https://vixra.org/pdf/2005.0200v1.pdf
or also available directly from this website:
https://theoryofeverything.org/TOE/JGM/3D_Polytope_Hulls_of_E8-421-241-142.pdf
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.
3rd largest hull of the 74 hulls in 142
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)
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 24 individual concentric hulls
b) In groups of 8 hulls
with vertex count in each hull and increasing opacity and
varied surface colors.
a) 74 individual concentric hulls
b) In groups of 8 hulls
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…
