Tag Archives: Math

E8, H4 and the McGee Group Graph

In a recent Azimuth post, John Baez discussed the McGee group and (3,7)-cage graph. A link to a conversation on MathOverflow describes its connection to the Fano plane and the octonions via the Heawood graph (which he had discussed on his Visual Insight blog here).

So I thought it would be fun to show how the alternate graph of the McGee group shown on WP is indeed the 24-cell with 64 of 96 edges removed being projected to the B4 Coxeter plane. So again, it is linked to E8=H42 (as 4 4D Left (L)/Right (R) golden ratio (φ) scaled 600-cell=24-cell+snub 24-cell from the E8 to H4 rotation or folding matrix). Another related post here shows how the E8 Petrie projection decomposes into the two sets of 4 rings as H4 and H4φ 600-cells.

User:Leshabirukov’s Alternate McGee graph is here and below with blue edges being the 4 dimensional axis:

Below are renderings from my hyperdimensional hypercomplex (quaternion/octonion/sedenion) VisibLie_E8 viewer:

McGee Group (3.,7)-cage graph as a 24-cell projected to the B4 Coxeter plane with 64 edges removed (and including 4 axis “edges” for a total of 24 vertices and 36 edges)

To get a view of how B4 projection is isomorphic to the Greg Egan animation, below is my animation overlaid on top of it:

The animation showing the isomorphism between Egan’s base image and the H4φ 24-cell projected to the B4 Coxeter plane.

The first frame is Greg Egan’s base image in the Baez article. Each frame is 2 seconds.

The second frame adds the XYZW axes annotations (4 pairs of red node links that are antipodal across the labeled axis letter (e.g. the nodes above/below the black X line are connected by Y axis endpoints with p#’s 56 & 201, the nodes above/below the black Y line are connected by X axis endpoints with p#’s 74 & 183). These red endpoints are all associated with the 16-cell (i.e. 4-orthoplex octahedron) within the 24-cell.

The third frame differentiates the the inner(cyan) / outer (purple) octagon rings. These two rings are each 8 of 16 vertices of the 8-cell (i.e. 4-cube Tesseract) within the 24-cell.

The 4th frame adds the vertex numbers (p and p*) split between #=1-128 and 257-# (as respectively palindromic 256-129) which are always antipodal in the X-Y axis of the B4 Coxeter plane projection (or any other Coxeter plane projection) as well as being antipodal within each of the 4 sets of 6 nodes in the Egan animation.

A static image for ease of analysis.
24-cell with all of its 96 edges projected to the B4 Coxeter plane. Vertex numbers are from a canonical sort of E8 in Pascal triangle order (when including the 8 generator and 8 anti-generator vertices as 2-9 and 248-255 (respectively).

For anyone who studies deeply the E8 to H4 connections as I do, you know the L/R H4 and H4φ 24-cells have the same palindromic L/R 4D elements in the E8 vertices as well. This means the E8 based McGee groups overlap identically, as shown here:

H4 and H4φ 24-cells in B4 Coxeter plane projection

So, here is the full E8 in the same Coxeter B4 projection with vertex coloring based on the overlap counts.

How the E8 H4 and H4φ snub 24-cells (as 4 π/5 rotations of each of the φ scaled 24-cells) fill in this E8 projection with more McGee groups is more interesting and complex. The patterns for the removal of the specific edges are also interesting and inform the possible physics of E8 based unification theories.

Comparing Coxeter’s Regular Polytopes Table V to Moxness’ Convex Hulls

This is a post to present a new visualization that algorithmically generates Coxeter’s Regular Polytopes Table V vertex & cell first 600-cell {3,3,5} & 120-cell {5,3,3} polytope sections and compares them with my convex hull projections. I have also visualized the face-first and edge-first versions which Coxeter never addressed. The convex hulls of these match those presented by Robert Webb’s Stella 4D.

