The number of different unfoldings of polyhedra

Abstract

Given a polyhedron, the number of its unfolding is obtained by the Matrix-Tree Theorem. For example, a cube has 384 ways of unfolding (i.e., cutting edges). By omitting mutually isomorphic unfoldings, we have 11 essentially different (i.e., nonisomorphic) unfoldings. In this paper, we address how to count the number of nonisomorphic unfoldings for any (i.e., including nonconvex) polyhedron. By applying this technique, we also give the numbers of nonisomorphic unfoldings of all regular-faced convex polyhedra (i.e., Platonic solids, Archimedean solids, Johnson-Zalgaller solids, Archimedean prisms, and antiprisms), Catalan solids, bipyramids and trapezohedra. For example, while a truncated icosahedron (a Buckminsterfullerene, or a soccer ball fullerene) has 375,291,866,372,898,816, 000 (approximately 3.75 ×10 20) ways of unfolding, it has 3,127,432,220, 939,473,920 (approximately 3.13 ×1018) nonisomorphic unfoldings. A truncated icosidodecahedron has 21,789,262,703,685,125,511,464,767,107, 171,876,864,000 (approximately 2.18 ×1040) ways of unfolding, and has 181,577,189,197,376, 045,928,994,520,239,942,164,480 (approximately 1.82 ×1038) nonisomorphic unfoldings. © 2013 Springer-Verlag.