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.
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.
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).
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.
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein
2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle
3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize
4) Math: Group Theory: Dynkin Diagram Algebra Create
5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize
6) Physics: Quantum Elements: Fundamental Quantum Element Select
7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity
8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays
9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation
10) Chemistry: Atomic Elements: 4D Periodic Table Element Select
11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select
12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding
13) Biology: Human Neurology: OrchOR Quantum Consciousness
14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori
15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria
16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier
17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life
18) CompSci: Human/Machine Interfaces: nD Human Machine Interface
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.
For more detail on the modules, see this blog post.
Please be patient, it is very large and can take 10 minutes to load, depending on your Internet connection, memory and CPU speed.
In case you’re interested, I just verified the demo works on the free Mathematica CDF player v.13 for Win10.
Just go to https://www.wolfram.com/player/ install, download and open the app:
https://theoryofeverything.org/TOE/JGM/ToE_Demonstration.nb
There is a ton of other cool interactive stuff in there. FYI – Some features don’t work without a full Mathematica license.
Enjoy.
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:
It is interesting to note that with a 30° rotation of my projection, the missing overlaps are reduced to 12*(15+2)=204.
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.
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.
Now for a few new visualizations that are not in the paper…
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!
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:
(* E8 4_21 vertices *)
e8421 = Union@Join[
Eperms8@{1, 1, 1, 1, 1, 1, 1, 1}/2,
perms8@{1, 1, 0, 0, 0, 0, 0, 0}];
The 2160 vertices of the Witting Polytope are generated using various permutations:
(* E8 2_41 vertices *)
e8241=Union@Join[
perms8[{1,0,0,0,0,0,0,0}4],
perms8[{1,1,1,1,0,0,0,0}2],
Eperms8[({2,0,0,0,0,0,0,0}+1)]]/4;
Another view shows just the 2160 Witting Polytope vertices. Click these links for a higher resolution PNG or the SVG version.
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:
