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

Abstract

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.