Three-periodic nets and tilings: regular and quasiregular nets.

Regular nets are defined as those with symmetry that requires the coordination figure to be a regular polygon or polyhedron. It is shown that this definition leads to five regular 3-periodic nets. There is also one quasiregular net with a quasiregular coordination figure. The natural tiling of a net and its associated essential rings are also defined, and it is shown that the natural tilings of the regular nets have the property that there is just one kind of vertex, one kind of edge, one kind of ring and one kind of tile, i.e. transitivity 1111. The quasiregular net has two kinds of natural tile and transitivity 1112.