The base (H4) objects are generated (from E8 actually) using a interactive group theoretic hypercomplex (octonion/quaternion) hyperdimensional (nD with n<9) Geometric Algebra (GA) / Space-Time Algebra (STA) capable symbolic engine which I created using Mathematica.

These are 4D rotated into vertex/cell/face/edge and then grouped & sorted palindromically using Coxeter’s numbering and scaling. See here, here and here for the PDF, interactive 3D Mathematica Notebook, and Wolfram Cloud (respectively). For best performance, I suggest using the free Wolfram Player if you don’t own a Mathematica license for local manipulation.

The hyper-dimensional (4D) vertices used in these panels were also used to generate Koca’s referenced dual snub-24-cell as visualized on the that Wikipedia page.

This is unique (AFAIK). Several of the Table V section entries are not identified by name. This is sometimes due to the irregular faces such that they lack a formal name (e.g. Coxeter labels section 15 of the vertex first {5,3,3} 120-cell as a 1+8 octahedron aka. an octahedron and an irregular truncated octahedron).

Other times it seems they were simply not recognized due to lack of visualization (e.g. while section 6 (& palindromically paired section 10) of the cell first {5,3,3} 120-cell are labeled as rhombicosidodecahedra, section 8 (and my hull #8) is also an irregular form of the rhombicosidodecahedron). Also on that object, sections 4, 5 (& 12, 11 respectively) are irregular truncated icosahedrons.

On the cell first {5,3,3} 600-cell, unidentified sections 5, 6 (& 11, 10) are seen to be truncated tetrahedra, and 4 (& 12) are irregular cuboctahedra.

As already referenced above, the most irregular is the vertex first {5,3,3} 120-cell with unidentified sections 2, 4 (& 28, 26) now seen as truncated tetrahedra, sections 6, 14 (& 24, 16) as irregular truncated octahedra, and sections 10 (& 20) as irregular cuboctahedra. Section 9 (& 21) is a pair of truncated tetrahedrons (shown below).

Interestingly, Coxeter’s labels for sections 3, 11 (& 27, 19) are “2 icosahedra” (aka. truncated octahedra). Coxeter’s partial label on 12 (& 18) is “Tetrahedron” for the 4 vertices with 24 others unidentified. These 24 are irregular hexagon (8) and rectangular (6) faces forming an irregular truncated octahedron.

Section 7 (& 23) have 24 vertices that are two irregular cuboctahedra each with triangular (8) and rectangle (6) faces.

Another notable section is found with 9-gon or nonagon faces in sections 13 (& 17), which are generated by the chiral left/right “chamfered tetrahedrons” (3 per corner).

This likely derives from the 8-simplex A8 SU(9) as a subgroup of E8 as shown in the E8 subgroup tree (with more detail in this post). This folds to the H4 family of polytopes via my E8 <-> H4 folding matrix.

Coxeter’s Regular Polytopes Table V-iii – Vertex First {3,3,5} 600-cell
Coxeter’s Regular Polytopes Table V-iv-a – Cell First {3,3,5} 600-cell
Coxeter’s Regular Polytopes Table V-iv-b – Cell First {5,3,3} 120-cell
Coxeter’s Regular Polytopes Table V-v – Vertex First {5,3,3} 120-cell

Since Coxeter never presented any {3,3,5} / {5,3,3} edge-first or face-first sectioning, here is my attempt at visualizing them. Please note the top section of the SVG output below now includes the XYZW vertex +/- & cyclic position permutation base values and other data collected in the process. There is also a 4D perspective projection below that uses a distance factor to change perspective. Note the differences between the edge-first (distance=1) vs. the face-first (distance=10) showing how the 4D perspective changes on both the {3,3,5} 600-cell / {5,3,3} 120-cell :

Face First {3,3,5} 600-cell
Edge First {3,3,5} 600-cell
Edge First {5,3,3} 120-cell
Face First {5,3,3} 120-cell

And just for more fun… here are combinations of the above displayed together producing 720 and 1200 vertex overall hulls that are nice and/or interesting!

