Symmetry

Symmetry concepts play a central role in physics [13]. The ► invariance (► covari-ance) properties of a system under specific symmetry transformations can either be related to the conservation laws of physics or be able to establish the structure of the fundamental interactions. This is the most essential aspect of symmetry as it concerns the basic principles of physics and the interactions themselves and not only the properties of a particular system [14].

In geometry, figures or bodies are called symmetrical when they possess common measures or proportions. Thus the Platonic bodies can be rotated and turned at will without changing their regularity. Similarity transformations, for example, leave the geometric form of a figure unchanged, i.e. the proportional relationships of a circle, equilateral triangle, rectangle, etc. are retained, although the absolute dimensions of these figures can be enlarged or decreased. Therefore one can say that the form of a figure is determined by the similarity transformations that leave it unchanged (invariant). In mathematics, a similarity transformation is an example of an automorphism [12]. In general an automorphism is the mapping of a set (e.g. points, numbers, functions) onto itself that leaves unchanged the structure of this set (e.g. proportional relations in Euclidean space). Automorphisms can also be characterized algebraically in this way: (1) Identity I that maps every element of a set onto itself, is an automorphism. (2) For every automorphism T an inverse automorphism T' can be given, with T∙T' = T' ∙ T = I. (3) If S and T are automorphisms, then so is the successive application S ∙ T. A set of elements with a composition that fulfils these three axioms is called a group. The symmetry of a mathematical structure is determined by the group of those automorphisms that let it unchanged (invariant).