  We use ab initio density functional calculations to study the formation and
structural as well as thermal stability of cellular foam-like carbon
nanostructures. These systems with a mixed $sp^2/sp^3$ bonding character may be
viewed as bundles of carbon nanotubes fused to a rigid contiguous 3D honeycomb
structure that can be compressed more easily by reducing the symmetry of the
honeycombs. The foam may accommodate the same type of defects as graphene, and
its surface may be be stabilized by terminating caps. We postulate that the
foam may form under non-equilibrium conditions near grain boundaries of a
carbon-saturated metal surface.
