Exact construction of periodic minimal surfaces : the I-WP surface and its isometries

HAL Id: jpa-00212410

https://hal.archives-ouvertes.fr/jpa-00212410

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Exact construction of periodic minimal surfaces : the

I-WP surface and its isometries

S. Lidin, S.T. Hyde, B.W. Ninham

To cite this version:

801

LE

JOURNAL

DE

_PHYSIQUE

Exact construction of

_periodic

minimal surfaces : the I-WP

surface and its isometries

S. Lidin

_(1),

S. T.

_Hyde

₍₂₎

and B. W. Ninham

₍₂₎

(1)

Department

of

_{Inorganic Chemistry}

2, Chemical Centre, Lund

_University,

Box 124, S-22100

Lund, Sweden

(2)

Department

of

_Applied

Mathematics, Research School of

_Physical

Sciences, Australian National

_University,

Box 4, Canberra, Australia

(Reçu

le 3 août 1989,

accepté

sous

_forme

_définitive

le 8

_janvier

₁₉₉₀₎

Résumé. 2014 On

analyse

en détail la

_géométrie

de la surface minimale infinie et

_périodique

I-WP,

découverte par Alan Schoen dans les années 60. En utilisant les

équations

de Weierstrass locales pour la surface, on obtient une

_{paramétrisation}

exacte à l’aide

d’intégrales

hyperelliptiques

modifiées. A l’aide de méthodes

_{d’intégrations}

_{numériques rapides,}

le _rapport surface/volume de

cette surface est calculé ; il diffère de la valeur

_conjecturée

par Anderson. De

plus,

on trouve _que

la famille de surfaces minimales

_{isométriques}

reliées à la surface I-WP

_possède

une nouvelle suite de structures

_qui

_reproduisent

la surface I-WP elle-même.

Abstract. 2014 We have

analysed

in detail the _geometry of the I-WP infinite

_periodic

minimal

surface, discovered

_by

Alan Schoen in the 1960’s. An exact

_{parametrisation}

has been

found,

using

the local Weierstrass

_equations

for the _surface,

_involving

modified

_{hyperelliptic integrals.}

We have used

_{rapidly converging integration}

_techniques

to calculate the surface to volume ratio of this surface, and found that it differs from the value

_conjectured

_by

Anderson. _Further, the

_family

of isometric minimal surfaces related to the I-WP surface has been found to exhibit a novel sequence of structures ; viz.

repeated

formation of the I-WP surface itself.

J.

_Phys.

France 51

₍₁₉₉₀₎

801-813 1er MAI _1990,

Classification

Physics

Abstracts 02.40 - 61.30 - 68.00

Introduction.

Until

_recently

infinite

_periodic

minimal surfaces

_(IPMS)

_provided

a

_quiet

backwater for

_study

that

_{belonged exclusively}

to _pure mathematicians. This is no

_longer

so. It is now clear that a

deeper understanding

of minimal surfaces is central to _{progress in} microstructural sciences where the

_ubiquity

of bicontinuous media is now

_firmly

established.

Minimal surfaces and related surfaces of constant mean curvature occur in abundance in condensed matter.

_Examples

within the solid state are the multitudinous

_shapes

taken up

_by

silicates,

proteins

and enzymes, and those surfaces formed

_by

condensed

_phases

of

_synthetic

macromolecules like block

_copolymers.

Within the

_liquid

state, minimal surfaces occur and

recur in cell

membranes,

e.g. the

chloroplasts

of

_plant

cells,

and in the blue

_phases

of

thermotropic liquid crystals.

The

_{phase diagrams}

formed

_{spontaneously by}

_{surfactant-water,}

and surfactant-water-oil mixtures are rich in cubic

_phases

that

provide

a continuum of

structures as model

_systems.

Awareness of the

_diversity

of

_physical

realisations of bicontinuous structures is new. It is

not so

surprising

then that there is a dearth of fundamental

_geometric

studies of allowed bicontinuous structures. An infinite number of IPMS are known to exist

_[1].

