to various 3D spaces Each 3D projection shown lists the projection name, the nu- meric basis vectors used, and the 421 & 142 overlap color coded vertex groups, and the projection with vertices (larger) & 6720 edges and the 142 vertices (smaller)FIG. 14: Concentric hulls of 241 in Platonic 3D projection with vertex count in each hull and increasing opacity and varied surface colors. a) 24 individual concentric hulls b) In groups of 8 hullsFIG. 15: Concentric hulls of 142 in Platonic 3D projection with vertex count in each hull and increasing opacity and varied surface colors. a) 74 individual concentric hulls b) In groups of 8 hullsFIG. 18: E8’s outer two hulls scaled to unit norms in Platonic 3D projection with vertex counts color coded by overlaps a) 54 vertex (42 unique) 421=241 icosidodecahedron (30 yel- low) & two overlapping icosahedrons (12 red) scaled 1.051 b) 100 vertex (80 unique) 142 non-uniform rhombicosidodeca- hedron (60 yellow) & two overlapping dodecahedrons (20 red) scaled 1.0092 c) 154 vertex (122 unique) combination of a & b d) 208 vertex (122 unique) combination same as c with color coded vertex counts for both 421 & 241 Note: The internal numbers of the image are the 8 axis (pro- jection basis vectors).
Now for a few new visualizations that are not in the paper…
Various 3D projections of 2_41Various 3D projections of 4_21
This is what I expect to be the first ever 3D visualization of the E8 1_42 polytope with 17280 vertices showing concentric hulls of Platonic solid related structures!
E8 1_42 polytope with 17280 vertices showing concentric hulls of Platonic solid related structures! Vertex colors represent the overlap counts.Each of 74 concentric hulls based on 3D Norm’d vertex positions (with varying opacity in sets of 8). Vertex counts in each hull listed above.9 sets of 8 concentric hulls plus the last 2 outer hulls, Vertex counts in each of 8 hull listed above. Notice this is a combination of two overlapped dodecahedrons (40) and a Nonuniform Rhombicosidodecahedron (60).
For the sake of completeness in visualization, see below for various projections to 2D. Click these links for a higher resolution PNG or the SVG version.
I read an interesting article about a pattern discovered by Warren D. Smith (discussed at length here):
“The sum of the first three terms in the Eisenstein E_4(q) Series Integers of the Theta series of the E8 lattice is a perfect fourth power: 1 + 240 + 2160 = 2401 = 7^4”
So I decided to visualize the 2401=1+240+2160 vertex patterns of E8 using my Mathematica codebased toolset based on some previous work I put on my Wikipedia talk page.
The image below represents various projections showing 6720 edges of the 240 E8 vertices, plus a black vertex at the origin, and the 2160 Witting Polytope E8 2 _ 41 vertices using the same projection basis (listed at the top of each image along with the color coded vertex overlaps). Click these links for a higher resolution PNG or the SVG version.
Some of the particular projections of the Witting Polytope may need 8D rotations applied to the basis vectors to find better symmetries with the Gosset, but this is a start using my standard set of projections.
The 240 vertices of the Gosset Polytope are generated using various permutations:
Another view shows just the 2160 Witting Polytope vertices. Click these links for a higher resolution PNG or the SVG version.
2160 Witting Polytope (vertices only)
Another great source of visualizations on E8 and this Witting Polytope is here.
Now visualizing in 3D the structure in 3D using rows 2-4 of the E8->H4 folding matrix, we get:
Witting Polytope with 2160 E8 2 _ 41 vertices sorted by Norm distance into 3D Platonic solid related concentric hull structures using rows 2-4 of the E8->H4 folding matrix. The last image shows overlap counts using colored vertices.
Sets of 8 concentric hulls with hull vertex counts above each.Full set of concentric hulls with color coded vertices showing overlap counts. Norm’d vertex groups in red.
Abstract: We introduce a unimodular Determinant=1 8×8 rotation matrix to produce four 4 dimensional copies of H4 600-cells from the 240 vertices of the Split Real Even E8 Lie group. Unimodularity in the rotation matrix provides for the preservation of the 8 dimensional volume after rotation, which is useful in the application of the matrix in various fields, from theoretical particle physics to 3D visualization algorithm optimization.
Visualization of the Icosahedral points generated recursively…
It is a Mathematica 11/12 Computable Document Format (CDF) web interactive app that has been purpose built for visualizing Solar System Orbital Mechanics
I’ve read some recent papers by Cohl Furey and was intrigued by the potential relationship between the octonion and Clifford group theoretic assignments to standard model particles. Since I have developed an extensive Mathematica notebook to perform symbolic analysis using these structures (derived primarily for E8 Lie group work), I decided to follow her suggestion… “The reader is encouraged to check that C⊗O forms the 64-complex-dimensional Clifford algebra Cl(6), generated by the set {i e1,…i e6} acting on f”.
So to that end, I created the this .pdf with some preliminary results of that analysis. I found a few minor issues, but so far the model seems to symbolically compute consistently. Very cool!
Please take a look at my latest paper based on my original work circa 1997-2007. JGM/010-MOND-CEG-TOE [pdf, cite] Title: A new ‘more Natural’ ToE model with Covariant Emergent Gravity as a solution to the dark sector Authors: J Gregory Moxness Comments: Apr 4 2018, 4 pages. Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics – Theory (hep-th)
For citations, remove the LaTex href tag structure if you don’t use the hyperref package.
They really are a bunch of holier than thou bureaucrats with limited perspectives. T. Ruin (sp) and JBL (like the speakers) and Dmcq – a cabal like no other.
Dedicated to the pursuit of beauty and Truth in Nature!
