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 鈥� 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鈥檛 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, at the University of Arkansas and his colleagues听have found a single tile shape 鈥� which they have called 鈥渢he hat鈥� 鈥� that does the same job.
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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.
鈥淵ou鈥檙e literally looking for like a one in a million thing. You filter out the 999,999 of the boring ones, then you鈥檝e got something that鈥檚 weird, and then that鈥檚 worth further exploration,鈥� he says Goodman-Strauss. 鈥淎nd then by hand you start examining them and try to understand them, and start to pull out the structure. That鈥檚 where a computer would be worthless as a human had to be involved in constructing a proof that a human could understand.鈥�
Until now, it wasn鈥檛 even clear whether such a single shape,听known as an einstein (from the German 鈥渆in stein鈥� or 鈥渙ne stone鈥�), could even exist. at Birkbeck, University of London, who wasn鈥檛 involved with the research, says that until now she thought it would be impossible. 鈥淭here are infinitely many possible candidate tiles, and even the existence of a solution feels quite counterintuitive,鈥� she says.
Despite evading mathematicians for decades, the newly discovered einstein isn鈥檛 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 .
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.
鈥淐ertain strange and wonderful types of crystalline structures called quasicrystals exhibit aperiodicity,鈥� she says. 鈥淚t may be that this new tiling may have applications to our understanding of the possible structures in quasicrystals.鈥�
at Williams College in Massachusetts says he was shocked at the simplicity of the solution, and that this was a problem that 鈥渄oes not easily yield to brute force鈥� computation. He is also keen to put it to practical use.
鈥淵ou鈥檙e going to see people putting these in a bathroom because it鈥檚 just cool. I would put it in my bathroom if I were tiling it听right now,鈥� he says.
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