The Search for Quasi-Periodicity in Islamic 5-fold Ornament

Abstract

T he Penrose tilings are remarkable in that they are non-periodic (have no translational symmetry) but are clearly organised. Their structure, called quasiperiodicity, can be described in several ways, including via self-similar subdivision, tiles with matching rules, and projection of a slice of a cubic lattice in R. The tilings are also unusual for their many centres of local 5-fold and 10fold rotational symmetry, features shared by some Islamic geometric patterns. This resemblance has prompted comparison, and has led some to see precursors of the Penrose tilings and even evidence of quasi-periodicity in traditional Islamic designs. Bonner [2] identified three styles of self-similarity; Makovicky [20] was inspired to develop new variants of the Penrose tiles and later, with colleagues [24], overlaid Penrose-type tilings on traditional Moorish designs; more recently, Lu and Steinhardt [17] observed the use of subdivision in traditional Islamic design systems and overlaid Penrose kites and darts on Iranian designs. The latter article received widespread exposure in the world’s press, although some of the coverage overstated and misrepresented the actual findings. The desire to search for examples of quasi-periodicity in traditional Islamic patterns is understandable, but we must take care not to project modern motivations and abstractions into the past. An intuitive knowledge of group theory is sometimes attributed to any culture that has produced repeating patterns displaying a wide range of symmetry types, even though they had no abstract notion of a group. There are two fallacies to avoid: • abstraction: P knew about X and X is an example of Y therefore P knew Y. • deduction: P knew X and X implies Y therefore P knew Y.