720 vertex combining vertex first 335 600-cell with cell first 533 120-cell
cell720p (or prime) 720 vertex combining cell first 335 600-cell with vertex first 533 120-cell
1200 vertex combining both vertex & cell first 335 600-cell and 533 120-cell
1200 vertex cell1200p by quaternion calculation:
octSimplify /@Flatten@prq[[DoubleStruckCapitalA], octExp[Alpha]Lsw, TpT]

A Visual Overview of how the H4 600-cell(s) embed into E8

Please see this Powerpoint for an interactive presentation (160 Mb).

2D projection of 8,16,24, snub-24 and 600 cells as part of E8 vertex Petrie projection

Below (and here and here in PDF and Mathematica Notebook form or here on the Cloud) are the I & I’ 600-cells based on quaternion construction from the T & T’ 24-cells. These are also documented in Coxeter’s book “Regular Polytopes” as {3,3,5} vertex first & cell first “sections” in Table V (respectively) vs. the projections shown here with one of the four dimensions projected to 0 such that the hulls are palindromic pairs of sections (e.g. for 9 sections, section 1-4 pairs with 9-6 in vertex counts equal to hull vertex counts 1-4 and the center section 5 = hull 5).

The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows pairs of:

1) points at the origin

2) icosahedrons

3) dodecahedrons

4) icosahedrons

5) and a single icosadodecahedron

for a total of 120 vertices and an overall outer 3D hull of the Pentakis Icosidodecahedron.
Alternate 600-Cell Convex Hulls. The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:

1) a pair of tetrahedrons (or single cube)

2) a pair of tetrahedrons (or single cube)

3) a pair of icosahedrons

4) truncated cube

5) rhombicuboctahedron

6) rhombicuboctahedron

7) a pair of tetrahedrons (or single cube)

8) cuboctahedron

for a total of 120 vertices and an overall outer 3D hull of a near-miss of the Pentakis Icosidodecahedron.

For completeness, here are the J’ (alternate) & J form of the 120-cells based on quaternion construction from the T & T’ 24-cells. These are also documented in Coxeter’s book “Regular Polytopes” as {5,3,3} vertex first & cell first “sections” in Table V (respectively) vs. the projections shown here with one of the four dimensions projected to 0 such that the hulls are palindromic pairs of sections (e.g. for 31 sections, section 0-14 pairs with 30-16 in vertex counts equal to hull vertex counts 1-15 and the center section 15 = hull 16).

Hull 1 has 2 points at the origin.
Hulls 2 & 6 are cubes.
Hulls 3 & 5 are rhombicuboctahedrons.
Hulls 4 & 12 are each pairs of truncated octahedrons.
Hulls 7 & 15 are truncated cuboctahedrons.
Hull 11 is a truncated cube.
Hulls 8, 9, 10, 13, 14, and 16 are unnamed solids.
Hulls 1, 2, & 7 are each overlapping pairs of dodecahedrons.
Hull 3 is a pair of icosidodecahedrons.
Hulls 4 & 5 are each pairs of truncated icosahedrons.
Hulls 6 & 8 are each rhombicosidodecahedrons.

Below is an interactive Mathematica 4D visualization of the 8,16, 24, 120, and 600 cells (including alternate forms 24p, 120p, 600p). Click here or on the image below to open in a new tab.

Group Theory in a Nutshell: Taming the Monster

For a paper with much of the content below, please see: theoryofeverything.org/TOE/JGM/E8_and_H4_in_QM_and_QC.pdf, and associated Mathematica notebook here.

Y555 Hasse diagram

The wreath product of the Monster group with Z2 (M ≀ Z2) is the BiMonster, a quotient of the Y555 Dynkin diagram which contains six E8(Y124) Dynkin diagrams and the 23rd Niemeier lattice E83 root system. This can be reduced to Y444 by applying the spider relation to E83 .

