An aperiodic monotile

Abstract

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

91 pages, 57 figures. Copyedited journal version of article. Significant editing of the exposition throughout, but particularly in Section 3. Overall the same results appear in the same order

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• Inflation rules for a chiral pentagonal quasiperiodic tiling of stars and hexes
• Dynamics and topology of the Hat family of tilings
• Diffraction of the Hat and Spectre tilings and some of their relatives
• Tilings of a bounded region of the plane by maximal one-dimensional tiles
• Colouring monohedral tilings: defects and grain boundaries