Crystallographic aspects of the Bonnet transformation for periodic minimal surfaces (and crystals of films)

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Crystallographic aspects of the Bonnet transformation for periodic minimal surfaces (and crystals of films)

C. Oguey, J.-F. Sadoc

To cite this version:

C. Oguey, J.-F. Sadoc. Crystallographic aspects of the Bonnet transformation for periodic mini- mal surfaces (and crystals of films). Journal de Physique I, EDP Sciences, 1993, 3 (3), pp.839-854.

�10.1051/jp1:1993166�. �jpa-00246761�

Classification

Physics

Abstracts

02.40 61.50K 68.00

Crystallographic aspects ^{of the Bonnet} transformation for

periodic minimal surfaces (and crystals of films)

C.

Oguey (*)

and J.-F. Sadoc

Laboratoire de Physique des Solides (**), Bit. 510~ Universitd de Paris-Sud, F-91405 Orsay, France

(Received 2

July

1992,

accepted

in final

form

29 October1992)

Abstract. The surfaces with

vanishing

_mean curvature, or minimal surfaces, _are linear

projections

of minimal surfaces imbedded in the

complex

3 dimensional space C~. For real minimal su~aces, the Bonnet transformations form _{a one} parameter group of isometries which correspond

to generalised ^{rotations in} the complex _or higher dimensional space. The translation symmetries of fully

periodic

minimal su~aces in both R~ and C~

are

investigated,

with emphasis _on the example of the P, D and G su~aces. The relevance of the Bonnet transformation for physical transitions is

then discussed in the

light

of

higher

dimensional crystallography.

1. Introduction.

Minimal surfaces _are characterized

by

^their mean curvature H

being

_zero. In

simple

_cases,

H

= 0 is the

Euler-Lagrange equation

of _a variational

problem

attributed _to Plateau _:

given

_a

regular

closed _{curve c} in

R~,

find the surface bounded

by

_c with minimal _area.

Locally,

smooth

surfaces _are

graphs

_{Over convex} domains and minimal surfaces _are

unique

solutions _to the

problem

of

minimizing

_area in this context.

Nowadays,

unbounded minimal surfaces _are also used to model the

shape

^{of films} and interfaces in micro-emulsions and in structured

phases appearing

in _some mixtures of

liquids,

the prototype

being

water + soap

(and eventually oil) [I].

In definite

regions

of the temperature concentration

diagram,

the

phase (a

so-called

mesophase)

has cubic _or

hexagonal

symmetry and the full space group, as found

by X-ray analysis,

coincides with _one of the groups of

known and classified minimal surfaces

[2].

In _{some cases,} Fourier transform of the diffraction

(*) Permanent address PST, Universit£ de

Cergy-Pontoise,

47 _av. des Genottes, F-95806

Cergy-

Pontoise, France.

(**) CNRS URA 02.

840 JOURNAL DE PHYSIQUE I N° 3

data _or direct electron

microscopy

_on

quenched samples

reveal local features close to some

minimal surfaces

[3].

In

mathematics,

minimal surfaces constitute _a classical

subject,

_so the literature is extensive

j4].

When the surface M is

parametrized by

^isothermal

coordinates,

the

equation

H

= 0 is

equivalent

to the

Cauchy-Riemann equations

for the standard first differential form _on M. So the natural

geometry

^{for such}

Objects

is conformal geometry ^the

mappings

_are

analytic.

In this

work,

_we

investigate

the

symmetries

of minimal surfaces.

Mainly

the translations in the _case of

periodic

surfaces _are considered.

The

equation

H

= 0 is _a non-linear second order

partial

differential

equation

for the coordinate functions which is

essentially

solved

by

the

Weierstrass-Enneper

formula

(WE)

Xl

z

I W~

x(z)

= x~ =

Re _a I

(I

+

w~) R(w)

dw

(I)

x~

~°

2 _w

This formula

gives

the cartesian coordinates of the

position

vector

x(z)

as a function of the

complex

parameter z

(equivalent

to two real _{ones, as} usual for _two dimensional

objects).

It involves _a

meromorphic

function R _so that R dw is _a

meromorphic

differential

[5, 6].

