This is the second volume in the series “Mathematics and the Built Environment,” edited by Kim Williams. Like the first book in the series, The

*Fractal Dimension of Architecture*, this book provides a mathematical approach to a topic in architecture and design. In both cases, the authors make an interesting contribution to the intersection of mathematics and the arts. In each case, the books make it clear where the authors’ programs have succeeded and where mathematics cannot tell the whole story. Consequently, both volumes will be of value to those who wish to carry on similar investigations.

The present volume seeks to describe Islamic geometric designs mathematically. It begins with six chapters by David Wade, which provide the cultural context for the designs that will be considered mathematically by Brian Wichmann in the second part of the book. Wade begins with a brief but useful history of the Islamic culture and its scientific contributions from the time of the revelations to Muhammad through the post-World War II period. I plan to require that my history of mathematics students read these first two chapters.

The third chapter considers the religious influences on the designs, exploring why there are no (or few) images of sentient beings by considering the Qur’ānic roots of design, based on perfection, light, and a sense of otherworldly esotericism. These roots contribute to a preoccupation with the dissolution of matter, hints of infinity, and a sense of space that artisans wished to create in their designs (pages 36–37.)

Chapters 4 and 5 give a brief sketch of the evolution of Islamic design as it grew out of the culture and its consistency across a variety of media, including wood, stonework, plaster, ceramics, textiles, brickwork, stucco, and paper. In particular, the introduction of design on paper made it easier to transmit designs across the Islamic world, from Spain in the west to northwest India in the east. There are three constituents of Islamic design: calligraphy, geometric designs, and arabesques. Not surprisingly, the focus of this book is the geometric designs. In chapter 6, Wade discusses Islamic design in various geographic regions; each region has its own history, impacting the variety of designs.

The second part of the book is written by Brian Wichmann. His goal is to provide mathematical descriptions, analyses, and organization of the Islamic geometric designs, with success measured in part by how accurately they can be reproduced from the mathematics by computer graphics. Some patterns can be successfully classified and completely analyzed this way, while others require the intervention of an artistic eye to choose certain parameters or to approximate what cannot be made geometrically exact.

Wichmann begins in Chapter 7 with a geometric introduction, introducing the rosettes with their central stars and accompanying (4-sided) kites and (6-sided) petals. Often, but not always, the four outer edges of the petals have the same length, and the two outermost petals are part of a regular polygon. Chapter 8 provides a worked out example of a pattern with both 16- and 8-pointed stars from the Sala del Mexuar in the Alhambra of Granada, Spain. The pattern was copied both by Owen Jones in 1837 and by Maurits Escher in 1936. Wichmann shows how the various segment proportions and angle measurements can be calculated, then displays a copy of the pattern using computer graphics.

In Chapter 9, Wichmann considers a set of over 140 patterns based on the khatem, an 8-pointed star with vertex angles that are right angles. These patterns share four properties that allow a complete mathematical analysis:

- The tiles are edge-to-edge, so that adjacent tiles share entire edges;
- The pattern can be colored by just two colors so that no tiles sharing an edge are the same color;
- At each vertex where four tiles meet, the edges form parts of just two lines;
- For each tile, the internal angles are multiples of 45 degrees (pages 76–77.)

Most, but not all, of the patterns in this set are from Morocco or Spain.

Chapter 10 adds the complications introduced by the inclusion of 16-, 24-, and 32-pointed stars. As one would expect, these make the mathematical analyses more complex. Here there are noticeable differences between the computer-generated patterns and the original designs, with some hand-adjustments being required that were introduced by the Islamic artisans. Wichman suggests that some of the physical patterns were created by the artisan laying out the tiles in a sandbox, making some adjustments to find the most aesthetically pleasing modifications, and finally fixing the tiles in place. (Such a technique is used today in both Morocco and Iran.)

The topic of Chapter 11 is a consideration of the edges in which the tiles meet. The artisans treated the edges as either: interlacing, with edges passing over then under at consecutive intersections; or borders, in which the edges have width rather than being just line segments; or banding, which are typically wide borders that include a polygon at the corners. In some cases, the width of the borders can be adjusted freely, while in other designs, the width of the edges are fixed or can fall only within a range of values. In interlacings, it can be interesting to trace the paths that are defined by the edges, some of which form loops, while others would be infinite if the pattern itself were continued. An additional complication mentioned in this chapter is that of a chelate, a tile that is not topologically a disc, but rather contains another polygon within it; in every example known, the included polygon is a kite.

In Chapter 12, Wichmann considers decagonal patterns, as contrasted with the octagonal patterns of Chapter 9. These patterns, primarily from Iran, Turkey, or Egypt, are characterized by the inclusion of a 10-pointed star. The analysis cannot be as complete in this case, as there is more variety possible in these sets of tiles. An unusual example from Fez contains 5-, 10-, and 20-pointed stars.

Chapter 13 discusses several patterns that cannot be fully determined mathematically. These include patterns copied by Jones from the Alhambra, two copied by Jules Bourgoin (one of which is likely from Cairo), and the front page of a Mamluk Qur’ān from Cairo. “In those cases in which a pattern is not fully determined, a complex analysis needs to be undertaken to give an acceptable result. Both mathematical and artistic aspects need to be considered.” (page 136.)

The analyses of patterns that have a symmetry rotation of order 6 is the subject of Chapter 14. There is quite a variety considered, many offering a choice of width proportions or angles when producing computer graphics from the physical designs.

The patterns considered in Chapter 15 are two-level, offering patterns within patterns. These are even more difficult to fully analyze mathematically. Attempts to conduct the analysis by laying the upper pattern over the lower pattern are not entirely satisfactory. Conversely, building the “tiles” of the upper pattern from the lower-level tiles typically cannot be accomplished solely by geometry. Instead, creative modifications must be made.

Examples of analyses of two Mamluk patterns occupy Chapter 16. The first pattern, which includes nearly-but-not-quite-regular heptagons, is analyzed first using trigonometry, which had proved very effective in the octagonal patterns, and then with ruler-and-compass techniques. Neither is completely satisfactory, but the computer graphic reconstruction via trigonometric analysis seems preferable. For the second pattern, the trigonometric analysis and making reasonable choices for some angles and lengths give a good fit to the high-resolution photograph of the design.

In Chapter 17, Wichmann concludes by considering what kinds of design methods might have been used by Islamic craftsmen across the centuries. All of the mathematics used by Wichmann in this book was known to Islamic scholars by the ninth century, but it would seem that visual inspection and an artistic eye were used more often than overly complex mathematical computations. After a review and summary of what was accomplished in the earlier chapters, the author provides a time-line showing how many new patterns appeared at various times and what kinds of patterns predominated in each era. Then he provides information on pattern types by geographic region.

Appendix A includes information on planar, frieze, and circular groups in terms of both the older notation used in such works as Grünbaum and Shephard’s

*Tilings and Patterns* as well as that of the more recent 2006 notation of the

*International Tables for Crystallography*. A final section provides an interesting table of the relative frequency of the appearance of each symmetry type among Islamic geometric patterns. Appendix B includes a handy time-line of dates in Islamic history. Appendix C offers a glossary of words from Islamic history or culture, though some geometric terms are also included.

I enjoyed this book. I am delighted that a book like this, which offers substantial contributions to the literature but does not completely exhaust the topic, has been published. It provides inspiration for others, including undergraduate mathematics students, who are seeking interdisciplinary topics for independent research.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.