We consider the geometries of hypothetical structures, derived from a graphite net by the inclusion of rings of seven or eight bonds, which may be periodic in three dimensions. Just as the positive curvature of fullerene sheets is produced by the presence of pentagons, so negative curvature appears with a mean ring size of more than six. These structures are based on coverings of periodic minimal surfaces, and surfaces parallel to these, which are known as exactly defined mathematical objects. In the same way that the cylindrical and conical structures can be generated (geometrically) by curving flat sheets so that the perimeter of a ring can be identified with a vector in the two-dimensional planar lattice, so these structures can be related to tessellations of the hyperbolic plane. The geometry of transformations at constant curvature relates various surfaces. Some of the proposed structures, which are reviewed here, promise to have lower energies than those of the convex fullerenes.

Introduction

The characteristics of the process of X-ray crystal structure analysis have led to an undue emphasis on classically crystalline materials to the neglect of organized structures which do not give simple diffraction patterns with sharp spots.

Gradually, even in the inorganic field, curved layers have become recognized as essential structural components. These were first recognized in asbestos and halloysite (Whittaker 1957; Yada 1971), where concentric cylinders and spiral windings of silicate sheets were disclosed. We can now begin to assemble the basic geometry of such curved structures under the rubric of ‘ flexi-crystallography’. This might be part of what de Gennes (1992) has called the study of ‘soft matter’, the main characteristics of which are complexity and flexibility.