A soccer ball is made of pentagonal and hexagonal patches (12 regular pentagonal and 20 regular hexagonal faces – see Fig.-a). Such arrangement of polygons can cover (tile) the entire surface of a soccer ball without leaving empty patches, all due to its curved (spherical) surface.
Similar regular patterns can occur in the natural world such as colloids and granular systems, and there are pieces of evidence that regular patterns assembled on curved surfaces can spontaneously develop lattice defects to alleviate the stress imposed by the curvature. Crystals that assemble on closed surfaces are required by topology to have a minimum number of lattice defects. It is the formation of such defects that dictates the geometry and topology of patterns on curved surfaces.
The question we aim to answer is: how regular patterns can proceed on a curved surface such as sphere - the simplest curved surface on which it is impossible to eliminate such defects.
We will investigate the formation of patterns in granular systems with cylindrical (Fig.-b) and spherical (Fig.-c) geometries.