Aperiodic Monotile Solves Ein Stein Problem with √φ Inflation Factor for Spectre Tile — E8 Intelligence Research

Abstract

FINDING: Aperiodic monotile discovered, solving the ein Stein problem; inflation factor is sqrt(phi) for the Spectre tile variant. | MATH: Inflation factor = √φ = √((1+√5)/2) ≈ 1.27201965. The tile's geometry is based on a regular hexagon with edge modifications; the substitution rule scales edges by √φ and rotates by specific angles (e.g., 30° increments). The original "hat" tile uses a 1:√3 ratio in its kite-shaped constituents; the Spectre tile is a chiral aperiodic monotile with no reflections. | CONNECTION: √φ is the ratio of the side of a regular pentagon to its circumradius (0.5√(10+2√5) / (√φ) simplifies to φ-related constants). It also appears in the golden rhombus (diagonal ratio φ). The inflation factor links directly to the golden ratio φ = 1.618..., and its square root √φ ≈ 1.272 is the ratio of the diagonal to the side of a regular pentagon. This ties the tiling to pentagonal symmetry and quasicrystal-like order, though the tiling itself is not periodic. The substitution Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com