Tag Archives: Physics

The H4 600-Cell in all its beauty

These models use a tetrahedral mesh generator package << TetGenLink` to analyze the cells created by the edges selected.

Note: The Mathematica auto-generated geometry statistics of 120 vertices, 720 golden-ratio length edges, 1200 faces, and 600 color-coded cells, with a surface area of 11.4 volume of 127.

The transparent shaded blue is the outer hull that forms the Pentakis Icosidodecahedron with a vertex hull having 30 vertices of Norm=1 from the center. All 120 vertices form concentric hulls in groups:

1) two points at the origin
2) two icosahedra (24)
3) two dodecahedra (40)
4) two larger icosahedra (24)
5) one icosidodecahedron (30)

H4 600-Cell with 3D vertex shapes representing Standard Model physics particle assignments and color-coded cells

H4 600-Cell with 1200 unit length edges, 2400 faces and 600-cells. 3D vertex shapes representing Standard Model physics particle assignments and color-coded cells are also shown

The H4 120-cell (J), dual of the 600-cell, showing the hull of the chamfered dodecahedron. It is also called a truncated rhombic triacontahedron, constructed as a truncation of the rhombic triacontahedron. It can more accurately be called an order-5 truncated rhombic triacontahedron because only the order-5 vertices are truncated. It is shown with physics particle assignments.

Same as above with interior color-coded cells displayed

3D Visualization of the outer hull of the Dual 120-Cell (600) J’ generated using T

The Dual H4 120-cell (J’) shown above (and more below).

An animated alternate form 120-cell (J’), which seems to be a new (near-miss Johnson solid?) constructed using D4′ (or T’) vs. D4 (T) above.

The H4 dual 120-cell (J’) build of outer edges and color-coded cells

The H4 dual 120-cell (J’) build of outer edges with physics particle assignments

There is yet another construction of a 120-cell that has the same outer hull as the 600-Cell (above), yet with the 600 vertices (5 on each of the 120 vertices of the 600-Cell. The exception to this is a set of 6 square faces.

The Isomorphism of 3-Qubit Hadamards and E8

Click here for a PDF of my latest paper. This has been accepted by the arXiv as 2311.11918 (math.GR & hep-th) with an ancillary Mathematica Notebook here. These links have (and will continue to have) minor descriptive improvements/corrections that may not yet be incorporated into arXiv, so the interested reader should check back here for those. The prior related paper on THE ISOMORPHISM OF H4 AND E8 is referenced in the blog post here.

Abstract: This paper presents several notable properties of the matrix U shown to be related to the isomorphism between H4 and E8. The most significant of these properties is that U.U is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, U.U-(U.U)-1 is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.

Info about my creative work: The Universe, Math, Physics, & Art

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 Moxness, Tucson AZ

Interactive Cloud VisibLie-E8 4D Periodic Table

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:

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

Cloud Based VisibLie_E8 Demonstration

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.

Link to the demonstration.

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

My VisbLie E8 demonstration system for Mathematica v13

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.

The free Wolfram CDF Player v. 13 works with my VisibLie E8 ToE demonstration on Win10

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.