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

Abstract

Symmetrical embeddings are given for multiply intergrown sets of some commonly occurring nets such asdia(diamond),qtz(quartz),pcu(net of primitive cubic lattice) andsrs(labyrinth net of theGminimal 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 netssql(square lattice) andhcb(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.