[Essays on the application of nonclassical symmetries to natural biomorphs]

The following symmetries are conventionally termed "nonclassical": conformal symmetry, inversion symmetry, Mikheev homology, curvilinear symmetry, colour symmetry, and antisymmetry. These symmetries are applicable to biomorphs differing in linear dimensions and proportions (such as the shells of bivalves, crustaceans, diatoms, etc.) and to such three-dimensional biomorphs as gastropod shells. These three-dimensional objects used to be substituted so far with two-dimensional images. The shift transformation causing inhomogeneous deformations is discussed in the framework of Mikheev homologies. Antisymmetry and colour symmetry are discussed by the example of flowers, duckweed plants, and the crustacean chela. Early stages of cleavage, like colloid crystals, may be represented as polyhedrons. They have Euler characteristics and face symbols, and, because of this, stages of cleavage may have symmetries of crystals. Mastering nonclassical symmetries may promote the progress of biosymmetrics. The potential of discrete (arithmetical) biomorphology in taxonomy and the potential of continuous (geomertical) biomorphology in biosymmetrics are discussed.