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.

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 gallery by Ken Brakke [7]
TPMS at the Minimal Surface Archive [8]
Triply periodic minimal balance surfaces with cubic symmetry [9]
Periodic minimal surfaces gallery [10]
3-periodic minimal surfaces without self-intersections [11]

References

http://epinet.anu.edu.au/mathematics/minimal_surfaces
Yuru Deng and Mark Mieczkowski. Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis. Protoplasma, 203(1-2):16–25, 1998.
Novel Morphologies of Block Copolymer Blends via Hydrogen Bonding. Jiang, S., Gopfert, A., and Abetz, V. Macromolecules, 36, 16, 6171–6177, 2003
Alan L. Mackay, "Periodic minimal surfaces", Physica B+C, Volume 131, Issues 1–3, August 1985, Pages 300–305
Hermann Karcher, Konrad Polthier, "Construction of Triply Periodic Minimal Surfaces", Phil. Trans. R. Soc. Lond. A 16 September 1996 vol. 354 no. 1715 2077–2104 [1]
William H. Meeks, III. The Geometry and the Conformal Structure of Triply Periodic Minimal Surfaces in R3. PhD thesis, University of California, Berkeley, 1975.
Martin Traizet: On the genus of triply periodic minimal surfaces. Journal of Diff. Geom. 79, 243–275 (2008) [2]
[3]
H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal Flachen", Akad. Abhandlungen, Helsingfors, 1883.
Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)[4]
[5]
H. Karcher. The triply periodic minimal surfaces of A. Schoen and their constant mean curvature compagnions. Man. Math., 64:291{357, 1989.
Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006
W. Fischer & E. Koch (1996b): Spanning minimal surfaces. – Phil. Trans. R. Soc. Lond. A354, 2105–2142.
H. Karcher, U. Pinkall, and I. Sterling. New minimal surfaces in S3. J. Diff. Geom., 28:169–185, 1988.
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.
Wojciech T. Góźdź and Robert Hołyst, Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys. Rev. E 54, 5012–5027 (1996)
Laurent Mazet, Martin Traizet, A quasi-periodic minimal surface, Commentarii Mathematici Helvetici, pp. 573–601, 2008 [6]
Qing Sheng and Veit Elser, Quasicrystalline minimal surfaces, Phys. Rev. B 49, 9977–9980 (1994)

Mathematics Encyclopedia