Negative Curvature Surfaces in Chemical Structures

Abstract

Regular tessellations of polygons are not only possible for flat planes (e.g., the {4,4}, {6,3}, and {3,6} tessellations) and the sphere (e.g., the {3,3}, {4,3}, {3,4}, {5,3}, and {3,5} tessellations corresponding to the regular polyhedra), but also for surfaces of negative Gaussian curvature (i.e., hyperbolic planes), of which the {7,3}, {8,3}, and {6,4} tessellations are of greatest actual or potential chemical interest. However, it is not possible to construct an infinite surface with a constant negative Gaussian curvature to accommodate such tessellations because the pseudosphere, the negative curvature “analogue” of the sphere, has an inconvenient cuspidal singularity that prevents it from being used to describe periodic chemical structures. However, patches of varying negative curvature and constant zero mean curvature can be smoothly joined to give various infinite periodic minimal surfaces (IPMSs), which have zero mean curvature and are periodic in all three directions. The unit cells of the simpl...