Quasiperiodic icosahedral graphite sheets and high-genus fullereneswith nonpositive Gaussian curvature

A graphite model obtained by decorating a quasiperiodic icosahedral surface with heptagonal and hexagonal rings of ${\mathrm{sp}}^2$ hybridized carbon is presented. We have generated closed compact fullerenes with icosahedral symmetry, genus 11, without dangling bonds and no pentagonal rings (no positive curvature). Additionally, cubic fullerenelike structures (${\mathrm{O}}_{\mathrm{h}}$ symmetry) with genus 5, without dangling bonds and no pentagonal rings have also been obtained. Extending this idea, closed perforated tubules with no pentagonal rings and genus higher than 5 are also generated. The geometry and stability of these nanostructures are discussed.