Hasse diagram of sporadic groups in their Monster group subquotient relationships.

Happy Family genarations:
Red=Mathieu groups (1st)
Green=Leech lattice groups (2nd)
Blue=other Monster subquotients (3rd)
Black=Pariahs

Table of sporadic group orders (with Tits group).
Niemeier lattices with group structure, Dynkin diagram, Coxeter number, and group orders.
Kneser neighborhood graph of Niemeier lattices with associated Mathematica Notebook here.

Each color coded node represents one of the 24 Niemeier lattices, and the lines joining them represent the 24-dimensional odd unimodular lattices with no norm 1 vectors. The Coxeter number of the Niemeier lattice is to the left. The red node index number indicates the row in the table above.

Table of sublattices of Conway groups (Green nodes in the Monster Group sub-quotients graph at the top of the page)
The Freudenthal magic square

”The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that ”the exceptional Lie groups all exist because of the octonions: G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space, see prehomogeneous vector space.”

E8 Subgroup Tree with associated Mathematica Notebook here.

This E8SubgroupTree.nb Mathematica Notebook (140Mb), viewable using the free viewable Player, which has interactive Tooltips on group nodes (with group data and Hasse diagram) and Tooltips on the subgroup edges (with its individual maximal embedding chart). Also in the notebook is one cell with a table of all of the above data plus the Dynkin diagram based maximal embedding charts and an example 2D or 3D polytope projection.

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.

One of my arXiv papers is featured as a Wolfram Community Editor’s Pick

Please take a look here to view a paper I wrote about in a recent blog post here. The paper registered in arXiv is here and titled: The_Isomorphism_of_H4_and_E8.

As I like to provide more than just type-set text and graphics, I include in the arXiv the Mathematica notebooks with code snippets and actual output w/interactive 3D graphics. This gets attached in the ancillary files and is also linked as “Papers with Code” using my GitHub account here.

That caught the eye of the Wolfram editors who graciously offered to help with this post. My hats off to them. Enjoy the paper, with a follow-on paper: The_Isomorphism_of_3-Qubit_Hadamards_and_E8 here, including the notebook 😉

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.

Concentric hulls of4_21 in Platonic 3D projection with numeric and symbolic norm distances
2D to the E8 Petrie projection using basis vectors X and Y from (6) with 8-polytope radius 4\sqrt{2} and 483,840 edges of length \sqrt{2} (with 53% of inner edges culled for display clarity
An alternative set of structure constant triads, octonion Fano plane mnemonic, and multiplication table, with decorations showing the palindromic multiplication.
1_42 projections of its 17,280 vertices
Visualization of the 144 root vertices of S’+T+T’ now identified as the dual snub 24-cell
Breakdown of E8 maximal embeddings at height 248 of content SO(16)=D8 (120,128′)
Archimedean and dual Catalan solids, including their irregular and chiral forms. These were created using quaternion Weyl orbits directly from the A3, B3, and H3 group symmetries listed in the first column
A3 in A4 embeddings of SU(5)=SU(4)xU1
These include the specified 3D quaternion Weyl orbit hulls for each subgroup identified
The Eigenvalues, Eigenvector matrix, and characteristic polynomial coefficients of the unitary form of U as e^{IU} showing a Tr@Re@e^{IU}~4 and a traceless imaginary part

Collections of 3D Surfaces, 3D Chaotic Attractors, 2D/3D Fractals, and 3D Cellular Automaton

Below are SVG images of output from Pane 1 (Chaos) and Pane 17 (AI) of the VisibLie_E8 visualization tool:

3D Parametric Surfaces
Chaotic Attractors
Fractals
Fractal (same as above) in 3D
Fractals
AI Cellular Autonatons 1_66
AI Cellular Automaton 67_132
AI Cellular Automaton 133_198
AI Cellular Automaton 199_253

Greg Moxness, Tucson AZ

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