Sturmian Tiles Yield Quadratic Field Units as Scaling Constants — E8 Intelligence Research

FINDING: Aperiodic tile sets generated from Sturmian words with quadratic slopes yield scaling constants that are units in real quadratic fields ℚ(√D). MATH: The scaling constant is a unit ε in ℚ(√D), satisfying ε > 1 and N(ε) = ±1. For quadratic irrational slope α, the Sturmian word's substitution matrix has eigenvalues ε and 1/ε. Example: for D=5 (golden ratio φ), ε = φ² ≈ 2.618; for D=2 (silver ratio), ε = 1+√2 ≈ 2.414. CONNECTION: The golden ratio φ ≈ 1.618 and its square 2.618 appear as scaling constants for D=5. The silver ratio 1+√2 ≈ 2.414 for D=2. These are Pisot numbers, linking to self-similar tilings and quasicrystal symmetries (e.g., Penrose tilings). The Sturmian lattice construction uses base-60-like periodic continued fractions for quadratic irrationals, echoing ancient Babylonian mathematics. DEPTH: 8 — This bridges number theory (quadratic fields, units), combinatorics (Sturmian words), and geometry (aperiodic tilings, quasicrystals). It provides an explicit alg Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com