Pisot Numbers and Pure Point Diffraction in Aperiodic Tilings — E8 Intelligence Research

FINDING: Pisot numbers as inflation factors generate pure point diffraction spectra in aperiodic tilings, linking algebraic number theory to crystallographic order. | MATH: Pisot–Vijayaraghavan (PV) numbers are algebraic integers >1 with all conjugates <1 in modulus. Smallest PV number is the plastic ratio ρ ≈ 1.324717957 (root of x³ − x − 1 = 0). For a 1D inflation tiling with PV unit λ, the diffraction spectrum's pure point part is computed via a Fourier matrix cocycle in internal space (arXiv:1907.11012v2). The plastic ratio satisfies ρ² = ρ + 1/ρ, and its reciprocal ≈ 0.754877666. | CONNECTION: The plastic ratio is NOT the golden ratio (φ ≈ 1.618), but it is the smallest Pisot number, and its inverse (0.7549) is close to 0.786 (√φ − 1 ≈ 0.7862) — a harmonic ratio used in Gann/geometric trading. More critically, PV numbers are exactly those whose powers approach integers exponentially fast, giving them a "quasi-crystalline" self-similarity. The Fourier cocycle method shows that the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com