Times Online UK: Amazing maths of the mosaic makers

Author: Mark Henderson, Science Editor
Date: February 23, 2007
Times Online UK

Medieval Islamic artists produced intricate decorative patterns using geometrical techniques that were not understood by Western mathematics until the 20th century, scientists have discovered.

The combinations of ornate stars and polygons that have adorned mosques and palaces since the 15th century were created using a set of just five template tiles, which could generate patterns with a kind of symmetry that eluded formal mathematical description for another 500 years.

The discovery, by Peter Lu, of Harvard University, published in the journal Science, suggests that the Islamic artisans who created these typical girih designs had an intuitive understanding of highly complex mathematical concepts, even if they had not worked out the underlying theory. “We can’t say for sure what it means,” said Mr Lu, who is studying for a PhD in physics. “It could be proof of a major role of mathematics in medieval Islamic art or it could have been just a way for artisans to construct their art more easily.

“It would be incredible if it were all coincidence. At the very least, it shows us a culture that we often don’t credit enough was far more advanced than we thought.”

Girih designs feature arrays of tessellating polygons of multiple shapes, and are often overlaid with a zigzag network of lines. It had been assumed that straightedge rulers and compasses were used to create them — an exceptionally difficult process as each shape must be precisely drawn.

From the 15th century, however, some of these designs are symmetrical in a way known today as “quasicrystalline”. Such forms have either fivefold or tenfold rotational symmetry — meaning they can be rotated to either five or ten positions that look the same — and their patterns can be infinitely extended without repetition. The principles behind quasicrystalline symmetry were calculated by the mathemetician Roger Penrose in the 1970s, but it is now clear that Islamic artists were creating them more than 500 years earlier.

Mr Lu, who designs physics experiments for the International Space Station, began wondering whether there were quasicrystalline forms in Islamic art after seeing decagonal artworks in Uzbekistan, which he visited after a trip to a space facility in Turkmenistan.

On returning to Harvard, he started searching the university’s vast library of Islamic art for quasicrystalline designs. He found several, as well as architectural scrolls that contained the outlines of five polygon templates — a ten-sided decagon, a hexagon, a pentagon, a rhombus and a bow-tie shape — that can be combined and overlaid to create such patterns.

There is no evidence that the template tiles were themselves attached to surfaces to create mosaics. Artists probably used holes in the templates to trace a design on to a surface, which would be made into a mosaic.

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