Yet to date

_only

about

_thirty

such structures have been elucidated

_[2-14].

Further,

nothing

is known of bicontinous structures of low

_symmetry

and little of surfaces of

_{high topology. Among}

the

global

properties

of these surfaces necessary to our

_{understanding}

of the

_physics

of bicontinuous media are : surface to volume

_ratios,

isometric transformations

_(which

alter the

global

geometry

of these surfaces without

_affecting

the local

_geometry)

and Gaussian

curvature variations over the surface.

The field is open and awaits

_exploration.

We cannot even write down

_equations

for the cartesian coordinates of

_{high topology}

IPMS. In

short,

advances in the broad domain of natural sciences

_underpinned

and characterised

_by

bicontinuous microstructures will continue

to be held back until our

_knowledge

of minimal surface

_geometry

_improves.

This _{paper is} one

_attempt

to

_push

our

_{understanding}

of these surfaces a little further. We do so

_{by focussing}

attention on a

_single

_IPMS,

the I-WP surface. It has

_recently

been observed in a condensed

_phase

of star block

_co-polymers

_[15],

and is

_conjectured

as the structure of a cubic

_phase

of a

_ternary

surfactant mixture

[22].

In contrast to the « classical »

_periodic

minimal surfaces of

_Schwarz,

the I-WP surface contains no

_straight

_lines,

which

_implies

that the

_labyrinths

on either side of the surface are

_{geometrically}

different. The surface defines

body-centred

cubic

_{labyrinths :}

the tunnels on one side radiate from the cube centre towards the

_eight

vertices,

while the tunnels on the other side are connected at each cube

_face,

_forming

crosses on each face

_{(Fig. 1).}

_{This surface has genus 4}

_(and

_{is very} different to those

previously studied).

Apart

from

_eludicating

the structure of a new

_surface,

our

_study

has more

general

interest for two reasons :

_First,

it demonstrates the

_utility

of a

specific

construction

Fig.

1. -

algorithm

for minimal surfaces which admits calculation of the intrinsic

_geometric

parameters

of the surface - the metric and Gaussian curvature -

as well as the cartesian coordinates of the surfaces and the surface to volume ratio. This

_algorithm

had

_previously

been established

only

for low

_topology

IPMS

_{(genus three) [6, 7].}

Second,

the isometric

_family

of minimal surfaces related to the I-WP surface exhibits hitherto unknown reflexive

_{properties :}

the isometric

_bending

of this surface results in the same surface

_repeatedly.

The outline of this paper is as follows : in the

_following

sections we outline the

background

mathematics. Next we

_apply

our mathematical formalism to the I-WP surface under

investigation,

in order to derive the surface to volume ratio. We then

_analyse

associate surfaces to the I-WP surface.

Theory :

characterisation of minimal surfaces.

Minimal surfaces are surfaces with

_vanishing

mean curvature.

_They

are

_{consequently equally}

concave and convex at all

_points

on the

surface,

i.e. saddle

shaped.

It can be shown

_[16]

that such a surface isba local critical

_point

of the area functional with

_respect

to some

_boundary,

hence the

_etymology

of the term « minimal surface ». A minimal surface element can be

analytically

continued

_beyond

its

_boundary

to

_yield

an infinite

surface ;

such a continuation is

particularly

simple

when the

_boundary

consists of

_straight

lines and

_plane

lines of curvature.

These curves constitute two fold axes of rotation and mirror

planes

[2].

In order to

_generate

open infinite

periodic

minimal surfaces we must choose the

_boundary

_judiciously

to

_yield

a

$tructure

which is free of self-intersection and does not

_densely

fill _space.

