A 13-sided tile called “the hat” forms a pattern that covers an infinite plane, yet it cannot repeat, making it a long-sought shape known as an “einstein.” A sample of that pattern is shown here.
Mathematicians are tipping their caps at the 13-sided shape known as "the hat."
It represents the first real instance of an "einstein," a singular form that creates a unique tiling of a plane: Similar to bathroom floor tile, it can completely cover a surface without any gaps or overlaps, but only with a single, unique pattern.
The mathematician Marjorie Senechal of Smith College in Northampton, Massachusetts, who was not involved with the discovery, claims that "everybody is astonished and is delighted, both." For fifty years, mathematicians had been looking for such a shape. Senechal claims that "it was not even clear that such a thing could exist."
Although the name "Einstein" conjures up the famous physicist, it actually derives from the German "ein Stein," which refers to the single tile and means "one stone." Einstein is caught in a strange limbo between order and chaos. The tiles can cover an infinite plane and fit together perfectly, but they are aperiodic, which means they cannot create a repeating pattern.
The tiles can be moved and still perfectly match their previous arrangement if the pattern is periodic. If you slide the rows over by two, for instance, an infinite checkerboard still appears the same. Other single tiles can be arranged in non-periodic patterns, but the hat is unique because it cannot be used to make a periodic pattern.
The hat is a polykite, which was discovered by David Smith, a nonprofessional mathematician who describes himself as a "imaginative tinkerer of shapes," and was reported in a paper posted online on March 20 at arXiv.org. According to Chaim Goodman-Strauss of the National Museum of Mathematics in New York City, one of a group of trained mathematicians and computer scientists that Smith collaborated with to study the hat, "that is a type of shape that had not been studied closely in the search for Einsteins."
It is a deceptively simple polygon. Before this work, if you asked Goodman-Strauss what an Einstein would look like, he would have drawn "some crazy, squiggly, nasty thing."
Non-repeating tilings involving multiple tiles of different shapes were previously known to mathematicians. In the 1970s, mathematician Roger Penrose discovered that only two different shapes could form a non-periodic tiling (SN: 3/1/07). "It was natural to wonder whether there could be a single tile that does this," says mathematician Casey Mann of the University of Washington Bothell, who was not involved in the research. That one has finally been discovered, and "it is massive."
Other shapes came close. Taylor-Socolar tiles are aperiodic, but they are a jumble of multiple disconnected pieces, rather than what most people consider to be a single tile. "This is the first solution without asterisks," says CNRS and École Normale Supérieure de Lyon mathematician Michal Rao.
Smith and colleagues demonstrated the tile's Einstein status in two ways. One idea came from observing how the hats arranged themselves into larger clusters known as meta tiles. These meta tiles then form larger supertiles, and so on indefinitely, in a type of hierarchical structure common in non-periodic tilings. This method revealed that the hat tiling could cover an infinite plane and that its pattern would never repeat.
The second proof was based on the fact that the hat is part of a shape continuum: The mathematicians were able to create a family of tiles that can take on the same non repeating pattern by gradually changing the relative lengths of the sides of the hat.
The team was able to demonstrate that the hat could not be arranged in a periodic pattern by considering the relative sizes and shapes of the tiles at the extremes of that family, one shaped like a chevron and the other like a comet.
While the paper has not yet been peer-reviewed, the experts interviewed for this article agree that the outcome appears likely to withstand close scrutiny.
Nonrepetitive patterns can have real-world applications. Dan Shechtman, a materials scientist, won the Nobel Prize in chemistry in 2011 for his discovery of quasicrystals, materials with atoms arranged in an orderly structure that never repeats, which are often described as analogues to Penrose's tilings (SN: 10/5/11). According to Senechal, the new aperiodic tile could spark further research in materials science.
Artists have been inspired by similar tilings, and the hat appears to be no exception. The tiling has already been artistically rendered as smiling turtles and a jumble of shirts and hats. It is only a matter of time before someone starts putting hat tiles on hats.
And it does not stop there. According to computer scientist Craig Kaplan of the University of Waterloo in Canada, a co-author of the study, researchers should keep looking for more Einsteins. "Hopefully, now that we have unlocked the door, other new shapes will appear."