Islamic geometric designs from the Topkapı Scroll II: a modular design system

The Topkapı Scroll is an important documentary source for the study of Islamic geometric ornament. Here we give a mathematical analysis of some of its exemplary designs, focussing on a modular construction in which designs are produced by assembling copies of six tiles. We show applications of this versatile system, including the introduction of non-standard design elements, a non-periodic pattern based on the Penrose rhombi, and several kinds of two-level design that feature complementary patterns on two scales simultaneously.