It’s been known for some time that carbon can form structures with positive curvature (fullerenes) and structures with zero curvature (graphene). Recently researches discovered a form of carbon with negative curvature (schwartzites). News story here.
Graphene is trivial, and fullerenes are pretty straightforward (i.e. tubes with possibly closed ends), but I’m having trouble visualizing the shape of large schwartzite molecules. The graphic above suggests a repeating pattern — is it just a bunch of hyperboloid structures that happen to meet at the edges in a way that aligns the bonds? Are there regions of positive curvature at the ‘joints’, or is it really constant negative curvature everywhere?