The

integration

is _meant

along

_a

path

from zo

(fixed)

to z

(running)

ⁱⁿ C.

By Cauchy's

theorem, the value of the

integral

does _not

depend

_on the

path

_as

long

_as the

paths

_are

homotopic

ⁱⁿ the definition domain of

R,

which will be _a Riemann surface. The module of the

complex

^number

a fixes the

length-scale

whereas its

phase provides,

_upon

variation,

different minimal surfaces.

Changing Arg (a )

in

(I)

is known _{as a} Bonnet transformation. Two surfaces related

by

such

a transformation _are in

general

different

(they

cannot be

brought

to coincidence

by

_any

euclidean

motion)

but

they

_are

locally

isometric _as surfaces. Besides their _own interest, the

Bonnet transformations

regarding triply periodic

minimal surfaces have been

proposed

to

model _some martensitic transitions in

metallurgy [7].

One of _{our purposes} is _to understand the

crystallographic

content of such

proposals.

When the parameter z is restricted _to vary within _a suitable

domain,

free from

singularities

of

R,

the WE formula

provides

_an

analytic

I I map between the domain and _a

patch

of the surface M. Fundamental

pieces

^for the space group can be constructed in this way

(for example by

numerical

evaluation).

However such _a restriction is

by

no means necessary.

By analytic continuation,

the

integration

_can be carried

along

_any

path avoiding

the

singularities (poles

^and

branch

points

in the _cases under

study).

Of _{course non}

homotopic paths

to some value

z of the parameter may

give

different values for the coordinates _x

(z ),

and _so, different

points

ⁱⁿ

R~.

_{The vector}

joining

two such

points

is _a

period

in mathematidal

parlance

-, that

is,

a

vector of the translation lattice of the surface. This is the way

periodic

surfaces _can be

expressed

in _a

single

formula. For the mathematical aspects ^of

periodic

minimal

surfaces,

_see

[8, 9].

The most well known

fully periodic

surfaces _are the P

(primitive),

D

(diamond

_or face

centered

F)

and G

(giroid, body centered)

cubic surfaces whose WE differential form is

provided by

R

(w

=

(1

+ 14

w~

₊

w~)~

^'/~

(2)

Albeit

wildly

^different

regarding

their

geometrical

_aspect

(different

_{space groups,} lattice

parameters etc...),

those three surfaces

only differ,

in formula

(I), by

the Bonnet

angle

which has value

0°,

90° and 51.985°,

respectively.

To

study

the Bonnet

transformation,

the most natural

thing

to do is to

investigate

the surface in

C~

which lies above all the

(three

in this

case)

members of _a Bonnet

family.

What the WE

Fig. I.

Singular

points of the WE integrant for the PDG su~ace. The multi-valued function

provided by

WE formula is _an inverse of the Gauss map followed

by stereographic projection

M

~ S~

~ C. The

lines _are images of lines of symmetry ^included in M. Such lines _are mapped onto

principal

lines edges and face

diagonals

of the cube, _as projected onto S~, so that what is

displayed

is _a

stereographic projection

of the

spherical

cube. The _zeros

w~ of the

polynomial

+ 14 w~ ₊ w~ _are the centers of 3-fold symmetries. _w~

=

(/

_1)/

/

= 0.518... if j _w 4, and

w~ =

(/

+ I)/

/

= 1.932... for j _m 5.

formula

provides is,

before

all,

_an immersion of the surface into

C~

m

R~

~

j

~2

z

z _- f

(z )

=

f~

₌ ^I

(I

₊

w~

_R

(w

dw

(3)

f~

zo

2 _w

The real surface is then Obtained

by

linear

orthogonal projection

down to the

physical

_space

x = Re

(at ).

Note

that, locally,

^this

projection produces

no

singularity

on the surface M since the metric _tensors of the

complex

surface

f

and of the real

projected

_one x are related

by

a

constant factor 1/2.

842 JOURNAL DE

PHYSIQUE

I N° 3

The translation

symmetries (')

of the real surface _are

projections

of the translations of the

complex

_one. Our first

Objective

will be to find and describe this symmetry lattice. For

example,

_on the Riemann surface

R(w)~~

= l +14

w~

₊ w~ the map

(3)

has six

possible complex periods [8, 9].

