Later On

A blog written for those whose interests more or less match mine.

Aperiodic tiling with a single shape

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A tiling of the plane with tiles of the same shape in four colors: dark blue, light blue, white, and grey.

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.

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? 

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:


Written by Leisureguy

21 March 2023 at 10:04 am

Posted in Daily life, Math

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