Islamic geometric designs from the Topkapı Scroll I: unusual arrangements of stars

The Topkapı Scroll is an important documentary source for the study of Islamic geometric ornament. Here we give a mathematical analysis of some its exemplary star patterns that illustrate a variety of methods of construction. Highlights include a design combining 9- and 11-point stars, and another with 13- and 16-point stars. We show that the practice of producing a design by replicating a template using reflections in its sides restricts the range of symmetry types produced. In particular, the lack of 3-fold symmetry is related to the exclusive use of rectangular templates. By transferring the distributions of stars traditionally seen in a square template to their equivalent arrangements in an equilateral triangle, we produce new 3-fold designs in the Islamic style.