minimales classiques : la surface du diamant cubique (surface D) et la surface de Scherk. Abstract. 2014 Infinite periodic minimal surfaces are constructed from ...
J.
Physique
48
1585-1590
(1987)
SEPTEMBRE
1987,
1585
Classification Physics Abstracts 61.30 61.50E 02.40 -
-
A construction S. Lidin
(*)
algorithm for
and S. T.
minimal surfaces
Hyde
(*) Inorganic Chemistry 2, Chemical Centre, P.O. (~) Department of Applied Mathematics, Research
Box
124, S-22100 Lund, Sweden Physical Sciences, P.O. Box 4, Canberra 2601,
School of
Australia
(Requ
le 25
mars
1987, accept6 le 7 mai 1987)
Résumé.2014 Nous construisons des surfaces minimales périodiques infinies à partir de fonctions complexes qui simplement reliées à l’orientation des points plats sur la surface. Nous engendrons deux familles tétragonales de surface, dont nous montrons qu’elles se réduisent dans des cas particuliers à des surfaces minimales classiques : la surface du diamant cubique (surface D) et la surface de Scherk. sont
Infinite periodic minimal surfaces are constructed from complex functions, which are simply Abstract. related to the orientation of flat points on the surface. Two tetragonal families of surfaces are generated, which are shown to reduce in special cases to classical minimal surfaces : the cubic diamond surface (D surface) and the Scherk surface. In all cases the construction algorithm for the complex functions yields the expected results, supporting the validity of the procedure. The algorithm can be used to determine new periodic minimal surfaces. 2014
1. Introduction.
In recent years interest in translationally periodic minimal surfaces has risen considerably. Minimal surfaces have been found to be useful for describing such diverse phenomena as lyotropic liquid crystalline phases [1, 2], crystal structures [3, 4] and ion mobility in the solid state [5]. Progress in all these fields has been seriously retarded by the conspicuous paucity of known infinite periodic minimal surfaces (IPMS). With the exception of the P and D surfaces (Schoen’s nomenclature [6]), found last century by Schwarz [7] and Schoen’s gyroid [6] or (G surface) whose Cartesian coordinates have been computed explicitly [8], the fifteen-odd other known IPMS have been described in terms of their approximate boundary curves, or, at best, in some implicit parametrisation. In order to confirm the natural occurrence of IPMS in liquid and solid phases, more detailed knowledge of the multiplicity of surfaces of various space groups and the intrinsic geometry of these solutions is required. Surface coordinates permit direct comparison of crystal structures with IPMS, while computation of the surface area, volume fraction and curvatures is required to match IPMS with X-ray scattering data from cubic phases of surfactant/water/oil mixtures [9].
Evidence is mounting that many more IPMS exist than those so far discovered [10]. It is likely that many solutions exist within each space group [11]. Considerable progress has been made towards developing techniques for deriving appropriate boundaries for IPMS of a required symmetry. A method for generating the exact coordinates of IPMS has been conjectured [9]. In this paper we present some examples of the application of this method to tetragonal IPMS which support this conjecture. 2.
Theory.
The translational periodicity of IPMS invariably results in the surface being non-analytic in R3. Weierstrass established that any minimal surface (except the plane) can be expressed as a line integral of elliptic functions [12]. The integral is calculated in the plane, which is related to real space (R3) by a pair of mappings. The surface is mapped into a unit sphere under the Gauss transformation, which associates each point on the surface in real space with a point on the surface of the sphere which corresponds to the point of intersection of the normal vector of the surface (at the point to be mapped), centred at the origin of the sphere, with the unit sphere. (Thus, for example, a horizontal portion of the surface is mapped into the
complex
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090158500
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north
pole of the sphere). sphere coordinates :
The Gauss map results in
where the surface is parametrized in real space by f (x, y, z ) Const. The sphere coordinates are then transformed into the complex plane by standard stereographic projection from the north pole, giving : =
where x" and y" are the real and respectively. These coordinates are to the single complex number,
imaginary axes usually reduced Fig.
1. - The T
surface, AP
Fig.
2. - The F
surface, T = 14.
