3-periodic minimal surfaces

A minimal surface in R³ is usually defined as a surface, the mean curvature H = (k₁ + k₂)/2 of which equals zero at each of its points. Accordingly, the two main curvatures k₁ and k₂ are equal in magnitude but opposite in sign, and the surrounding of each point is saddle shaped. In the exceptional case of k₁ = k₂ = 0 the curvature is zero in any direction and the saddle has more than two valleys and two ridges. Such points, known as flat points, are isolated points except for the plane.

Well-known examples of minimal surfaces are the catenoid and the helicoid. The catenoid is aperiodic, and the helicoid periodic in one direction. The first examples of 3-periodic minimal surfaces were given by Schwarz (1890), who originally presented his results in 1867 to the Berlin Academy, and by his pupil Neovius (1883). Later, remarkable pioneering work in the field was done by Stessmann (1934) and especially by Schoen (1970) (cf. Karcher 1989). Schoen made use of the two reflection principles of Schwarz (1890): (1) If part of the boundary of a minimal-surface patch is a straight line, then the minimal surface is smoothly continued across this line by 2-fold rotation. (2) If a minimal-surface patch meets a plane at right angles, then the minimal surface is smoothly continued across this plane by mirror reflection.

More recently, crystallographers became interested in 3-periodic minimal surfaces because of their relation to certain crystal structures (Andersson 1983; Andersson et al. 1984; Hyde & Andersson 1984, 1985; Mackay 1985). Based on Schwarz's first reflection principle and on crystallographic knowledge of the mutual orientation and position of 2-fold rotation axes in space groups, a systematic treatment of 3-periodic minimal surfaces containing straight lines was developed and new surfaces were derived (Fischer & Koch 1987, 1989a-c, 1990, 1992, 1996a,b, 2001; Koch & Fischer 1988, 1989a,b, 1990, 1993a,b, 1999; Koch 2000a,b, 2001 ).

A 3-periodic minimal surface in R³ is either free of self-intersections or may intersect itself in a more or less complicated way. Each 3-periodic minimal surface without self-intersections is orientable, i.e. two-sided, and subdivides R³ into two infinite connected disjunct regions. These two regions are not simply connected and interpenetrate each other in a complicated way. They are known as the labyrinths of the surface. If the two labyrinths are congruent, i.e. if there exist symmetry operations mapping one labyrinth onto the other, then the surface is termed a minimal balance surface.

On the other hand, a 3-periodic minimal surface with self-intersections may be either orientable or non-orientable, i.e. one-sided like a Möbius strip or a Klein bottle. Only 3-periodic minimal surface with self-intersections along straight lines have been considered so far in detail. Regardless of its orientability, such a surface also subdivides R³ into spatial subunits, but these may be either two or more 3-periodic labyrinths, or infinitely many 2-periodic labyrinths ('flat labyrinths'), 'tubes' or 'polyhedra'.

Schwarz (1894) was the first to pay special attention to 3-periodic minimal surfaces containing 2-fold axes. A characteristic feature of such surfaces is that in many cases their existence is guaranteed without sophisticated individual proof because of the general solution of the Plateau problem for the spanning of frames by minimal surfaces. For the sake of brevity 3-periodic minimal surfaces containing straight lines will be called spanning minimal surfaces. Note that spanning minimal surfaces without self-intersections are special cases of minimal balance surfaces. The gyroid surface derived by Schoen (1970) is an example of a minimal balance surface which is not a spanning surface.

The Euler characteristic c is a measure for the topological complexity of any surface in R³. c = f - e + v can be derived from any arbitrary tiling of the surface, where f is the number of faces (tiles), e the number of edges and v the number of vertices in the tiling. As c becomes infinite for any 3-periodic surface, it is necessary to refer c not to the entire surface, but to calculate c only for one primitive unit cell.

References

Andersson, St. (1983) Eine Beschreibung komplexer anorganischer Kristallstrukturen. Angew. Chem. 95, 67-80.

Andersson, St., Hyde, S. T. & Schnering, H. G. von (1984) The intrinsic curvature of solids. Z. Kristallogr. 168, 1-17.

Brakke, K. (1992) The Surface Evolver. Experimental Mathematics 1, 141-165.

Fogden, A., Haeberlein, M. & Lidin, S. (1993) Generalizations of the gyroid surface. J.Phys. I France 3, 2371-2385.

Fogden, A. & Hyde, S. T. (1992) Parametrization of triply periodic minimal surfaces. II. Regular class solutions. Acta Crystallogr. A48, 575-591.

Hyde, S. T. & Andersson, St. (1984) A systematic net description of saddle polyhedra and periodic minimal surfaces. Z. Kristallogr. 168, 221-254.

Hyde, S. T. & Andersson, St. (1985) Differential geometry of crystal structure descriptions, relationships and phase transformation. Z. Kristallogr. 170, 225-239.

Karcher, H. (1989) The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. Manuscripta Math. 64, 291-357.

Karcher, H. & Polthier, K. (1990) Die Geometrie von Minimalflächen. Spektrum der Wiss. 10, 96-107.

Mackay, A. L. (1985) Periodic minimal surfaces. Nature 314, 604-606.

Mackay, A. L. & Klinowski, J. (1986) Towards a grammar of inorganic structure. Comp. & Maths. with Appls. 12B, 803-824.

Neovius, E. R. (1883) Bestimmung zweier speciellen periodischen Minimalflächen. Helsingfors: Akad. Abhandlungen.

Schoen, A. H. (1970) Infinite periodic minimal surfaces without self-intersections. NASA Techn. note no. D-5541.

Schwarz, H. A. (1890) Gesammelte mathematische Abhandlungen, Band 1. Berlin: Springer.

Schwarz, H. A. (1894) Zur Theorie der Minimalflächen, deren Begrenzung aus geradlinigen Strecken besteht. Sitz.-Ber. Königl. Preuss. Akad. Wiss. Berlin, Phys.- Math. Cl. 1237-1266.

Stessmann, B. (1934) Periodische Minimalflächen. Math. Zeit. 38, 417-442.

Links

Last update: June 2002