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.
I’ve published a version of these demonstrations in the cloud, so no Mathematica installation is needed.
Please click here for the interactive Hindu Sri Yantra demonstration shown below.
Image of the User Interface (UI) of the 3D Sri Yantra demonstration
Please click here for the interactive Buddhist (Tao/Confucian) I-Ching demonstration shown below.
Image of the User Interface (UI) of the 2D I-Ching demonstration.
The Mathematica Cloud version doesn’t work in 3D as well as using a local .NB version with the free Mathematica CDF player. You can File->Download the local demonstration and use it after installing the CDF player found here.
Of course, the full licensed Mathematica Notebook works the best for this. See this file here for the Hindu 3D Sri Yantra and here for the Buddhist 2D I-Ching notebooks.
There is even a full pane 15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria here.
Here is a list of other Interactive Demonstrations within my VisibLie-E8 package:
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
This is a link to the free cloud Mathematica demonstration. (Note: You need to enable “Dynamic Behavior” aka. interactivity in the upper left corner).
Please bear in mind that this demonstration is written for a full Mathematica licensed viewer. The cloud deployments are limited in interactivity, especially those that involve 3D and significant computation. Also, be patient – it takes a minute to load and more than a few seconds to respond to any mouse click interactions.
The utility of the cloud demo of this 4D (3D+color) Periodic Table is in visualizing it in 2D or 3D (from the left side menu) and building up n=1 to 8. Select the Stowe vs. Scerri display for different 3D models. The explode view slider helps distribute the lattices in the model.
The 2D/3D electron density representations for each atom’s orbitals are too slow for the cloud, so they don’t show anything. The isotope and list-picker of internet curated element data also does not function.
For an explanation of this pane #10 in the suite of 18 VisibLie-E8 demonstrations, please see this link.
High resolution 4D (3D+color) images of the demonstration.High resolution 2D images from the demonstration.
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
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.
I’ve updated an analysis I did on the work of Cohl Furey’s papers from several years ago. Since then, she added another paper: https://arxiv.org/abs/1910.08395v1
The short pdf version of my analysis (with some detail cells collapsed) is here (34 pages), and a longer version is here (no collapsed cells and 51 pages). These pdf’s are a direct output from my Mathematica (MTM) Notebook. I will follow up with a LaTex paper on the topic soon.
This notebook has code built in to operate symbolically on native MTM reals, complexes, and quaternionic forms, as well as my custom code to handle the octonions, and now the bi-octonions (which doesn’t assign the octonion e1 to be equivalent to the complex imaginary (I)). That change also applies to the native quaternion assignments where of e1=I, e2=J, and e3=K in order to work with quater-octonions. This was a fairly trivial change to make since it simply involves removing the conversion of complex (and quaternion) operators from being involved in the octonionic multiplication.
Please note that my previous analysis here (from Feb. 2019) made the mistake of not commenting out these operations. As such, it was operating on octonions (not complexified bi-octonions), so some of my concerns were resolved based on correcting that error.
The bottom line is that I did validate much of the work presented in the referenced papers, with the exception of some 3 generation SM charge (Q) assignments in that latest paper (Oct. 2019).
I am very interested here in the suggestion at the very end of that paper [5] in the Addendum Section IX(B/C) on Multi-actions splitting spinor spaces, Lie algebras/groups, and Jordan algebras. I suspect having the ability to create a machine (i.e. a symbolic engine such as MTM) to operate on and visualize these structures as hyper-dimensional physical elements is critical to making progress in understanding our Universe more thoroughly.
While I have had some success in replicating quark color exchange, as well as flavor changes (e.g. green u2 to d3 quark exchange using g13), there doesn’t seem to be a complete description of how to construct each of those color and flavor exchange actions from the examples given. So for reference I present all possible combinations of these actions across the particle/anti-particle definitions (see the image linked in the last paragraph of this post). This comment about limited examples also applies to replicating the 3 generation charge (Q) calculation using the sS constructs mentioned above.
I welcome any help or advice or additional examples.
Below is an example image of the 3 generation SM from the 2019 paper built from bi-octonions (with my octonion multiplication table reductions applied. The anti-particles (not shown) are simply the complex-conjugate of these. While I show in string form of Q, I am not showing the commutations based evaluations for them due to the questions / issues I have on how to get it to work.
The image below shows more detail of the 3 generation SM from 2014 with my code implementing the reductions. This leaves off the charge (Q) which was not defined as above in 2014 (AFAIK).
The image below shows a simple construction of the 0-V to 6-V splitting of the Mf Clifford algebraic structures, which I generated using MTM Subsets:
The rather large (long) image here checks all SM particle color and flavor changing actions and includes the anti-particles. The output is extensive and given my open questions on the formalism presented, the accuracy likely deviates from the intent of [5], but it is interesting to show how everything transforms. If no transform is found for a particular action, it outputs an * for that action. If a color or flavor changing transformation action is found, it identifies that action with the list of particles to which the transformation applies. Note: it only identifies a transformed particle if the source particle has a non-zero reduced value and the resulting match is exact (red) or a +/- integer factor of that particle (blue).
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
Dedicated to the pursuit of beauty and Truth in Nature!
