Space fullerenes: a computer search for new Frank–Kasper structures

Abstract

A Frank-Kasper structure is a 3-periodic tiling of the Euclidean space E3 by tetrahedra such that the vertex figure of any vertex belongs to four specified patterns with, respectively, 20, 24, 26 and 28 faces. Frank-Kasper structures occur in the crystallography of metallic alloys and clathrates. A new computer enumeration method has been devised for obtaining Frank-Kasper structures of up to 20 cells in a reduced fundamental domain. Here, the 84 obtained structures have been compared with the known 27 physical structures and the known special constructions by Frank-Kasper-Sullivan, Shoemaker-Shoemaker, Sadoc-Mosseri and Deza-Shtogrin.