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"The Hat", a shape that tiles but never makes a repeating pattern (infosec.pub)

submitted 5 days ago by silverchase@sh.itjust.works to c/wikipedia@lemmy.world

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Image by Cmglee, CC BY-SA 4.0, via Wikimedia Commons

A square can tile a plane but can form a repeating pattern. Is there a single shape that can tile but never repeats? That's what's called the "einstein problem".

Link to the article

In 2010, the first never-repeating tile was discovered: the Socolar-Taylor tile. But it's a bit weird, having several separated, disconnected bits.

In 2022, "The Hat" (shown in pic) was discovered, and it's a lot less weird. It only has 13 sides and nice angles that are multiples of 30°.

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[–] orbituary@lemmy.dbzer0.com 17 points 5 days ago (1 children)

Is it just me or is the radial pattern not apparent to others? Starting with the red "hat" top center, work outward in a spiral. It's not bilaterally symmetrical, but it appears to be chiral.

Going to read more about this.

[–] silverchase@sh.itjust.works 27 points 5 days ago (1 children)

Here's the paper that proves the aperiodicity of the shape.

And here's a site about this shape by Craig S. Kaplan, one of the mathematicians that contributed to the proof.

[–] orbituary@lemmy.dbzer0.com 7 points 4 days ago

cool. a zoomed out version of the image is all I needed.