![]() ![]() “The idea of identifying two einsteins back-to-back seemed too good to be true,” the researchers wrote.īy mid-January, Smith and Kaplan had enlisted two more researchers: Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics and the University of Arkansas, and Joseph Samuel Myers, a software engineer in Cambridge, England, with a doctorate in combinatorics. Meanwhile, to Kaplan’s shock, Smith made another discovery: a second tile, shaped like a turtle, that also appeared to be aperiodic. It filled 16 rings with hats before Kaplan told it to stop, figuring they had enough data to work with. This time, the program kept going and going. Apart from tiles that create repeating patterns, which have infinitely many rings, no one had ever found a tile that could keep going for more than six rings. So after Smith told Kaplan about the hat tile, Kaplan turned to a program he’d written that simply places copies of a tile around an initial seed tile in ever-growing rings. It’s impossible to create an algorithm that can determine, for every possible collection of tiles, whether they tile the plane (let alone whether they are aperiodic). “The real art is finding a shape that will allow you to tile the whole plane, but won’t let you do it in a periodic way,” Socolar said. You can, for instance, use a couple of vertical dominoes while otherwise filling the plane with horizontal dominoes. It’s easy to make tilings that aren’t periodic from tiles that also form periodic tilings. Perhaps hobbyists, unlike mathematicians, are “not burdened with knowing how hard this is,” Senechal said. And then there was Joan Taylor’s discovery of the Socolar-Taylor tile. Marjorie Rice, a California housewife, found a new family of pentagonal tilings in 1975. Robert Ammann, who worked as a mail sorter, discovered one set of Penrose’s tiles independently in the 1970s. “I’m not used to this kind of thing.”īut this is far from the first time a hobbyist has made a serious breakthrough in tiling geometry. The excitement the tiles have generated has felt “a bit surreal,” said Smith, who lives in the coastal town of Bridlington in northern England. In the days since the announcement, mathematicians and tiling hobbyists have rushed to get their hands on the new tiles, making paper cutouts, 3D-printing them, and making hat quilts and cookies. The hat tile, Senechal said, shows that periodic and aperiodic tiles are more closely linked than mathematicians had realized. With just a little work, anyone with a magic marker and a hexagonally tiled bathroom floor can trace out a hat tiling. This divides every hexagon into six “kites.” Each hat is made of eight adjacent kites, combined from neighboring hexagons. ![]() To get a hat tiling from a hexagonal tiling, first connect the midpoints of the opposite sides of the hexagons. What its tilings do have is a deep relationship with a particular periodic tiling: the honeycomb lattice of hexagons. The hat, by contrast, has no symmetry and is “almost mundane in its simplicity,” the authors wrote. What’s more, they realized, the hat is one of infinitely many different tiles of this type. The hat tile embodies “enough complexity to forcibly disrupt periodic order at all scales,” the researchers wrote in their paper. Mathematicians call such a tile, or set of tiles, “aperiodic,” in contrast to shapes like squares or hexagons that can cover the plane in a repeating (or periodic) fashion. On March 20, Smith and Kaplan, together with two more researchers, announced that the hat tile was something mathematicians have been seeking for more than five decades: a single tile whose copies can fill the entire plane, but only in patterns that don’t consist of a repeating block of tiles. “It’s a tricky little tile.” He sent a description of his tile to Craig Kaplan, an acquaintance and computer scientist at the University of Waterloo in Canada, who immediately started investigating its properties. “I noticed that it was producing a tessellation that I had not seen before,” he said. Smith cut out 30 copies of the hat on cardstock and assembled them on a table. Usually when he created tiles, they would either settle into some repeating pattern or fail to tile much of the screen. Now he was experimenting to see how much of the screen he could fill with copies of that tile, without overlaps or gaps. Using a software package called the PolyForm Puzzle Solver, he had constructed a humble-looking hat-shaped tile. In mid-November of last year, David Smith, a retired print technician and an aficionado of jigsaw puzzles, fractals and road maps, was doing one of his favorite things: playing with shapes.
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