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

For some, tiles are rarely thought of unless it’s time for home renovations, but for mathematicians they present plenty of mysteries – and a clever team has come up with an incredibly challenging puzzle. The researchers identified a shape previously only theoretical: a 13-sided configuration called a “hat” that can tile a surface without repeating.

A hat is what is known as an acyclic monotile, which means that a single shape can tile the surface without any transitional symmetry, or without repeating its pattern at all. famous Penrose tiling is an example of aperiodic tiling, where the pattern is aperiodic but uses two different shapes.

Hat tiling uses only one shape, “einstein,” a German term for “one stone,” which makes the pattern an irregular monolith. The 13-sided hat is a polyhedral shape, consisting of eight kites attached at its 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’re literally looking for one thing in a million. You filter out 999,999 boring items, and then you get something weird, and then it’s worth further exploration,” said co-author Chaim Goodman-Strauss, a mathematician at the National Mathematical Museum. new world. “And then you start manually examining it and trying to make sense of it, and you start pulling the structure off. This is where a computer is going to be worthless as a human has to be involved in building evidence that a human can understand.”

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For mathematicians, the discovery appears to answer a long-standing engineering question. But for the rest of us, perhaps it represents a funky new bathroom tile option.