On the exotic fishes given to... geometry

Polyhedral forms are extremely widespread both in animate and inanimate nature. Thus, crystals occur as polyhedra only. Besides, these forms are quite common with various primitive organisms, i.e. icosahedral viruses, radiolaria and algae. Here we discuss the cases of exotic Boxfish and Porcupinefish. The specific morphology of the Boxfish reveals in polygonal osseous blades, covering its body. As for the Porcupinefish, its polyhedral approximation was observed via certain geometrical techniques applied. Namely, their spine bases were considered the Delaunay point (R, r)-systems. Consequently, the respective Dirichlet tiling proved to be quasifullerenes and analogous to the Boxfish morphology. This unexpected geometrical dualism of the two families corroborates their taxonomic affinity within the Tetradontiformes order. The above biometrical method is highly recommended as a means of characterization of the Tetraodontiformes specimens in terms of the Delaunay (R, r)-systems

Crystal morphology of spherical viruses

The article discusses modern views on the structure of spherical virus capsids, which have the shape of icosahedrons (icosahedral viruses). Each face of icosahedron is composed of a single-layer closest packing of protein globules, which can have different orientation relative to the edges of icosahedron. If the lines of globules are parallel to the edges of icosahedron, then the capsid has a point symmetry group Ih (with symmetry planes), if they are not parallel – the symmetry group I (without planes). From a mathematical point of view, in both symmetry groups there are series that unite equally (up to similarity) arranged capsids. They are connected pairwise by transitions to dual forms (homologous series). A hypothesis is formulated that the largest spherical viruses can have even more diverse and complex capsid structures. Along with icosahedron, their basic forms can be any simple shapes, allowed in Ih and I symmetry groups (8 in total). A suggestion is made that transitions within similarity series and between homologous series have a phylogenetic significance. There are known spherical viruses of both symmetry groups. For example, the SARS-CoV-2 coronavirus has a symmetry group Ih and belongs to a well-known series. The crystallographic approach allows to construct a strict morphological classification of spherical viruses. This is important for their early recognition and separate examination. The article demonstrates practical application of crystal morphology in the study of viral systems – an urgent problem of geoecology and life protection

Petrographic structures and Hardy – Weinberg equilibrium

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data

On double bonds in fullerenes

Various distributions of double carbon bonds in the fullerenes have been considered in the paper from the point that they are absent in the pentagonal rings. The appropriate classification of the fullerenes has been built. The results may be used when modeling the fullerenes of a given topology and calculating their physical-chemical propertie

On the age of sediments from the Sredni, Rybachy Peninsulars and Kildin Island (the Kola region) in connection with the finding of strata stromatolites

Substantiation of the age of Kildin, Volokov, Eina and Bargout formations from the Sredny and Rybachy Peninsulas and Kildin Island has been presented in the paper. The traces of microbial life as filamentous and ribbon-like forms were detected using the scanning laser microscopy in stratiform stromatolites of the Sredny Peninsula. It has been proved that the studied stratiform stromatolites cannot be referred to the formal Early Riphean species Stratifera flexurata Komar 196