The Cartesian coordinates of a minimal surface can be determined

_by

the local Weierstrass

equations [17]

Here w is a

_complex

valued

_variable,

_{R (w )}

is

_analytic

_except

at isolated

_points

and the

mapping

from the minimal surface to the

_w-plane

is a

_composite

_map

consisting

of a Gauss

(normal)

map from the minimal surface to the unit

_sphere

followed

_{by stereographic}

projection

onto the

_«plane.

This is

_clearly

a local

_{representation,}

since it

_requires

_unique

normal vectors and _{many different}

_points

on the _{minimal surfaces may have the} same normal

vector. Ifi the case of an IPMS every normal direction on a surface re-occurs an infinite number of times - at least

once _{every unit cell.}

The

_{general problem}

of

_deriving

new IPMS is to find the function

_R(w)

that

_yields

a

specific

minimal surface.

It has been shown

_by

Bonnet

_[18]

that if

_{R(w )}

is

_{replaced by}

ei 6

_{R (w ),}

the surfaces

generated by

different values of 0 are all isometric. This means that the Gaussian curvature is

independent

of 0,

and that two surfaces related in this _way can be transformed into each other

by

a

_{simple bending}

transformation. Two surfaces related to each other

_through

the isometric

property

of the Bonnet Transformation are said to be associated. If the

_angle

of association

(0 )

is a

_multiple

of

_{TT /2}

_(which

_corresponds

to

_multiplying

the

_integrals

in

_Eq.

₍₁₎

_by

the

From the work of Schwarz

[2]

one of us has

_conjectured

that

R (w )

could be

determined

from the flat

_points

of an

IPMS,

where the flat

_points

of a surface are

_points

with

_vanishing

Gaussian curvature

_{[19]. (The}

flat

_points

are

_locally

isometric to a

plane.)

The

_conjectured

_algorithm

is :

where

_wi

are the

_images

in the

_{complex plane}

of then flat

_points

of the surface. The

_degree,

bi,

of the Gauss map at the flat

_{point j}

is

_equal

to the ratio between the

_angle

of intersection between any two

_geodesics

_through

the flat

_point

and the _{Gauss map}

_image

of the same

_angle.

(The

degree

is one

_greater

than the branch

_point

order.)

The

_product

in

_equation

(2)

is taken over the

primitive

unit cell of the

surface ;

viz. the smallest

portion

of the IPMS which forms the full IPMS under translation alone. The

_complex

constant

K == r . ei6

_represents

a combination of a

_{simple scaling}

and a Bonnet transformation.

The choice of the

_{exponents - llbj}

_guarantees

that the _{Gauss map is} conformal at every

point.

Note that this does not

_require

the numerator of the

_exponent

in

_{equation (2)}

to be

_1,

any

integer

_Ni

such that

_{Ni bj}

and

_{Nj, bBi}

coprime

is sufficient.

In two _{earlier papers} we have demonstrated the

_utility

of the

_algorithm

_{[6, 7].}

Some constraints on the

_universality

of this

_algorithm

which had not been

_recognised

_previously

deserve note.

Firstly,

the Gaussian curvature,

K,

can be

_expressed

in the Weierstrass

_{parametrization}

as

Now if a

_point

on the IPMS

corresponding

to the north

_pole

on _{the Gauss map is} a

_regular,

non flat

_{point, K}

will have a

_finite,

non zero,

negative

value

_{( K ).}

This situation can

_always

be achieved

_by

a suitable orientation of the surface since flat

_points

are isolated. This means that :

(where

c is non

_positive

real

_numbers)

So,

the sum over the

_exponents

in

_{equation (2)}

is :

where

_Ni

are the numerators of the

_exponent.

A second constraint is that the Riemann surface must be a

_{regular covering}

of the Gauss

sphere.

In other

_words,

the unit cell must not contain

_points

with identical normal vectors and different Gaussian curvatures. If these constraints are met, the Riemann surface of the function

_{R (úJ )}

will

_correspond

to the

_primitive

unit cell of the IPMS.

topology.

