Version of Zones and Zigzag Structure in Icosahedral Fullerenes and Icosadeltahedra

A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For odd-sized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are chosen alternately. Zigzag (Petrie) circuits in cubic (= trivalent) polyhedra correspond to alternating zones in their deltahedral duals. With these definitions, a full analysis of the zone and zigzag structure is made for icosahedral centrosymmetric fullerenes and their duals. The zone structure provides hypercube embeddings of these classes of polyhedra which preserve all graph distances (subject to a scale factor of 2) up to a limit that depends on the vertex count. These embeddings may have applications in nomenclature, atom/vertex numbering schemes, and in calculation of distance invariants for this subclass of highly symmetric fullerenes and their deltahedral duals.