3-periodic minimal surfaces with self-intersections along straight lines

In contrast to 3-periodic minimal surfaces free of self-intersections, 3-periodic minimal surfaces with self-intersections attracted little attention for a long time, probably because there exists an infinite variety of such surfaces. Most of them show too many self-intersections to be of general or even crystallographic interest. A special set of such surfaces stands out, however, namely those that intersect themselves exclusively along straight lines. Surfaces of that kind are relatively rare. 82 families of them could be derived and characterized using crystallographic methods: 78 families of minimal surfaces with disc-like surface patches (Koch & Fischer 1999, Koch 2000a,b, 2001) and 4 with catenoid-like surface patches (Fischer & Koch 2001). It turned out that these surfaces show some interesting properties:

(1) Some of them contain branch points, i.e. points around which the surface winds twice until it closes up. A branch point is generated at each vertex of a surface patch where the edges meet under 120°. Then the corresponding 2-fold axes are lines of self-intersection on one side of the branch point but run through the spatial subunit on the other side.

(2) Most of them are non-orientable like a Möbius strip.

(3) Though most of them subdivide R³ into two 3-periodic mutually interpenetrating labyrinths a variety of other spatial subunits occur. There also exist surfaces separating two different kinds of spatial subunits.

(4) For the generation of surface patches not only 2-fold, but also 4- and 6-fold rotation axes may be used which then become lines of self-intersection.

For each such surface the two sides of each surface patch may be coloured in such a way that the inside of each spatial subunit gets its own colour. This colouring scheme has been used for all corresponding figures. In the following, normal edges of the surface patches are drawn in black, lines of simple self-intersection in red, and lines where three, four or six parts of the surface intersect are marked in yellow.

Symbols for self-intersecting minimal surfaces have been constructed that give a rough classification. Each symbol starts either with OR for orientable or NO for non-orientable. Next the peridiocity of the spatial subunits is specified by one number (with a superscript if equal subunits run in different directions) or two numbers separated by a comma if the subunits differ. After a hyphen a small letter states the crystal system (c = cubic, h = hexagonal, t = tetragonal, o = orthorhombic), followed by a running number.

(1) Surfaces with disc-like surface patches

Most 3-periodic minimal surfaces with self-intersections along straight lines have been derived from disc-like surface patches spanning skew polygons. Examples are given by Stessmann (1934), Schoen (1970) and Karcher (1989) , but only Schoen described the geometrical properties of the one surface he had derived. He claimed that his surface is non-orientable and subdivides R³ into two labyrinths as surfaces without self-intersection do. The lack of any proof in his paper stimulated the development of methods to deduce the geometric properties of the surfaces from the crystallographic characteristics of their generating polygons (Koch & Fischer 1999).

(1.1) Surfaces forming 3-periodic labyrinths

Most of the new surfaces separate two 3-periodic mutually interpenetrating labyrinths from each other. Non-orientable as well as a few orientable surfaces of this kind have been found. In addition some non-orientable surfaces subdividing R³ into 4 or even into 8 labyrinths have been derived. Surfaces with 3 or 6 labyrinths seem not to exist.

(1.1.1) Non-orientable surfaces with 2 labyrinths

As for surfaces without self-intersection, the symmetry group U_lab of a labyrinth is a subgroup with index 2 of the symmetry group G of the surface. The symmetry of the surface with differently coloured labyrinths may be described by the group-subgroup pair G-U_lab or by the corresponding black-white space group.

(1.1.2) Orientable surfaces with 2 labyrinths

Different colouring of the two sides of such a surface exactly corresponds to colouring the two labyrinths differently. Accordingly, the symmetry situation is analogous to that of surfaces without self-intersection: The symmetry group S of the oriented surface, i. e. of the surface with two differently coloured sides, and the symmetry group U_lab of a single labyrinth are identical. The symmetry of the coloured surface may be described by the group-subgroup pair G-S or by the corresponding black-white space group.

