About discrete groups of symmetry of similarity of Euclidean space

TL;DR

This paper visualizes the action of discrete symmetry groups of similarity in Euclidean space using computer models, focusing on quasicrystals and their algebraic properties, including tilings and conformal mappings.

Contribution

It introduces computer models for visualizing discrete similarity symmetry groups, including quasicrystal tilings and algebraic properties of homothety coefficients.

Findings

Homothety coefficients for 2D quasicrystals are algebraic integers.

Models of colored symmetry groups reveal complex tiling patterns.

Conformal mappings link stereocyclic projections to inverse radii coordinates.

Abstract

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.