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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

18 comments fedilink hide all child comments

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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[–] probable_possum@leminal.space 22 points 5 days ago (3 children)

Can these tiles actually be bought? For the bathroom floor. Or for paving the driveway. :)

[–] IAmNorRealTakeYourMeds@lemmy.world 16 points 5 days ago (2 children)

i 3d printed a lot of them and made a mural.

it's actually hard to tessalate them.

[–] bitfucker@programming.dev 4 points 3 days ago (1 children)

You know what? If you want to be a prick you could hire a contractor to tesselate it lol. "Hey, I already have the tile, can you assemble it for me?"

[–] IAmNorRealTakeYourMeds@lemmy.world 3 points 3 days ago

they'll be smart and figure out a mathematical tool to efficiently predict where the next tile goes. and win the fields medal award

[–] tal@lemmy.today 4 points 4 days ago* (last edited 4 days ago) (1 children)

I'd imagine that one could have software generate a tesselation.

[–] bitfucker@programming.dev 1 points 3 days ago

I think due to the fact that we know for sure the patterns are non repeating, the algorithm would never terminate for an infinitely large plane. But for a bounded plane, then yeah, that's doable

[–] silverchase@sh.itjust.works 12 points 5 days ago

If you find a contractor patient enough to cut tiles like this

[–] Hackworth@piefed.ca 6 points 5 days ago* (last edited 5 days ago)

Best I could find is plastic or balsa wood, but I bet you could use one of those clay 3d printers to make some.