{
  "id": "2311.01486",
  "version": "v1",
  "published": "2023-11-02T15:31:15.000Z",
  "updated": "2023-11-02T15:31:15.000Z",
  "title": "The Isomorphism of $H_4$ and $E_8$",
  "authors": [
    "J. G. Moxness"
  ],
  "categories": [
    "math.GR",
    "hep-th"
  ],
  "abstract": "This paper gives an explicit isomorphic mapping from the 240 real $\\mathbb{R}^{8}$ roots of the $E_8$ Gossett $4_{21}$ 8-polytope to two golden ratio scaled copies of the 120 root $H_4$ 600-cell quaternion 4-polytope using a traceless 8$\\times$8 rotation matrix $\\mathbb{U}$ with palindromic characteristic polynomial coefficients and a unitary form $e^{\\text {i$\\mathbb{U}$}}$. It also shows the inverse map from a single $H_4$ 600-cell to $E_8$ using a 4D$\\hookrightarrow$8D chiral left$\\leftrightarrow$right mapping function, $ \\varphi$ scaling, and $\\mathbb{U}^{-1}$. This approach shows that there are actually four copies of each 600-cell living within $E_8$ in the form of chiral $H_{4L}$$\\oplus$$\\varphi H_{4L}$$\\oplus$$H_{4R}$$\\oplus$$\\varphi H_{4R}$ roots. In addition, it demonstrates a quaternion Weyl orbit construction of $H_4$-based 4-polytopes that provides an explicit mapping between $E_8$ and four copies of the tri-rectified Coxeter-Dynkin diagram of $H_4$, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored $E_8$-based Grand Unified Theories or GUTs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-02T15:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B30",
      "22E46",
      "51M15",
      "52B20"
    ],
    "keywords": [
      "isomorphism",
      "palindromic characteristic polynomial coefficients",
      "8d chiral left",
      "golden ratio scaled copies",
      "grand unified theories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}