The Crystallography of the Hyperbolic Plane and Infinite Periodic Minimal Surfaces

Infinite periodic minimal surfaces are being now introduced to describe some complex structures with large cells, formed by inorganic and organic materials, which can be considered as crystals of surfaces or films. Among them are the spectacular cubic liquid crystalline structures built by amphiphilic molecules in presence of water. We study here the crystallographic properties of these surfaces, from an intrinsic point of view, using operations of groups of symmetry defined by displacements on their surface. This approach takes advantage of the relation existing between these groups and those characterizing the Wigs of the hyperbolic plane. First, the general bases of the particular crystallography of the hyperbolic plane are presented. Then, the translation sub-groups of the hyperbolic plane are determined in one particular case, that of the tiling involved in the problem of cubic structures of liquid crystals. Finally, it is shown that the infinite periodic minimal surfaces used to describe these structures can be obtained from the hyperbolic plane when some translations are forced to identity. This is indeed formally analogous to the simple process of transformation of an Euclidean plane into a cylinder, when a translation of the plane is forced to identity by rolling the plane onto itself. Thus, this approach transforms the 3-D problem of infinite periodic minimal surfaces into a 2-D problem and, although the latter is to be treated in a non-Euclidean space, provides a relatively simple formalism for the investigation of infinite periodic surfaces in general and the study of the geometrical transformations relating them.