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

Olaf Delgado-Friedrichs, Michael O’Keeffe, Davide M. Proserpio, Michael M.J. Treacy

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)192-202
Number of pages11
JournalActa Crystallographica Section A: Foundations and Advances
Issue numberPt 2
StatePublished - Feb 13 2023


  • 3-periodic nets
  • 3-periodic tilings
  • essential rings
  • nets
  • tilings

ASJC Scopus subject areas

  • Structural Biology
  • Biochemistry
  • General Materials Science
  • Condensed Matter Physics
  • Physical and Theoretical Chemistry
  • Inorganic Chemistry


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