He and Dr. Steinhardt recognized that the artisans in the 13th century had begun creating mosaic patterns in this way. The geometric star-and-polygon girihs, as quasi crystals, can be rotated a certain number of degrees, say one-fifth of a circle, to positions from which other tiles are fitted. As such, this makes possible a pattern that is infinitely big and yet the pattern never repeats itself, unlike the tiles on the typical floor.

This was, the scientists wrote, “an important breakthrough in Islamic mathematics and design.”

Dr. Steinhardt said in an interview that it was not clear how well the Islamic designers understood all the elements they were applying to the construction of these patterns. “I can just say what’s on the walls,” he said.

Image A pattern taken from a Turkish mosque. Credit... W. B. Denny

Mr. Lu said that it would be “incredible if it were all coincidence.”

“At the very least,” he said, “it shows us a culture that we often don’t credit enough was far more advanced than we ever thought before.”

From a study of a few hundred examples, Mr. Lu and Dr. Steinhardt determined that the technique was fully developed two centuries later in mosques, palaces, shrines and other buildings. They noted that “a nearly perfect quasi-crystalline Penrose pattern” is found on the Darb-i Imam shrine in Isfahan, Iran, which was built in 1453. The researchers described how the architects there had created overlapping patterns with girih tiles at two sizes to produce nearly perfect quasi-crystalline patterns.

In the report, Mr. Lu and Dr. Steinhardt said the examples they had studied so far “fall just short of being perfect quasi crystals; there may be more interesting examples yet to be discovered.”

In a separate article in Science, some experts in the math of crystals questioned if the findings were an entirely new insight. In particular, Emil Makovicky of the University of Copenhagen in Denmark said the new report failed to give sufficient credit to an analysis he published in 1992 of mosaic patterns on a tomb in Iran.