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

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 allotropes 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 z z ( – ) 1 2 7 , which can also described by the homogeneous quartic polynomial 3 3 3 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 4 4 4 0. The symmetry of the Fermat quartic relates to the automorphism group of the Dyck tessellation of order 96.