The

_relationship

between the

_bj’s

and the _{genus per unit cell} of the IPMS is

_given

as

[19] :

where X

is the Euler characteristic of a unit cell of the surface.

If we consider the

_simplest

case

_only,

where all flat

_points

are of the same

_type

(all

Ni

are

_equal

and all

bi

are

_equal)

we

_get

the

_{following possible}

combinations of flat

point

order and surface

_topology

which are consistent with

_equations

(4)

and

(5) :

Table 1.

and so on

Thus,

this

_algorithm

admits direct

_computation

of the IPMS

_topology,

which is a crucial

« index » of the surface.

Further,

we can calculate the surface to volume ratio and associate surfaces from the

algorithm.

The normalized surface to volume

ratio,

A/V2J3

is another

_important

index of the surface. This

_quantity

can be determined to an

_{arbitrary degree}

_{of accuracy}

_using

our

parametrisation.

To

compute

this

_entity

we

_{employ a}

result

_{by Smyth}

[20].

A minimal

surface, S,

with

_boundary

on a

_simplex,

M

_obeys

the

_following

rule :

Where r is the inradius

₍₁₎

of

M,

and P and A are the

_perimeter

and area of S

respectively.

In

_general,

the

_boundary

of S in our

_{parametrisation}

consists of a set of

_plane

lines of

curvature.

Instead of

_calculating

the

_perimeter

of S

_directly,

we calculate that of the

adjoint

surface element S *.

Since S and S * are

isometric,

_{corresponding}

curves have

_{equal length,}

and so the

perimeters

of S and S * are

_equal.

Note however that the

quadrirectangular

tetrahedra that bound S and S * are of different size. Thus the inradius must be

_computed

from the

_original

cell

_{(bounding S).}

(1)

The inradius is defined to be the radius of a

_sphere

which touches each face of the

_bounding

The 1-WP surface.

With this

_background

established we

_proceed

to

_investigate

a

_single

IPMS,

known as the I-WP surface. This IPMS was first described

_by

Schoen

_[5]

who derived it as the

_adjoint

surface of a self

intersecting

IPMS

_{given by}

Stessmann

_[4].

It has been described since then

_by

von

Schnering

and

_Nesper

_[9]

in connection with the structure of

_BaCu02

and

_by

_Karcher,

who has derived a

_{global parametrization}

of the surface

[13].

The I-WP surface can be

_generated

by

the

_{propagation (using}

mirror

_reflections)

of an element inscribed in a

_{quadrirectangular}

tetrahedron.

_{(Fig. 2a),}

which is an almost

_asymmetric

unit for the space _group of the I-WP surface - Im3m.

Fig.

2. -

a)

Fundamental element of the I-WP

_periodic

minimal _surface, confined within a

quadrirec-tangular

tetrahedron. The faces of this

_{(kaleidoscopic)}

cell are mirror

_planes

for the minimal surface. Successive reflections in these faces and all new faces result in the

_complete

I-WP minimal surface,

which _separates two distinct

_labyrinths.

_b)

Fundamental element of the

_adjoint

minimal surface to the I-WP surface, first realised

_by

Stessman

_[4].

This unit is bounded

_by

four

_{straight edges}

of the

quadrirectangular

tetrahedron. The

_complete

_{(self-intersecting)}

surface is formed

_by

rotation of 180°

about each

_straight

_edge.

_c)

Coordinate axes for all

_geometric

calculations.

This surface element contains a two fold axis of rotation

_{perpendicular}

to the

surface,

but no

_straight

lines in the surface. This element will now be referred to as the

[undamehfal

element. The

_primitive

unit cell is

_composed

of 24

_equivalent

elements. Note that it takes 48 fundamental elements to build the

_body

centered cubic. unit cell of the I-WP surface.

tetrahedron is

_{TT /4}

-

an

_eighth

of the

_complete

flat

_point

-

so the

_primitive

unit cell contains 24. 2/8 = 6

flatpoints,

all

equivalent

due to the

_symmetry

of the surface.

We now

_proceed

to

_explore

_{the Gauss map in order} to determine whether the

_algorithm

described

_by

_equation

₍₂₎

can be valid and if so, to determine the function

_{R (w ).}

The shaded

_region

in

_figure

3 is _{the Gauss map of the} fundamental element. As is evident from the

_figure,

the

_images

of the dihedral

_angles

at the

_flatpoints

are both 3

TT /4,

so the

degree

of the Gauss _map at the flat

_point

is 3.

Fig.

3. -

Gauss map of the surface elements

depicted

in

_figures

la, lb

_(surface

normals

_mapped

onto

the unit

_sphere).

A

_mapping

of the whole

_primitive

unit cell onto the

_sphere,

results in the six flat

_points

being

distributed on the vertices of a

_regular

_octahedron,

so that the

_sphere

is covered three times.

Applying

equations (3)

and

₍₄₎

we find that

_Ni

= 2 and that the genus of the I-WP surface is

4.

_(This

is an

_example

of the second case in Tab.

_I.)

In order to

_{perform stereographic}

_projection,

the Gauss _{map is rotated}

_by

_7r/4

around the y axis thus

removing

the flat

point

from the north

pole.

The

image

subsequently

produced by

projection

onto the

_urplane

is

_given

in

_figure

4.

With the flat

_{point images}

at ± i and ±

à

± 1 we then have

This function indeed

_yields

the I-WP surface which follows

using

equation

(1)

so the

algorithm

holds for this case.

We now deduce some

_{geometric properties}

of the I-WP surface from the Weierstrass

equations.

We

_{firstly proceed}

to calculate the normalised surface to volume ratio. For reasons outlined

above,

we calculate the surface area of the surface element

_adjoint

to the I-WP element. The

_adjoint

_element,

S *

_belongs

to the Stessmann surface - a surface

containing

self-intersections

_{(Fig. 2b).}

The

side f

of the cube in which the

_{quadrirectangular}

tetrahedron

_(M)

_bounding

S is inscribed can be determined from the coordinates of the

_points

marked

₍₂₎

and

₍₄₎

of

Fig.

4. -

The Gauss map of a fundamental element of the I-WP surface

_projected

onto the

complex

plane

(after

rotation of the element shown in

_Fig.

_2),

in order to remove the branch

point

at

_infinity.

The

_perimeter

of S * is determined

_by

the coordinates of the

_point

marked

₍₂₎

in

_figure

_2b,

which is

By

applying

elementary

linear

_algebra

to the

_geometry

_depicted

in

_figure

2a we find

The inradius of M is determined in terms of

f

_by

_considering

a

_projection

of M onto its basal

Fig.

5.

_{- Projection}

of a

_{quadrirectangular}

tetrahedron onto its basal

_plane.

It is evident that the centre of the

_insphere

must lie on the two fold axis of the tetrahedron. This

_gives

us

The

_perimeter

of S * is

_{given by}

Applying equation (6)

we have

The area of one

(bcc)

unit cell of surface is

while the volume is

The normalised surface to volume ratio then follows as

This

_yields

the normalized surface to volume ratio as

This value does not

_{significantly}

differ from 2

f

and this leads us to

_conjecture

that it is indeed

analytically

equal

to that value. The

_computation

refutes the

_conjecture

made

_by

Anderson

_[22]

for the surface to volume ratio. This result is

_significant,

since it means that

normalised

surface to volume ratio

is,

in

_general,

not obtained from

_{complete elliptic}

integrals.

In

fact,

this is not

_{surprising, given}

that the Weierstrass

_equations

for an

_arbitrary

IPMS are not

_readily

reduced to

_elliptic

_integrals

_(whereas

those for the

P-,

F- and

_gyroid

surfaces

_are).

Associate surfaces to the I-WP.

