The wreath product of the Monster group with Z2 (M ≀ Z2) is the BiMonster, a quotient of the Y_{555} Dynkin diagram which contains six E8(Y_{124}) Dynkin diagrams and the 23rd Niemeier lattice E8^{3} root system. This can be reduced to Y_{444} by applying the spider relation to E8^{3} .

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

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.

”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).”

The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.

Please see the STA Wikipedia page or Lasenby, A. (2022), Some recent results for SU(3) and Octonions within the Geometric Algebra approach to the fundamental forces of nature, doi:10.1002/mma.8934 for more information on STA based octonion products.

Or if you prefer a more visually attractive form using the Hodge star

I have updated my comprehensive Fano plane list of 480 octonion triad sets. This new file adds the SpaceTime Algebra (STA) octonion representation on the nodes.

There are 7 sets of split octonions for each of the 480 “parent” octonions (each of which is defined by 30 sets of 7 triads and 16 7 bit “sign masks” which reverse the direction of the triad multiplication – detail of that below). The 7 split octonions are identified by selecting a triad. The complement of {1,2,3,4,5,6,7} and the selected triad leaves 4 elements which are the rows/colums corresponding to the negated elements in the multiplication table (highlighted with yellow background). The red arrows in the Fano Plane indicate the potential reversal due to this negation that defines the split octonions. The selected triad nodes are cyan, and the other 4 are yellow. Similar to STA, the split octonions allow for the simplification of Maxwell’s four equations which define electromagnetism (aka.light) into a single equation.

It also shows each octonion’s “Derivation” of the G2 automorphism group (the octonions). This means it analyzes each of the 480 octonions (and the 7 split-octonions with one per triad) in terms of the 7×7 “Nullspace” of the 21 upper-triangular vectors (14 null vectors that define the dimension of G2 (2*6 roots + 2 generators) and the 7 non-null vectors (shown). The first column (next to the green highlighted “missing e_{n}“) is the Derivation of the (non-split) parent octonion. The other 3 columns are the unique non-null Derivation entries which are not the same as the parent octonion’s entry. These are identified by the triad-based split numbers from which they were produced. Notice too that like the “missing e_{n}” for each row, there is also one of the four columns having a blank upper Derivation entry.

In total, these updated PDF files are 480 11×17 landscape 12pt font pages sorted by the assigned octonion to its E8 root vertex number based on a canonical sort from the 9th row of the Pascal triangle. As such (excluding the 8 positive and 8 negative “excluded generator roots” (#2-9 & #248-255) which don’t produce unique octonions. For each split real even E8 vertex, the algebra root, weight and height are listed along with the Clifford/Pascal binary and physics rotation coordinates.

It is now in two sets of files that show the STA detail multiplication table in each 240 + 240 sets of E8 indices based on whether the flag triad1Rev=False or True is set. This determines how the triad arrow directions are set using the 8 sign-masks (more detail on that below). This flag can be thought of as choosing which of two non-commutative multiplication options is used for defining the quaternions (IJ=K or right-handed or JI=K left-handed) as either True or False (340Mb each). There is a corresponding Mathematica notebooks with the same content in case you want to interact (e.g. rotate in 3D the Fano plane) here (280 Mb).

My original Fano plane / split Fano plane posts from 10+ years ago are here and here.

A bit about octonion multiplication tables: The 480 possible octonion table permutations are generated with a list of 30 sets of triples (3 numbers from 1-7 with no two triples with two numbers in common).

These can be visualized as a Fano plane. The Fano plane is easily constructed by taking the first 3 (of 7) triads and deleting the 4th and 7th (of the 9) numbers. This gives a “flattened” list of numbers (1-7) in some permutation order. While there are 7!=5040 possible, only 480 create proper octonions. But how to pair up the 480 to mesh with E8?

There are 16 possible +/- sign reversal (or sign mask (sm) bits also shown as hex number) permutations available in each of the 30 Fano planes (30*16=480). These sm bits also “invert” like the “anti” bits in E8 (which sorts it into a top half/bottom half canonical list built using the 9th row of the Pascal triangle and Clifford algebras). So the 16 sm possibilities split into two sets of 8. This changes the direction of the Fano plane arrows (one bit for each of the 7 triads in the chosen canonical triad, with the least significant hex bit (LSB) unused).

The algorithm to produce the 480 triad sets from the above sign masks and 30 canonical sets of 7 triads comes from a discussion by D. Chesley on twisted octonions and a paper “Octonion Multiplication and 7-Cube Vertices” from his personal website (1988).

In it he says: “Each representation of the octonions arranges the seven imaginary unit elements into seven triads, with no two distinct triads having a pair of elements in common. There is a natural mapping between the 30 distinct sets of triads and the 30 distinct versions of B7 which contain the vertex (0,0,0,0,0,0,0). Assigning a chirality to each triad of a given octonion representation then reveals that not all 128 possible chirality assignments result in a normed division algebra. In fact, for each of the 30 possible ways of grouping the elements, each of 16 chirality assignments (again corresponding to a B7) gives a distinct representation of the octonions for a total of 480 representations, and each of the other 112 (corresponding to D7) gives a distinct representation of the twisted octonion algebra.”

It turns out with some skillfull pattern matching in constructing the sign masks, the 8 sm bits pair up into two sets of four {1,4,6,7} and {2,3,5,8} giving a kind of “spin” match to E8. These pairs of 4 also operate as a left-right symmetry pattern seen in the structure of E8.

I would also like to show what I think is the most beautiful octonion due to its symmetric patterns. This was introduced in my last few papers here and here. It happens to be the first canonical triad (with Fano plane index fPi=1) with 16 sign-mask set of sm=5 taking the first sign-mask in that set being hex 08H (which reverses triad 4 from 246→264). This not only naturally produces a palindromic RCHO Standard model with E8 vertex counts, the STA Hodge star elements are in the reverse diagonal of e_{7}, its non-null Derivation has e_{n} with n= {1,2,3,4,5,6,7} with the unique split derivations being the 7 triads in order. It is the only octonion of the 480 that has that property! There are only 6 others (out of 48 total starting with the canonical quaternion first triad of {1,2,3}) that have the triad derivations, but the triad node orders are mixed along with the multiplication table +/- signs in the last row being mixed across {0,1,2,3} and {4,5,6,7}. Here is an animated Powerpoint build of the diagram below:

Below are a few of the relevant Mathematica function definitions that are used in the calculations above.

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

The H4 120-cell (J) showing another layer of outer edges. Note:These edges are at a different Norm radius and show a similar pattern to the dual 120-cell (J’) 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

The following “paper with code” in viXra is a comprehensive analysis of Furey’s work on the hyper-complex algebra of RCHO. Please see the ancillary notebook for the full detail.

In my research that has produced two arXiv papers on the topic of the isomorphisms of E8, I found a sequence derived from the sum of coefficients in Hadamard matrices that relates to certain rows of the Pascal triangle and E8. Please see that OEIS database entry here.

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 😉

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.