Convex and discrete geometry Tilings Theorem

Einstein problem

Builds on Tessellation Periodic tiling Aperiodic set of prototiles

Statement

There is a single closed topological disk $T$ in the plane $\R^2$ with the following two properties.

The plane has a tiling in which every tile is congruent to $T$ .
No tiling of the plane in which every tile is congruent to $T$ is periodic.

In other words, the one-element set $\set{T}$ is an aperiodic set of prototiles, and $T$ is called an aperiodic monotile or einstein.

Remarks

The question whether one tile can be aperiodic, with no matching rules, is the einstein problem; the name plays on the German "ein Stein", one stone. David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss answered it in 2023 with the "hat", a polygon made of eight kites with angles $60^\circ, 90^\circ, 120^\circ, 90^\circ$ . Tilings by the hat use both the hat and its mirror image. The same authors later found the "spectre", which tiles the plane only non-periodically even when only rotations and translations are allowed. The proof groups the tiles of any hat tiling into larger clusters that again form a tiling of the same kind. This hierarchical structure at all scales rules out periods.

Going deeper

For a student

Aperiodic sets of prototiles had shrunk from Berger's huge set of tiles in 1966 to the two Penrose tiles in the 1970s. The einstein problem asked whether the number can be $1$ . The hat solves it if reflections are allowed, and the spectre solves it even with rotations and translations only. The key technique, as for Penrose tilings, is a substitution: tilings are built from clusters that copy the structure on a larger scale.

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Tag	`F456KQGPFV`
URI	`m2://F456KQGPFV`
Kind	statement, theorem
Writing	W0 stub
MSC	52C23, 52C20, 05B45
Status	This copy of the site has no build data. Run `m2 status F456KQGPFV`.
About	Aperiodic set of prototiles
Uses	Tessellation, Periodic tiling, Aperiodic set of prototiles
Same as	Wikidata Q18204561 Wikipedia Einstein problem

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