Triply periodic minimal balance surfaces
with cubic symmetry

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Surfaces whose mean curvature H is everywhere zero are minimal surfaces; any sufficiently small patch cut from a minimal surface has the least area of all surface patches with the same boundary. The shapes taken by soap films are minimal surfaces.

The triply periodic minimal surfaces (TPMS) are particularly fascinating. A triply periodic minimal surface is infinitely extending, has one of the crystallographic space groups as its symmetry group and, if it has no self-intersections, it partitions space into two labyrinthine regions. Its topology is characterised by two interpenetrating networks - its 'labyrinth graphs'.

The first TPMS to be discovered and investigated was reported by Schwarz in 1856. He considered a soap-film across a quadrilateral frame, the edges of which are four of the six edges of a regular tetrahedron and realised that such a surface could be smoothly continued by joining the pieces edge to edge, the edges becoming two-fold [diad] axes of symmetry of the resulting infinite object. The surface is known as the D surface because its labyrinth graphs are 4-connected 'diamond' networksBy an ingenious application of a formula of Weierstrass, Schwarz was able to obtain an analytic expression for the D surface, and also for the P surface, whose labyrinth graphs are networks consisting of the vertices and edges of a primitive cubic lattice. His student Neovius discovered C(P), - the 'complement' of P, so-called because P and C(P) have the same symmetry group.

The next development in TPMS did not take place until the 1960s when A. H. Schoen investigated for NASA whether surfaces of this type might be of use as space structures and found more than a dozen new examples. Those surfaces with cubic symmetry are called (following Schoen's rather eccentric notation) IWP, FRD, OCTO, C(D), and G ('the gyroid').

An intensive search for new possibilities was taken up by Fischer and Koch from 1987. In a remarkable sequence of papers they systematically investigated the various ways in which frameworks of diad axes of the space groups can be spanned by minimal surfaces. The two labyrinths of a TPMS obtained in this way are necessarily congruent, since a 2-fold rotation about a diad axis embedded in the surface interchanges the two sides of the surface. Triply periodic surfaces with congruent labyrinths are called balance surfaces. Their symmetry properties are described by two space groups: G, the symmetry of the surface, and H (a subgroup of G of index 2), the symmetry of a single labyrinth. The gyroid is unique; it is a balance surface with no embedded diad axes - the transformations that interchange the labyrinths are inversion centres lying on the surface. Koch & Fischer listed exhaustively all pairs of space groups G/H which might have associated balance surfaces and found a large number of new triply periodic minimal balance surfaces (TPMBS). Those with cubic symmetry are called S, C(Y), ±Y, C(±Y), C(P)/H, C(D)/H and C(Y)/H.

In 1990 Gozdz and Holyst discovered two more cubic TPMS which they called BFY ('the Butterfly') and CPD. Karcher & Polthier [23] indicated how in certain cases more complicated variants of known surfaces could arise by inserting extra tunnels in their labyrinths.

It has now become apparent that the nomenclature for minimal surfaces is getting out of hand. What is needed is a way of naming a surface so that the structure of the name reveals unambiguously which surface is being referred to - a system analogous to the Hermann-Mauguin symbols for space groups, or the 'inorganic gene' that employs Delaney-Dress symbols to specify triply periodic networks.

Fundamental units and surface patches

A 'fundamental region' or 'asymmetric unit' of a given space group is a polyhedron which, when copied by applying all the transformations of the group, produces a tiling of space, such that the only transformation of the group that leaves a tile invariant is the identity. The International Tables for Crystallography list the vertices and faces of a unit for each group. Obviously, the whole group is generated by the set of transformations that relate a unit to the units immediately surrounding it, sharing a face or part of a face. The importance of this concept lies in the fact that an asymmetric unit for a space group G contains a smallest repeating unit of any triply periodic structure with symmetry G. A crystal structure is described by giving the positions of atoms in such a unit. Here we insert into the asymmetric unit of a particular cubic space group a patch or element of surface of zero mean curvature, which is then repeated round, as in a kaleidoscope, to give a non-self-intersecting surface dividing all space into two regions. Inserting a piece of tube would generate 3-D knots or weaves which should, as well as the surfaces, have engineering applications. The patch of surface may be inserted in several different ways and may or may not be capable of refinement so that its mean curvature should become everywhere zero. A number of configurations may have to be tested to see whether they refine to a TPMS. Some candidates may become stuck in a stable local minimum and others may collapse in various ways.

We have asked ourselves the following question:

What possibilities exist for TPMBS with cubic symmetry, generated by structural units that span finite boundaries all of whose edges are straight?

