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.
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.
The Dual Snub 24-Cell with less opacity
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 CellVarious 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 Visualization of the outer hull of the 120-Cell (600) generated using T3D 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.
The cloud deployments don’t have all the needed features as the fully licensed Mathematica notebooks, so I included a few of the panes that seem to work for the most part. Some 3D and animation features won’t work, but it is a start. Bear in mind that the response time is slow.
The newer version of the VisibLieE8-NewDemo-v13.nb (130 Mb) will work with those who have a full Mathematica v13 license. It is backward compatible to earlier versions. There are a few bug fixes from the older version of ToE_Demonstration.nb (130 Mb), which should work on v13 and older versions as well.
This post is an analysis of a June 2013 paper by Mehmet Koca, Nazife Koca, and Ramazan Koc. That paper contains various well-known Coxeter plane projections of hyper-dimensional polytopes as well as a new direct point distribution of the quasicrystallographic weight lattice for E6 (their Figure 3), as well as the quasicrystal lattices of B6 and F4.
What is interesting about this projection is that it precisely matches the point distribution (to within a small number of vertices) from a rectified E8 projection using a set of basis vectors I discovered in December of 2009, published in Wikipedia (WP) in February of 2010 here.
Rectification of E8 is a process of replacing the 240 vertices of E8 with points that represent the midpoint of each of the 6720 edges. In this projection, there are overlaps which are indicated by different colors in the color-coded WP image linked above.
The image below is an overlay of the above images highlighting the 12*(9+3+26+7)=540 points that are not overlapping:
Given the paper’s explanation for the methods using E6 (720) with 6480 edges as a projection through a 4D 3-sphere window defined by q1 and q6, it may be insightful to study my projection basis for E8’s triality relationships with the Koca/Koc paper’s defined 4D 3-sphere.
For more information on why my projection basis is called the E8 Triality projection, see this post.
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
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