Three-periodic nets and tilings: regular and related infinite polyhedra

The six infinite regular (flag-transitive) polyhedra with finite faces of Grunbaum and Dress are described as tilings of the P and D periodic minimal surfaces. The three polyhedra formed by analogous tiling of the G surface are also described. The nets of these polyhedra are identified. It is shown how these polyhedra, and the nets they carry, could be found by mining the EPINET database of structures. The nets of regular three-periodic polyhedra with infinite helical or zigzag faces are also identified.