Triply periodic minimal surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations.

Schwarz H surface

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.

Properties

Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).[5]

All connected TPMS have genus ≥ 3,[6] and in every lattice there exist orientable embedded TPMS of every genus ≥3.[7]

Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.[8]

History
Schwarz P surface

The first examples of TPMS were the surfaces described by Schwarz in 1865, followed by a surface described by his student E. R. Neovius in 1883.[9][10]

In 1970 Alan Schoen came up with 12 new TPMS based on skeleton graphs spanning crystallographic cells.[11] [12] While Schoen's surfaces became popular in natural science the construction did not lend itself to a mathematical existence proof and remained largely unknown in mathematics, until H. Karcher proved their existence in 1989.[13]

Using conjugate surfaces many more surfaces were found. While Weierstrass representations are known for the simpler examples, they are not known for many surfaces. Instead methods from Discrete differential geometry are often used.[5]

Families

The classification of TPMS is an open problem.

TPMS often come in families that can be continuously deformed into each other. Meeks found an explicit 5-parameter family for genus 3 TPMS that contained all then known examples of genus 3 surfaces except the gyroid.[6] Members of this family can be continuously deformed into each other, remaining embedded in the process (although the lattice may change). The gyroid and lidinoid are each inside a separate 1-parameter family.[14]

Another approach to classifying TPMS is to examine their space groups. For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.[8][15]

Generalisations

Periodic minimal surfaces can be constructed in S3[16] and H3.[17]

It is possible to generalise the division of space into labyrinths to find triply periodic (but possibly branched) minimal surfaces that divide space into more than two sub-volumes.[18]

Quasiperiodic minimal surfaces have been constructed in ℝ2×S1.[19] It has been suggested but not been proven that minimal surfaces with a quasicrystalline order in ℝ3 exist.[20]

External galleries of images
  • TPMS at the Minimal Surface Archive
  • Periodic minimal surfaces gallery

References
  1. "Mathematics of the EPINET Project".
  2. Deng, Yuru; Mieczkowski, Mark (1998). "Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis". Protoplasma. Springer Science and Business Media LLC. 203 (1–2): 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. American Chemical Society (ACS). 36 (16): 6171–6177. Bibcode:2003MaMol..36.6171J. doi:10.1021/ma0342933. ISSN 0024-9297.
  4. Mackay, Alan L. (1985). "Periodic minimal surfaces". Physica B+C. Elsevier BV. 131 (1–3): 300–305. Bibcode:1985PhyBC.131..300M. doi:10.1016/0378-4363(85)90163-9. ISSN 0378-4363. S2CID 4267918.
  5. Karcher, Hermann; Polthier, Konrad (1996-09-16). "Construction of triply periodic minimal surfaces" (PDF). Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. The Royal Society. 354 (1715): 2077–2104. arXiv:1002.4805. Bibcode:1996RSPTA.354.2077K. doi:10.1098/rsta.1996.0093. ISSN 1364-503X. S2CID 15540887.
  6. William H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975.
  7. Traizet, M. (2008). "On the genus of triply periodic minimal surfaces" (PDF). Journal of Differential Geometry. International Press of Boston. 79 (2): 243–275. doi:10.4310/jdg/1211512641. ISSN 0022-040X.
  8. "Without self-intersections". Archived from the original on 2007-02-22.
  9. H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
  10. E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal Flachen", Akad. Abhandlungen, Helsingfors, 1883.
  11. Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)"Infinite periodic minimal surfaces without self-intersections by Alan H. Schoen" (PDF). Archived (PDF) from the original on 2018-04-13. Retrieved 2019-04-12.
  12. "Triply-periodic minimal surfaces by Alan H. Schoen". Archived from the original on 2018-10-22. Retrieved 2019-04-12.
  13. Karcher, Hermann (1989-03-05). "The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions". Manuscripta Mathematica. 64 (3): 291–357. doi:10.1007/BF01165824. S2CID 119894224.
  14. Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006
  15. Fischer, W.; Koch, E. (1996-09-16). "Spanning minimal surfaces". Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. The Royal Society. 354 (1715): 2105–2142. Bibcode:1996RSPTA.354.2105F. doi:10.1098/rsta.1996.0094. ISSN 1364-503X. S2CID 118170498.
  16. Karcher, H.; Pinkall, U.; Sterling, I. (1988). "New minimal surfaces in S3". Journal of Differential Geometry. International Press of Boston. 28 (2): 169–185. doi:10.4310/jdg/1214442276. ISSN 0022-040X.
  17. K. Polthier. New periodic minimal surfaces in h3. In G. Dziuk, G. Huisken, and J. Hutchinson, editors, Theoretical and Numerical Aspects of Geometric Variational Problems, volume 26, pages 201–210. CMA Canberra, 1991.
  18. Góźdź, Wojciech T.; Hołyst, Robert (1996-11-01). "Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions". Physical Review E. American Physical Society (APS). 54 (5): 5012–5027. Bibcode:1996PhRvE..54.5012G. doi:10.1103/physreve.54.5012. ISSN 1063-651X. PMID 9965680.
  19. Laurent Mazet, Martin Traizet, A quasi-periodic minimal surface, Commentarii Mathematici Helvetici, pp. 573–601, 2008
  20. Sheng, Qing; Elser, Veit (1994-04-01). "Quasicrystalline minimal surfaces". Physical Review B. American Physical Society (APS). 49 (14): 9977–9980. Bibcode:1994PhRvB..49.9977S. doi:10.1103/physrevb.49.9977. ISSN 0163-1829. PMID 10009804.
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