Usually when he created tiles, they would either settle into some repeating pattern or fail to tile much of the screen. But the hat tile seemed to do neither. Smith cut out 30 copies of the hat on cardstock and assembled them on a table. Then he cut out 30 more and kept going. "I noticed that it was producing a tessellation that I had not seen before," he said. "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. 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. Mathematicians call such a tile, or set of tiles, "aperiodic," in contrast to shapes like squares or hexagon