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Medieval Islamic artists made elaborate mosaics of tiles arranged in well-ordered patterns that never repeated themselves. But could such a pattern occur at the molecular level?

Thirty-two years ago, the Israeli scientist Dan Shechtman found that it could. During a sabbatical in the United States, at the National Bureau of Standards, he melted together aluminum and manganese and cooled the mixture rapidly. He fired a beam of electrons at the sample and found that the pattern bouncing off it could not be explained by a lattice of atoms stacked in the usual manner.

Paul J. Steinhardt, then a physics professor at the University of Pennsylvania, and Dov Levine, a graduate student, came up with a possible explanation and a term to describe the new structures: quasicrystals.

Dr. Shechtman’s quasicrystals were imperfect, however, and the data could not rule out other explanations, like “twinning,” in which two periodic crystals are fused at an angle.

Some leading scientists scoffed at the whole idea. “There is no such thing as quasicrystals,” said Linus C. Pauling, the winner of two Nobel Prizes, “only quasi scientists.”

Then a Japanese scientist, An-Pang Tsai, at Tohoku University, discovered a new alloy made of iron, copper and aluminum — “the first example of a true, bona fide quasicrystal,” said Dr. Steinhardt, now at Princeton.

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