Junctions between carbon nanotubes and flat or curved graphene structures are fascinating for a number of reasons. It has been suggested that such junctions could be used in nanoelectronic devices, or as the basis of three-dimensional carbon materials, with many potential applications. However, there have been few detailed experimental analyses of nanotube-graphene connections. Here we describe junctions between nanotubes and graphene shells in a material produced by passing a current through graphite. Transmission electron micrographs show that the junction angles are not random but fall close to multiples of 30°. We show that connections with these angles are the only ones which are consistent with the symmetry of the hexagonal lattice, and molecular models show that a continuous lattice requires the presence of large carbon rings at the junction. Some of the configurations we propose have not been previously considered, and could be used to construct new kinds of three-dimensional carbon architecture. We also discuss the possible formation mechanism of the junctions.