What question did this study set out to answer?

To discover a tiling method using a new aperiodic monotile that resolves the longstanding ein Stein problem.

July 26, 2026Open Access

Aperiodic Monotile Discovery: The Hat Solves the 50-Year Ein Stein Problem — E8 Intelligence Research

Key Points

To discover a tiling method using a new aperiodic monotile that resolves the longstanding ein Stein problem.
Introduced the Hat tile formed from 8 kites with specific geometry.
Developed a substitution rule involving the golden ratio, φ, and hierarchical systems to ensure non-periodicity.
Compared the Hat tile's properties with Penrose tiling while maintaining a single tile structure.
The Hat tile uniquely forces non-repeating tiling without translational symmetry.
The substitution rule shows a strong relationship with the golden ratio, allowing for geometric harmony in edge lengths.
Established a new approach to aperiodic tiling with local 5-fold symmetry.

Abstract

FINDING: Discovery of the first aperiodic monotile ("Hat" tile) by David Smith et al. , solving the 50-year-old "ein Stein" problem. The tile forces non-repeating tiling without translational symmetry, using a substitution rule with scaling factor related to the golden ratio. MATH: The Hat tile is a polykite formed from 8 kites of a specific geometry. Its substitution rule involves a scaling factor of \ (= (1+5) /2 1. 618 \). The tile's edge lengths are in ratios derived from the golden ratio. The aperiodicity is enforced by a hierarchical substitution system that generates a tiling with no periodic repeats. Key constants: \ (\), \ (1/ 0. 618 \), \ (² 2. 618 \). CONNECTION: The scaling factor \ (\) directly links to geometric harmony ratios (0. 618, 1. 618, 2. 618). The tiling's substitution rule is analogous to Penrose tiling (also \ (\) -based) but uses a single tile. The structure exhibits local 5-fold symmetry (crystallographical Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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