Crystallographic Aspects of Minimal Surfaces

Symmetry properties of 3-periodic minimal surfaces subdividing R3 into two congruent regions are discussed. The relation between the order of a flat point and its site symmetry is established. Explicit formulae are given for the calculation of the genus of such a surface depending on the kind of surface patches that build up the surface. Making use of 2-fold axes that have to be embedded in a surface with given symmetry new families of minimal balance surfaces have been derived. Corresponding lists are referred to the different kinds of surface patches. 1 INTRODUCTION A minimal surface in R3 is defined as a surface the mean curvature H of which is zero at each of its points: H = 2 (kl + kz) = 0. 2 ~ h u s the two main curvatures k~ and kz are equal in magnitude but opposite in sign for each point of a minimal surface. From the crystallographic point of view those minimal surfaces are especially interesting which are periodic in three independent directions and, therefore, show space-group symmetry. Among them mainly those surfaces which are free of self-intersections seem to be of practical significance, e. g. as biological membranes or amphiphilic films. Each 3-periodic (minimal) surface without self-intersections subdivides R3 into two disjunct regions. Each such region is connected but not simply connected and forms a 3-periodic infinite labyrinth. The two labyrinths interpenetrate each other with the (minimal) surface as their common interface. Both labyrinths may be either congruent or different, examples for both being known for a long time past /1,2,3/. In the former case the (minimal) surface has special symmetry properties which may be used to derive new kinds of such surfaces. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990713 C7-132 COLLOQUE DE PHYSIQUE 2 SYMMETRY OF MINIMAL BALANCE SURFACES An intersection-free 3-periodic (minimal) surface that subdivides R3 into two congruent labyrinths has been called a (minimal) b a l a n c e s u r f a c e /g/. Each balance surface is uniquely related to a group-subgroup pair G-S of space groups with the following properties: G describes the full symmetry of the surface. Then an isometry of G either maps each side of the surface and each labyrinth onto itself or it interchanges the two sides and the two labyrinths. S is a uniquely defined subgroup of index 2 containing all those isometries of G that do not interchange the two sides of the surface and the two labyrinths. If the two sides of a balance surface are envisioned in different colours it is obvious that certain black-white space groups may also be used to describe the symmetry of such a surface /5/. The following considerations show that not all space-group pairs with index 2 are compatible with balance surfaces: If G-S describes the symmetry of a given balance surface then each symmetry operation g& G with g 4 S necessarily interchanges the two sides of the surface and, as a consequence, if there exist fixed points of g, the surface has to run through all these fixed points. That means, the corresponding symmetry element (rotation axis, mirror plane, rotoinversion point) in total has to be embedded in the surface. Especially, the following rules hold: (1) If giG, g d S is a mirror reflection, the corresponding mirror plane must be contained in any surface with symmetry G-S. As an intersection-free 3periodic surface cannot comprise an entire plane, G-S cannot be the symmetry of a balance surface. (2) If g,: G, g& S is a %-fold, 4-fold or 6-fold rotation (a 3-fold rotation cannot occur because of the subgroup index 2), the corresponding rotation axis must be entirely embedded within any surface with symmetry G-S. As a consequence, the surface shows self-intersection along this axis in case of a 4-fold or 6-fold rotation and again G-S cannot describe the symmetry of a balance surface. A 2-fold rotation axis, however, does not give rise to selfintersections and may be contained within a balance surface. As already Schwarz /l/ had shown, each straight line that is embedded in a minimal surface is a 2-fold rotation axis of this surface. (3) If g& G, g 4 S is a 7, 3-, or 2-operation, the corresponding rotoinversion point has to lie on each surface with symmetry G-S. 6operations can be excluded because 63 gives a mirror reflection [cf. (1) 1 . The compatibility with balance surfaces has been studied for all 1156 types(') of group-svbgroup pairs of space groups with index 2. For the reasons described above 509 of these types are incompatible with balance surfaces. The remaining 547 types are distributed on the crystal families (referred to G) as follows: 34 cubic, 67 trigonal/hexagonal, 168 tetragonal, 231 crthorhombic, 44 monoclinic and 3 triclinic ones / 6 / . For 352 of these 547 types G contains 2-fold rotations that do not belong to S, for 88 types-such 21fold rotations do not exist but G contains additional I-, 3-, or 4-operations in comparison with S. For the 107 types left there do not exist symmetry operations with fixed points that belong to G but not to S. (')Two group-subgroup pairs of space groups are assigned to the same type if the two groups as well as the two subgroups are mapped onto each other by conjugation with the same affine mapping. Enantiomorphic pairs are not distinguished. 3 FLAT POINTS OF MINIMAL BALANCE SURFACES A minimal surface in R3 fulfills the conditions k l+kz=O for each of its points (kl, k z main curvatures). For most points this defining condition holds with k i , k z + O . Then the area around the considered (ordinary) point has a saddlelike shape. For exceptional points, however, k i = k z = O holds. These points are called the flat points of the surface, As for any point of a minimal surface the relation I k ~ l = l k z l is fulfilled the set of all flat points coincides with the set of all points with zero Gaussian curvature k i k z . In the surrounding of a flat point a minimal surface shows j>2 valleys separated by j ridges. If a tiling on the surface is constructed such that all flat points lie on vertices and the edges are defined by lines of curvature that connect the flat points, then at least six tiles meet at each flat point. The best known example for a flat point is the "monkey saddle" with j = 3 . It has already been observed for the