On Commensurability and Symmetry

Abstract

Commensurability and symmetry have diverged from a common Greek origin. We review the history of this divergence. In mathematics, symmetry is now a kind of measure that is different from size, though analogous to it. Size being given by numbers, the concept of numbers and their equality comes into play. For Euclid, two magnitudes were symmetric when they had a common measure; also, numbers were magnitudes, commonly represented as bounded straight lines, for which equality was congruence. When Billingsley translated Euclid into English in the sixteenth century, he used the word commensurable for Euclid's symmetric magnitudes; but the word had been used differently before. Symmetry has always had also a vaguer sense, as a certain quality that contributes to the beauty of an object. Today we can precisely define the symmetry of a mathematical structure as the automorphism group of the structure, or as the isomorphism class of that group. However, when we consider symmetry philosophically as a component of beauty, we can have no foolproof algorithm for it.