Aperiodic Monotile Discovery: Hexagon-Derived Hat Shape with Hierarchical Substitution — E8 Intelligence Research

July 20, 2026Open Access

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Key Points

The research aims to discover a single aperiodic monotile with unique hierarchical substitution rules.
Identified the tile as a 13-sided polygon derived from hexagonal geometry.
Developed substitution rules using scaling factors related to the golden ratio.
Explored the aperiodicity and chirality to prevent periodic arrangements in tiling.
Demonstrated the aperiodic monotile's geometry emerges from hexagonal lattice deformation.
Established the substitution scaling factor as approximately 2.618, the square of the golden ratio.
Solved a 50-year-old problem regarding the existence of a single aperiodic tile.

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Finding reveals a single aperiodic monotile in mathematical geometry, suggesting new insights into tiling theory.

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Andrew Stewart Caldin (2026) studied this question.

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