Base-60 Harmonic Structure Links Reciprocal Lattices and Coxeter Groups — E8 Intelligence Research

FINDING: Reciprocal lattice construction and Coxeter group classification are linked via root system duality and the Plimpton 322 reciprocal pair method, revealing a hidden base-60 harmonic structure in crystallographic symmetry classification. MATH: - Reciprocal lattice vectors: \( \mathbf{a}^*_i = 2\pi \frac{\mathbf{a}_j \times \mathbf{a}_k}{\mathbf{a}_i \cdot (\mathbf{a}_j \times \mathbf{a}_k)} \) (cyclic permutations) - Coxeter group classification: finite Coxeter groups correspond to crystallographic root systems (types Aₙ, Bₙ, Cₙ, Dₙ, E₆, E₇, E₈, F₄, G₂) - Plimpton 322 reciprocal pairs: \( (p, q) \) with \( p/q \) generating Pythagorean triples via \( (p^2 - q^2, 2pq, p^2 + q^2) \); reciprocal pairs satisfy \( 1/p + 1/q = 1 \) in base-60 sexagesimal fractions - Involution product condition: \( w = xy \) with \( x^2 = y^2 = 1 \), minimal length \( \ell(x) + \ell(y) - \ell(w) \) relates to Coxeter element factorization CONNECTION: - Reciprocal lattice construction mir Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com