3-periodic minimal surfaces without self-intersections

Each 3-periodic minimal surface without self-intersections subdivides R³ into two 3-periodic spatial subunits, its labyrinths, interpenetrating each other in a more or less complicated way. Such a surface necessarily is orientable, i.e. its two sides may be coloured differently. The two labyrinths may either be congruent or they may differ. In the first case the surface is called a (minimal) balance surface, otherwise a (minimal) non-balance surface. These two cases will be treated separately. Most of the surfaces are illustrated by figures calculated by means of the computer program SURFACE EVOLVER (Brakke 1992).

(1) Balance surfaces

If the two labyrinths of a 3-periodic (minimal) surface without self-intersections are congruent, then there exist symmetry operations mapping one labyrinth onto the other. The symmetry of such a surface is adequately described not merely by a space group G but by a group-subgroup pair G-S of space groups with index 2: G is the group of all symmetry operations which map the surface onto itself, while S contains only those symmetry operations which map each labyrinth onto itself. The coset G\S then consists of all symmetry operations interchanging the two sides of the surface and the two labyrinths (Fischer & Koch 1987). This symmetry situation may also be described with the aid of black-white space groups (Mackay & Klinowski 1986).

If a symmetry operation g out of G belongs to the coset G\S, it follows that all fixed points of g must lie on the surface. As an intersection-free minimal surface cannot contain embedded mirror planes or 3-, 4- or 6-fold rotation axes, all types of space-group pairs G-S where G\S contains mirror reflections, 3-, 4- or 6-fold rotations or 6-fold roto-inversions are incompatible with minimal balance surfaces (Fischer & Koch 1987). This rules out 609 of the 1156 types of space-group pairs G-S (members of enantiomorphic pairs are not counted separately) as possible symmetry for a balance surface. 2-fold rotation axes, however, may lie in the surface which then spans in some way a frame of such axes. Such a surface, therefore, has been called a spanning minimal surface. According to the first reflection principle of Schwarz (1890), a minimal surface without self-intersections and which contains straight lines must be a minimal balance surface.

(1.1) Spanning surfaces

The possible symmetries for intersection-free spanning minimal surfaces are further restricted because the coset G\S must contain 2-fold rotations. There remain 352 types of space-group pairs. A closer inspection of these space-group pairs results in 52 patterns of 2-fold axes, called line configurations, which may be used as frameworks to be spanned by such minimal surfaces (Koch & Fischer 1988, 1993b). Especially two kinds of line configurations have been used for the crystallographic derivation of 3-periodic intersection-free minimal surfaces:

(1) 18 line configurations consist of 2-fold axes in three linearly independent directions which intersect each other such that the entire line configuration forms a three-dimensionally connected net with skew polygons as meshes.

(2) 12 line configurations consist of parallel plane nets of 2-fold axes. In 9 cases all nets are congruent, whereas 3 cases each contain two different kinds of alternating nets.

A line configuration may be spanned by a minimal surface. Depending on the connectivity of the line configuration, different kinds of surface patches may be used as starting units for the generation of infinite minimal surfaces by application of Schwarz's first reflection principle.

(1.1.1) Surfaces with disc-like surface patches

Three-dimensionally connected line configurations give rise to the simplest kind of surface patches, because they contain skew polygons, i.e. closed circuits of 2- fold axes, which may be spanned disc-like. According to the general solution of the Plateau problem, such a spanning by a patch of a minimal surface always exists. Therefore, any skew polygon found in a line configuration may be used to generate a 3-periodic minimal surface, but self-intersection will necessarily occur if too large a skew polygon is chosen. In order to avoid self-intersection two conditions must be fulfilled: (i) Each vertex angle of the skew polygon must be chosen to be as small as possible. Especially, no angle may be larger than 90°. (ii) None of the 2-fold axes out of the line configuration, i.e. no 2-fold axis not belonging to S, may run through the skew polygon.

In the following table the symmetry of the infinite surface generated from a skew polygon is described by a group-subgroup pair G-S. If S refers to a larger conventional unit cell than G, the cell enlargement is specified in parentheses. 2a, 2b and 2c mean a doubling of the corresponding lattice parameter, whereas v stands for the transformation a' = a - b, b' = a + b. It has to be noticed that the same minimal surface may be generated with the aid of different skew polygons referring to different space-group pairs. In such a case, there always exists a smallest surface patch, and all further surface patches result from uniting some of the smallest patches. As a consequence, there exists a subgroup relationship between the corresponding space-group pairs. Only the highest possible symmetry is given in the table. The minimal surface is identified by a symbol in the first column. These symbols come from different sources and therefore reflect different properties of the surfaces or their labyrinths.

