This paper will explore the Topkapı Scroll sketch known as Catalog Number 30, the only repeat unit, that, when replicated using symmetry operations, creates an overall pattern consisting of “nearly regular” thirteen-pointed stars, regular sixteen-pointed stars as well as irregularly-shaped pentagonal stars. Since there are no known written instructions explaining how this repeat unit was generated, we set out to see if we could recreate this highly unusual design using only the simplest of Euclidean construction techniques that may have been known to, and used by, master builders during medieval times.

Read More: The Topkapi Scroll’s Thirteen-Pointed Star Polygon Design

The Islamic world has a rich artistic tradition of creating highly geometric and symmetric ornamentation. Over the centuries, the process of creating Islamic tilings was refined from the 15th century ornamentation in the Alhambra Palace in Granada, Spain to the exquisite tilings, which are seen in mosques, mausoleums and minarets throughout the world today. The contemporary mathematics of group theory and knot theory combined with computer programs provide tools for creating modern day variations of these historical tilings. The title of this paper is motivated by the 10th century treatise On Those Parts of Geometry Needed by Craftsmen written by the Khorasan mathematician and astronomer Abu’l-Wafā who described several constructions made with the aid of straightedge and ‘rusty compass’, a compass with a fixed angle. He was one of a long line of Islamic mathematicians who developed geometric techniques that proved useful to artisans in creating the highly symmetrical ornamentation found in architecture around the world today. In this paper, Raymond Tennant looks at the geometry of Abu’l-Wafā with an eye toward determining geometric methods for reproducing Islamic tilings with students in the classroom.

Read More: Islamic Constructions, by Raymond Tennant