In a general way the term symmetry returns to the existence, in an unspecified figure, of a geometrical operation which does not modify this figure. One can make correspond to each point of the figure another point, without modification of the general figure.

In mathematics, a symmetry is a geometrical Transformation which is involutive, i.e. applied twice at a stretch to a figure, it leaves this unchanged figure. These symmetries are described in the article Symétrie (geometrical transformation).

A system is symmetrical when one can simultaneously permute all his elements without modifying his structure. Symmetries translate a kind of equality of the system with itself, or of uniformity of its structure. The concept of Automorphism, or Isomorphism intern, exposed further, makes it possible to specify this definition.

Examples

A butterfly, for example, is symmetrical: one can exchange all the points of the left half of the body with all the points on the right half without the appearance of the butterfly being modified. The majority of the Animal , including the Human S, present a Symétrie planes, i.e. that their right and left half are symmetrical compared to a Symmetry plane which one calls Median plane in Anatomie. This characteristic defines the Clade Bilateria among the animals. In addition, one finds also morphologies with radiate Symétrie like the Méduse S, these animals form the clade Radiata. In the Vegetable reign , the plants with Fleur S often have a floral Pièce with radiate symmetry: the daisy is one of the example most known of this type of flowers, known as Actinomorphe S.

In Mathematical, one can define very many types of symmetries. There is as much as there are ways simultaneously of permuting the parts of a system: symmetries compared to a Axis or a Plan, Rotation S, translation S, unit Homothety S, like all their combinations, inter alia.

The Euclidean Espace in its entirety is one of the most symmetrical systems, with the direction where the whole in the ways simultaneously of permuting all its points without modifying its structure, its group of symmetries, is one of largest, among the groups of geometrical symmetries. All the points of space are similar. They do not have other quality to only be a point. They have all the same relations with the remainder of space. Principal symmetries of Euclidean space are the Isométrie S. That all the points are similar is expressed then by the fact that any point can be transformed into any other by a isometry. If one breaks the symmetry of space by introducing a Sphère, then all the points are not similar any more: there are points on the sphere, others inside and others outside. On the other hand, all the points of the sphere are similar. Any of them can be transformed into any other by a isometry: a rotation around the center of the sphere.

When a system is symmetrical, the permutable parts are necessarily similar inside a model, and almost identical in the physical world, since the system is not modified by their permutation.

What an automorphism?

The concept of automorphism makes it possible to specify the definition of symmetries. What wants to say “without modifying its structure”? A system is defined like a model. It is necessary to determine
  • the unit U, finished or infinite, of its elements, its elementary points, its atoms or its components. It is the field of existence associated with the system or the studied universe.
  • the unit, in general finished, of the fundamental Predicate S, basic properties of the elements and relations between them.
  • the unit, in general empties or finished, of the Opérateur S, or function S, which determines the structure of the system more.

A geometrical Transformation T is a Automorphisme, a Isomorphisme intern or a symmetry, for a binary relation R when it is an invertible function, or Bijection, of U in U such as

for any X and there, X R there if and only if tx R ty

What is true X and there, to satisfy the relation R, is also true of tx and ty.

X is similar to tx, there with ty.

This definition of an automorphism spreads easily with the unary predicates and all the relations, whatever the number of their arguments. For a unary predicate P, a transformation T is an automorphism when

for any X, Px if and only if Ptx

In the example of the butterfly, symmetry between the left and the line are an automorphism for the properties (unary predicates) of color. A point with the same color as its symmetrical point.

A transformation T is an automorphism for a binary operator + when for all X and there, T (x+y) = (tx) + (ty) This definition of an automorphism spreads easily with all the operators, whatever the number of their arguments. T is an automorphism for a unary operator when

for any X, T (- X) = - T (X)

In other words, a transformation is an automorphism for a unary operator, or function of only one variable, when it commutates with him. When transformations commutate between them, they all are of the automorphisms the ones with respect to the others, with the direction where any structure defined by a transformation is preserved by all the others. With a binary operator +, one can associate a ternary relation definite by x+y=z. It is seen whereas the definition of an automorphism for an operator is a particular case of the definition of an automorphism for the relations.

Groups of symmetries

The group of symmetries is the whole of all the automorphisms of the system. There are the following properties.

For all automorphisms T and U, t°u is an automorphism and the reverse of T is an automorphism. The identical tranformation (which always associates X with X) is an automorphism.

In other words,

  • if a structure is preserved by two transformations carried out separately, it is also preserved when one carries out the two transformations one following the other. It is simply the Transitivité of the equality of the structure.
  • if a structure is preserved by a transformation, it is also preserved by the reverse transformation.
  • moreover, there exists always an identical transformation, which does not transform anything, which is thus always an automorphism, since it cannot modify anything.

These three properties make of the whole of the automorphisms of a system a group for its Law of composition interns natural.

The Théorie of the groups is the principal theoretical tool of study of symmetries.

See too

Simple: Symmetry

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