Crystallography uses the most elementary expression of symmetry, the visual one of geometry whose elements are the isometries of the Euclidian space: inversion, rotation, reflexion in a mirror, plus the translation in space, ideal crystal being supposed infinite. Mathematically, an object is symmetric if it remains invariant under the application of a set of isometries called symmetry group.

The drawings of Escher [*1*] are crystallographic classics in teaching two-dimensional space groups. Due to the biological nature of the interlocking objects in them, they are most appealing in the cases of the lower-symmetry groups. The other man-made two-dimensional periodical patterns suitable for the study of plane groups are primarily the Roman, Byzantine and Romanesque mosaics, and the gothic. Especially suitable, however, is the large group of Islamic geometric patterns, executed in tilework, brickwork, stucco, wood, marble and metal [2].

In Islamic ornaments, we encounter two types of symmetry groups:

1. Finite group: cyclic group C_{n} (Figure 1) and dihedral D_{n} (6 <= n <= 96) (figure 2) called groups of rosettes,

2. Infinite space groups: group of friezes in one-dimensional space (1D) and wallpaper group or crystallographic planar groups in two-dimensional space (2D).

The Islamic geometric patterns best suited for teaching come from linear drawing, the multi-coloured and interlaced patterns (Figure 2). Introduction of colour and interlaces introduce additional elements of complexity and subtlety in the decorative patterns. To rigorously describe the intertwined and coloured patterns, large number of groups are deduced from the parent plane groups [3]. Moreover, the order-disorder and quasiperiodic structures encountered in Islamic ornaments may facilitate the approach of structures in crystallography.

[1] M. C. Escher: Art and Science, H. S. M. Coxeter, M. Emmer, R. Penrose, M. Teuber, eds. Amsterdam: North Holland, 1986.

[2] Y. Aboufadil, A. Thalal & M.A. Elidrissi Raghni - Symmetry groups of Moroccan geometric woodwork patterns, J. Appl. Cryst. 2013.

[3] E. Makovicky, Symmetry through the eyes of old masters, Walter de Gruyter GmbH, Berlin/Boston, 2016.

Keywords: symmetry, teaching, Islamic ornament.