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.

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关于一些水生生物的对称性转化的群论分析

摘要—在原始BioPsLeaf和BioPsFlower中首次进行了共生藻类，芽孢杆菌属，棘皮动物和小行星类水生生物二维图像假对称性的群理论分析以及棘皮动物个体发育中的对称性转化。软件，以及分析结果如下。已出版的材料，包括Haeckel的《自然之书》中的图形插图是这项研究中使用的水生生物二维图像的来源。水生生物的选择在很大程度上取决于居里原理，该原则考虑到特定的栖息地，对对称生物群施加了限制。对所考虑类别的生物进行的分析表明，可以由Schoenflies系统的C _nv组操作大致描述的生物对象的不变性（对称性）通常可以通过两个数值参数来表征，即度伪对称的两个之间的所有其局部最大值为转弯操作（η的_[R ）和镜反射（η _b）。紫ster分析如显示一个例子，该完整海星变态可以通过对称变换在以下系列的形式来表示：С _{4 v} → С _{2 v} → ç_小号→ Ç _{5 v}由于生物在不同发育阶段的生物学特性，它反映了从旋转对称到双侧的自然过渡，再到旋转。该系列与居里原理一致：在外部影响下的系统以这样的方式更改其点对称性，即仅保留与影响的对称操作相同的对称操作。要强调的是，正是群论使对象相对于空间变换（即对称）的不变性得以表征。反过来，将不变式识别为某一类对象可以确定其结构基础，从而有助于找到变量中的不变式。