When math and art overlap it seems that the role of and relative complexity of the math involved becomes exaggerated. It's as though the art can't be good enough without a complex theory. Witness the never ending false attribution of powers of the golden ratio to art.
I'm aware of a couple of controversies surrounding Islamic patterns and math. There is the general problem of how big a role mathematicians played in the development of patterns. Also, there's the small controversy over whether or not an Islamic pattern predicted the Penrose quasi-crystal pattern. (See Peter Lu and Paul Steinhardt's answer to Emil Makovicky here. )
Previously, I referred to the fact that my Girih Extended drawings are developed largely through trial and error. In A.J. Lee's 1987 paper, "Islamic Star Patterns", he addresses the part played by mathematicians in the development of Islamic patterns, and shows how many of the patterns could be developed simply. So simply, in fact, that little more than trial and error was needed. Lee states:
"It may be an advantage for a modern author to develop a systematic analysis of Islamic patterns in purely mathematical terms, but a knowledge of pure mathematics or geometry is unnecessary for those who wish merely to draw Islamic patterns or invent new ones. A theoretical background will often allow the artist to see a number of combinatorial possibilities more quickly than the use of trial-and-error methods, but it forms no substitute for true creativity." Trial and error can take you a long way. My Girih Extended patterns are not truly Islamic. They only resemble some of the historical patterns. However, they are new art, not a mathematical analysis, cataloging, or repetition of Islamic patterns.
I should add that there’s no simple or recursive definition at work in the Girih Extended designs. After settling on a general goal, I reach a graphic solution through trial and error. Though digital tools were used and custom programs were developed, the designs are not a programmatic approach to drawing. It is still drawing with polygons.
Reza Sarhangi's paper presented to the Bridges 2004 conference provides the best insight into the possible role of the mathematician in Islamic design. He cites a a treatise written by Buzjani in the 10th century. Sarhangi writes:
"Buzjani wrote in On Those Parts of Geometry Needed by Craftsmen that he participated in meetings between artisans and mathematicians. 'At some sessions, mathematicians gave instructions on certain principles and practices of geometry. At others, they worked on geometric constructions of two- or three- dimensional ornamental patterns or gave advice on the application of geometry to architectural construction.'" So Islamic mathematicians instructed artisans on basic geometry. Did a 15th century Islamic mathematician understand principles behind quasicrystals? (See "The Tiles of Infinity" by Sebastian Prange. ) Or, is it possible that the singular example, the Darb-i Imam shrine in Isfahan, Iran, was created by chance, that is by trial and error?
1. Lu, Peter J., et al. Response to Comment on "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture". Science 318, 1383 (2007). [The full comment is available at www.sciencemag.org. here.]
2. Lee, A.J. 1987. "Islamic Star Patterns. In Muqarnas IV: An Annual on Islamic Art and Architecture". Oleg Grabar (ed.) Leiden: E.J. Brill. [The full document is available from ArchNet.]
3. Sarhangi, Reza. "Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations". Presented at the Bridges 2004 conference. [The full document is available here.]
4. Prange, Sebastian. (September/October 2009). "The Tiles of Infinity". Saudi Aramco World: 24–31. [The full document is available here.]