We will _see that _{a set} of generators ^{for the PDG surface is}

r is is 0 _r is _r is

(tj,

_{t~, t~, vi, r~,}

r~)

=

is _r is _r is 0 _r is

,

is is _{r r} is _r is 0

where _r

= 2,156.. and _s =1.686.., are the cubic parameters ^{of the P and F surfaces}

respectively.

Note that this is not the cartesian

product

of the lattices of the two

conjugated

surfaces P and F. The

global scaling

factor

is,

of _course,

arbitrary

_; here and in the

following,

it is set to

la

₌ I. The ratio of the _two

parameters

r and _s

is, however,

not

arbitrary

because it

has to fulfill the

isometry

condition _:

being mapped

onto each Other

by

the Bonnet

transformation,

fundamental

patches

enclosed in fundamental cells of their

respective

lattices of the _two surfaces _must have the _{same area.} As _we will _see, this fixes

r/s to be 1.279.. _a value close to, but

definitely

^different from,

2'/~

= l,2599..

,

the ratio _one would get ^{if the lattices P and FCC had}

equal

volume per unit cell.

The paper is outlined _as follows. When the WE differential is known _as it is for _a number of

examples [1, 4, lo,

I

I] including

the PDG the basic

periods

_can be found

by evaluating

the WE

integral along

closed

loops.

^This is

briefly explained

in section 2.

Section 3

proceeds

to a more detailed

investigation

of the

algebra

of the 6 D lattice and of the various 3 D sublattices involved in the

projections.

The

starting point

is the

hyperbolic plane H~

_{and its}

tiling (honey-comb)

_groups

[12].

It _serves here _{as a} universal

covering

of the surfaces with genus g m 3

(counted

_per unit cell for

periodic surfaces).

The

complex

surface itself is _an intermediate

covering,

_common to all the members of _a Bonnet

family

and this

allows to solve the

algebra.

The various lattices in 3 and 6 D, as well _as their

algebraic relationships,

will be derived

by analysing (commutative) quotients

of the

hyperbolic

translation

subgroup

^of

[6, 4].

The 6 D

crystallography

is settled down in section 4 _as far _as translations _are concemed

(the point symmetries

will be studied in _a

forthcoming paper).

The

geometrical

counterpart ^of the

quotient operation

is

projection.

A first

step

_maps the

hyperbolic

space

H~

_{onto the}

complex

surface in

C~.

_The second

projection, yielding

the real surface, is linear and the geometry ^{of the} lattices is solved

by

linear

algebra.

This issues of the Bonnet transformation _to account for real processes

regarding crystals

of films _{or even} solid state

transitions,

_as well _as the

problems

met in such

applications,

_are

postponed

to section 5.

2.

Analytic approach

:

elliptic integrals.

The

example

of the PDG surface will suffice _to present ^{the main lines of the method.}

Changing

variables

(see

Ref.

[4], 85,

_pp. 77-78 for

details)

in the

integrals (3)

with R

given

in

(2) yields (take

_zo

=

0)

lf(2 ^z/[I

⁺

^z~])

f(z)

=

f(2 iz/[I z~]) ₍₄₎

f([z~ I]/[z~

₊

_{I])+ f(1)}

(1) We will only consider orientation

preserving

translations. If orientation is not taken into account, the symmetry ^{lattices of the P and D surfaces become}

respectively

^{FCC and P, both}

containing,

with

index 2, the former orientation

preserving

lattices. For the G su~ace, _as ^well as for the C _one, the motions reversing orientation _are screws (rotation + translation).

where

f(u)

=

l(I _v~

+

v~)~'/~

dv is _an

elliptic integral

^{of the first} kind. The main 2

properties

of

elliptic

functions _are listed in _most tables of

special

functions

[13].

Whereas

trigonometric (or circular)

functions

(2)

_are I

-periodic,

the

elliptic

functions have two

periods generating

a whole lattice in the

complex plane.

In canonical

form,

f (u

= 1/2 exp

(-

I ar/6 sn~ ^'

(u exp(I ar/6),

k

(5)

The second argument

k,

called the modulus, is fixed here _to k

= exp

(-

I ar/3

).

