The palindromic E8, STA Hodge star Octonion/BiQuaternions, and [CU;AG] Codon GenoMatrix can be integrated into a possible isomorphism between an E8 based ToE (i.e. unifying a 3 generation Grand Unified Theory (GUT) of the Standard Model (SM) with General Relativity (GR)) and the biochemistry of Life itself. It does this from a single mathematical 240+8 dimensional Exceptional Lie Algebra / Group / Lattice / Polytope in one Grand Objective Design structure with a Trinity of triads and triality.
The above figure is a matrix integration of my E8-H4 Folding (Rotation) matrices, the STA Hodge Star Octonion/BiQuaternion multiplication matrices, and the [CU;AG] Codon GenoMatrix. The GenoMatrix is a modified form of that presented by Sergey V. Petoukhov, Head of Laboratory of Biomechanical System, Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow in a 2011 paper arXiv:1102.3596 “The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts”.
The H4 Cell-First {5,3,3} 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.
The linkage between the GenoMatrix and E8 is not surprising when understood in the light of how E8 folds to 4 copies of the H4 4D duals of the 600/120-cells, which have the convex hull of the chamfered dodecahedron. This 3D shape is similar to the truncated icosahedron (aka. soccer ball) used in the (2003 abandoned patent) mnemonic created by Mark White and documented in 2007 arXiv here “The G-Ball, a New Icon for Codon Symmetry and the Genetic Code”. There is an interesting blog post in American Scientist about that here. The germ of this idea (pardon the pun) seems to have come from Petoukhov in 1981 (as per his list of works here) in a paper ” Investigations on Non-Euclidean biomechanics. Projective geometry, Fibonacci numbers and the kinematic scheme of human body.”
It is interesting to note that the particular octonion (or Hodge Star BiQuaternion) multiplication matrix used by Petoukhov is the same one as highlighted in my recent papers and blog posts regarding the most beautiful of the 480 octonion symmetry pattern, which I introduced in my “Isomorphism of H4 and E8” arXiv paper here and discussed in a bit more detail here “Space-Time-Algebra (STA) Octonionic Illustration”, with this summary:
This was originally introduced in my (always updated/corrected) last few papers here and here (or if you prefer to get the originals off arXiv 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 made up of e7‘s, its non-null Derivation has en 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}.
For completeness, we show below the 4D projection of the Atomic Element Periodic Table of Chemistry below:
