A construction algorithm for minimal surfaces

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A construction algorithm for minimal surfaces

S. Lidin, S.T. Hyde

To cite this version:

S. Lidin, S.T. Hyde. A construction algorithm for minimal surfaces. Journal de Physique, 1987, 48

(9), pp.1585-1590. �10.1051/jphys:019870048090158500�. �jpa-00210592�

A construction algorithm for minimal surfaces

S. Lidin (*) and S. T. Hyde

(*) Inorganic Chemistry 2, ^Chemical Centre, ^{P.O. Box 124, S-22100} Lund, ^Sweden

(~) Department ^of Applied Mathematics, Research School of Physical Sciences, ^{P.O. Box 4, Canberra 2601,}

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

sont 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.

Abstract.

Infinite periodic minimal surfaces are constructed from complex functions, ^{which are} simply

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.

Classification

Physics ^Abstracts

02.40

61.30

61.50E

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 crystal-

line 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 infi- nite periodic minimal surfaces (IPMS). ^{With the} exception of the P and D surfaces (Schoen’s ^nomenc-

lature [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 devel-

oping 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 complex ^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}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090158500

1586

north pole ^{of the} sphere). The Gauss map results in

sphere coordinates :

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 projec-

tion from the north pole, giving :

where x" and y" ^are the real and imaginary ^axes respectively. These coordinates are usually ^reduced

to the single complex number,

i denoting ^the complex ^constant.

The R3 coordinates of the minimal surface are

related to the coordinates of the surface in the

complex plane by Weierstrass’ equations [12] :

where Re refers to the real part ^{of the} complex integrals.

These integrals ^are generally ^not analytically ^sol- vable, ^{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 can 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 con-}

struct 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 :}

Fig. 1. - The T surface, AP

^50.

Fig. 2. - The F surface, T ^{= 14.}

where ^K is a constant, Wa ^are the images of the flat

points ^{on the IPMS} (in ^the complex plane) ^and ba ^are 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 n-}

th 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

Fig. 3. - The T surfaces

^3.

Fig. ^{4. - The T} surface,

^2.01.

simple ^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 :

(i) ^{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.

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 ^1-

4 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. ^5.

The normal map of the flatpoints ^{to the} general

T surface.

The positions ^{of the} flatpoint images ^{are of the}

form :

giving ^{rise to a} Weierstrass function :

-1 1

Substituting 11 ^{for the} quadric ^{term, we write this}

as :

For symmetry ^{reasons the} ba will all be equal. ^Two special ^{cases are} of interest. The first emerges when A is set to B/2/ J3, giving ^a Weierstrass function of :

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

1588

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.

This 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}

form :

These result in Weierstrass functions :

Here ^we also encounter two interesting special

cases. When the axial ratio approaches infinity ^the

Fig. 6. - The CLP surface, 41

^0.

Fig. 7. - The CLP surface, 11 = 1.95, ^{K = 1.}

Fig. 8. - The CLP surface,1/’= 1.95, K

^i.

plane ^{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/ B/2), ^{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}

Fig. ^9.

The normal map of the flatpoints ^{to the} general

CLP surface.

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} tetragon- al distortion. Numerical calculations have confirmed this result. Indeed, setting ^{A to} (B/2 - 1 )/2 ^B/2

results in a surface which is perfectly self-adjoint.

This suggests that the surface generated ^from

K = 1, ^A =1 / J2 is identical to K

i, ^A

0, ^so

that this surface is adjoint ^to the Scherk surface.

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.

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 determining ^{the nature of}

the surface :

0 > ’ > - 2, ^K = 1 or i results in the CLP

surface,

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, knowing only the orientation of the flat points

^a parameter which is often constrained by the sym- metry of the surface.

Consequently, ^{we are at} last able to produce

IPMS conforming ^to space group symmetries ^exhi-

bited 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.

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

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[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.

[6] SCHOEN, A., NASA Technical Report ^{No. D-5541} (1970).

[7] ^{SCHWARZ, H. A.,} Gesammelte Mathematische Abhandlungen (Springer) ^1890.

[8] HYDE, ^{S. T. and} ANDERSSON, S., ^Z. Kristallogr. ¹⁷⁰

(1985) ^225-239.

1590

[9] HYDE, ^S. T., ^{to be} published.

[10] RUBINSTEIN, H., private communication.

[11] MEEKS, W., ^{Lectures on Plateau’s} Problem, ^Escole

de Geometria Differencial, Universidade Feder- al do Ceara (1978).

[12] NITSCHE, ^{J. C.} C., Vorlesungen ^über Minimalflächen (Springer Verlag, Berlin) ^1975.

[13] SPIVAK, M., ^A Comprehensive Introduction to Diffe-

rential Geometry (Publish ^{or Perish} Inc., ^Ber- keley) 1979, ^vol. IV, p. 400.

[14] WILLMORE, T. J., An Introduction to Differential Geometry (Oxford University Press) ^5th Ed., Delhi, 1985, p. 107.

[15] ^{LIDIN, S. and} ANDERSSON, S., ^{to be} published.