Such _« multi- valued _»

integrals

are to be considered _as Riemann surfaces

[14].

Above any definite value of the parameter z

(hence,

of

u)

the

multiple

values of the

integral

are

provided by

the

symmetries

of the inverse function. In this _case, the Jacobi _sn function satisfies

sn

(2

_~y + nj K + n~

iK'), k)

₌

(-

_1)~~ sn

(2

_y,

k)

where nj, n~ are two

integers

and K

=

K(k)

= sn

(I, k),

K'

=

K((I k~)'/~)

= sn

(I, (I k~)'~~)

are two

complete elliptic integrals.

Taking

into _account the

phase conjugation,

_{we can} set

kj=exp(-iw/6)K,

k~ = exp

(-

I w/6 iK'

which,

after evaluation,

yields kj

=

1.078 0.843 I, k~ =

1.686 I. So

the full _set of values taken

by f

is

(-

1)~~ ~y + nj

kj

+ n~ k~

)

_{; nj, n~ in}

Z),

where

y is a

particular

determination of the

integral.

Notice that the _same

integral,

with the _same translation lattice

[2 kj, k~],

appears in all three coordinates in

(4), (this

could have been shown _a

priori

_{as a} direct consequence of the 3-fold symmetry ^{of the}

cube).

We conclude that the group L of the translation

symmetries

of the surface in

C~

is _a lattice included in the

product

of the three lattices

appearing

in the coordinate

planes.

^{This is}

only

an inclusion because the symmetry ^of _any

object

is smaller than the

product

of the

symmetries

of its linear

projections,

here the C-coordinates. In _a shorter way,

we will write this _as L _«

[2 kj,

_{k~]fl~~ («}

[...]

_» is the lattice

generated by..

_{; « « » means}

«

subgroup

of _», also written _«

< » when the inclusion is

strict).

To find the actual translation group

L,

two ways are

possible.

The first _one is _to

proceed by

direct evaluation of the

integrals;

to do _so, one has _to find a set of generators ^{of the} fundamental group of the Riemann surface associated _to the Weierstrass differential R this surface is built

by progressive analytic

continuation of the differential

form, taking

into

account and

combining together

the various branches then the basic

translations,

the _so-

called

periods,

are the

integrals

of the differential _on these

loops.

^{In this}

specific example,

there _are 8 branch

points

of order 2

(necessarily mapped

onto flat

points

in

M),

the genus of the surface is 3 _so that the number of

independent loops generating

the fundamental group is 6

(Fig. 2). By,

_e.g., the

change

of variables

proposed above,

the

integrals along

these six

loops

reduce to finite combinations of

complete elliptic integrals (the kj,

_k~ defined

above).

3.

Algebra

_: the translations reduce to commutative groups.

The other way

requires

_some

algebraic

considerations based _on the fact that the

complex

surface is _a

covering

^of the members of the Bonnet

family.

The universal

covering

in the

u

(2) Non-experts _may refer to the analogy with the

integral

Arcsin (u)

= (I

v~)~

^'/2 dv.

844 JOURNAL DE

PHYSIQUE

I N° 3

ti

Fig.

2. Fundamental

loops

_on which the WE

integral provides

the basic translations

(tj,

...,

r~).

(A

picture

^of a genus 3 surface _can be found, for _ex., in Hilbert D. and Cohn-Vossen S.

Geometry

and the

imagination, ^{Chelsea Pub]. Co.} 1983,

chap.

VI 45).

hyperbolic plane ^H~

was

analysed

in

[15].

We

only give

here _a summary

suitably

extended _to

our purposes.

I)

The three surfaces P, D, G admit _{a common} universal

covering

which is the,

hyperbolic plane ^H~.

The

projection ^H~

- M is conformal. More

precisely,

each surface M

=

P,

G _or F is

conformally equivalent

to a

quotient H~/T~

of the

hyperbolic plane by

_a

specific

discrete group of pure

(hyperbolic)

translations.

Actually,

the entire space groups of the three surfaces _can be

derived _as

quotients

of the _same

subgroup

of the group of

automorphisms

of

H~, namely

the

honeycomb (6, 4).

