BY analogy with the positively curved carbon networks that comprise the fullerenes1-3, it has been suggested4 that negative curvature might be possible in graphitic carbon sheets, giving rise to extended structures corresponding to periodic minimal surfaces5 that divide space into two disjoint labyrinths. Whereas the positive curvature of fullerenes results from the presence of five-membered rings, negative curvature would derive from seven-membered rings. Here we present calculations of the cohesive energy and bulk moduli of two such hypothetical, negatively curved carbon networks. We find that both have a cohesive energy smaller than that of graphite but significantly greater than that of C60, even though the proportion of odd-membered rings is comparable. We therefore suggest that it is worth scrutinizing the insoluble residue generated in the carbon-arc preparation of fullerenes6 for possible evidence of fragments of negatively curved graphitic carbon. © 1992 Nature Publishing Group.