The basics of the theory of symmetry


"The extent and versatility of the theory of symmetry can be partially observed by recognizing the scientific and art disciplines in which it has a significant role: mathematics, physics, crystallography, chemistry, biology, esthetics, visual arts, theory of proportion, architecture, etc." (Jablan, 2002). Because of its universality and synthetic role in the sciences, several authors have assigned to it the status of a philosophical category which expresses the fundamental laws of nature (Akopyan, 1980). The symmetry of natural laws and the products of human creativity (material and intellectual) is an expression of the symmetry of nature. A.V. Shubnikov, who treats symmetry as a "structural law of construction" agrees with this (Shubnikov and Koptsik, 1972).

The existence of a term for symmetry in all languages speaks about its significance: Greek and Latin (simmetria, symmetria), Sanskrit (sammita), Hebrew (ketzev, toam), Chinese ( dey-cheng), Japanese (taisho, kinsei) (Nagy, 1990).

In the European culture, the meaning of the concept of symmetry has its roots in Greek philosophy and esthetics. Primarily interpreted in the sense of commensurability, balance, proportion and regularity, this term points to a specter of synonymous philosophical and esthetic terms (harmony, consistency, wholeness, etc.), which have been used throughout the history of human thought.

The original meaning of the word "symmetry", common measure, points directly to the main problem of Greek mathematics - the question of the commensurability of line segments. Two line segments are commeasurable if a line segment (measuring unit) is contained m times in the first line segment and n times in the second line segment. The ratio between the two measured line segments is expressed by the fraction [m/n]. In this manner we make a direct transition from the geometric theory of proportions to the algebra of rational numbers (fractions).

The other central problem of Greek geometry, the basics of the theory of regular geometric figures (regular polygons and polyhedra, Platonic solids, semi-regular Archimedean polyhedra) has a natural link to the concept of symmetry. All the other derived meanings of the term ▀ymmetry" (harmony, consistency, proportion, balance, etc.) express, to a lesser or greater extent, the mentioned ideal of Greek mathematics - the tendency to construct a universal system of measurement based on rational numbers and a tendency towards regular forms.

In the following period the primary meaning of the concept of symmetry is significantly reduced and narrowed down to its simplest form, bilateral symmetry (mirror reflection) and plane symmetry (plane reflection).

The appearance of symmetrology as an independent scientific area of research is a result of the aspiration towards uniting the various aspects of symmetry that appear in all scientific and art works and subjecting them to a common multidisciplinary analysis. The theory of symmetry which originated in the 19th and 20th centuries as part of the exact sciences (mathematics, crystallography, etc.), aspires towards achieving the level of generality that existed in the original meaning of the term "symmetry" (simmetria).