**71**, 2, pp. 265–283, 2023

**10.24423/EngTrans.2261.20230531**

### Symmetry

The concept of symmetry was introduced already by the ancient Greeks in relation to spatial (geometric) systems. They understood it as commensurability and proportionality and linked it with the aesthetic categories of harmony and beauty. A spatial system

(object) was considered symmetric if it consisted of regular, repeatable parts of comparable size, creating a coordinated, ordered, larger whole. Only two thousand years later, in the twentieth century, the essence of the concept of symmetry was identified. Symmetry is invariance (stability, durability, constancy) of a feature (geometric, physical, biological, informational, etc.) of an object (an object can here be a geometric system, a material thing, but also a natural phenomenon, physical law, social relation, etc.) after subjecting it to a set of transformations (transformations can be shifts, reflections, rotations, permutations, etc.), with respect to which symmetry is considered. The above observation led to the discovery of the universal nature of the concept of symmetry, which in a broader sense can be understood as a philosophical category, one of the fundamental regularities of mathematical character in the organization of the Universe. The contemporary understanding of symmetry has led to significant and nonobvious conclusions. For example, it turned out that the invariance (symmetry) of the laws of motion with respect to the shift in time is equivalent to the necessity of the existence of the principle of conservation of energy, the invariance (symmetry) of the laws of motion with respect to the shift in physical space proves to be equivalent to the existence of the principle of conservation of momentum. The Report provides an outline of the general formal language of symmetry applicable to the study of any situation in which this concept appears. The key elements of the mathematical apparatus of the algebraic theory of symmetry are defined and discussed, the notions of Γ-sets, orbits, orbital markers, invariants, and invariant functions. They provide versatile tools enabling the analysis of all types of symmetries. The Report concisely presents important results of the theory of symmetry, such as: the ornament principle – expressing the most straightforwardly the innermost property of complex symmetrical objects, the representation theorem for symmetric objects, the theorem on the symmetry of causes and effects of physical laws, the theorem on invariant extension of any function. [*Editorial note:* Abstract by Andrzej Ziółkowski.]

**Keywords**: formal language of symmetry, concept of Gamma-set, orbits as subsets (classes) of symmetric (similar, mutually reachable, invariant, stable, permanent) elements, subgroup of operators, orbit markers, invariants, ornament principle, motif,

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DOI: 10.24423/EngTrans.2261.20230531