Mathematicians are tipping their caps at the 13-sided shape known as “the hat.”
It represents the first real instance of a “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.
Mathematician Marjorie Senechal of Smith College in Northampton, Massachusetts, who was not involved with the discovery, claims that “everyone is startled and is thrilled, both.” For fifty years, mathematicians had been looking for such a shape. Senechal claims that “it wasn’t even evident that such a creature could exist.”
Although the word “Einstein” brings up the famous physicist, it actually derives from the German ein Stein, which refers to the single tile and means “one stone.” The Einstein is caught in a strange limbo between order and chaos. The tiles can cover an endless plane and fit together perfectly, but they are aperiodic, which means they cannot create a repeating pattern.
The tiles can be moved and still completely match their prior arrangement if the pattern is periodic. If you push the rows over by two, for instance, an infinite checkerboard still seems the same. Other single tiles can be arranged in non-periodic designs, but the hat is unique since it cannot be used to make a periodic pattern.
The hat was discovered by David Smith, a self-described “imaginative tinkerer of shapes” and amateur mathematician, and described in a paper published online on March 20 at arXiv.org. The hat is a polykite, which is a collection of smaller kite shapes adhered together. Chaim Goodman-Strauss of the National Museum of Mathematics in New York City, one of a group of qualified mathematicians and computer scientists Smith teamed up with to analyse the hat, claims that that is a form of shape that hadn’t been studied carefully in the quest for einsteins.
It’s a really straightforward polygon. According to Goodman-Strauss, if you had asked him to sketch an einstein before this work, he “would’ve drawn some strange, squiggly, horrible thing.”
There were previously known nonrepeating tilings in mathematics that had many tiles of various forms. Mathematician Roger Penrose established in the 1970s that a tiling with only two distinct shapes could not be periodic (SN: 3/1/07). Therefore, according to mathematician Casey Mann of the University of Washington Bothell, who was not involved with the research, “it was natural to question, might there be a single tile that does this.” It’s finally been located, and “it’s massive.”
There have been similar shapes. Although Taylor-Socolar tiles are aperiodic, they are not what most people consider to be a single tile; rather, they are a collection of several scattered fragments. According to mathematician Michal Rao of the CNRS and École Normale Supérieure de Lyon in France, “This is the first solution without asterisks.”
The tile’s Einstein status was established by Smith and colleagues in two ways. One comes from seeing how the hats self-organize into larger groups known as metatiles. These metatiles subsequently form a type of hierarchical structure that is typical for nonperiodic tilings, arranging into even bigger supertiles, and so on endlessly. The hat tiling could fill an endless plane using this method, and it was discovered that its pattern would not repeat.
The second argument centred on the hat’s place in a continuum of shapes: The mathematicians created a family of tiles that can adopt the same non-repeating design by progressively varying the relative lengths of the hat’s sides. The team was able to demonstrate that the hat couldn’t be organised in a periodic pattern by taking into account the relative sizes and shapes of the tiles at the extremities of that family — one fashioned like a chevron and the other resembling a comet.
The experts consulted for this article concur that the outcome appears likely to hold up to careful examination even though the work has not yet undergone peer review.
Non-repeating patterns can be connected to the outside environment. For his discovery of quasicrystals, which are materials with atoms arranged in an orderly arrangement that never repeats and are sometimes compared to Penrose’s tilings, materials scientist Dan Shechtman was awarded the 2011 Nobel Prize in Chemistry (SN: 10/5/11). Senechal believes that further studies in materials science may be inspired by the new aperiodic tile.
Artists have been influenced by similar tile patterns, and the hat appears to be no different. Already, the tiling has been creatively depicted as a mess of shirts and hats and grinning turtles.
The hat isn’t the end, either. Craig Kaplan, a computer scientist and study co-author from the University of Waterloo in Canada, advises researchers to keep looking for new einsteins. “Perhaps, additional new shapes will appear now that we’ve unlocked the door.”