What symmetry groups are present in the Alhambra?

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Grünbaum, Branko: What symmetry groups are present in the Alhambra? Notices AMS 2006.



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 na- ture of Islamic design, incorporating complex sym- metries, has been well-explored from a mathe- matical 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 ap- peared; 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 ornamenta- tion 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 wall- paper groups among the ornaments of the Al- hambra.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 ...

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Used References

[1] R. FENN, Review of [6], Math. Reviews, MR 0915761 (89d:57016).

[2] B. GRÜNBAUM, The Emperor’s clothes: Full regalia, G string, or nothing? Math. Intelligencer 6, No 4, (1984), 47–53.

[3] ——— , Periodic ornamentation of the fabric plane: Lessons from Peruvian fabrics, Symmetry 5 1 (1990), 45–68. Reprinted6 in Symmetry Comes of Age, The Role of Pattern in Culture, (eds. D. K. WASHBURN and D. W. CROWE), Univ. of Washington Press, Seattle, 2004, pp. 18–64.

[4] B. GRÜNBAUM, Z. GRÜNBAUM, and G. C. SHEPHARD, Symmetry in Moorish and other ornaments, Computers and Math- ematics with Applications 12B (1986), 641–653 = Symmetry, Unifying Human Understanding, (ed. I. Hargit- tai), Pergamon, New York, 1986, pp. 641–653.

[5] J. JAWORSKI, A Mathematician’s Guide to the Alhambra, 2006 (unpublished).

[6] J. M. MONTESINOS, Classical Tessellations and Three- Manifolds, Springer, New York, 1987.

[7] E. MÜLLER, Gruppentheoretische und Strukturanalytis- che Untersuchungen der Maurischen Ornamente aus der Alhambra in Granada, Ph.D. thesis, Universität Zürich, Baublatt, Rüschlikon, 1944.

[8] ——— , El estudio de ornamentos como aplicación de la teoría de los grupos de orden finito, Euclides (Madrid) 6 (1946), 42–52.


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