Mathematicians discover shape that can tile a wall and never repeat

Mathematicians have discovered a single shape that can be used to cover a surface completely without ever creating a repeating pattern. The long-sought shape is surprisingly simple but has taken decades to uncover – and could find uses in everything from material science to decorating.

Simple shapes such as squares and equilateral triangles can tile, or snugly cover a surface without gaps, in a repeating pattern that will be familiar to anyone who has stared at a bathroom wall. Mathematicians are interested in a more complex version of tiling, known as aperiodic tiling, which involves using shapes that don’t ever form a repeating pattern.

The most famous aperiodic tiles were created by mathematician Roger Penrose, who in the 1970s discovered that two shapes could be combined to create an infinite, never-repeating tiling. Now, Chaim Goodman-Strauss at the University of Arkansas and his colleagues have found a single tile shape – which they have called “the hat” – that does the same job.

Goodman-Strauss says that both finding and proving the tile to be aperiodic involved the use of powerful computers and human ingenuity. The team used computers to eliminate large numbers of options, then applied their experience to finding a shape and developing a proof.

“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,” he says Goodman-Strauss. “And then by hand you start examining them and try to understand them, and start to pull out the structure. That’s where a computer would be worthless as a human had to be involved in constructing a proof that a human could understand.”

Until now, it wasn’t even clear whether such a single shape, known as an einstein (from the German “ein stein” or “one stone”), could even exist. Sarah Hart at Birkbeck, University of London, who wasn’t involved with the research, says that until now she thought it would be impossible. “There are infinitely many possible candidate tiles, and even the existence of a solution feels quite counterintuitive,” she says.

Despite evading mathematicians for decades, the newly discovered einstein isn’t a convoluted or complex shape. It features just 13 sides. The shape also retains its aperiodic qualities when varying the lengths of the sides, meaning that the solution is actually a continuum of similar shapes.

Much of the difficulty in finding an einstein is proving that it really can tile aperiodically, without throwing up unusual counterexamples. The team discovered two proofs for the tile, with one being based on computer code that has been publicly released.

Hart says that knowledge of aperiodic tile shapes could help us design materials that are stronger or have other useful properties. Repeating patterns like tiles are also seen in crystal structures, where they can lead to fault lines along which material tends to break.

“Certain strange and wonderful types of crystalline structures called quasicrystals exhibit aperiodicity,” she says. “It may be that this new tiling may have applications to our understanding of the possible structures in quasicrystals.”

Colin Adams at Williams College in Massachusetts says he was shocked at the simplicity of the solution, and that this was a problem that “does not easily yield to brute force” computation. He is also keen to put it to practical use.

“You’re going to see people putting these in a bathroom because it’s just cool. I would put it in my bathroom if I were tiling it right now,” he says.