The term “symmetry” derives from the Greek words sun (meaning ‘with’ or ‘together’) and metron (‘measure’), yielding summetria, and originally indicated a relation of commensurability (such is the meaning codified in Euclid's Elements for example). It quickly acquired a further, more general, meaning: that of a proportion relation, grounded on (integer) numbers, and with the function of harmonizing the different elements into a unitary whole. From the outset, then, symmetry was closely related to harmony, beauty, and unity, and this was to prove decisive for its role in theories of nature. In Plato's Timaeus, for example, the regular polyhedra are afforded a central place in the doctrine of natural elements for the proportions they contain and the beauty of their forms: fire has the form of the regular tetrahedron, earth the form of the cube, air the form of the regular octahedron, water the form of the regular icosahedron, while the regular dodecahedron is used for the form of the entire universe. The history of science provides another paradigmatic example of the use of these figures as basic ingredients in physical description: Kepler's 1596 Mysterium Cosmographicum presents a planetary architecture grounded on the five regular solids.
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. 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. 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.
The group-theoretic notion of symmetry is the one that has proven so successful in modern science. Note, however, that symmetry remains linked to beauty (regularity) and unity: by means of the symmetry transformations, distinct (but “equal” or, more generally, “equivalent”) elements are related to each other and to the whole, thus forming a regular “unity”. The way in which the regularity of the whole emerges is dictated by the nature of the specified transformation group. Summing up, a unity of different and equal elements is always associated with symmetry, in its ancient or modern sense; the way in which this unity is realized, on the one hand, and how the equal and different elements are chosen, on the other hand, determines the resulting symmetry and in what exactly it consists.
The definition of symmetry as “invariance under a specified group of transformations” allowed the concept to be applied much more widely, not only to spatial figures but also to abstract objects such as mathematical expressions — in particular, expressions of physical relevance such as dynamical equations. Moreover, the technical apparatus of group theory could then be transferred and used to great advantage within physical theories.
When considering the role of symmetry in physics from a historical point of view, it is worth keeping in mind two preliminary distinctions:
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The first is between implicit and explicit uses of the notion. Symmetry considerations have always been applied to the description of nature, but for a long time in an implicit way only. As we have seen, the scientific notion of symmetry (the one we are interested in here) is a recent one. If we speak about a role of this concept of symmetry in the ancient theories of nature, we must be clear that it was not used explicitly in this sense at that time.
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The second is between the two main ways of using symmetry. First, we may attribute specific symmetry properties to phenomena or to laws (symmetry principles). It is the application with respect to laws, rather than to objects or phenomena, that has become central to modern physics, as we will see. Second, we may derive specific consequences with regard to particular physical situations or phenomena on the basis of their symmetry properties (symmetry arguments).
