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

TL;DR

This paper presents a method for creating Penrose-type partitions of a sphere to model icosahedral quasicrystals, including detailed polyhedron models and boundary analysis in spherical space.

Contribution

It introduces a new construction of Penrose-type partitions on the sphere and links them to quasicrystal modeling with high-fidelity polyhedron models.

Findings

Constructed limit series of Penrose-type partitions on the sphere

Modeled quasicrystals with icosahedral symmetry Ih

Established boundaries of chromatic number in spherical space

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.