![]() ![]() An infinite checkerboard, for example, looks just the same if you slide the rows over by two. With a periodic pattern, it’s possible to shift the tiles over and have them match up perfectly with their previous arrangement. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats. The einstein sits in a weird purgatory between order and disorder. “It wasn’t even clear that such a thing could exist,” Senechal says.Īlthough the name “einstein” conjures up the iconic physicist, it comes from the German ein Stein, meaning “one stone,” referring to the single tile. Mathematicians had been searching for such a shape for half a century. “Everybody is astonished and is delighted, both,” says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats. ![]() A 13-sided shape known as “the hat” has mathematicians tipping their caps. ![]()
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