Molecular Platonic Solids Realized via Finite Coxeter Groups Including Non-Crystallographic Icosahedral H3 — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Platonic solids are realized in molecular chemistry via finite Coxeter groups (A3, B3, H3), with non-crystallographic H3 (icosahedral) appearing in actual molecules, bridging discrete geometry and chemical structure. | MATH: Coxeter groups: A3 (tetrahedron, order 24), B3 (octahedron/cube, order 48), H3 (icosahedron/dodecahedron, order 120). Euler characteristic: V−E+F=2. Duality: tetrahedron self-dual; cube↔octahedron; dodecahedron↔icosahedron. Non-crystallographic constraint: H3 lacks invariant lattice under reflection (no 5-fold translational symmetry). | CONNECTION: H3 contains golden ratio φ=1.618 and its inverse 0.618 in edge ratios of icosahedron/dodecahedron; dihedral angles yield 0.786 (cos⁻¹(1/√5) related) and 2.618=φ² appears in icosahedral vertex coordinates. Base-60 not directly implicated, but 5-fold symmetry (pentagon) ties to Babylonian sexagesimal fractions (1/5=0.2, 1/3=0.333 in base-60). | DEPTH: 7 — solid mathematical grounding (Coxeter classification) but f
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026molecular,
title = {{Molecular Platonic Solids Realized via Finite Coxeter Groups Including Non-Crystallographic Icosahedral H3 — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490833},
url = {https://zenodo.org/records/23490833},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Molecular Platonic Solids Realized via Finite Coxeter Groups Including Non-Crystallographic Icosahedral H3 — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490833
UR - https://zenodo.org/records/23490833
AB - FINDING: Platonic solids are realized in molecular chemistry via finite Coxeter groups (A3, B3, H3), with non-crystallographic H3 (icosahedral) appearing in actual molecules, bridging discrete geometry and chemical structure. | MATH: Coxeter groups: A3 (tetrahedron, order 24), B3 (octahedron/cube, order 48), H3 (icosahedron/dodecahedron, order 120). Euler characteristic: V−E+F=2. Duality: tetrahedron self-dual; cube↔octahedron; dodecahedron↔icosahedron. Non-crystallographic constraint: H3 lacks invariant lattice under reflection (no 5-fold translational symmetry). | CONNECTION: H3 contains golden ratio φ=1.618 and its inverse 0.618 in edge ratios of icosahedron/dodecahedron; dihedral angles yield 0.786 (cos⁻¹(1/√5) related) and 2.618=φ² appears in icosahedral vertex coordinates. Base-60 not directly implicated, but 5-fold symmetry (pentagon) ties to Babylonian sexagesimal fractions (1/5=0.2, 1/3=0.333 in base-60). | DEPTH: 7 — solid mathematical grounding (Coxeter classification) but f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -