Mathematicians discover a new 13-sided shape called a “hat”

For some people, tiles are seldom thought of unless it’s time for a home renovation, but for mathematicians they present a lot of mystery – and a clever team has come up with a particularly challenging puzzle. The researchers identified a shape that was previously only theoretical: a 13-sided configuration called a “cap” that can form a surface without repetition.

Hats are what’s known as acyclic monotiles, meaning that a single shape can overlay a surface with no transitional symmetry, or without repeating the pattern at all. famous Penrose tiles is an example of aperiodic tiling, where the pattern is aperiodic but takes two different shapes.

The cap tiles use only one shape, “einstein”, the German term for “one stone”, which makes the pattern an irregular monolith. The 13-sided hat is polyhedral in shape, consisting of eight kites attached to the ends. The existence of a non-periodic monotyl was purely theoretical until a research team led by mathematician David Smith and colleagues demonstrated its existence in prepress sheet Posted online this month.

“You really are looking for one thing in a million. You sift through 999,999 boring items, and then you get something odd, and that’s worth exploring further,” said co-author Chaim Goodman-Strauss, a mathematician at National Mathematical. Museum. new world. “And then you start going through it manually and trying to make sense of it, and you start unplugging the structure. This is where computers become worthless because humans have to be involved in constructing proofs that humans can understand. ”

For mathematicians, the discovery appears to answer a long-standing engineering question. But for the rest of us, it’s probably the funky new bathroom tile choice.