=
50.
the complex constant. coordinates of the minimal surface are related to the coordinates of the surface in the complex plane by Weierstrass’ equations [12] : i
denoting The R3
where Re refers to the real part of the
complex
integrals. These integrals are generally not analytically solvable, but once the IPMS is rendered into this form the Cartesian coordinates can be computed to any degree of accuracy. Thus the problem of determining . the geometry of the IPMS lies with a suitable technique for generating the « Weierstrass func-
tion », R (co ). This function
be written as a composition of two maps, one of which is the inverse of the Gauss map of the surface [13]. We suggest here that the Weierstrass function is simply related to the Gauss map of the surface on the sphere, considered as a Riemann surface. The Weierstrass equations imply that the function R (w ) diverges at a flat point, for which the equations are invalid [12]. We thus construct a complex function whose Riemann surface is equivalent to the Gauss map of the IPMS, with branch points corresponding to flat points on the surface. The prescription is : can
constant, Wa are the images of the flat the IPMS (in the complex plane) and the order of the branch points, the product taken over all flat point normal vectors in the IPMS. The branch point order is simply determined from the Gauss map in the vicinity of the flat point. If the normal vector traces out (n + 1) loops around a flat point on the sphere while traversing a single loop on the real surface in R3, the flat point gives rise to an nth order branch point. In some cases it is difficult to establish the presence of flat points. By combining various topological techniques this can be overcome [9]. A where
points ba are
K
is
on
a
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3. The T surface.
This surface is the earliest discovered IPMS. The boundary problem was proposed by Gergonne in 1816. The straight line boundary shown in figures 14 results in flat points as shown in figure 5 (by criteria (i)). As the vertical lines are stretched, the normal vectors to these flat points change smoothly, resulting in a distinct Weierstrass function for each tetragonal axis ratio. The Gauss map for a generic member of this family of surfaces is shown in figure 5.
Fig.
3. - The T
surfaces
=
3.
Fig. 5. The normal map of the T surface. -
The form :
positions
giving
rise to
of the
flatpoints
to the
flatpoint images
are
general
of the
Weierstrass function :
a
-11
Fig.
4. - The T
surface,
=
2.01.
Substituting 11 rule of thumb is that both asymptotic lines and lines of curvature on a minimal surface are everywhere orthogonal, except at flat points [14]. Whence :
for the
quadric term,
we
write this
as :
simple
If a straight line (an asymptote) intersects a plane line of curvature (curve of intersection of the surface with a mirror plane of the boundary) at right angles, the point of intersection is a flat point, and (ii) If two straight lines or two plane lines of curvature meet obliquely, their common point is a flat point. These criteria suffice to determine flat points for the IPMS described in this paper.
(i)
For
symmetry
special
A is set to of :
the ba will all be equal. Two of interest. The first emerges when giving a Weierstrass function
reasons
cases are
B/2/ J3,
If b = 1, this function reduces to the Weierstrass parametrisation for the P, D and G surfaces. Since the D surface is a special case of the T surface (with all three cell axes equal), all other cases of the T
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surface are homeomorphic to the D surface. Thus b =1 for general T surfaces, of arbitrary tetragonal axis ratios. When A is set to unity, the Weierstrass, function reduces to : .
which, for b
set to
unity, simplifies
to :
This is the Scherk surface, a celebrated analytic minimal surface, which represents the T surface whose tetragonal axis ratio tends to infinity. 4. The CLP surface. was discovered by Schwarz [7]. Once again, we can stretch the linear boundary unit along the tetragonal axis, forming distinct Weierstrass functions for each tetragonal axial ratio. Figures 6-9 show the resulting units of IPMS formed by varying this ratio, together with the general positions of the flat point normal vectors. The positions of the flat points will be of the
This surface
Fig.
7. - The CLP surface, 11 = 1.95,
Fig.
8. - The CLP surface,1/’= 1.95, K
K
= 1.
form :
These result in Weierstrass functions :
Here cases.
also encounter two interesting special When the axial ratio approaches infinity the
we
=
lines of curvature perpendicular to the stretched axis in figure 6 tend to straight lines and the surface approaches the Scherk surface. As can be seen from figures 6-9, the parameter A tends to zero, resulting in the Scherk surface Weierstrass function. At the other extreme (A = 1/ the Gauss map is identical to the vanishing A case, except for a rotation, which is due to different orientations of the surfaces. Two surfaces sharing the same Weierstrass function can only be distinguished through the multiplicative constant of the Weierstrass function, K. For varying real K, a multiplicity of surfaces are
plane
_
B/2),
Fig.
6. - The CLP
surface, 41
=
0.
i.
