High-symmetry embeddings of interpenetrating periodic nets. Essential rings and patterns of catenation.

Symmetrical embeddings are given for multiply intergrown sets of some commonly occurring nets such as dia (diamond), qtz (quartz), pcu (net of primitive cubic lattice) and srs (labyrinth net of the G minimal surface). Data are also given for all known pairs of nets which have edge-transitive self-dual tilings. Examples are given for symmetrical polycatenation of the 2-periodic nets sql (square lattice) and hcb (honeycomb). The idea that the rings that are the faces of natural tilings form a complete basis set (essential rings) is explored and patterns of catenation of such rings described.