Minimal surface	Space-group pair G-S	Polygon	Reference
P	Im-3m - Pm-3m	4-gon	Schwarz 1890, Fischer & Koch 1987
C(P)	Im-3m - Pm-3m	8-gon	Neovius 1883, Fischer & Koch 1987
D	Pn-3m - Fd-3m (2a)	4-gon	Schwarz 1890, Fischer & Koch 1987
C(D)	Pn-3m - Fd-3m (2a)	12-gon	Schoen 1970, Fischer & Koch 1987
S	Ia-3d - I-43d	12-gon	Fischer & Koch 1987
C(Y)	I4132 - P4332	9-gon	Fischer & Koch 1987
HS1	P6222 - P6122 (2c)	8-gon	Koch & Fischer 1988
HS2	P6222 - P3212	8-gon	Koch & Fischer 1988
CLP	P42/mcm - P42/mmc (v)	6-gon	Schwarz 1890, Koch & Fischer 1988
tD	P42/nnm - I41/amd (v, 2c)	5-gon	Schwarz 1890, Koch & Fischer 1988
oCLP	Pccm - Cccm (2a, 2b)	6-gon	Koch & Fischer 1988
oDa	Pnnn - Fddd (2a, 2b, 2c)	6-gon	Schoen 1970, Koch & Fischer 1988
oDb	Cmma - Imma (2c)	8-gon	Koch & Fischer 1988

(1.1.2) Surfaces with catenoid-like surface patches

If a line configuration consists of congruent parallel plane nets and if the polygon centers of neighbouring nets lie directly above each other, then a ring-like surface patch similar to a catenoid may span two such polygons. An example of a minimal surface generated from such a surface patch is the H surface of Schwarz (1890). Such a spanning exists in general as a solution of the anular Plateau problem, but only as long as the distance between the polygons is not too large compared to their size. This restriction corresponds to a limit for the axial ratio(s). For example, within the permissible range of the axial ratio there exists a whole family of H surfaces, rather than a unique 'H surface'.

Strictly speaking, the spanning of two circles by a minimal surface leads to two solutions if the distance between the circles is smaller than the limiting value. The catenoid with the smaller waist, however, is physically unstable and, therefore, cannot be formed by a soap film (or calculated by the SURFACE EVOLVER). Likewise there should exist an unstable catenoid-like surface patch spanning two polygons in addition to the stable one.

(1.1.3) Surfaces with branched catenoids as surface patches

If a line configuration can be decomposed into parallel plane nets of two different kinds, the corresponding polygons have different areas. The area of the larger polygon is an integer multiple, say n-fold, of the area of the smaller polygon. A ring-like surface patch spanning the gap between a small and a large polygon would result in a minimal surface with self-intersections. In all three cases of such alternating nets it is possible, however, to stretch a surface patch from one polygon of the wider net to the union of n polygons of the finer net. These n polygons share one vertex lying directly above or below the centre of the larger polygon. The resulting surface patch resembles a catenoid at one end, but it branches into n openings at the other end. Again, branched catenoids can exist only if an upper limit for the distance of neighbouring nets is not exceeded.

(1.1.4) Surfaces with multiple catenoids as surface patches

Multiple catenoids are also ring-like surface patches, but they are branched at both ends. A multiple catenoid, therefore, connects two congruent concave polygons, the self-contacting points of which must lie directly above each other. It may be visualized as resulting from the fusion of up to six neighbouring catenoids. Such surface patches can only be formed between congruent plane nets stacked directly one upon another, provided the distance between the nets is sufficiently small. Most of these surfaces have been derived independently by Karcher (1989) and Koch & Fischer (1989a).

(1.1.5) Surfaces with infinite strips as surface patches

If the fusion of two catenoids is repeated infinitely many times along a straight line, a channel bounded by two infinite strips results. The first boundary of such a strip may either be a straight line, a zigzag or a meander line, whereas the second boundary must be a zigzag or a meander line. Most such strips, however, may be subdivided by additional 2-fold axes into disc-like surface patches and, therefore, refer to previously known families of minimal surfaces. Only two strip-like surface patches give rise to new families. By contrast to catenoids, branched catenoids and multiple catenoids, for which the distance between the parallel nets must not exceed a certain limit, no such restriction applies to infinite strips: the larger the distance between the nets, the more the surface patch resembles a plane with two undulating rims.

(1.1.6) Surfaces consisting of catenoid-like surface patches with spouts

By analogy to the C(P) surface of Neovius (1883), Schoen (1970) derived a new family of minimal surfaces. He designated it as C(H) because such a surface spans the same line configuration as the H surface. The relationship between an H surface and a C(H) surface may be visualized as follows. The line configuration spanning an H or a C(H) surface consists of parallel triangular nets stacked directly upon each other. In an H surface, catenoid-like surface patches are located in every second trigonal prism between two adjacent nets. The corresponding C(H) surface results from an H surface by pulling three spouts out of each catenoid and uniting every three of them into a three-armed handle inside those trigonal prisms which are left empty by the H surface. The surface patch of a C(H) surface may therefore be considered as a catenoid with three attached spouts. In contrast to all other surface patches described so far, it is not entirely bounded by straight lines, but the spouts end on mirror planes, i.e. in plane lines of curvature.

