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ELABORATE SYMMETRICAL tile work on medieval Islamic buildings, such as this 15th-century Turkish mosque, seem to have been constructed from sets of decorated tiles.
COURTESY OF W. B. DENNY

Medieval Islamic artisans seem to have developed a procedure for creating jigsawlike mosaics that ultimately led them to an exotic pattern that mathematicians would discover nearly half a millennium later. Researchers report that 15th-century buildings in Iran feature tiles arranged in a so-called quasicrystal, which is symmetric but does not repeat itself regularly.

"Here is evidence it [the pattern] was being used, if not understood, 500 years ahead of when we had any idea what was going on with [it] in the West," says physics graduate student Peter J. Lu of Harvard University. Lu began poring over photos from Iran, Iraq, Turkey and Afghanistan after seeing hints of the pattern while traveling in Uzbekistan. The Islamic artisans seem to have spun a wide variety of symmetric traceries from a set of five shapes, according to a report Lu co-authored, published online February 22 in Science.

Medieval Islamic mosques, palaces and other buildings were routinely covered in ornate tile work, called girih, that inscribes stars and other shapes [see images above and below]. Scholars believe that workers drew many of these patterns with compass and straightedge. But some of the shapes could only have been accurately constructed using a set of five "girih tiles," Lu and Princeton University physicist Paul Steinhardt, a quasicrystals expert, say in their study.

The proposed tiles—a bow tie, pentagon, diamond, elongated hexagon and a large decagon—were decorated with outlines of other shapes, including stars and bent triangles. Placing the tiles together would link the outlines, forming interlocking figures [such as the blue lines in this image (see note at end of text)]. "It gives you a very easy way to generate a large number of complex patterns by just stacking the tiles [like puzzle pieces] and keeping the decoration," Lu says.

Lu and Steinhardt found that the shapes of the tiles actually appear on a 15th-century Islamic scroll documenting architectural practices. They note that the tiles seem to have come into use as early as the 12th century to make regularly repeating, or periodic, patterns. But by the 15th century, artisans, perhaps driven to increasingly complex artistic feats, seem to have reached a new level of sophistication.

In particular, the Darb-i Imam shrine in Isfahan, Iran, which dates to A.D. 1453, is covered in a symmetric pattern of pentagons and 10-sided stars. If extended indefinitely in all directions, the researchers say, it would never repeat itself—the hallmark of a quasicrystal.

"You want to have a very complicated tiling that's just magnificent to anyone who walks by," Lu says. "My take on it is, they just wanted to make something that looked really cool."

The researchers note that the pattern is equivalent to the most famous example of a quasicrystal, discovered in the 1970s by famed mathematician and physicist Roger Penrose, who showed how to construct it by fitting two types of "Penrose tiles" edge to edge according to certain rules.

Lu says Islamic designers seem to have utilized one of the two methods discovered by Penrose: they assembled the tiles into larger versions of themselves. He says the pattern on the shrine contains a few slight errors that probably occurred during construction or repair, because a flaw in the pattern itself would have led to large, obvious flaws.

Early Middle Eastern mosques display all of the periodic patterns that can be cobbled together on a flat surface, says mathematician John O'Connor of the University of St. Andrews in Scotland. "They understood very well how things had to be fitted together."

O'Connor says that the heyday of Islamic mathematics occurred between the ninth and 13th centuries. "By the 15th century, the Renaissance was going on in Europe," partly stimulated by the introduction of mathematical concepts from the Islamic world, including trigonometry and algebraic symbols, he says. "If they [quasicrystals] had been out there the whole time, it's rather nice that everyone had overlooked them."