In the beauty and geometric complexity of tile mosaics on walls of medieval Islamic buildings, scientists have recognized patterns suggesting that the designers had made a conceptual breakthrough in mathematics beginning as early as the 13th century.

A new study shows that the Islamic pattern-making process, far more intricate than the laying of one’s bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago.

The findings, reported in the current issue of the journal Science, are a reminder of the sophistication of art, architecture and science long ago in the Islamic culture. They also challenge the assumption that the designers somehow created these elaborate patterns with only a ruler and a compass. Instead, experts say, they may have had other tools and concepts.

Two years ago, Peter J. Lu, a doctoral student in physics at Harvard University, was transfixed by the geometric pattern on a wall in Uzbekistan. It reminded him of what mathematicians call quasi-crystalline designs. These were demonstrated in the early 1970s by Roger Penrose, a mathematician and cosmologist at the University of Oxford.

Mr. Lu set about examining pictures of other tile mosaics from Afghanistan, Iran, Iraq and Turkey, working with Paul J. Steinhardt, a Princeton cosmologist who is an authority on quasi crystals and had been Mr. Lu’s undergraduate adviser. The research was a bit like trying to figure out the design principle of a jigsaw puzzle, Mr. Lu said in an interview.

In their journal report, Mr. Lu and Dr. Steinhardt concluded that by the 15th century, Islamic designers and artisans had developed techniques “to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before discovery in the West.”

Some of the most complex patterns, called “girih” in Persian, consist of sets of contiguous polygons fitted together with little distortion and no gaps. Running through each polygon (a decagon, pentagon, diamond, bowtie or hexagon) is a decorative line. Mr. Lu found that the interlocking tiles were arranged in predictable ways to create a pattern that never repeats — that is, quasi crystals.

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