"The Hat", a shape that tiles but never makes a repeating pattern
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".
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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Can these tiles actually be bought? For the bathroom floor. Or for paving the driveway. :)
i 3d printed a lot of them and made a mural.
it's actually hard to tessalate them.
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?"
they'll be smart and figure out a mathematical tool to efficiently predict where the next tile goes. and win the fields medal award
I'd imagine that one could have software generate a tesselation.
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
If you find a contractor patient enough to cut tiles like this
Best I could find is plastic or balsa wood, but I bet you could use one of those clay 3d printers to make some.
When these went viral in 2022 I read the research paper and found out that not only do they form a non-repeating pattern, but that non-repeating pattern relies on the occasional tile being reversed. That inspired me to 3D print a bunch of these that were a different color on each side and try assembling them. It's very interesting, because you have a lot of options for how to put them together, but occasionally you'll hit a point where the pattern itself forces you to put one in upside down, even though it's non-repeating. Also, it's possible to put it together "wrong" where at one edge you can't add any more tiles in either orientation and have to disassemble part of it to continue. Very interesting to mess with.
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.
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.
cool. a zoomed out version of the image is all I needed.
That's a surprising prooerty for a shape that's just 8/3 of an equilateral triangle
The way you said this make it sound like you're trying to start a bar fight with a geometric shape
I don't really get it, don't the colors highlight the repeating patterns? How can it not have repeating patterns with clearly repeating patterns like that? They're probably using some useless definition of "repeating", right?
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.
I wonder if there's a way to apply this pattern to create some specific kind of cellular automaton.
Honestly idk enough about this stuff to even know if that's a dumb question, but at the very least, the image reminded me of that.
without a doubt, they would me awesome for some kind of board game
is this the pattern that was found on the walls of a 1000 year old chapel in like Uzbekistan?
according to the scriptures of that particular sect it was called "the pattern of life" or something.
doubt "x"