{
  "abstract": "FINDING: Platonic solids are governed by the finite Coxeter groups A₃, B₃, H₃ (crystallographic and non-crystallographic), whose root systems and Weyl groups encode all five regular polyhedra; recent work extends this to molecular realization (e.g., Platonic hydrocarbons) and even to convergence in AI latent spaces. | MATH: The five Platonic solids correspond to rank-3 Coxeter groups: tetrahedron = A₃ (order 24), cube/octahedron = B₃ (order 48), icosahedron/dodecahedron = H₃ (order 120, non-crystallographic). Key invariants: Schläfli symbols {3,3}, {4,3}, {3,4}, {5,3}, {3,5}. Dual pairs: (cube, octahedron), (icosahedron, dodecahedron), tetrahedron self-dual. Coxeter numbers: h(A₃)=4, h(B₃)=6, h(H₃)=10. Golden ratio φ = (1+√5)/2 = 1.618… appears explicitly in H₃: coordinates of icosahedron vertices involve φ; dodecahedron edge/radius ratios involve φ. The non-crystallographic nature of H₃ means no full lattice in 3D has 5-fold symmetry, but molecular realizations (e.g., dodecahedrane C₂\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55zwez3za8yn6.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23525936",
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  "id": "55zwez3za8yn6",
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  "landing_url": "https://zenodo.org/records/23525936",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55zwez3za8yn6.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:21:53Z",
      "native_id": "23525936",
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      "url": "https://zenodo.org/records/23525936"
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  "subjects": [],
  "title": "Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research",
  "title_text": "Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:53Z",
  "url": "https://research.draconitic.com/paper/55zwez3za8yn6",
  "venue": "Zenodo",
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}