%0 Journal Article
%T Three-periodic nets, tilings and surfaces. A short review and new results.
%A Delgado-Friedrichs O
%A O'Keeffe M
%A Proserpio DM
%A Treacy MMJ
%J Acta Crystallogr A Found Adv
%V 79
%N 0
%D Mar 2023 1
%M 36862044
%F 2.331
%R 10.1107/S2053273323000414
%X 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.