A similar procedure has been applied to other catenoid-like surface patches resulting in five new families of spanning minimal surfaces. Rigorous proofs are lacking, however. It is remarkable, that attaching four spouts to the catenoids of a tetragonal P surface results in tetragonal symmetry of the surface even if the metric is cubic.

(1.1.7) Surfaces consisting of disc-like surface patches with spouts

In a C(P) surface there exist pairs of disc-like surface patches which share every second vertex and show a comparatively small distance between their middle regions, i.e. between their centring flat points. Accordingly, it is possible to connect such two discs by a handle. The resulting surface patch is a disc with one spout attached at its middle, again only partly bounded by straight lines. A similar situation occurs for C(D) and C(Y) surfaces. Like in the narrow catenoid the handle connecting two polygons has a rather slim waist and the surface patches are physically unstable, therefore.

(1.1.8) Surfaces with other surface patches

Catenoids and discs with spout-like attachments are examples of surface patches not entirely bounded by straight lines. There are other spanning minimal surfaces which also cannot be subdivided into finite surface patches with all boundaries formed by 2-fold axes.

(1.1.8.1) VAL surfaces
VAL surfaces are related to PT surfaces: In a PT surface a catenoid is continued across its two spouts and this procedure is repeated ad infinitum. The resulting infinite surface patch looks like a perforated tube. Each hole in the tube is formed by a rectangle of 2-fold axes and the holes are located pairwise at the opposite sides of the tube. A VAL surface likewise may be dissected into perforated tubes spanned between ribbons of rectangles, but here the holes on the two sides alternate. Accordingly, the symmetry of a VAL surface (Cmma-Cmma(2c)) differs from that of the corresponding PT surface (Fmmm-Cmmm), although both span the same line configuration. A simple finite surface patch of a VAL surface is a disc with eight vertices. Four of its edges are straight lines and four are plane lines of curvature. An infinite VAL surface is generated from such a surface patch by means of the two reflection principles of Schwarz (1890).

(1.1.8.2) Surfaces spanning isolated 2-fold axes
In the case of isolated 2-fold axes running in three linearly independent directions, surfaces have been constructed which contain these axes and seem to be minimal surfaces. Such a situation occurs for the group-subgroup pairs Ia-3 - Pa-3 and Ibca-Pbca. If a cube with 0 < x,y,z < ½ (referred to the conventional setting of Ia-3) is considered, then six skew 2-fold axes bisect its faces and run parallel to its edges. Inside such a cube a disc-like surface patch span the six 2-fold axes. In addition, the patch has six curved borderlines alternating with the straight boundaries. If such a surface patch is continued across the 2-fold axes, the resulting infinite surface acquires triangular holes (with curved edges) arranged in pairs. It is possible, however, to close up the two holes of each pair, either by a ring-like patch giving rise to the family of  ^±Y surfaces, or by two discs resulting in the family of C(^±Y) surfaces. Instead of generating an infinite surface from two different surface patches, one may use larger disc-like surface patches of only one kind, namely skew 18-gons with alternately one straight edge and two curved edges. ^±Y surfaces as well as C(^±Y) surfaces can be orthorhombically distorted resulting in o^±Y surfaces and in oC(^±Y) surfaces. Calculations with the SURFACE EVOLVER indicate that only C(^±Y) and oC(^±Y) surfaces are physically stable.

(1.2) Non-spanning surfaces

Stable non-spanning minimal balance surfaces are rare. The reason for this is that the two labyrinths in this case must be mapped onto each other by inversion or by 3- or 4-fold rotoinversion with the inversion centres on the surface. Such surfaces so far have not been derived using crystallographic arguments. For a long time the only example was the so-called gyroid surface (G surface or Y* surface) discovered by Schoen (1970). This surface is intermediate between the P and the D surface in the course of the Bonnet transformation for their Weierstrass functions. Described from the crystallographic point of view the symmetry of a gyroid surface is Ia-3d - I4₁32 and a 4-gon with alternating rotoinversion points -3 and -4 as vertices and congruent curved edges may be used as a surface patch. The two labyrinths are enantiomorphic. Schoen supposed that this type of minimal surfaces is the only one with enantiomorphic labyrinths. Fogden, Haeberlein and Lidin (1993) described a family of rhombohedrally distorted gyroid surfaces with symmetry R-3c - R32 and claimed the existence of further low-symmetry variants.

(2) Non-balance surfaces

Schoen (1970) also used the second reflection principle of Schwarz (1890) to derive 3-periodic minimal surfaces: Inside a 'kaleidoscopic cell', i.e. inside a polyhedron the faces of which correspond to mirror planes of a space group, a disc-like surface patch is fitted such that - along all its edges - it runs perpendicular to the faces of the polyhedron. Then the surface is generated by successive mirror reflections and also subdivides R³ into two mutually interpenetrating labyrinth. These labyrinths are different, however, unless a twofold axis or a rotoinversion center lies in the surface patch.

Last update: February 2002