Miller Indices, Reciprocal Lattices, and Quaternions in Crystallographic Texture Analysis — E8 Intelligence Research

FINDING: Crystallographic lattice geometry is indexed by integer Miller triples (hkl) whose reciprocal-space construction encodes plane spacing and symmetry equivalence classes; quaternions supply the rotation algebra for texture analysis. | MATH: Plane spacing d_hkl = a/√(h²+k²+l²) (cubic); reciprocal lattice vector G = h b₁ + k b₂ + l b₃; interplanar angle cosθ = (h₁h₂+k₁k₂+l₁l₂)/(√(h₁²+k₁²+l₁²)·√(h₂²+k₂²+l₂²)); unit-cell lattice points: SC=1, BCC=2, FCC=4; quaternion q = w + xi + yj + zk, ‖q‖=1, rotation via q v q⁻¹. | CONNECTION: Cubic symmetry group O_h (order 48) = 24 rotations (isomorphic to S₄) × inversion; FCC/BCC are the A₁/D₃ root lattices — FCC = A₃ root system, BCC = D₃, both appearing in the ADE classification. The 60° and 90° interplanar angles in {111}/{100} families reflect the hexagonal-cubic twinning relationship (60° = base-60 residue). | DEPTH: 6 FINDING: The search returned no content on 60-smooth numbers; the lattice material is standard crystallography, not smo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com