But there's a catch: you need to use both this shape and its reflected version. Is this necessary? Or is there a single shape whose translated and rotated versions can tile the plane, but only aperiodically? This is an open question!
Penrose's famous 'dart' and 'kite' are a *pair* of tiles whose translated and rotated versions tile the plane, but only aperiodically.
In 1966, Robert Berger showed there is a collection of shapes that can tile the plane if and only if there's a proof of Goldbach's conjecture in Zermelo-Fraenkel set theory.
But actually he did much more: for any Turing machine he constructed a finite set of shapes that can tile the plane if and only if that Turing machine halts!
For more on the new result:
• An aperiodic monotile, https://cs.uwaterloo.ca/~csk/hat/
For more on Berger's result:
• Wang tile, https://en.wikipedia.org/wiki/Wang_tile
There is no shape in the plane, homeomorphic to a disk, whose translated (not rotated) versions tile the plane, but only aperiodically. But there's been a recent surprise in higher dimensions:
• Rachel Greenfeld, Terence Tao, A counterexample to the periodic tiling conjecture, https://arxiv.org/abs/2211.15847
Thanks go to npub1dvt0rhv4vx34pzzt9fqcszg39cpjxnxfpwgwl4g3e4tt4kt2q25svxl9mf (npub1dvt…l9mf) and npub1yxxu3fkku2jfpuuhesaexzuhh6z5al0rjjs292kcvcryxc35v02qenp7am (npub1yxx…p7am) for correcting an error in a previous version of this post.
