2023/11/02 by J Gregory Moxness, Moxness, J. G.
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #15B30 #22E46 #51M15 #52B20 #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #FOS: Mathematics #FOS: Physical sciences #Fractal and DNA sequence analysis #Group Theory (math.GR) #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2311.01486
openalex publication_date 2023/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper gives an explicit isomorphic mapping from the 240 real ℝ8 roots of the E8 Gosset 421 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 \mathbbU with palindromic characteristic polynomial coefficients and a unitary form e^\text i\mathbbU. It also shows the inverse map from a single H4 600-cell to E8 using a 4D\hookrightarrow8D chiral left↔right mapping function, φ scaling, and \mathbbU-1. This approach shows that there are actually four copies of each 600-cell living within E8 in the form of chiral H4L⊕φH4L\oplusH4R⊕φH4R 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.