Schoen discovered the I-WP surface

_by

Bonnet

_transforming

the self

_intersecting

IPMS first realised

_by

Stessman. Schoen also showed that there exists a

_countably

infinite number of

periodic

associates _{in every}

_family

of surfaces related

_by

the Bonnet transformation

_[5].

An

interesting

question

is if any of these surfaces are free of self

_{intersection,}

_yielding

new _open IPMS.

In

_general,

the issue is difficult to resolve.

_However,

this

_question

can be

investigated

for the I-WP

_{family by monitoring}

the

_positions

of the fourfold axes in a

projection

of the surface

during

the transformation.

A necessary condition for an associate minimal surface to the I-WP surface to be

_periodic

is that the two fourfold axes must coincide

_{(Fig. 6) ;}

i.e. the x- and z-coordinates must be the same for both axes.

It is well known that the coordinates of an associate minimal

surface, S’,

can be

_expressed

in terms of two

_adjoint

_surfaces,

_S,

_S*,

in the same

_{Bonnet-family by}

If S denotes the I-WP surface and S * the Stessmann

_{surface, then,}

_using

_equation

₍₁₇₎

when the two fourfold axes coincide.

Now

_{equation (18)}

is

_always

satisfied for the

_points

_chosen,

but

_equation

₍₁₉₎

is

_only

valid for certain discrete values of the

_angle

of

association,

8.

We have

with

_I1-

_I4

defined as above

According

to the

_{previous conjecture}

the

_angle

of association is

_exactly

_{TT /6,}

and indeed that appears to be correct, to seven decimal

_places.

Solving equation (1)

for this

_angle

of

_association,

we

_generate

a new

periodic

minimal surface. It turns out that this associate surface is non self

_{intersecting.}

_Surprisingly

this new surface is identical with the I-WP surface itself.

Thus,

the

_family

of associate surfaces to the I-WP surface has a

_period

of

_7r/3

in contrast to the usual

situation,

where a transformation

Wierstrass

_{parametrisation}

it is

_apparent

that the

_algorithm

cannot be

_universally

valid. In

particular,

we are restricted to _{surfaces whose Gauss maps}

_yield

a

_{regular covering}

of the

sphere.

In its

present

form the

_algorithm

cannot deal with IPMS

_{containing points}

that share the same normal vectors but whose local

_geometry

_{(curvature, metric)}

differs.

_(This

occurs,

for

_example,

in the Neovius

_surface.)

Work is

_underway

to resolve this

_problem.

We have demonstrated here the

possibility

of

_computing

the surface to volume ratio from the local Weierstrass

_generating

functions for the IPMS. It is clear that this

_global

ratio is not

simply

related to

_{complete elliptic integrals,}

a result that refutes the

conjecture

of Anderson.

Perhaps

the most

_surprising

result of our

_analysis

lies in the

_{self-similarity}

of

_periodic

associate minimal surfaces to the I-WP surface. In this case, the Bonnet transformation -which links new surfaces of identical local

_geometry

-

yields

the I-WP surface

_repeatedly,

rather than another IPMS

_globally

distinct from the I-WP surfaces. This

_{phenomenon implies}

a

_{unique stability}

property

of the I-WP surface in

_comparison

to all the IPMS studied so far. Local isometric distortions of I-WP surface result in the surface «

folding

back » into itself

repeatedly

rather than

_transforming

to new

_global

_geometry.

This

_property

_{may indicate} a

physical preference

for interfaces

_subject

to natural fluctuations to

_adopt

the I-WP

_geometry.

It is

_important

at this

_stage

not be become fixated on a

_single

IPMS.

_Accordingly,

this paper aims to demonstrate the

_utility

of our

_{generating algorithm}

for minimal

_surfaces,

as much as

revealing

the

_surprising

behaviour of the I-WP associate

_family

of IPMS.

_Many,

_many more

exist,

and this

_proof

of more

_{general validity}

of the

_algorithm

is critical.

