Islamic tiles and modern mathematics

Islamic art has always had a relation with complex mathematical patterns. Now two researchers have found that some of these patterns from medieval times reflect mathematical relations discovered by mathematician Roger Penrose in the 1970's. One the interesting features of the pattern is that it appears regular, but never repeats itself.

Islamic tiling patterns were put together not with a compass and ruler, as previously assumed, but by tessellating a small number of different tiles with complex shapes, say Peter J. Lu of Harvard University in Cambridge, Massachusetts and Paul Steinhardt of Princeton University in New Jersey.

The researchers think that this technique was developed around the start of the thirteenth century. By the fifteenth century, it was sophisticated enough to make complex patterns now described as quasi-periodic.

These patterns were 'discovered' in 1973 by the British mathematical physicist Roger Penrose. In 1984, they were found in metal alloys called quasi-crystals that seemed to break the geometric rules of atomic packing

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And here is the abstract of the paper published in Science (Feb 23, 2007; vol 315, no. 5815, p 1106):

Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
Peter J. Lu & Paul J. Steinhradt

The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons ("girih tiles") decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns,five centuries before their discovery in the West.

The tile work on Darb-i Imam shrine in Isfahan, Iran (1453 C.E.) has patterns that reflect quasi-crystalline structure worked out by mathematician Roger Penrose in 1973

It appears that the artisans didn't know the mathematical theory behind their creation, but they indeed had an intuitive sense regarding the design. It reminds me of the mathematical analysis of Jackson Pollock's work, which shows that he was working with fractal patterns before the actual development of the fractal theory. Both the tile works and Pollack's paintings are amazing even without any mathematical theories...but this just adds some extra oomph!