Islam Got It First; The tiling in medieval Islamic architecture turns out to embody a mathematical insight that Westerners thought they had discovered only 30 years ago.

Byline: Mary Carmichael

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 ...