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*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)

    Abstract

    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
    Pages (from-to)192-202
    Number of pages11
    JournalActa Crystallographica Section A: Foundations and Advances
    Volume79
    Issue numberPt 2
    DOIs
    Publication statusPublished - 13 Feb 2023

    Fingerprint

    Dive into the research topics of 'Three-periodic nets, tilings and surfaces. A short review and new results'. Together they form a unique fingerprint.

    Cite this