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.
More curvature posts:
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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?