An 'einstein' tile? Mathematicians discover pattern that never repeats

This culminates a decades-long effort in search of "a one-in-a-million" shape.
Chris Young
Part of the aperiodic tile.
Part of the aperiodic tile.

Craig Kaplan / Twitter 

Computer scientists finally discovered a new tile they've dubbed "einstein" after decades of trying. It is one shape, with 13 sides, that can cover an entire plane without ever repeating a pattern.

The type of shape, known in the world of mathematics as an "aperiodic monotile", is a shape that has real-world applications in material science. Mathematicians are hailing this discovery as a breakthrough in the history of aperiodic tiles.

Mathematicians finally discover the 'einstein' tile

Though the tile type is known as "einstein", it isn't named after the famous physicist. Instead, it comes from the German words ein stein, meaning "one stone", referring to the fact it is one tile. The researchers presented their findings in a paper in preprint server arXiv.

"In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constrains applied via matching conditions," Craig Kaplan, a computer science professor from the University of Waterloo and one of the four authors of the paper, wrote in a statement. "We prove that this shape, a polykite that we call 'the hat,' must assemble into tilings based on a substitution system."

Kaplan added in a thread on Twitter that the first aperiodic sets were made up of over 20,000 tiles. "Subsequent research lowered that number, to sets of size 92, then six, and then two in the form of the famous Penrose tiles [back in 1974]."

"Since then," Kaplan explained, "others have constructed sets of size two, but nobody could find an 'einstein,' a single shape that tiles the plane aperiodically. Could such a shape even exist?"

A pattern that never repeats

Incredibly, the international team of computer scientists was able to discover the unique shape. In their new paper, they outlined how they proved the nature of the shape through computer modeling. Essentially, they showed that the einstein tile can cover a surface completely without ever exhibiting a repeat pattern.

"We finally got down to one!" Kaplan wrote.

In an interview with New Scientist, Chaim Goodman-Strauss, a member of the team and professor at the University of Arkansas, said "you're literally looking for like a one-in-a-million thing. You filter out the 999,999 of the boring ones, then you've got something that's weird, and then that's worth further exploration. And then by hand you start examining them and try to understand them and start to pull out the structure."

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