With the discovery of C-60 or Buckminsterfullerene a new kind of materials with important applications has emerged. Orginary graphite is composed of flat hexagonal layers. If pentagons are introduced, the graphite sheets start to curve in such a way that 12 pentagons are needed to close the structure and form fullerenes. Starting from the properties of 2D manifolds or surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfacds can be constructed. The D, G, P, H and I-WP type surfaces have been obtained from graphite-like sheets. In terms of the Gaussian curvature K, ordinary and cylindrical graphite have K=O, Fullerenes have K>O and triply periodic surfaces decorated with graphite have K<O. The stability of several negatively curved graphite structures has been measured using an empirical potential. The results show that these structures are more stable than C-60 Geometric and physical properties of hypothetical periodic graphite foams will be discussed.