An icosahedral fullerene may be considered as a tessellation of the sphere specified by an ordered pair ofintegers, or as a tightly wound spiral of faces. Explicit analytical relations for interconverting the tworepresentations are given, enabling the canonical spiral code to be constructed for an icosahedral fullereneof any size. Analogous relations hold for the octahedral square + hexagon polyhedra that have been mentionedas possible candidates for boron−nitride “fullerenes”.