What Escher might have done?

There are many ways in which a tiling can be coloured systematically. the most satisfactory colourings, both from an artistic and a mathematical point of view are those which are proper and perfect. all Escher’s colourings, with one or two possible exceptions, have both these properties which we shall now describe.

A colouring of a tiling is proper if no two adjacent tiles have the same colour. This condition is familiar in the colouring of maps, where adjacent countries are usually coloured differently. A colouring of a tiling is perfect if every symmetry of the tiling is associated with a colour symmetry. H.S.M. Coxeter explains, in his contribution to this volume, the concept of a colour symmetry. Here we shall confine ourselves to giving some examples. The simplest is that of the chequerboard colouring with two colours. This can be applied to any tiling in which each tile has four adjacents, and four tiles meet at each vertex. Such a colouring is clearly proper. To see that it is perfect we need only observe that every symmetry of the tiling either leaves each tile the same colour or inter changes the two colours, and so yields a colour symmetry of tiling. Most, but not all, Escher’s colourings of tilings using two colours are of this kind.

A perfect colouring with three colours; every symmetry of the tiling is associated with a permutation of the colours and so with a colour symmetry.A perfect colouring of this same tiling by two  colours is also possible, namely the chequerboard colouring mentioned above.

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