What is Symmetry

"Symmetry, as though is wide or narrow we did not perceive this word, there is the idea, with the help of which a man attempted to explain and to create the order, beauty and perfection".

What is "symmetry"? When we look in the mirror we can see in it our reflection; this is the example of the "mirror" symmetry. The mirror reflection is the example of so-called "orthogonal" transformation varying orientation.

In the most general case "symmetry" in mathematics is perceived as such transformation of space (plane), at which each point of M passes in other point of M' regarding to some plane (or straight line) a, when the line segment of MM' is perpendicular to the plane (or to the straight line) a and is divided by it in halves. The plane (or the straight line) a is called as the plane (or axis) of symmetry.

The plane of symmetry, symmetry axis, center of symmetry are fundamental concepts of symmetry. The plane of symmetry  is called such plane, which divides the figure into two mirror-equal parts arranged from each other as some subject and its mirror reflection. For example, the isosceles triangle €‚‘ shown in Fig.1 at the left is divided by the altitude BD into two mirror- equal halves ABD and BCD; thus the altitude BD is the "track" of the plane of symmetry , perpendicular to the triangle plane. In Fig. 1 on the right the rectangular parallelepiped (match boxes) is shown; this one has three orthogonal planes of symmetry 3. It is easy to establish, that the cube has nine planes of symmetry - 9.

Figure 1. Symmetries of triangle and parallelepiped.

The symmetry axis L is called such straight line, around of which the symmetrical figure can be turned around some times in such manner that each time the figure is "combined" with itself in space. A number of such turns around of the symmetry axis is called the order of the axis. For example, the equilateral triangle has the symmetry axis L₃, that is, there are three ways of turn of the triangle around of the axis, at which there is its "self-alignment". It is clear, that the square has the symmetry axis L₄ and the pentagon - L₅. The cone also has the symmetry axis, and, as a number of turns of the cone around of the symmetry axis resulting in "self-alignment" is infinitely, speak, that the cone has the symmetry axis of the type .

At last, the symmetry center C is called such singular point inside the figure that any straight line draw through the point C and on the equal distances from the center C, meets identical points of the figure. "Ideal" example of such figure is a sphere, which center is its center of symmetry.

The symmetry widely meets in the objects of the alive and nonliving nature. For example, the symmetry in chemistry is reflected in the geometrical configuration of molecules. So, for example, the molecule of methane ‘H₄ has symmetry of the tetrahedron. The concept of "symmetry" is central at research of crystals. Thus the symmetry of exterior forms of crystals is determined by symmetry of its atomism, which causes also symmetry of physical characteristics of the crystals.

Concept of "symmetry" with reference to the physical laws is used widely in modern physics. If the laws, establishing relations between values or determining a change of these values in the course of time, do not vary at definite operations (transformations), to which the system can be subjected, speak, that these laws have symmetry (or are invariant) concerning to the given transformations. For example, the law of gravitation acts in any points of space, that is, it is invariant regarding to carry of a system as the whole in space.

In opinion of academician Vernadski, the outstanding Russian scientist, "the symmetry encompasses properties of all fields of physics and chemistry".

Still the Pythagoreans paid attention to the phenomenon of symmetry in the alive nature in connection with development by them of the harmony doctrine. It is established, that in the nature two kinds of symmetry, the "mirror" and "radial" symmetry, are most widespread. The butterfly, the leaf, and the beetle (Fig.2-a) have the "mirror" symmetry and often such kind of symmetry is called as the "leaf symmetry" or the "bilateral symmetry". The mushroom, the chamomile, the pine tree (Fig.2-b) have the "radial" symmetry and often such kind of symmetry is called as the "chamomile- mushroom" symmetry.

Figure 2. The natural forms with the "bilateral" ( ) and the "radial" (b) symmetries.

Still in the 19th century the researches in this area resulted in the conclusion that the symmetry of the natural forms largely depends on influence of the Earth gravitation forces, which have the symmetry of the cone in each point. In outcome it was found the following law, to which the forms of natural bodies are subjected:

"Everything that grows or moves in vertical direction, that is up or down regarding to the Earth surface is subjected by the "radial" ("chamomile- mushroom") symmetry. Everything that grows and moves horizontally or sloping regarding to the Earth surface is subjected by the "bilateral" symmetry, the "leaf symmetry".

The principle of "symmetry" is used widely in art. The borders used in architectural and sculptural works, the ornamental designs used in an applied art are examples of symmetry usage.

The concept of symmetry is used often in the art works together with the "golden section" principle. Rafael's picture "Betrothing of Mary" (Fig.3) is such example.

Figure 3. Harmonic analysis of Rafael's picture "Betrothing of Mary".

In modern science the interest in symmetry and its diverse applications to the nature, science and art increased extremely and establishment in 1989 of the International Society for interdisciplinary study of symmetry (ISIS-Symmetry) "became the beginning of considerable intellectual motion".