Golden Ratio and Caspar–Klug Geometry in Viral Capsid Architecture — E8 Intelligence Research

Abstract

FINDING: Viral capsid geometry is governed by the Caspar–Klug triangulation number T = h² + hk + k², with the golden ratio emerging from pentagonal symmetry constraints, not from arbitrary decoration. | MATH: T = h² + hk + k² (h,k non-negative integers, not both zero). Icosahedral symmetry group: A₅ (order 60). Pentagonal face diagonal / side = φ = (1+√5)/2 ≈ 1.618. For a regular pentagon, diagonal² = φ² = φ + 1 = 2.618. The 12 vertices of an icosahedron correspond to the 12 pentagons in the dual dodecahedron; each capsid protein subunit sits on a lattice point of the hexagonal (triangular) tiling folded onto the icosahedron. The allowed T-numbers (1, 3, 4, 7, 9, 12, 13, 16, 19, 21, …) are precisely the integers representable by the Eisenstein norm form h² + hk + k², which is the norm in the Eisenstein integers ℤ[ω], ω = e^(2πi/3). | CONNECTION: The golden ratio φ appears directly: the icosahedron's 12 vertices can be placed at (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1) — coordinates involv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com