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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.



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    I only skimmed the paper, but this looks amazing! After Penrose’s darts and kites, its really surprising to see such a relatively complex monotile and that it fits so neatly onto a regular hex grid.

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      I didn’t think this would be solved in my lifetime. Very exciting. The candidate tile is pleasingly simple compared to previous efforts.

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        I can’t wait for the toy. Im impressed with how simple the tile is.

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          I look forward to an updated variant of the penrose(6) screensaver.

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            This is a fantastic finding! What about engineering a chemical with this crystal structure?