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The T and CLP surfaces can be described using the same Weierstrass function. This function is also suitable for describing the Scherk surface and a further analytic surface, probably the adjoint to the Scherk surface. The nature of the D surface as a special case of the family of T surfaces is clearly illustrated by the construction algorithm for minimal surfaces. The common Weierstrass function is :
where 11 is a parameter the surface :
0 > ’ > - 2,
K
determining
the nature of
= 1 or i results in the CLP
surface, The normal map of the 9. CLP surface.
Fig.
-
flatpoints to
the
general
produced which are related to each other by a scaling factor. If K is complex, varying the argument (without changing the modulus) results in distinct, isometric associate » surfaces. In particular, if K is purely imaginary, the resultant surface is said to be adjoint to the surface parametrised by the purely real K of the same modulus. This operation of changing the argument of K is known as the Bonnet transformation [15]. Studies of the CLP surface reveal that the surface formed by setting A equal to 1/ B/2 is distinct from «
«
»
the Scherk minimal surface. Schoen states that the CLP surface is self-adjoint [6], except for a tetragonal distortion. Numerical calculations have confirmed this result. Indeed, setting A to results in a surface which is perfectly self-adjoint. This suggests that the surface generated from K = 1, A is identical to K i, A 0, so =1 / that this surface is adjoint to the Scherk surface.
(B/2 - 1 )/2 B/2
J2
=
=
W = - 2, K = 1 or i gives the Scherk or adjoint Scherk surface, 41 - 2, K = 1 : gives the T family of surfaces (AP = 14 forms the D surface).
The validity of these calculations suggests that we have a powerful constructive procedure for IPMS. The detailed geometry of the surface is dependent only on the orientation and order of the flat points on the surface ; all other points on the surface are uniquely constrained by these singularities. The defining property of the IPMS flat points enables the generation of more complicated IPMS, a knowing only the orientation of the flat points which is often constrained the parameter by symmetry of the surface. Consequently, we are at last able to produce IPMS conforming to space group symmetries exhibited by solid and lyotropic liquid crystals, without recourse to previous « hit and miss » techniques. We expect this construction procedure to yield a plethora of new IPMS, and thus increase our poor vocabulary of translationally ordered surface structures. -
Acknowledgments. 5. Conclusions. The construction procedure for the Weierstrass function results in IPMS and analytical minimal surfaces (periodic in two dimensions) which are consistent with the expected results.
We thank the Swedish Research Council for financial support, which enabled us to carry out this work. One of us (S.T.H.) is grateful to this body for assistance as a visiting researcher in Lund. Thanks to Prof. Sten Andersson for useful discussions.
References
[1] HYDE, S. T., ANDERSSON, S., ERICSSON, B. and LARSSON, K., Z. Kristallogr. 168 (1984) 213-219. [2] HYDE, S. T., ANDERSSON, S. and LARSSON, K., Z. Kristallogr. 174 (1986) 237-245. [3] ANDERSSON, S., HYDE, S. T. and VON SCHNERING, H. G., Z. Kristallogr. 168 (1984) 1-17. [4] ANDERSSON, S., Angew. Chem. Int. Ed. Engl. 22 (1983) 69-81.
[5] ANDERSSON, S., HYDE, S. T. and BOVIN, J.-O., Z. Kristallogr. 173 (1985) 97-99. SCHOEN, A., NASA Technical Report No. D-5541 [6] (1970). [7] SCHWARZ, H. A., Gesammelte Mathematische Abhandlungen (Springer) 1890. [8] HYDE, S. T. and ANDERSSON, S., Z. Kristallogr. 170 (1985) 225-239.
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[9] HYDE, S. T., to be published. [10] RUBINSTEIN, H., private communication. [11] MEEKS, W., Lectures on Plateau’s Problem,
[13] SPIVAK, M., A Comprehensive Introduction to Differential Geometry (Publish or Perish Inc., Berkeley) 1979, vol. IV, p. 400. [14] WILLMORE, T. J., An Introduction to Differential Geometry (Oxford University Press) 5th Ed.,
[12]
Delhi, 1985, p. 107. S. and ANDERSSON, S., LIDIN, [15]
Escole de Geometria Differencial, Universidade Federal do Ceara (1978). NITSCHE, J. C. C., Vorlesungen über Minimalflächen (Springer Verlag, Berlin) 1975.
to be
published.
