What do we know about three-periodic nets?

Abstract An account is given of various classifications of three-periodic nets. It is convenient to classify nets according to the nature of their maximum-symmetry embeddings. Other classifications, particularly in terms of the tilings that carry the nets, are also discussed. Although there is an infinity of possible nets, for certain types the number of possibilities is limited—there are for example exactly five regular nets. An account is given of the enumerations of various types of special structures such as sphere packings, the nets of simple tilings and self-dual tilings. Some databases of relevant structures and computer programs are described.

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