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°.
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.