1. Introduction (Mathematics and Aesthetics)

In the course of our very pleasant correspondence with Professor Denes Nagy about our contribution to the Proceedings of The Third International Conference on Symmetry, held in Washington D. C. in August 1995, the idea took shape of our writing two articles about the symmetry of geometrical figures, one of a practical nature, the other of a more theoretical nature. Thus this article is a companion to the article Symmetry in Practice (in this issue), which describes very practical ways of constructing regular polygons and polyhedra. We subtitle that article "Recreational Constructions" - and refer to it henceforth as [Rec] - because the constructions, involving the use of colored paper, have an undoubted recreational flavor. However, it is our conviction, based on many years' experience, that the execution of such model constructions can play a vital role in enlivening and enriching the study of geometry, especially if the mathematical theory underlying the constructions features prominently. Thus it is our strong hope that readers of [Rec] will be encouraged to move on to this more theoretical sequel, to learn why the constructions work and better to understand the nature of symmetry. We also set the mathematical development in its historical context and show explicitly how the geometry is related to other parts of mathematics - real analysis, number theory, group theory, combinatorics. Such connections should, in our view, form an integral part of the teaching and learning of any part of mathematics. We will refer to the present article, briefly, as [Math].

In Section 2 we link the practical instructions of [Rec] to a mathematical discussion of the parameters of the polygons constructed. Thus we answer two questions which stand in a converse relation to each other, namely,

  • (i) given the folding instructions for our tape, when will we be able to produce a regular convex polygon and how many sides will it have, and
  • (ii) given a number p, what folding instructions will produce a regular p-sided polygon (or p-gon)?

    Having learnt in Section 2 how to construct certain regular figures, we turn in Section 3 to the question of just what we should understand by the symmetry of a geometrical figure, and how it should be measured. From a mathematical point of view it makes very little sense to say that a given figure A is symmetrical, 1 but we have a precise idea of its group of symmetries, that is, of the subgroup of the group of Euclidean movements of the ambient space of A under which A is invariant. Based on this idea, we can give meaning to the statement that figure A is more symmetrical than figure A. However, we need to bear in mind that the symmetry group of A depends on our convention as to what is the ambient space of A. Thus if A is a circle, then its symmetry group as a subset of the plane depends on whether we allow reflexions of the plane or not (note that a reflexion of the plane cannot be achieved by a movement in the plane, but only by a movement in 3-dimensional space).

    Another important aspect of symmetry arises when one considers actual physical models of geometrical configurations. Suppose we have constructed a model M of the figure A by braiding together colored strips; A may be a regular dodecahedron, say. Our model cannot have more symmetry than A itself - but it may well have less. For to every symmetry g of A we have a movement of the model M which may create an image Mg recognizably different from M because of the arrangement of colors. Thus the symmetry group of M may only be a subgroup of the symmetry group of A; and aesthetics come into the story here by requiring the symmetry group of M to be as large as possible. Thus can mathematics contribute to the study of aesthetics!

    It turns out (not surprisingly!) that, if B is a subset of A and if GA is the symmetry group of A, then the set of images of B under the action of elements of GA is the set of homologues of B in the sense of George Pólya; we explain this in Section 4. Actually, Pólya never wrote down his work on homologues (which, so far as we know, he only discussed in the case where A is a Platonic solid), but, when he was a very old man, he asked us to write it down for him, and we are proud and happy to have this opportunity to do so (see Figure 9 of for the only extant copy of his original notes on the subject).

    In Section 5 we explain Pólya's famous Enumeration Theorem, one of the most important theorems of that branch of mathematics known as combinatorics. We apply it to the symmetries of geometrical figures, where parts of the figures are colored in prescribed ways, and again recover the notion of homologue from the formulation of the theorem.

    The final section is an informal epilogue, describing our relationship with George Pólya.  We are grateful to Denes Nagy for inviting us to write these two articles, [Rec] and [Math], and for persuading us to include some personal reflections on our good fortune in knowing that remarkable man so well.

    1 We might perhaps say that A is symmetrical if there is a non-trivial Euclidean movement sending A to itself. A classification of symmetricality due to Kepler is to be found in [C2].

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