In theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group - i.e. a topological group with a discrete topology whose elements form a finite or a countable set. Theoretical physics employs mathematical models in an attempt to understand Nature. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, a countable set is a set with the same cardinality (i. ...
The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant.
Discretesymmetry groups come in two types: finite point groups, which include only rotations and reflections - they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections.
The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups.
The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in the concept of symmetry in physics that took place in the twentieth century.
The discretesymmetries C, P and T are connected by the so-called CPT theorem, demonstrated by Lüders in 1952, which states that the combination of C, P, and T is a general symmetry of physical laws.
Symmetries may be used to explain (i) the form of the laws, and (ii) the occurrence (or non-occurrence) of certain events (this latter in a manner analogous to the way in which the laws explain why certain events occur and not others).
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