Triply periodic minimal surface

Schwarz H surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in that is invariant under a rank-3 lattice of translations.

These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.[1]

TPMS are of relevance in natural science. TPMS have been observed as biological membranes,[2] as block copolymers,[3] equipotential surfaces in crystals[4] etc. They have also been of interest in architecture, design and art.

  1. ^ "Triply Periodic Minimal surfaces". Mathematics of the EPINET Project. Archived from the original on 2023-02-28.
  2. ^ Deng, Yuru; Mieczkowski, Mark (1998). "Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis". Protoplasma. 203 (1–2). Springer Science and Business Media LLC: 16–25. doi:10.1007/bf01280583. ISSN 0033-183X. S2CID 25569139.
  3. ^ Jiang, Shimei; Göpfert, Astrid; Abetz, Volker (2003). "Novel Morphologies of Block Copolymer Blends via Hydrogen Bonding". Macromolecules. 36 (16). American Chemical Society (ACS): 6171–6177. Bibcode:2003MaMol..36.6171J. doi:10.1021/ma0342933. ISSN 0024-9297.
  4. ^ Mackay, Alan L. (1985). "Periodic minimal surfaces". Physica B+C. 131 (1–3). Elsevier BV: 300–305. Bibcode:1985PhyBC.131..300M. doi:10.1016/0378-4363(85)90163-9. ISSN 0378-4363. S2CID 4267918.