The classification of football patterns

Abstract

We prove that every spherical football (also known as a spherical soccer ball) is a branched cover, branched only in the vertices, of the standard football made up of 12 pentagons and 20 hexagons. We also give examples showing that the corresponding result is not true for footballs of higher genera. Moreover, we classify the possible pairs (k,l) for which football patterns on the sphere exist satisfying a natural generalisation of the usual incidence relation between pentagons and hexagons to k-gons and l-gons.