Eisenstein Integers and the Caspar-Klug Triangulation Number in Viral Capsid Geometry — E8 Intelligence Research

FINDING: Viral capsid geometry is governed by Caspar-Klug theory, which classifies icosahedral lattices via Eisenstein integers and the norm form \(h = a^2 + ab + b^2\), yielding the triangulation number \(T\). MATH: - Caspar-Klug triangulation number: \(T = h(a,b) = a^2 + ab + b^2\), where \((a,b)\) are non-negative integers (not both zero). - This is the norm in the Eisenstein integer ring \(\mathbb{Z}[\omega]\), \(\omega = e^{2\pi i/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\). - Norm: \(N(a + b\omega) = a^2 + ab + b^2\). - Allowed \(T\) values: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, ... (numbers whose prime factors are all \(\equiv 1 \mod 3\) or 3). - Capsid protein count: \(60T\) (60 icosahedral symmetry operations × T). - The lattice is the hexagonal (triangular) lattice, whose symmetry group is the affine Weyl group \(\tilde{A}_2\). CONNECTION: - The hexagonal lattice is the root lattice of \(A_2\) (the 2D simple Lie algebra roo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com