Golden Rectangles Unfold Icosahedron and Dodecahedron into Teichmüller Curves — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The five Platonic solids are uniquely determined by vertex/face regularity; the icosahedron and dodecahedron are constructible from three mutually orthogonal golden rectangles, and their unfoldings generate lattice surfaces with computable Teichmüller curves. | MATH: - Euler: V − E + F = 2; for regular {p,q}: 1/p + 1/q = 1/2 + 1/E → only (3,3), (3,4), (4,3), (3,5), (5,3). - Icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) where φ = (1+√5)/2 = 1.618…, and φ⁻¹ = 0.618…, φ² = 2.618… - Dodecahedron vertices: (±1, ±1, ±1) and (0, ±φ⁻¹, ±φ), (±φ⁻¹, ±φ, 0), (±φ, 0, ±φ⁻¹) — dual to icosahedron. - Lattice surface result (arXiv:1811.04131): unfolding each Platonic solid yields a translation surface; all are lattice surfaces (Veech surfaces) — Teichmüller curves computed explicitly. | CONNECTION: - Golden ratio φ appears directly in vertex coordinates of both icosahedron and dodecahedron — the only two Platonic solids with 5-fold symmetry, which is impossible in peri Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7219733468
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