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°.
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.