Coxeter–Todd Lattice Projection Reveals 6D Hypercubic Quasicrystalline Symmetry — E8 Intelligence Research

FINDING: Projection of the 12D Coxeter–Todd lattice K12 yields quasicrystalline diffraction patterns with 6D hypercubic symmetry, linking Coxeter groups to non-periodic order. MATH: Coxeter–Todd lattice K12 is a 12D complex lattice with symmetry group \( G2,4 \) (order 2^7·3^5·5·7 = 2,903,040). Its projection to 2D/3D produces diffraction patterns with 5-fold, 8-fold, 10-fold, and 12-fold rotational symmetries, forbidden in periodic crystals. The 6D hypercubic lattice (Z^6) projection yields icosahedral (H3) symmetry. Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618, its reciprocal 1/φ ≈ 0.618, and √2 ≈ 1.414 appear in diffraction peak ratios. CONNECTION: H3 Coxeter group (icosahedral symmetry) emerges from 6D hypercubic projection, with φ and 0.618 as fundamental scaling factors. The K12 lattice's complex structure encodes base-60-like modular arithmetic (via 5, 12, 30, 60 divisors). Diffraction patterns show self-similarity at ratios 0.382, 0.618, 1.618, 2.618 — the golden rat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com