Quasi-Periodicity in Islamic Geometric Design

For nearly 150 years, scholars have analyzed the symmetries of Islamic ornamental designs, constituting the most highly developed chapter in cultural symmetry studies. Yet these studies hardly exhaust the mathematically significant properties of Islamic designs. Over the past 30 years, mathematicians have given increasing attention to “quasi-periodic” geometric structures (such as the Penrose tilings) which exhibit infinite repetition of their bounded subparts and crystallographically forbidden symmetries, occupying an important niche between random and highly ordered, periodic structures. This chapter provides a critical review of the literature and argues that the dualization of quasi-periodic tiling fragments from two designs from twelfth- and fifteenth-century Iran helps to inform their aesthetic complexity.