Coloring Symmetrical Objects Symmetrically

Abstract

Color symmetry-the symmetrical distribution of colors in regular patterns-is as old as ornamental art itself. Beautiful examples from many cultures can be found in the colored plates of Owen Jones' classic Grammar of Ornament [12] and, in our time, in the tessellations of M. C. Escher. Also striking are the patterns of identity and difference [2] that are found in nature, for example in the arrangements in crystals of different atoms, or of magnetic spins [8]. Colors are often used in structure models to represent such nongeometric characteristics. In this article, we give an introduction to the mathematical theory of color symmetry that has been developed in recent years. This theory complements and extends the usual characterization of the symmetry of an object by describing the ways of coloring it that are consistent with its symmetry. In addition to being of interest in its own right and for its applications, color symmetry provides a simple way of illustrating concepts in elementary group theory, such as subgroup, coset, normality, conjugacy, and so forth; we hope it will turn out to be a useful topic in introductory algebra courses. More advanced students will find that the theory of permutation groups provides a unified framework for examining various aspects of color symmetry; reformulating in more abstract terms the theory outlined here is an instructive exercise. As an example of the problem of coloring an object symmetrically, suppose we wish to color each face of an octahedron with one of two colors, say, black and white. Intuitively we expect that in a symmetrical distribution of colors, four faces should be black and four white. There are many