Three-periodic nets, tilings and surfaces. A short review and new results

A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.