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More than a thousand years ago, Islamic artisans began to adorn architectural surfaces with geometric patterns. As the centuries passed, this practice developed into a rich system of intricate ornamentation that followed the spread of Islamic culture into Africa, Europe, and Asia. The ornaments often took the form of a division of the plane into star-shaped regions, which we will simply call ``Islamic star patterns''; a typical example appears on the right. To this day, architectural landmarks in places like Granada, Spain and Isfahan, Iran demonstrate the artistic mastery achieved by these ancient artisans.

Lurking in these geometric wonders is a long-standing historical puzzle. The original designers of these figures kept their techniques a closely guarded secret. Other than the finished works themselves, little information survives about the thought process behind their star patterns.

Many attempts have been made to reinvent the design process for star patterns, resulting in a variety of successful analyses and constructions. Grünbaum and Shephard [9] decompose periodic Islamic patterns by their symmetry groups, obtaining a fundamental region they use to derive properties of the original pattern. Abas and Salman apply this decomposition process to a large collection of patterns [2]. Elsewhere, they argue for a simple approach tied to the tools available to designers of the time [1]. Dewdney proposes a method of reflecting lines off of periodically-placed circles [5]. Castera presents a technique based on the construction of networks of eightfold stars and ``safts'' [7].

This paper presents a technique described by Hankin [10], based on his experiences seeing partially-finished installations of Islamic art. It also incorporates the work of Lee [11], who provides simple constructions for the common features of Islamic patterns. Given a tiling of the plane containing regular polygons and irregular regions, we fill the polygons with Lee's stars and rosettes, and infer geometry for the remaining regions. We have implemented this technique as the Java applet Taprats, available on the web for experimentation. Taprats was used to produce the examples in this paper.

The rest of this report is organized as follows. Section 2 presents constructions for the common features of Islamic patterns: stars and rosettes. Section 3 shows how complete designs may be built using repeated copies of those features. Techniques for creating visually appealing renderings of the designs are given in Section 4. Some results appear in Section 5. The paper concludes in Section 6 by exploring some opportunities for future work.

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Craig Kaplan 2000-08-16