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"The Hat", a shape that tiles but never makes a repeating pattern (upload.wikimedia.org)
submitted 7 months ago by 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°.

Example tiling of "The Hat"
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[–] 9 points 7 months ago

Local repetition exists but global repetition does not. Think of it like wallpaper. There is no way to put this "pattern" on a wallpaper in such a way that two identical strips of wallpaper fully match each other at the edges.

Look at a more zoomed out version to see it clearly. It always looks close to being repeating but then you see a part that's just a little bit off.

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