Introduction

One practical aim in Part I is to equip the reader to build a pattern-generating computer engine. The patterns we have in mind come from two main streams. Firstly the geometrical tradition, represented for example in the fine Moslem art in the Alhambra at Granada in Spain, but found very widely. (See Figure 1.1.)

Less abundant but still noteworthy are the patterns left by the ancient Romans (Field, 1988). The second type is that for which the Dutch artist M. C. Escher is famous, exemplified in Figure 1.2, in which (stylised) motifs of living forms are dovetailed together in remarkable ways. Useful references are Coxeter (1987), MacGillavry (1976), and especially Escher (1989). In Figure 1.2 we imitate a classic Escher-type pattern.

The magic is due partly to the designers' skill and partly to their discovery of certain rules and techniques. We describe the underlying mathematical theory and how it may be applied in practice by someone claiming no particular artistic skills.

The patterns to which we refer are true plane patterns, that is, there are translations in two non-parallel directions (opposite directions count as parallel) which move every submotif of the pattern onto a copy of itself elsewhere in the pattern. A translation is a movement of everything, in the same direction, by the same amount. Thus in Figure 1.2 piece A can be moved to piece B by the translation represented by arrow a, but no translation will transform it to piece C. A reflection would have to be incorporated.