Aperiodic Monotiles from a Golden Ratio Substitution Rule — E8 Intelligence Research

FINDING: The hat monotile and its chiral relative the Spectre tile are aperiodic monotiles constructed from a continuum of polygon shapes, with direct substitution rules enabling hierarchical aperiodic order. | MATH: The hat tile is a polykite shape formed from 8 kites; its tiling is generated by a substitution rule with inflation factor \(φ^2 = (1+√5)^2/4 ≈ 2.618\) (the square of the golden ratio). The correlation decay in such aperiodic tilings is governed by the subdominant eigenvalue of the substitution matrix, which for the hat tile is \(λ_2 = φ⁻² ≈ 0.382\) (the golden ratio conjugate squared). | CONNECTION: The eigenvalues \(φ^2\) and \(φ⁻²\) are directly linked to the golden ratio \(φ = 1.618...\), with \(φ⁻² = 0.382\) and \(φ^2 = 2.618\). These are the dominant and subdominant eigenvalues of the substitution matrix, controlling scaling and correlation decay. The tiling also exhibits 6-fold rotational symmetry in its diffract Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com