classical 3-periodic minimal surfaces of Schwarz /l/. For any point of an intersection-free minimal surface the degree of its flatness may be characterized by a non-negative integer 8 , called its order. Let PO be a point of a minimal surface and no the normal vector at that point. According to Hyde /7/ the order p of PO may be derived as follows: A second point P is moved on the surface around PO and the direction of its normal vector n is considered during this notion. If PO is an ordinary point, n rotates once around no during one revolution of P around PO. If, howaver, PO is a flat point, n rotates p>l times around no per one revolution of P. The order p of PO is then defined as P=p-l. Accordingly, an ordinary point has order P=0, whereas the order of a flat point may be any positive integer. For 3-periodic minimal surfaces flat-point orders up to 4 have been observed / 8 / . The number of valleys or ridges surrounding a flat point of order p is given by j=p+2. For (flat) points of given order $ 2 4 the geometrical situation is illustrated in Figs. 1 to 5. The diagrams refer to the maximal site symmetry compatible with the respective order. The left part of each figure shows the surrounding of the (flat) point. The arrows represent projections of the normal vectors of the surface. The right part illustrates in a stereographic projection the change of the direction of the normal vector along the closed path indicated at the left. The relation betwee% p and the maximal site sym~etry can easily be expressed if rotoreflection~ N insLead of rotoinversions N are considered: The maximal site symmetry is Nm2 or Nm for P even or 8 odd, respectively, with N=2j=2$+4. For 3-periodic minimal surfaces this maximal site symmetry can only be realized for $=0 or P=l. Flat points with any higher value of $ and maximal site symmetry occur, for example, on a special kind of l-periodic minimal surfaces, called saddle towers / g / . Fig. 1 Ordinary point with P=O and maximal site symmetry 4m2-2mm. COLLOQUE DE PHYSIQUE Fig. 2 Flat point with order 1 (monkey saddle) and maximal site symmetry 3m-3m. Fig. 3 Flat point with order 2 and maximal site symmetry zm2-4mm. Fig. 4 Flat point with order 3 and maximal site symmetry Sni-5m. The relation between the orders of (flat) points of minimal surfaces and their possible site symrnetries are summarized in Fig. 6. In analogy to the symmetry description of minimal balance surfaces with group-subgroup pairs of space groups, group-subgroup pairs of point groups are used, if the site symmetry of a (flat) point contains symmetry operations that interchange the sides of the surface. If, however, all site-symmetry operations preserve the sides of the surface, only one symbol is given. For minimal surfaces that subdivide R3 into two non-congruent regions only the latter case may occur. Fig. 5 Flat point with order 4 and maximal site. symmetry Em2-6mm. Fig. 6 Group-subgroup diagram showing the maximal site symmetry and all possible crystallographic site symmetries for (flat) points with order p 1 4 As Fig. 6 shows, most site symmetries of points on intersection-free minimal surfaces in R3 necessarily enforce these points to be flat points of some minimal order. On the other hand, only points with site symmetry 4m2-222, 4-2, 222-2, 2mm, 2, m or 1 can be ordinary points on minimal (balance) surf aces. The flat points of all minimal balance surfaces described so far have been tabulated by Koch & Fischer / a / . 4 GENERA OF MINIMAL BALANCE SURFACES A non-periodic surface in R3 has genus g, if it may topologicalLy be deformed into a sphere with g handles. Consequently, the genus of a 3-periodic minimal surface necessarily has to be infinite. Therefore, a modified definition for the genus of such a surface has been introduced /3/ counting only the number of handles per unit cell. In other words, the surface is embedded in a flat 3torus to get rid of all translations, and then the conventional definition of the genus may be applied. This procedure corresponds to identifying all opposite faces of a primitive unit cell. In case of a minimal balance surface obviously the unit cell used has to refer to the subgroup S, if the symmetry of the surface is described by the space-group pair G-S. Otherwise the C7-136 COLLOQUE DE PHYSIQUE identification process would not take into account the existence of two different sides of the surface and of two labyrinths. The genus of a 3-periodic minimal surface may be calculated in different ways, two of which will be dj.scussed in the following: (1) As has been proposed by Schoen /3/ for each of the two labyrinths associated with an intersection-free 3-periodic minimal surface a labyrinth graph may be constructed: each labyrinth graph is entirely located within its labyrinth; each branch of a labyrinth contains an edge of its graph; each circuit of a labyrinth graph encircles an edge of the other graph. Then, any of the two labyrinth graphs may be used to represent the surface. As each circuit of the graph corresponds to a handle of the surface, the number of circuits per unit cell (with respect to S in case of a minimal balance surface) or the number of circuits within the finite graph embedded in the 3torus has to be counted to get the genus. In a modification of a procedure proposed by Hyde /7/ a connected subgraph containing no tranlationally equivalent vertices may be separated from the labyrinth graph. Then, the genus may be calculated as r g = S + 2' where r is the number of edges connecting the finite subgraph to the rest of the infinite labyrinth graph and q gives the number of edges that must be omitted to make the subgraph simply connected. As r equals at least 6 the genus of a 3-periodic surface without self-intersection must be at least 3. Keeping in mind the embedding in the flat torus, a more crystallographic formula may be derived. The number of edges of the finite embedded labyrinth graph may be calculated from 1 e = Zmiei, 2 i where mi is the multiplicity of the ith kind of vertices (referred to a primitive unit cell of S), ei is the number of edges meeting in such a vertex and i runs over all symmetrically inequivalent kinds of vertices of the labyrinth graph. Then holds, where es is the number of edges in a finite, simply connected graph with the same number of vertices v. From