# Higher Math From Medieval Islam

Ancient, closely held religious secrets; messages encoded on the walls of Middle Eastern shrines; the divine golden ratio—readers of a recent issue of the journal Science must have wondered if they'd mistakenly picked up "The Da Vinci Code" instead. In stretches of intricate tiling on several 500-year-old Islamic buildings, Peter Lu and Paul Steinhardt wrote, they'd spotted a large fragment of a mathematical pattern that was unknown to Western science until the 1970s. Islam gave the world algebra, from the Arabic al-jabr, a term referring to a basic equation. But this pattern is far from basic; it comes from much higher math. "The ridiculous thing is, this pattern has been staring Westerners in the face all this time," says Keith Critchlow, author of the book "Islamic Patterns." "We simply haven't been able to read it." Now that we can, though, it is serving as a startling indication of how accomplished medieval-era Muslims may have been.

No one knows what the architects of the complex pattern in the tiles named it a half millennium ago. Today, scientists call it a "quasiperiodic crystal with forbidden symmetry." It's forbidden not for any religious reason, of course, but because at first glance it appears impossible to construct. Take a pattern of triangular tiles, rotate it one third the way around, and the resulting pattern is identical. The same goes for rectangular tiles (which look the same rotated one fourth the way around) or hexagonal tiles (one sixth the way around). But a grid made purely of pentagons simply can't exist. The five-sided shapes don't fit together without leaving gaps, and there's no way to put them in a pattern that looks the same when turned one fifth the way around.

The breakthrough that took the "forbidden" out of that "forbidden symmetry" was to use two shapes, not one, to build a fivefold-symmetrical grid. In 1973, having given up on pentagons, mathematician Sir Roger Penrose designed a fivefold pattern with shapes he called "kites" and "darts." He was the first Westerner (and at the time, he thought, the first person) to do so, and his creation turned out to have fascinating mathematical properties. Any given fragment of it, containing a finite number of kites and darts, could be infinitely divided into a never-repeating pattern of smaller kites and darts.

As the number of small shapes in the pattern increased, the ratio of kites to darts approached the "golden ratio," a number practically sacred to mathematicians. Discovered by Pythagoras, the golden ratio is irrational, which means it extends to an infinite number of decimal places. (The actual number is 1.618033989 ... and so on.) It is linked to the famous Fibonacci sequence and cited in the writings of astronomer Johannes Kepler and, yes, Leonardo da Vinci. It is also found at the atomic level. In the 1980s, Steinhardt, a physicist at Princeton, armed with Penrose's insight, found that some chemicals had their atoms arranged in a "quasicrystalline" shape like that of the fivefold grid.

Medieval Muslims apparently figured out at least some of this math. On the wall of one shrine in Iran, Lu found, two types of large tiles are divided into smaller tiles of the same shapes, in numbers that approximate the golden ratio. The builders certainly knew about the ratio, having inherited all the Greek science and curated it, says Critchlow. "The human creation was imitating, in abstract fashion, the wondrous creation of God," says Gulru Necipoglu, a professor of Islamic art at Harvard. Some geometric patterns, for instance, evoked the planets and stars. And throughout the medieval era and onwards, says Steinhardt, Muslims "were fascinated by fivefold symmetry and were always trying to incorporate it into their designs. Where the patterns ended up with gaps, they would cleverly place a door or a windowsill there so you couldn't tell." In the buildings examined by Lu, they succeeded.

Although the Penrose-patterned tiles date to the 14th and 15th centuries, the same shapes of tiles "were used all over the medieval Islamic world to generate all sorts of patterns" for hundreds of years before and after that, says Lu. The Topkapi scroll, a Persian artifact from the late 15th or early 16th century, lists many such designs. There may also be clues to ancient Muslims' mathematical prowess in other tiling on mosques in Iran and Turkey, madrassas in Baghdad and shrines in Afghanistan and India. They would fit nicely into the increasingly common image of the medieval Islamic world as an advanced society. Scholars now know that Muslims of that era could solve equations with variables to the power of 3 and above, which are harder than the classic quadratic "x2" ones fundamental to algebra. They also had mechanical "computers" and knew considerably more about medicine and astronomy than Europeans of the time.

What has not yet been found, unfortunately, is any record of how early Muslims designed the fivefold patterns and conceptualized the math lurking in them, since few Muslim scholars wrote down their discoveries for wide dissemination. "You absolutely do not have to understand the higher math to be able to do it," says David Salesin, a computer scientist at the University of Washington. Lu agrees that there's no need to project a modern understanding of quasicrystals onto an ancient culture—but he also says the pattern design was no accident. "No matter how it was constructed," he adds, "it's a stunning achievement." Particularly now that the world has eyes to see it.