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.