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°.
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.