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Two-Dimensional Periodic Patterns

Plane groups can be illustrated using a great variety of 2-dimensional periodic patterns. Unexcelled classics are the widely known prints by M. C. Escher¹ 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 (c2mm). 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 p1 and p2) is by no means characterized by $a \neq b$ and $\gamma \neq 90^{\circ}$, a frequent misconception in `defining' crystal systems. A sheet of stamps, for example, has symmetry p1 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.

 

Figure 1: Example of two-dimensional periodic pattern having symmetry p1 (note oblique system despite `rectangular' unit cell).

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.)

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