Aperiodic tiling with a single shape

Periodic tilings of the plane — a pattern repeated indefinitely that covers the plane — are common: the checkerboard pattern using squares, the public-restroom-floor tiling of black and white hexagons, various tilings of equilateral triangles, and so on. And you can force aperiodicity — nonrepeating — on those by using colors at varying intervals. But what about a tiling that is necessarily aperiodic (nonrepeating)?

Craig S. Kaplan in a Mastodon thread writes:

How small can a set of aperiodic tiles be? The first aperiodic set had over 20000 tiles. Subsequent research lowered that number, to sets of size 92, then 6, and then 2 in the form of the famous Penrose tiles. https://youtu.be/48sCx-wBs34

I have blogged that video before, and it’s fascinating — I just watched it again.

Kaplan continues:

Penrose’s work dates back to 1974. Since then, others have constructed sets of size 2, but nobody could find an “einstein”: a single shape that tiles the plane aperiodically. Could such a shape even exist? https://en.wikipedia.org/wiki/Einstein_problem 

The whole thread is worth reviewing, and he includes some interesting links.

Alo definitely read this post by Jason Kottke, which goes further into it — and includes this fascinating video: