Crystal structures built from highly symmetrical units

(Received 13 January 1992, accepted in final form 18 February 1992)

Abstract
Condensed matter physicists have recently recognised that systems as different as metal elements and metallic alloys with dense atomic packings, semiconductors with covalent tetracoordinated bonds, water clathrates with hydrogen bonds, liquid crystals of cholesteric molecules with double twist and liquid crystals of amphiphilic molecules at liquid/liquid interfaces, build crystal structures exhibiting particular characteristics : namely cubic symmetries with no directly apparent relation to local order cannot propagate over long distances in these systems, because of the filling requirements of Euclidean space. The conflict between those two equirements must be relaxed by defects and it is the organization of defects which builds the crystal structures. The common features of these structures can be understood, if it is recognised that in each case ideal structures without defects can be built in the same curved space, the hypersphere S3, the packing requirements of which are compatible with the different local orderings of these various systems.