The Isomorphism of H4 and E8

Click here for a PDF of my recent paper. This has been accepted by the arXiv as 2311.01486 (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. Another related follow-on paper on THE ISOMORPHISM OF 3-QUBIT HADAMARDS AND E8 is referenced in the blog post here.

The abstract: This paper gives an explicit isomorphic mapping from the 240 real R^8 roots of the E8 Gossett 4_{21} 8-polytope to two golden ratio scaled copies of the 120 root H4 600-cell quaternion 4-polytope using a traceless 8×8 rotation matrix U with palindromic characteristic polynomial coefficients and a unitary form e^{iU}. It also shows the inverse map from a single H4 600-cell to E8 using a 4D<->8D chiral L<->R mapping function, phi scaling, and U^{-1}. This approach shows that there are actually four copies of each 600-cell living within E8 in the form of chiral H4L+phi H4L+H4R+phi H4R roots. In addition, it demonstrates a quaternion Weyl orbit construction of H4-based 4-polytopes that provides an explicit mapping between E8 and four copies of the tri-rectified Coxeter-Dynkin diagram of H4, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E8-based Grand Unified Theories or GUTs.

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

A4 Group Orbits & Their Polytope Hulls Using Quaternions

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:

Comparing two different 5-Cell Cartesian Coordinate sets

This is to support an analysis of the 4D polytope called the 5-Cell. For more of the context, there is discussion on the Wikipedia Talk page for the 5-Cell here. I created a visualization of the these. Click here for the Mathematica Notebook or see the image below:

Greg Moxness

The (diminished) 120-Cell and it’s relationships to the 5-Cell (A4), the 600-Cell (H4), the two 24-Cells (D4), the Dual Snub 24-Cell, and of course E8!

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).

3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin ​Geometric Group Theory

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

Visualizing the Quaternion Generated Dual to the Snub 24 Cell

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).

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.

A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:

10) Chemistry: Atomic Elements: 4D Periodic Table Element Select

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

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.

Mapping the fourfold H4 600-cells emerging from E8: A mathematical and visual study

My latest paper is now available here or on Vixra.

{abstract}
It is widely known that the E8 polytope can be folded into two Golden Ratio (Φ) scaled copies of the 4 dimensional (4D) 120 vertex 720 edge H4 600-cell. While folding an 8D object into a 4D one is done by applying the dot product of each vertex to a 4×8 folding matrix, we use an 8×8 rotation matrix to produce four 4D copies of H4 600-cells, with the original two left side scaled 4D copies related to the two right side 4D copies in a very specific way. This paper will describe and visualize in detail the specific symmetry relationships which emerge from that rotation of E8 and the emergent fourfold copies of H4. It will also introduce a projection basis using the Icosahedron found within the 8×8 rotation matrix. It will complete the detail for constructing E8 from the 3D Platonic solids, Icosians, and the 4D H4 600-cell. Eight pairs of Φ scaled concentric Platonic solids are identified directly using the sorted and grouped 3D projected vertex norms present within E8. Finally, we will show the relationship of the Beordijk-Coxeter Tetrahelix emerging from the Petrie projection’s concentric rings of 30 vertices of the H4 600-cells.

This paper builds off of some recent work here, and here.

Below are a few figures from that paper.

FIG. 3: a) 24-cell highlighting the 16-cell (red on-axis vertices) and 8-cell (blue off-axis vertices), b) Snub 24-cell highlighting four pi/5 rotations of the 24-cell (black) in red, green, blue, yellow.

FIG. 4: The 5 24-cell decomposition of one 600-cell, each with 2D and 3D vertex numbered projections including edges; a)
24-cell, b-e) Snub 24-cells from four pi/5 rotations of the 24-cell in a)

FIG. 6: Vertex numbered Petrie projection of the rotated E8, showing the 96 edges of 8D norm l = 2Φ which links the Snub 24-cell H4 and H4Φ (L<->R) vertices (Note: the 8 excluded positive E8 8-Orthoplex (or equivalently, the H4 and H4Φ 4-Orthoplex) “generator vertices” are shown as larger gray labeled axis dots which overlap their darker black E8 vertices)

FIG. 7: Symbolic analysis using MathematicaTM comparing the Cartan matrix before and after rotating the simple roots matrix (E8srm) used to create it (Note: the resulting Cartan matrix is not precisely that of E8, even after applying the Dechant’s Φ=0 trick)

FIG. 8: a) Sorted list of the vertex norms with their grouped vertex counts. b) 3D surface models for each of the 7 hulls of vertices c) Combined 3D surface model with increasing transparency for each successive hull.

FIG. 11: 3D projection using an orthonormal basis on H4 and H4Φ vertices (i.e. the rotated E8)

FIG. 12: The Beordijk-Coxeter Tetrahelix emerging from the 8 concentric rings of 30 E8 and H4+H4Φ Petrie projection vertices; a) H4 inner and outer ring in 3D, b) with vertex size, shape, color assigned from physics model, c-d) inner and outer ring in orthonormal projection of H4Φ

FIG. 15: MetabidiminishedIcosahedron (with its negated vertices gives the GyroelongatedPentagonalPyramid) Crystal Projection Prism projecting E8 a) the crystal prism geometry and vertices, b) the selected basis vectors, c) the individual 3D projected concentric hull objects, d) the combined set of concentric objects with progressively increasing transparency and tally of vertex norms.

APPENDIX A: LIST OF E8 AND ITS ROTATED H4 AND H4Φ VERTICES
This table splits the E8 and its rotated H4 vertices into L/R 4D elements.

1. Table structure
The first column is an SRE E8 vertex index number derived from sorting the E8 vertices by their position based on the 256=2^8 binary pattern from the 9th row of the Pascal triangle (1, 8, 28, 56, 70, 56, 28, 8, 1) and its associated Cl8 Clifford Algebra. This construction is described in more detail in [3]. The odd groups (1,3,5,7,9) with (1,26,70,26,1) elements (respectively) are the 128 1/2 integer vertices. The even groups (2,4,6,8) with (8,56,56,8) elements (respectively) are the 112 integer vertices along with the 16 excluded 8 generator (and 8 anti-generator) vertices (2-9 and 248-255) with permutations of (1; 0; 0; 0; 0; 0; 0; 0) (a.k.a. the 8-Orthoplex), such that they indicate the basis vectors used for projecting the polytope.

Only the first half of the 240 vertices is shown, since the last half is simply the reverse order negation of the vertices
in the first half (e.g. the E8 vertex n=10 has as its’ negation 257-n=247). The middle column labeled L<->R indicates the vertex reference number that contains the same L as the R (and equivalently, the same R as the L, interestingly enough).

2. Table color coding
The E8 L/R columns’ green color-coded elements indicate that the 4D vertex rotates into the smaller H4 600-cell.
Conversely, the E8 L/R columns’ black color-coded elements indicate the 4D vertex rotates into the larger H4Φ
600-cell.

The H4 L/R columns’ red color-coded rows are 24-cell elements that always self-reference and are always members of the scaled up 600-cell H4Φ. The H4 L/R columns’ orange color-coded rows are 24-cell elements that always reference the negated elements in the range of 129-256 and are always members of the smaller 600-cell H4.