In their 1996 paper Koch & Fischer completed their classification of minimal surfaces "generated from skew polygons that are disc-like spanned, from catenoid-like surface patches, from branched catenoids, from multiple catenoids, or from infinite strips." They went on to say: "It may, however, be possible to find 3-periodic minimal surfaces without self-intersection which contain straight lines and are generated from surface patches other than those described..." The present work is an exploration of these other possibilities.

The area of surface per cubic unit cell of unit edge characterises the various surfaces uniquely. In practice, calculations with the Surface Evolver programme (downloadable from Brakke's website) gives values for the area about 1 in 1000 parts larger than the exact values.

Eric A Lord & A L Mackay 2003

G-H symmetry Surface name References
Im3m-Pm3m (1) P Schwarz, H. Über Minimalflächen, Motatsber. Berlin Akad., Apr 1865; Gesammelte Mathematische Abhandlungen vol.1, Springer, Berlin 1890.
(2) C(P) Neovius, E. R., Bestimmung zweier speciellen Minimalflächen, Akad. Abhandlungen, Helsingfors 1883.
(3) Pa 'Schoen's Batwing'; see Brakke's website
(4) C(P)aGozdz & Holyst's BFY ('Butterfly'): Gozdz, W. and Holyst, R., From the Plateau problem to periodic minimal surfaces in lipids and diblock copolymers, Macromol. Theory Simul.,1996, 5, 321-332; 'Schoen's Manta'
(5) Pb 'Schoen's C21(P)'
(6) C(P)b Fischer & Koch's C(P)/H: Fischer, W. and Koch, E., Spanning minimal surfaces, Phil. Trans. R. Soc. Lond. A, 1996, 354, 2105-2142.
Pn3m-P43m
(7) P3a NEW
Pm3n-Pm3
(8) P2b NEW
Pn3m-Fd3m
(9) D Schwarz
(10) C(D) Schoen, A. H., Infinite periodic minimal surfaces without self-intersections, NASA Technical Report TN D-5541, Washington DC 1970.
(11) Da NEW
(12) C(D)a Brakke's Starfish31
(13) D3a NEW
(14) D2a NEW
(15) Dc NEW
(16) C(D)c Fischer & Koch's C(D)/H: Fischer, W. and Koch, E., Spanning minimal surfaces, Phil. Trans. R. Soc. Lond. A, 1996, 354, 2105-2142.
P4332-F4132
(17) D2c NEW
P43m-F43m
(18) F K Brakke
Fd3m-F43m
(19) DPa NEW
(20) PDa NEW
I4132-P4332
(21) C(Y) Koch, E. and Fischer, W., On 3-periodic minimal surfaces, Z. Kristallogr., 1987, 179, 31- 52.
(22) Yb NEW
(23) C(Y)b Fischer & Koch's C(Y)/H: Fischer, W. and Koch, E., Phil. Trans. R. Soc. Lond. A, 1996, 354, 2105-2142.
Ia3d-I43d
(24) S Koch, E. and Fischer, W., Z. Kristallogr., 1987, 179, 31-52.


Links

Surfaces

Werner Fischer
Elke Koch
The contributions of Fischer & Koch to mathematical crystallography. Periodic minimal surfaces, sphere packings, Dirichlet domains, normalizers, lattice complexes, description of crystal structures, International Tables. The information on TPMS in particular is very thorough and extensive.

Ken Brakke's website. Surface Evolver can be downloaded. An introduction to triply periodic minimal surfaces is given, with numerous illustrations, and source codes for use with Surface Evolver.

Parametrisation of the Costa surface, for Mathematica.

Graphitic triply periodic minimal surfaces.

Gozdz & Holyst's home page. Triply periodic surfaces.

Stephen Hyde's home page. List of publications. Downloadable reprints. Differential geometry, topology and non-euclidean geometry of 2d surfaces, and applications to condensed atomic and molecular crystals and liquid crystals, etc.

Indiana University Minimal Surface Gallery. 29 artistic ray traced images.

Konrad Polthier's home page. Books on mathematical visualization. Mathematical visualization software. Videos on minimal surfaces, soap films, and geodesics.

An introduction to minimal surfaces, by Hermann Karcher and Konrad Polthier.

Stewart Ramsden. Mesoscale research: Crystal frameworks and hyperbolic tilings; embeddings of 3D nets in triply periodic surfaces.

University of Granada. Minimal surfaces, constant mean curvature and related topics.

PDF and HTML versions of Wohlgemuth et al.Triply periodic bicontinuous cubic microdomain morphologies by symmetries. Macromolecules 34 (2001) 6083-6089.