References

[1]

PITTS J. P., RUBINSTEIN J. R.,

Application

of minimax to minimal surfaces and the

_topology

of

3-manifolds,

Proceedings

of the Centre for Mathematical

_Analysis,

Australian National

University,

Canberra

_(1986).

[2]

SCHWARZ H. A., Gesammelte Mathematische

_{Abhandlungen (Springer)}

1890.

[3]

NEOVIUS E. R.,

Bestimming

Zweier

_Speciellen

Periodische Minimalflächen

_(J.

C. Frenckel &

Sohn)

Helsinki,

1883.

[4]

STESSMAN B., Periodische Minimalflächen, Math. Zeit. 38

₍₁₉₃₄₎

417-442.

[5]

SCHOEN A. _H., Infinite

_periodic

minimal surfaces without self-intersections, NASA Techn.

Rep.

D-5541

_(1970).

[6]

LIDIN S., HYDE S. T., A construction

_algorithm

for minimal surfaces, J.

_Phys.

France 48

₍₁₉₈₇₎

15-20.

[7]

LIDIN S.,

Ring-like

minimal surfaces, J.

_Phys.

France 49

₍₁₉₈₈₎

421-427.

[8]

ANDERSSON S., HYDE S. T., LARSSON K., LIDIN S., Minimal surfaces and structures: from

inorganic

and metal

_crystals

to cell membranes and

_biopolymers,

Chem. Rev. 88

₍₁₉₈₈₎

221-242.

[9]

V. SCHNERING H. G., NESPER R., How Nature

_adapts

chemical structures to curved _surfaces,

Angew.

Chem. Int. Ed.

_Engl.

26

₍₁₉₈₇₎

1059-1080.

[10]

FISCHER W., KOCH E., On

_3-periodic

minimal

surfaces,

Z.

_Kristallogr.

179

₍₁₉₈₇₎

31-52.

[11]

KOCH E., FISCHER W., On

_3-periodic

minimal surfaces with non-cubic _symmetry, Z.

_Kristallogr.

183

₍₁₉₈₈₎

129-152.

[12]

KOCH E., FISCHER W., Acta

_Cryst.,

in press.

[13]

KARCHER H., Minimal surfaces, the

Morgan Principle

and the Riemann

mapping theorem,

manuscript (Bonn

and

_I.H.E.S.)

1987.

[14]

SADOC J. F., CHARVOLIN J., Infinite

_periodic

minimal surfaces and their

_{crystallography}

in the

[15]

ANDERSON D. M., STRÖM P.,

Polymerisation

of

_lyotropic

_liquid

_crystals,

in

_Polymer

Association

Structures, ACS

_Symposium

Series # 384, M. A.

_El-Nokaly

Ed.

_(American

Chemical

Society,

Washington)

1989.

[16]

LAGRANGE J. L., Essai d’une nouvelle méthode pour déterminer les maxima et les minima des formules

_intégrales

indéfinies, Miscellanea Taurinensia 2

_(1760-1761)

173-193.

[17]

WEIERSTRASS K., Math. Werke, Band 3

_(Mayer

u.

_Mayer,

_Berlin)

1903.

[18]

BONNET O. Note sur la théorie

_générale

des surfaces, C. R. Acad. Sci. Paris 37 529-532.

[19]

HYDE S. T., The

_topology

and geometry of infinite

_periodic

minimal _surfaces, Z.

_Kristallogr.

187

(1989)

165-185.

[20]

SMYTH B.,

Stationary

minimal surfaces with

boundary

on a

_simplex,

Invent. Math. 76

₍₁₉₈₄₎

411-420.

[21]

LYNESS J. _N., NINHAM B. W., Num Math. 8

(1966)

444 ; Math.

_Comp.

21

₍₁₉₆₇₎

162; Math.

Comp.

23

₍₁₉₆₉₎

71.