Group-Theoretical Analysis of Symmetry Transformations on the Example of Some Aquatic Organisms

Abstract—Group-theoretical analysis of the pseudosymmetry of two-dimensional images of aquatic organisms of the classes Conjugatophyceae, Bacillariophyceae, Acantharia, and Asteroidea and the symmetry transformations in the ontogeny of echinoderms has been performed for the first time in the original BioPsLeaf and BioPsFlower software, and the results of the analysis are presented below. Published materials, including graphic illustrations from Haeckel’s book Künstformen der Natur were the sources of the two-dimensional images of aquatic organisms used in the study. The choice of aquatic organisms was largely determined by the Curie principle, which imposes restrictions on the symmetry groups of living organisms with consideration of the specific habitat. Analysis of the organisms from the considered classes showed that the invariance (symmetry) of a biological object that can be roughly described by the C_nv group of operations of the Schoenflies system could be generally characterized by two numerical parameters, i.e., the minimum values of the degrees of pseudosymmetry both among all of its local maxima for turn operations (η_r) and mirror reflections (η_b). Analysis of Asterina amurensis as an example showed that the complete starfish metamorphosis could be represented by symmetry transformations in the form of the following series:
                                                                        С_4v → С_2v → C_s → C_5v,
which reflects the natural transition from rotational symmetry to bilateral and again to the rotational due to the biological characteristics of the organism at different stages of development. This series is consistent with the Curie principle: a system under external influence changes its point symmetry in such a way that only the symmetry operations in common with the symmetry operations of the influence are preserved. It is emphasized that exactly the group theory enables the characterization of an object’s invariance with respect to spatial transformations—in other words, its symmetry. In turn, the identification of invariants as a certain class of objects makes it possible to determine their structural basis and thus can help to find the invariable in the variable.