The

corresponding

fundamental

patches

_are

triangular

orthoschems.

2)

The group

(6, 4) contains,

_as a pure translation

subgroup,

a

representation

of the

fundamental group vi of g = 3 surfaces

(3).

A

presentation

for this group is

(~)

(tl,

_{t2, t3, ~l> ~2> ~3 ~3 t2 ~l t3 ~2 tl ~3 t2 ~l t3 ~2 tl}

~

l) (6)

and its most

symmetrical

fundamental domain is _a

semi-regular dodecagon

A

(Fig. 4).

3)

The translation groups

Tp, T~, T~

involved in the

quotients

of

H~

are

subgroups

of

(this representation of~

_vi. In

particular,

the

quotient

of vi itself

by

_any of these

subgroups

is

isomorphic

to

Z~

_and

corresponds,

in each _{case, to} the Bravais lattice

(=

translation

symmetries)

of the surface in

R~.

_{The P and the F surfaces}

are

particularly

useful here because

they correspond

to

complementary

values of the Bonnet

angle (0°

and

90°).

As shown in reference

[15],

the

hyperbolic

_group

_T~

is the smallest normal

subgroup

of _arj

containing

the

following

elements _:

tj

_vi t~, t~

r/

^' tj

',

₊

_cyclic permutations

of 1,

2,

3

(7)

(3) _vi is

actually

the fundamental group of the surface M taken modulo its periods L~, I-e-

projected

into the torus T~.

(4) Notice _a

slight change

of conventions wrt. reference [15] the r~ ^here

correspond

to

r/

^{' there the}

t~ ^are the _same.

fi«)

riot

-i 1

Fig.

3. Polar

plot

of the

elliptic

function

f

(as this function is the

identity

_up to second order in the argument 2 near 0, and in I/z _{near z}

= aJ, it has not been drawn in these

regions).

The

periodicity

in the two directions is

clearly

visible.

When the

quotient by Tp

is taken

(an operation

which _{amounts to «} force the elements of

Tp

to the

identity »),

it

immediately

follows that the

r)

_are combinations of

t) (t)

denotes the elements of the

quotients,

I,e.

equivalence

classes _:

t)

= t~

Tp

=

Tp tj

ⁱⁿ

arj).

So the symmetry lattice of the P surface is P

=

art/Tp

=

[t(, t(, t(]. _Similarly,

the lattice of

periods

of the F surface is F

=

arj/T~

=

[r), r(, r(],

where

T~

is the normal

subgroup

of w~

generated by

r~ tj

vi ', _vi

^' _t[ ^'

r~, ⁺

cyclic permutations

of 1,

2,

3

(8)

Both P and F _are abelian free groups with 3 generators

(Appendix A),

that

is,

3 dimensional lattices.

4)

The three groups

Tp, T~, T~

are different but their intersection is not

empty.

^{It contains} at

leas the commutator

subgroup Q

of vi

(Appendix A).

As the

only

relation

(6) defining

vi can be

expressed

as a

product

of commutators, it follows that

wj/Q

is a six generators abelian free group, that is _a 6 D lattice

(5)

L

=

[tj,

t~, t~, vi, r~,

r~].

(5) ^We _use the _same notations for elements in H~~

corresponding

elements in the

quotient by Qi

and their geometrical realizations in C~ however the non-commutative group

operation

in H~ _{is written}

as a

multiplication

whereas the

corresponding operation

in the

quotient,

which is abelian, is written additively, _as usual.

~46 JOURNAL DE PHYSIQUE I N° 3

Fig. 4. The dodecagonal fundamental cell A of

wj in the

hyperbolic

plane H~ (Poincar£

representation).

The generators (tj,

..,

r~) _are ^{also shown.} The smaller (by a factor 2 in area)

dodecagon

corresponds to the patch displayed in figure 5.

This is the lattice _{we are}

looking

for _:

by

construction, it is the symmetry ^{lattice of} a surface

H~/Q

which is _a

covering

_common to the surfaces P, F, G. We claim that this

surface, together

with its lattice L, ^is

optimal

in the _sense that there is _no

hyperbolic subgroup larger

than

Q

which is

simultaneously

contained in

Tp, T~, T~.

