The construction of graphite and fullerene structures from networks of trigonal sp(2) carbon atoms of zero and positive curvature can be extended to a fourth form of carbon, provisionally called schwarzites, consisting of networks of trigonal sp(2) carbon atoms decorating infinite periodic minimal surfaces (IPMS’s) of negative curvature and topological genus 3 such as the so-called D and P surfaces. The carbon networks of the simplest schwarzite structures contain only six- and seven-membered rings analogous to the fullerene networks containing only six- and five-membered rings. The stable structures of both fullerenes and schwarzites contain enough six-membered rings so that no two five-membered rings in the fullerene structures or no two seven-membered rings in the schwarzite structures share edges leading to unstable pentalene and heptalene units, respectively. The smallest unit cell of a viable schwarzite structure of this type contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C-60 fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C-168 schwarzite unit cell has local O-h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C-60 fullerene considered as the isomorphous PSL(2,5) group. The porosity of the IPMS’s on which the schwarzite structures are based leads to predictions of unusually low density for this type of carbon allotrope.