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 think the most well-known (2D) image I’ve created so far (above) is found on Wikipedia’s (WP) math/geometry pages. It is what has been called the Public Relations (PR) or publicity photo for E8, which is an 8D (or 248 dimensional as the math guys count it) geometric object. It has been used in books, papers, math/science conference promotional materials, etc. That WP page also has what many used to consider “uninteresting” – my 3D versions of the same object. E8 is used in various theories to understand how the Universe operates way below the atomic level (i.e. the quantum stuff). I think it is responsible for what I call the shape of the Universe. As others have said, “who knew the Universe had a shape?”
In (very) layman’s terms, that 2D image of E8 looks sort-of-like how the lowly 4D Tesseract cube would look if it grew up into an 8D object, but a bit more interesting. The Tesseract was made (more) popular in the modern Avengers movies. In 2015 before I had ever seen any of those movies, I created a laser etched 3D projection of E8 within a jewel cut optical crystal cube and put a photo of that object lit from below by a blue LED on my website’s main homepage. Someone mentioned the resemblance and I was surprised at the likeness, motivating me to go see the movies.
Another of my more notable math/physics images has to do with the discovery of QuasiCrystals made by a guy, Dan Shechtman, who was ostracized from academia for the “impossible” idea of crystals having rotational symmetries beyond 2,3,4 and 6. He is now a Nobel Laureate. The WP image above is on that QuasiCrystal page as an overlay of part of E8 projected over a picture made from shooting a beam of x-rays at an icosahedral Ho-Mg-Zn quasicrystal. I didn’t actually do that experiment with the x-ray beams, but I did similar ones in my college days on a particle accelerator called the Cockroft-Walton Kevatron – see below for a picture of me (the one with the beard ;–) in the physics lab assembling a moon dust experiment. One of the professors had gotten the accelerator out of Germany after having worked on the Manhattan project.
E8 and its 4D children, the 600-cell and 120-cell (pages on which I have some work, amongst others) and its grandkids (2 of the 3D 5 Platonic Solids, one of which is the 3D version of the 2D Pentagon) are all related to the Fibonacci numbers and the Golden Ratio. So that kind of explains why most of my 2D art, 3D objects and sculptures (e.g. furniture like the dodecahedron table below), and 4D youtube animations all use the Golden Ratio theme.
Greg
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:
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.
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).
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.
Please click here for the interactive Buddhist (Tao/Confucian) I-Ching demonstration shown below.
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:
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
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
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.
