Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026platonic,
title = {{Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23525936},
url = {https://zenodo.org/records/23525936},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Platonic Solids as Rank-3 Coxeter Groups: From Root Systems to AI Latent Spaces — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23525936
UR - https://zenodo.org/records/23525936
AB - 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 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -