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.Read the full story here
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
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.