NEGATIVE CURVATURE SURFACES IN CHEMICAL STRUCTURES

Regular tessellations of polygons are not only possible for flat plane s (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 correspond ing to the regular polyhedra), but also for surfaces of negative Gauss ian curvature (i.e., hyperbolic planes), of which the {7,3}, {8,3}, an d {6,4} tessellations an of greatest actual or potential chemical inte rest. However, it is not possible to construct an infinite surface wit h a constant negative Gaussian curvature to accommodate such tessellat ions 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, pat ches of varying negative curvature and constant zero mean curvature ca n be smoothly joined to give various infinite periodic minimal surface s (IPMSs), which have zero mean curvature and are periodic in all thre e directions. The unit cells of the simplest IPMSs have genus 3 so tha t the unit cells of the {7,3}, {8,3}, and {6,4} tessellations on such IPMSs can be shown by a generalization of Euler's theorem to contain 2 4 heptagons, 12 octagons, and 8 hexagons, respectively. The {7,3} and {8,3} tessellations on suitable IPMSs can be used to derive possible s tructures of low-density polymeric carbon allotropes. Crystallography in the hyperbolic plane based on the {6,4} tessellation embedded in si milar IPMSs has been used by Sadoc and Charvolin to model the properti es of bilayers in liquid crystal and micellar structures.