Group Theory in a Nutshell: Taming the Monster
Category Archives: Physics
Space-Time-Algebra (STA) Octonionic Illustration
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.
Space-Time Algebra (STA) octonionic representation
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 here. It is a large PDF at 80Mb. It is sorted by the E8 root vertex numbers based on a canonical sort from the 9th row of the Pascal triangle. As such, it has 16 excluded roots that are the 8 positive and 8 negative generator roots (which don’t produce unique octonions).
There are now also two sets of files that show the STA detail data in each 240 + 240 sets of E8 indices splitting of the 480 (one with the first triad that defines the non-commutative multiplication by either Right (IJ=K) & Left (JI=K) quaternions. These are also 80Mb files. There are corresponding Mathematica notebooks with the same content in case you want to interact (e.g. rotate in 3D the Fano plane). They are here and here (respectively).
My original Fano plane / split Fano plane posts are here and here.
A bit about octonions:
The 480 octonion 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.
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.
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
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
Mathematica Analysis of an RCHO Standard Model
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.
Improved Niemeier Lattices Neighborhood graph
Please see this link for the Mathematica Notebook used to create this graph.
My E8 Petrie projection on Chevalley’s “Theory of Lie Groups” book cover
My E8, Hadamard, and Pascal Triangle related OEIS Integer Sequence A367629
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.
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 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.
My patent for Rapid 3D prototyping and fabricating of slow-wave structures, including electromagnetic meta-material structures, for millimeter-wavelength and terahertz-frequency high-power vacuum electronic devices
A few years ago and 5 1/2 years after I retired, the patent I worked on with Andrey and Maysa finally made it through the US Patent Office!
