Riemann Surfaces as Descriptors for Symmetrical Negative Curvature Carbon and Boron Nitride Structures

Sa？etak Leapfrog transformations starting with the genus 3 Klein and Dyck tessellations consisting of 24 heptagons and 12 octagons, respectively, can generate possible highly symmetrical structures for allo-tropes of carbon and the isosteric boron nitride, (BN)x. The Klein tessellation, alternatively described as a platonic {3,7} tessellation, corresponds to the Riemann surface for the multi-valued function w = 7√(z(z-1)2), which can also described by the homogeneous quartic polynomial ξ3n + η3ω + ω3ξ = 0. The symmetry of this polynomial is related to the heptakisoctahedral automorphism group of the Klein tessellation of order 168. Similarly the Dyck or {3,8} tessellation can be described by a Riemann surface which corresponds to the homogeneous Fermat quartic polynomial ξ4 + η4 + ω4 = 0. The symmetry of the Fermat quartic relates to the automorphism group of the Dyck tessellation of order 96