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Plane groups can be illustrated using a great variety of 2-dimensional periodic
patterns. Unexcelled classics are the widely known prints by M. C. Escher^{1}
which have found their way into many texts on crystallographic symmetry. Other
examples are provided by decoration and gift wrapping paper, various fabrics and
highly artistic designs including Arabic geometrical patterns.^{2}

A standard introductory exercise in crystallographic symmetry is the
determination of the plane group of such patterns, e.g. that of an ordinary
brick wall (*c*2*mm*). The best approach to such a problem is to begin with the
point symmetry (extended by glide lines if necessary) which will define the
system of axes to be adopted. Thus, the oblique system (comprising plane groups
*p*1 and *p*2) is by no means characterized by and , a frequent misconception in `defining' crystal systems. A sheet of
stamps, for example, has symmetry *p*1 in general,^{3} and therefore belongs to
the oblique system. The rectangular form of the stamps is not required by
symmetry, it simply happens to be convenient. Another example of this sort is
presented in Fig. 1.

Some knowledge of and practice in recognizing plane groups is not necessarily a prerequisite but a good basis from which to start an introduction to space group symmetry as described in the following sections. The pattern in Plate (i), if considered to represent a 2-dimensional structure, serves as an example for a preliminary exercise on plane groups. (Enter the symmetry elements and unit cell, and determine the plane group of the pattern.)

**Copyright © 1984, 1998 International Union of
Crystallography**