Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles

Michael F. Whittaker; Tinka Bruneau

Planar aperiodic tile sets: from Wang tiles to the Hat and Spectre monotiles

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arxiv 2310.06759 v2 pith:7V7HO73M submitted 2023-10-10 math.MG

Tinka Bruneau , Michael F. Whittaker This is my paper · ORCID

classification math.MG

keywords tileaperiodicsetsspectrediscoverymonotilemonotilesplanar

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abstract

A brief history of planar aperiodic tile sets is presented, starting from the Domino Problem proposed by Hao Wang in 1961. We provide highlights that led to the discovery of the Taylor--Socolar aperiodic monotile in 2010 and the Hat and Spectre aperiodic monotiles in 2023. The Spectre tile is an amazingly simple monotile; a single tile whose translated and rotated copies tile the plane but only in a way that lacks any translational periodicity. We showcase this breakthrough discovery through the 60$+$ years that aperiodic tile sets have been considered.

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Quasilattices of the Spectre monotile
physics.gen-ph 2025-02 conditional novelty 5.0 of 10

Decorating every Spectre tile with the same point yields a wide variety of non-periodic quasilattices, including sparse, clustered, and near-hexagonal examples.

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