The Planar Crystallographic Groups Represented at the Alhambra

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B. Lynn Bodner: The Planar Crystallographic Groups Represented at the Alhambra. In: Bridges 2013. Pages 225–232



The geometric patterns that cover the walls, ceilings and floors of the Alhambra in Granada, Spain have been the subject of study by many mathematical researchers ever since 1944, when E. Müller first reported the number of distinct planar crystallographic groups found there. In subsequent years, the count has varied, sometimes due to inconsistencies in the ways in which the patterns were chosen and analyzed. As such, contradictions and controversy still surround the issue. After a brief introduction, this paper will provide the criteria used by this author in an on-site visit in 2003 in order to catalog different crystallographic groups represented by patterns in the Alhambra. Each group identified is illustrated by a representative pattern.

Extended Abstract


Used References

[1] E. A. Müller, Gruppentheoretische und Strukturanalytische Untersuchungen der Maurischen Ornamente aus der Alhambra in Granada, University of Zürich, 1944.

[2] B. Grünbaum, A. Grünbaum, G. C. Shephard. Symmetry in Moorish and Other Ornaments. Comp. & Maths with Applications, Vol. 12B, No. 3 / 4, pp. 641 – 653, 1986.

[3] R. Perez-Gomez, The Four Regular Mosaics Missing in the Alhambra, Comput. Math. Applic., v. 14, no. 2, 1987, 133-137.

[4] B. Grünbaum. What Symmetry Groups are Present in the Alhambra? Notices of the AMS, June/July 2006 issue (Volume 53, Number 6) and also available at

[5] W. K. Chorbachi. In the Tower of Babel: Beyond Symmetry in Islamic Art. Computers & Mathematics with applications, Vol. 17b, pp. 751 – 189, (1989).

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[7] D. K. Washburn and D.W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press, 1988.

[8] S. J. Abas and A. S. Salman. Symmetries of Islamic Geometrical Patterns. World Scientific, 1998.

[9] The Geometer’s Sketchpad software program, available at


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