From the Cube to the Dyck and Klein Tessellations: Implications for the Structures of Zeolite-like Carbon and Boron Nitride Allotropes

Abstract

Geometrical transformations can be described which have the effect of multiplying the numbers of vertices in a trivalent polyhedron by three, four, or seven. Tripling the cube by the so-called leapfrog transformation gives the truncated octahedron. Quadrupling the cube followed by identifying the square faces to give a genus 3 surface gives the Dyck surface of 12 octagons. Septupling the cube by the so-called capra transformation followed by identifying the square faces to give a genus 3 surface gives the Klein surface of 24 heptagons. These geometrical transformations relate to the construction of low-density zeolite-like structures for carbon and boron nitride allotropes based on a cubic lattice