![]() ![]() 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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