# Chia-Chin Tsoo & Bih-Yaw Jin

We are chemists who are interested in topologically nontrivial structures inspired by fullerenes and graphene. Using the angle-weave technique of mathematical beading, we realize that it is possible to construct robust models of approximate 3D curved surfaces for arbitrary sp2-hybridized graphitic structures with only beads and strings. We show here two bead sculptures, the first model corresponds the 3D embedding of Felix Klein's all-heptagon network and the second model the hyperbolic soccerball.

The sculpture represents a periodic graphitic structure which approximates a genus-3 negative curvature D-surface decorated with the Felix Klein's open network consisting only of full helptagons. A hypothetical carbon allotrope based on this structure can be called a D56 protoschwarzite since there are 24 heptagons and 56 carbon atoms in a unit cell.

Buckyball or molecular soccerball is a spherical fullerene molecule with the formula C60, in which two neighbored pentagons are separated by a single carbon-cabon bond. Similarly, the hyperbolic soccerbal, or the D168 Schwarzite, is a hyperbolic graphitic structure, in which two neighbored heptagons are also separated by a single carbon-carbon bond. The hyperbolic soccerball is related to the D56 Schwarzite structure by a leapfrog transformation consisting of omnicapping followed by dualization.