(1.1.3) Non-orientable surfaces with 4 labyrinths

Each of these surfaces subdivides R³ into 4 congruent labyrinths. Accordingly, the symmetry group U_lab = I4₁22 of a single labyrinth is a subgroup with index 4 of the symmetry group G = P422 of the surface. There exist, however, two conjugate subgroups I4₁22 of P422. One of them maps e.g. labyrinths 1 and 3 onto themselves and interchanges labyrinths 2 and 4, whereas the other subgroup I4₁22 interchanges 1 and 3 and is the symmetry of labyrinth 2 as well as of 4. Any of the four labyrinths is adjacent to the two labyrinths that have the other labyrinth group. The symmetry of the surface with differently coloured labyrinths may be described by a four-colour space group. The respective permutation group of the colours has the order 8 and is isomorphic to the crystallographic group 422.

(1.1.4) Non-orientable surfaces with 8 labyrinths

Each of these surfaces subdivides R³ into 8 congruent labyrinths. Accordingly, the symmetry group U_lab of a single labyrinth is a subgroup with index 8 of the symmetry group G of the surface. Though the symmetry groups G differ in the two cases, the eight labyrinths of an NO3⁸-t1 as well as those of an NO3⁸-t2 surface correpond to four conjugate labyrinth groups U_lab, i.e. four times two of these labyrinths have identical labyrinth symmetry. In both cases, the symmetry of a surface with differently coloured labyrinths may be described by an eight-colour space group. The respective permutation group of the colours has the order 64 and is not isomorphic to any crystallographic group in R³.

(1.2) Surfaces forming 2-periodic (flat) labyrinths

In 14 cases the minimal surfaces with straight self-intersections subdivide R³ into infinitely many 2-periodic spatial subunits, called flat labyrinths. The symmetry of a single flat labyrinth is a layer group U_flab. All these surfaces turned out to be non-orientable. In all cases, the flat labyrinths of such a surface are congruent and all their labyrinth groups are conjugate subgroups of the symmetry G of the surface. Two different situations with respect to the mutual orientation of the labyrinths have to be distinguished.

(1.2.1) Surfaces with parallel flat labyrinths

In these cases, all the flat labyrinths run parallel. Each labyrinth is interwoven with its two neighbours.

(1.2.2) Surfaces with perpendicular flat labyrinths

In the other cases, there exist two infite subsets of flat labyrinths. Any two labyrinths of the same subset run parallel whereas any two labyrinths from different subsets run perpendicular to each other. Each surface patch separates labyrinths from different sets.

(1.3) Surfaces forming tubes or/and polyhedra

Some of the surfaces subdivide R³ into 1-periodic spatial subunits (called tubes), into finite spatial subunits (called polyhedra) or into a combination of both. Among these surfaces there are as well non-orientable as orientable ones. If self-intersection takes place along all the edges of the generating polygon then two different kinds of spatial subunits are produced. The symmetry of a single tube is a rod group U_t, that of a single polyhedron a point group U_p.

(1.3.1) Non-orientable surfaces

Surfaces NO1-hi and NO1-ti subdivide R³ into parallel congruent tubes that are branched, i.e. interwoven with three or four neighbouring tubes. By way of contrast, the congruent tubes of NO1³-h1 and of NO1²-t1 surfaces run in three or two symmetrically equivalent directions, respectively. The tubes of NO1³-h1 surfaces are unbranched, whereas those of NO1²-t1 surfaces are branched and interwoven. NO0-c1 surfaces are the only ones giving rise to congruent finite spatial subunits. Each such polyhedron is topologically equivalent to a sphere with 11 handles and is interlinked with six neighbouring polyhedra sharing the centre of the original polyhedron as common vertex. An NO0,0-c1 surface produces two different kinds of polyhedra, large and small ones.

(1.3.2) Orientable surfaces

All these surfaces subdivide R³ into two kinds of spatial subunits, either into two different kinds of tubes or into tubes and polyhedra or into two different kinds of polyhedra. Though these surfaces are orientable, different colouring of the two sides of a surface does not result in a uniform colouring inside all spatial subunits.

(2) Surfaces with catenoid-like surface patches

All minimal surfaces with catenoid-like surface patches result from spanning pairs of polygons from parallel Laves nets. If the polygons have vertices with an angle of 120° then the generated surface intersects itself along all polygon edges and possesses branch points. Each such surface is orientable and subdivides R³ into parallel flat labyrinths and tubes perpendicular to them. If the two sides of the surface are coloured differently then the outside of all catenoids show the same colour in one layer and different colours in adjacent layers. On the inside of each tube the colour changes from layer to layer in the contrary sequence.

Last update: March 2002