arXiv subjects

Alexander S. Prokhoda

Publications and source records attributed to Alexander S. Prokhoda.

5 recordsLinked to original sources

arXiv2024

Structural Features of Quasicrystalline Nanoparticles With Seventh-Order Rotational Symmetry

In this study, we present atomic models of nanoparticles exhibiting seventh-order rotational symmetry. We established that the point group symmetry of these objects corresponds to the dihedral group D7v. To gain a deeper understanding of their structure, we performed calculations of the pair radial distribution function of the atoms. This data, along with the diffraction patterns derived from dual quasicrystals, indicates that quasicrystals have a distinct and pronounced diffraction pattern, confirming their crystalline nature and atomic-level orderliness. Furthermore, analysis of the spatial arrangement of atoms from the center to the periphery revealed that the atomic density within these nanoparticles is inhomogeneous. Specifically, there is a noticeable decrease in atomic density as one moves from the center of the crystal towards its periphery.

cond-mat.mtrl-sci↗

arXiv2019

About quasiperiodic tilings, possessing five-fold symmetry

A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral group of symmetry D5, special attention is paid to those that, can be obtained by one universal set, with consists five different tiles. Geometric characteristics of tiles from this set are determined. It is shown, by this set of tiles it is possible to carry out topologically different tilings of the plane, possessing rotating symmetries of both the fifth and tenth orders.

math.GM↗

arXiv2019

About discrete groups of symmetry of similarity of Euclidean space

Starting from the classical results of Shubnikov and Zamorzayev, computer models of shapes are implemented, which allow to visualize the action of discrete subgroups of continuous topological groups. The action is visualize by performing partitions of the shapes into the fundamental domains of the discrete symmetry groups of similarity, the definition of which was given by Zamorzaev. Particular attention is paid to the models of quasilattice, with the help of which such tiling of figures are constructed, that their multicolored coloring allows us to investigate, including the action of colored groups of symmetry of similarity. It is shown that for two-dimensional quasicrystals (quasi-lattices) the homothety coefficients are integers algebraic numbers of the quadratic expansion of the field of rational numbers. Some conformal mappings of crystal sets have been studied, which made it possible to establish the correspondence between the stereocyclic projection and the coordinate system of inverse radii.

math.MG↗

arXiv2018

Geometrical modeling of dendritic structures

Given recipe of qualitative, kinetic modelling by geometric methods of three-dimensional dendritic crystals. Characteristic features of the perturbations appearing on the surface of a spherical body, leading to different scenarios of the same equilibrium shape receiving of the same of crystal, are established. Fulfilled computer embodiment of such dendritic crystals which are growing both in conditions without convective flow and in the oncoming flow of liquid. Special attention is paid to topological characteristics of models with non-crystallographic point groups of symmetry. Surfaces of various genus, including 270, were investigated.

cond-mat.mtrl-sci↗

arXiv2018

About tilings of the type of Penrose of the two-dimensional sphere, which modellings quasicrystals

The problem of constructing a limit series of Penrose type partitions of a two-dimensional sphere is solved, which makes it possible to model quasicrystals possessing a point icosahedral group symmetry Ih. Images of polyhedron models are given (the number of faces for which F> 5000). Based on certain spherical isohedral polyhedra, a recipe is described for constructing spherical polyhedra of Plato, Archimedes, Catalan and Johnson. The boundaries of the chromatic number in the space S2 are established.

cond-mat.mtrl-sci↗