What Symmetry Groups Are Present in the Alhambra?

2 NOTICES OF THE AMS VOLUME 53, NUMBER 6 On the occasion of the approaching International Congress of Mathematicians (ICM) (Madrid 2006) it is appropriate to renew the enjoyment of the arts—modern as well as historical—that grace many locations in Spain. The cover of the February 2006 issue, and the article by Allyn Jackson (starting on p. 218) are helpful, as is the note of Bill Casselman (on p. 213). Two sentences in the latter made me curious. Casselman states that “The geometric nature of Islamic design, incorporating complex symmetries, has been well-explored from a mathematical point of view. A fairly sophisticated discussion, referring specifically to the Alhambra, can be found in the book Classical Tessellations and Three-manifolds by José Maria Montesinos.” I had visited the Alhambra more than twenty years ago and had seen Montesinos’ book soon after it appeared; that’s a long time ago, and I had forgotten the details. I was about to get the book from our library, but before that I checked the Math Reviews. There I found an assertion that ran counter to my memories; so I eagerly started looking at the book itself and recovering old papers and notes on the topic. The question which of the seventeen wallpaper groups1 are represented in the fabled ornamentation of the Alhambra has been raised and discussed quite often, with widely diverging answers. The first to investigate it was Edith Müller in her 1944 Ph.D. thesis at the Universität Zürich, written under the guidance of Andreas Speiser.2 In her thesis [7] Müller documents the appearance of twelve wallpaper groups among the ornaments of the Alhambra.3 (She also investigates other kinds of groups, but this is not relevant for our discussion at this time.) Due to a misunderstanding of Müller’s comment that minor changes would have yielded two additional groups, several writers claimed that she found examples of fourteen groups. Some later