The Mathematics of Zeolite-like Possible Carbon and Boron Nitride Allotropes

摘要

Structures for low-density zeolite-like possible carbon and boron nitride allotropes can be constructed from networks of trigonal (sp~2) carbon atoms containing hexagons, heptagons, and/or octagons. The most symmetrical such structures are obtained by leapfrog transformations of either the Dyck tessellation of 12 octagons or the Klein tessellation of 24 heptagons. The Dyck and Klein tessellations can be obtained by subjecting a cube to quadrupling (chamfering) and septupling (capra) transformations, respectively. Both the Dyck and Klein tessellations can be embedded into genus 3 infinite periodic minimal surfaces of negative curvature. The automorphism group, ~4O, of the Dyck tessellation is a solvable group of order 96, the structure of which relates to the dual of the Dyck tessellation as the regular tripartite graph K_(4,4,4). However, the automorphism group, ~7O, of the Klein tessellation is a simple group of order 168, which can be generated from the prime number 7 in a way analogous to the generation of the icosahedral rotation group I (≈ the alternating group A_5) from the prime number 5.