Miller Indices and Quaternion Rotations Link Lattice Geometry to Texture Analysis — E8 Intelligence Research

FINDING: Crystallographic lattice geometry is indexed by Miller indices (hkl) — integer triples reciprocal to plane intercepts — and the same rotational algebra (quaternions) that generates lattice symmetry also governs texture analysis, linking discrete lattice geometry to continuous rotation groups. | MATH: Plane intercepts (a/h, b/k, c/l); interplanar spacing d_hkl = 2π/|G_hkl| with reciprocal lattice vector G_hkl = h·b₁ + k·b₂ + l·b₃; reciprocal basis b_i·a_j = 2πδ_ij; structure factor F_hkl = Σ f_j exp(2πi(hx_j+ky_j+lz_j)); Bravais lattices = 14; point groups = 32; space groups = 230; quaternion q = w + xi + yj + zk, |q|=1, rotation R = qvq⁻¹, double cover SU(2)→SO(3). | CONNECTION: Crystallographic symmetry is the finite subgroup structure of SO(3) and its double cover SU(2) — the same group manifold underlying the 0.618/1.618 golden-ratio rotations (icosahedral group I, order 60, contains 5-fold axes whose rotation angle 2π/5 has cos = 0.309 = (φ−1)/2, i.e. golden-ratio structur Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com