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.]
B.1. Lang S., Algebra [in Polish], PWN, Warsaw, 1973.
B.2. Berger M., Geometry [in French: G´eom´etrie], CEDIC, Nathan, Paris, 1978.
B.3. Weyl H., Symmetry, Princeton University Press, Princeton, 1952.
B.4. Szubnikow A.W., Selected works on crystallography [in Russian: Izbrannyje trudy po kristałłografii], Nauka, Moscow, 1975.
B.5. Wigner E., Symmetries and Reflections, Indiana University Press, Bloomington, 1970.
B.6. Weyl H., The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, 1946.
B.7. Wigner E., Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.
B.8. Lubarskij G. Ja., Group theory and its application in physics [in Russian: Tieorija grupp i jejo primienienije v fizikie], Fizmatgiz, Moscow, 1958.
B.9. –, Group Theory and Physics [in Russian: Tieorija grupp i fizyka], Nauka, Moscow, 1986.
B.10. Elliot J.P., Dawber P.G., Symmetry in Physics, Macmillan, London, 1979.
B.11. Szubnikow A.W., Kopcik W.A., Symmetry in science and art [in Russian: Simmietrija v nauke i iskusstvie], Nauka, Moscow, 1972.
B.12. Zełudiew I.S., Symmetry and its applications [in Russian: Simmietrija i jejo priłozenija], Energoatomizdat, Moscow, 1983.
B.13. Park D., Resource letter SP-1 on symmetry in physics, American Journal of Physics, 36(7): 577–584, 1968, doi: 10.1119/1.1975017.
B.14. Rosen J., Resource letter SP-2: Symmetry and group theory in physics, American Journal of Physics, 49(4): 304–319, 1981, doi: 10.1119/1.12504.
B.15. Rychlewski J., The symmetry of cause and effect [in Polish: Symetria przyczyn i skutków], WN PWN, Warsaw, 1991.
B.16. Peano G., On the definition of function [in Italian: Sulla definizione di funzione], Atti Real Accad. Linzei, 20: 3–5, 1911.
B.17. Kuratowski K., Mostowski A., Set Theory, North-Holland and PWN, Amsterdam–Warsaw 1967.
B.18. Curie P., Sur la sym´etrie dans les ph´enom`enes physiques, sym´etrie d’un champ ´el´ectrique et d’un champ magn´etique, Journal de Physique Th´eorique et Appliqu´ee, Paris, 3(1): 393–415, 1894; reprinted in: Oeuvres de Pierre Curie, pp. 118–141, Gauthier-Villars, Paris, 1908.
B.19. Skłodowska-Curie M., Pierre Curie [in Polish], PWN, Warsaw 1953.
B.20. Szubnikow A.W., On the works of Pierre Curie in the field of Symmetry [in Russian: O rabotach Pierra Curie w obłasti simmietrii], Uspiechi Fiz. Nauk, 59: 591–602, 1956; also in: Selected works on crystallography [in Russian: Izbrannyje trudy po kristałłografii], Nauka, Moskwa, 1975, pp. 133–144.
B.21. Sirotin J.I., Szaskolskaja M.P., Fundamentals of crystal physics [in Russian: Osnowy kristałłofiziki], Nauka, Moscow 1979.
B.22. Truesdell C., Rational Thermodynamics, Springer, New York 1984.
B.23. Kizel W.A., Physical causes of dissymmetry of life structures [in Russian: Fiziczeskije pricziny dissimietrii zywych sistiem], Nauka, Moscow 1985.
B.24. Łochin W.W., Siedow L.I., Nonlinear functions from multiple tensorial arguments [in Russian: Nieliniejnyje funkcji ot nieskolkich tienzornych argumientow], Prikł. Mat. Mech., 27: 393–417, 1963.
B.25. Liu I-S., On representations of anisotropic invariants, International Journal of Engineering Science, 20(10): 1099–1109, 1982, doi: 10.1016/0020-7225(82)90092-1.
B.26. Boehler J.P., A simple derivation of representations for nonpolynomial constitutive equations in some cases of anisotropy, Journal of Applied Mathematics and Mechanics, 59(4): 157–167, 1979, doi: 10.1002/zamm.19790590403.
B.27. Zhang J.M., Rychlewski J., Structural tensors for anisotropic solids, Archives of Mechanics, 42(3): 267–279, 1990.