This is constructed from VisibLie_E8 and uses the Mathematica package Mathgroup. A comprehensive PowerPoint/PDF that gives an overview of the underlying theory and VisibLie_E8 tool capability is in this post.
I discovered an error in the WP article on the Disdyakis Triacontahedron. It seemed the scaling factor for determining the coordinates of the Icosidodecahedron hull was added and changed without any citation references. Since the factor seemed off, I analyzed it using the Koca, Mehmet’s paper and Weisstein, Eric W. “Disdyakis Triacontahedron.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/DisdyakisTriacontahedron.html
Please see this Mathematica Notebook for the results here or the PDF here.
Below is the SVG image of the notebook.
Greg Moxness, Tucson AZ
Updated: 05/11/2023
This post is a Mathematica evaluation of A4 Group Orbits & Their Polytope Hulls Using Quaternions and nD Weyl Orbits. This is a work-in-process, but the current results are encouraging for evaluating ToE’s.
Please see the this PDF or this Mathematica Notebook (.nb) for the details.
The paper being referenced in this analysis is here.
Of course, as in most cases, this capability has been incorporated into the VisibLie_E8 Demonstration codebase. See below for example screen shots:
This is an example .svg file output from the same interactive demonstration:
More code and output images below:
Updated: 05/03/2023
This post is a Mathematica evaluation of important H4 polytopes involved in the Quaternion construction of nD Weyl Orbits.
Please see the this PDF or this Mathematica Notebook (.nb) for the details.
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).
Click here to view the Powerpoint presentation.
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 :
Or scroll down to see the .svg images here and here.
Greg Moxness, Tucson AZ
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).
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.
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
