Opening, stellation and handle replacement of edges of the cube, tetrahedron and hexagonal prism: application to 3-connected three-dimensional polyhedra and (2, 3)-connected polyhedral units in three-dimensional nets

Abstract

Opening, stellation or handle replacement of edges of the regular cube, the regular tetrahedron and the semi-regular hexagonal prism (minimum point group symmetry 2/m) yields totals of 144, 11 and 205 geometrically distinct configurations respectively. Edge opening is restricted to a vertex connectivity of two or three. Sixty-five of the (2,3)-connected polyhedral units were found in real three-dimensional (3D) framework structures. Low-symmetry configurations are as abundant as high-symmetry ones. With minor exceptions, those (2,3)-connected patterns with the fewest converted edges are observed in each point group. The 113 polyhedra derived by double-stellation or double-handle replacement are uniformly three-connected. The ones observed in frameworks have a high symmetry. This study extends the general theory of 3D polyhedra and can be further expanded to other polyhedra and geometrical transformations. It may provide insight on the nature and growth of some seemingly complex framework structures that cannot otherwise be easily described topologically. The new polyhedral units are useful for classification of known framework structures in zeolites and related materials. New hypothetical nets generated from linkage of the units may solve unknown framework structures.