With a New, Improved ‘Einstein,’ Puzzlers Settle a Math Problem

“We call this a ‘weakly chiral aperiodic monotile,’” Dr. Kaplan explained on social media. “It’s aperiodic in a reflection-free universe, but tiles periodically if you’re allowed to use reflections.”

The adjective “chiral” means “handedness,” from the Greek “kheir,” for “hand.” They called the new aperiodic tiling “chiral” because it is composed exclusively of either left- or right-handed tiles. “You can’t mix the two,” said Chaim Goodman-Strauss, a co-author and outreach mathematician at the National Museum of Mathematics in New York.

The team then went one better: They produced a family of strong or “strictly chiral aperiodic monotiles” through a simple modification of the T(1,1) tile: They replaced the straight edges with curves.

Named “Spectres,” these monotiles, owing to their curvy contours, only allow nonperiodic tilings, and without reflections. “A left-handed Spectre cannot interlock with its right-handed mirror image,” said Dr. Kaplan.

“Now there is no quibbling about whether the aperiodic tile set has one or two tiles,” Dr. Berger said in an email. “It’s satisfying to see a glazed ceramic einstein.”

Doris Schattschneider, a mathematician at Moravian University, said, “This is more what I would have expected of an aperiodic monotile.” On a tiling listserv, she had just seen a playful “Escherization” (after the Dutch artist M.C. Escher) of the Spectre tile by Dr. Araki, who called it a “twinhead pig.”

“It’s not simple like the hat,” Dr. Schattschneider said. “This is a really strange tile. It looks like a mistake of nature.”

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