^This amounts to say that the intersection

coincides with the _commutator group :

Tp

n

T~

=

Q.

We

already

know the intersection contains the commutator :

Q

<

Tp

n

T~.

Since

Q

is normal in vi, it is normal in any

subgroup

of _arj

(provided

it contains

Q)I

_{so we} can take

quotients

and _we will show that

(Tp

n

T~)/Q

is trivial

(reduces

to the neutral element

(0)).

First, by appendix

B,

(Tp

n

T~)/Q

=

(Tp/Q)

n

(T~/Q). Next,

let

Kp

₌

Tp/Q

and

K~

₌

T~/Q

_;

Kp

is the abelian group

generated by

0 0

(vi

₊ t~ t~, _r~ + t~ tj, _r~ + tj t~ ) =

(tj,

t~, t~, vi, r~,

r~) ~~ ₍₉₎

0 0

0 0

This is

naturally

a sublattice of L

=

arj/Q. Similarly, K~

is the sublattice of L

generated by

l

^{0 0}

0 0

(tj

_r~ + r~, t~ _r~ + vi, t~ vi +

r~)

₌

(tj,

t~, t~, vi, r~,

r~) (10)

0 0

Now,

_{to see} that the intersection

Kp

n

K~

^is

trivial,

it suffices to show that the two lattices _are

complementary.

But this is _so for

Kp

and

K~,

both of dimension

3, together

span a 6 D

lattice,

as

immediately

deduced from

(9)

and

( lo)

put

together,

these

equations

_mean that the lattice

K~

+

Kp corresponds

to the

following

matrix

i

^{o o o i i}

o i o i o

-~i _~i~

^~~~~

i o i o i o

i i o o o

which is

non-singular

because its determinant is 4.

5)

From these

equations

_we also get ^that

Kp

and

K~

are maximal

[16]

in L. The basis of

Kp

_can be

completed by adjoining (ti,

t2,

t3)

to

(9)

_{so as} to build _a basis of L. Indeed the

resulting

matrix is modular

((det(

= I

)

:

i

^{o o o -i i}

o i o i o -i

o o i -i i o

o o o i o o

o o o o i o

o o o o o

The basis

( lo)

of

K~ provides,

as

well,

a basis of L when

completed by (

_{vi, r~, r~}

).

It then follows that the symmetry ^lattices of the P and F surfaces _are

regular

lattices without torsion

[17]

and that the

complex

surface is _a

regular covering

of each of the real surfaces M

=

P _or D. Similar conclusions could be reached for the G surface. The symmetry ^lattice

L~

of any of them is _a

quotient

of the type

L/K~.

4.

Crystallography

_: the

settings

in 6D.

All this has _a

geometrical

counterpart : the lattice L is

naturally

imbedded in

C~

as a discrete

subgroup

of the translations

R~.

_{The maximal} sublattices _are the _ones which span, and therefore coincide

with,

intersections of L with linear

subspaces

of

R~.

Thus any

quotient

of L

by

_a maximal sublattice is

equivalent

to a linear

projection

_; the kemel is

equal

to the

subspace spanned by

the sublattice.

If,

_moreover, the

projectors

_are

agreed

to be

orthogonal,

the kemel

entirely

fixes the

projector. Finally, consistency

with the

complex

structure insures that the

projector locally

acts as a non

degenerate

map on the surface and that it _can be

represented

_as a

real

part

^of the coordinates. For

example,

_{if pp} denotes the

projector yielding

the P

surface,

Ker

~pp)

^is

nothing

but the

subspace spanned by Kp,

since all its vectors

project

onto 0 :

pp(

_vi + t2

t~)

₌

(r)

₊

_t~ t()

=

0,

etc.

If, similarly,

_p~ is the

projector

with Ker

~p~) spanned by K~,

^the two

projectors

_pp and p~ are

complementary and, according

to the metrics

underlying

the Bonnet _« rotation _»,

orthogonal.

