 From a modern perspective, the regular figures used in Plato’s and Kepler’s physics for the mathematical proportions and harmonies they contain (and the related properties and beauty of their form) are symmetric in another sense that does not have to do with proportions. In the language of modern science, the symmetry of geometrical figures — such as the regular polygons and polyhedra — is defined in terms of their invariance under specified groups of rotations and reflections. Where does this definition stem from? In addition to the ancient notion of symmetry used by the Greeks and Romans (current until the end of the Renaissance), a different notion of symmetry emerged in the seventeenth century, grounded not on proportions but on an equality relation between elements that are opposed, such as the left and right parts of a figure. Crucially, the parts are interchangeable with respect to the whole — they can be exchanged with one another while preserving the original figure. This latter notion of symmetry developed, via several steps, into the concept found today in modern science. One crucial stage was the introduction of specific mathematical operations, such as reflections, rotations, and translations, that are used to describe with precision how the parts are to be exchanged. As a result, we arrive at a definition of the symmetry of a geometrical figure in terms of its invariance when equal component parts are exchanged according to one of the specified operations. Thus, when the two halves of a bilaterally symmetric figure are exchanged by reflection, we recover the original figure, and that figure is said to be invariant under left-right reflections. This is known as the “crystallographic notion of symmetry”, since it was in the context of early developments in crystallography that symmetry was first so defined and  applied.[1]  The next key step was the generalization of this notion to the group-theoretic definition of symmetry, which arose following the nineteenth-century development of the algebraic concept of a group, and the fact that the symmetry operations of a figure were found to satisfy the conditions for forming a  group.[2]  For example, reflection symmetry has now a precise definition in terms of invariance under the group of reflections. Finally, we have the resulting close connection between the notion of symmetry, equivalence and group: a symmetry group induces a partition into equivalence classes. The elements that are exchanged with one another by the symmetry transformations of the figure (or whatever the “whole” considered is) are connected by an equivalence relation, thus forming an equivalence  class.[3]