The Symmetry of Polyhedral Cages Made from Crystals

Laure Bourgeois¹, Leslie Bursill² and Yoshio Bando¹

¹National Institute for Research in Inorganic Materials, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
²School of Physics, The University of Melbourne, Parkville 3052, Victoria, Australia
E-mail: laure@nirim.go.jp

(Received November 19, 1999; Accepted January 18, 2000)

Keywords: Keywords: Aperiodic Crystal Structures, Mapping, Polyhedra

Abstract. Abstract. Geometrical and crystallo-chemical criteria for mapping real crystal structures onto various surfaces such as cylinders, cones, saddles and polyhedra are presented. In particular the requirements for retaining some of the original crystal symmetry elements when constructing cones and polyhedra from crystals are given, using real structures made of boron nitride, graphite, and aluminosilicates as illustration. This is used as the basis for building several polyhedral shell models for the aluminosilicate mineral allophane. Other possible shell structures are shown to exist provided some of the mapping requirements are relaxed.