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

摘要

The Penrose tilings are remarkable in that they are non-periodic (have no translational symmetry) but are clearly organised. Their structure, called quasi-periodicity, can be described in several ways,includingvia self-similar subdivision, tiles with matching rules, andprojection of a slice of a cubic lattice in R5. The tilings arealso unusual for their many centres of local 5-fold and 10-fold rotational symmetry, features shared by some Islamicgeometric patterns. This resemblance has promptedcomparison, and has led some to see precursors of thePenrose tilings and even evidence of quasi-periodicity intraditional Islamic designs. Bonner [2] identified threestyles of self-similarity;Makovicky [20] was inspired todevelop new variants of the Penrose tiles and later, withcolleagues [24], overlaid Penrose-type tilings on traditionalMoorish designs; more recently, Lu and Steinhardt [17]observed the use of subdivision in traditional Islamicdesign systems and overlaid Penrose kites and darts onIranian designs. The latter article received widespreadexposure in the world's press, although some of thecoverage overstated and misrepresented the actualfindings.