How did medieval craftsmen lay out patterns of such staggering intricacy on the walls of mosques and shrines, without algebra and without measuring instruments beyond the compass and straightedge? For the earliest girih work, the classical tools suffice. But in 2007 the physicists Peter Lu and Paul Steinhardt argued in the journal Science that by around 1200 CE the designers had made a conceptual leap: they stopped drafting each star anew and began assembling patterns from a small kit of prefabricated shapes, which Lu and Steinhardt named girih tiles. There are five: a regular decagon, an elongated hexagon, a bowtie, a rhombus and a regular pentagon. All have edges of the same length, and every edge is crossed at its midpoint by decorating lines at fixed angles of 72 or 108 degrees. Fit the tiles together edge to edge and the decorating lines join automatically into a continuous strapwork, so a craftsman could build an enormous, flawless pattern the way a paver lays a floor, without ever computing an angle.
The masterpiece of the method is the Darb-i Imam shrine in Isfahan, dated 1453, where large girih tiles are subdivided into smaller copies of the same shapes. Carried on indefinitely, that subdivision produces a pattern that never repeats, a quasi-crystalline order of the kind Western mathematics first described with Penrose tilings in the 1970s. The claim is not that the builders held the modern theory, but that their craft system reached, in practice, a structure the West would need another five centuries to name.