848 JOURNAL DE

PHYSIQUE

I N° 3

To the resolution I

= pp + p~

corresponds

the

following decomposition

of the _vector space :

C~

m

Ep

@

E~

^with

Ep

₌ Ker

~p~)

₌ ^Ran

~pp)

and

E~

=

Ker

~pp)

=

Ran

~p~), (12)

both

subspaces being

of real dimension 3. In

particular,

the _vectors in the space

containing

the F surface are now identified to vectors of Ker

~pp).

Thus the intersection lattice

Kp

₌

E~

n L becomes a

sublattice,

with index 4

(the

above

determinant,

_see also

appendix C),

of the

projection

lattice F

= p~

(L ).

Indeed the lifted vectors vi + t~ t~

(+ cyclic permutations

of the

indices) which,

_as

generators

^{of the} intersection lattice

Kp,

coincide with their

complementary projection p~(.. )

can be

expressed

in _terms of the basis

(r), r(, r()

of the lattice F _: vi + t~ t~ =

r)

₊

r(

₊

r(.

This last combination of FCC generators

(6)

is well known to

provide

_a

primitive

cubic vector, which _we write

(2,0,0) (in

the cubic basis of

P~

introduced

below).

So in the F space the intersection lattice is

simple

cubic and of index 4 in the

projection

lattice F _; this is the classical

primitive

lattice of index 4 in the face centered F.

t2

/

ti ^'

t~

Fig.

5. Projections, in the directions yielding the P, G and D surfaces (from left _to right), ^of the six generators of the

complex

lattice L

together

with _a small

dodecagonal patch

of the surface this

dodecagon

is _a fundamental domain for the

improper

translation groups of the P and F surfaces.

A similar

reasoning

in the P space

Ep

shows that the intersection lattice

K~

=

L n

Ep

is

generated by

tj r~ + r~ =

t(

₊

t(

₊

t( _{(+ cyclic}

index

permutations).

This is _{now a} BCC

sublattice,

with index

4,

of the

simple

cubic

projection

P

=

pp(L).

The lattice L is the

product

of _a section, in L, over P and of the intersection

Kp (Appendix C).

So all what _we need is _a lift of 3 generators ^{of P. We chose}

t(, t(, t(

whose lift is found

by matching

to their

projection

in F. To describe

things

in

C~,

we can take _as reference the semi-cubic lattice P @

P~

where

P~

₌ F U

(F

₊

(- r)

₊

r(

₊

r()/2).

In these semi-cubic

coordinates,

the

resulting

basis of L

(6) Here, _{we are}

referring

to the known fact that the lattice of the F surface is FCC (and

similarly,

that the _one of P is

simple cubic).

These features cannot be deduced from the

purely

modular relations

presented

so far (Sects. 3 and 4), but they ^follow once the metric related to the WE integrals is taken into account, as done at the end of section 4. Nevertheless, it should be mentioned that the entire space group

can be derived

algebraically by keeping

in hands the entire

honeycomb

_group

(6, 4)

rotations included

throughout

the

analysis

of the

quotients.

is

given by

the

following

matrix _:

l

^{0 0 0 0 0}

0 0 0 0 0

0 0 0 0

0,

0 -1 2 0

0'

-1 0 0 2 0

-1 0 0 0

the first three columns _are

t~ while the last _{ones are} the

simple

cubic vectors of

Kp

contained in

E~.

A

simple change

of basis in L

(amounting

to

complete

the lifted _set

tj, t~, t~

by

lifted vi, r~,

r~) yields

the

following equivalent

set of generators

l

^{0 0 0}

0 0 0

(tj,

_{t~, t~, vi, r~,}

r~)

₌

~ j~ (13)

0 0

0 0

Notice that the determinant is

8,

a value which agrees with the value 4 of the determinant of

(I

I

)

times the value 2 of the index of P @ F is in the semi-cubic lattice.

Restoring

the metrics

and

identifying

^{the last} three cartesian coordinates _to

imaginary

parts

(P

@

P~

is

nothing

but

[2 kj,

_k~]fl~^~

=

[r,

is_]fl~~), we get

Re

kj

_{k~ k~} 0 2

kj

2

kj*

(tj,

t~, t~, vi, r~,

r~)

= k~ ^{2 Re}

kj

_k~ 2 kj* ^{0 2}

kj (14)

k~ k~ 2 Re

kj

2

kj

2

kf

0

With _r

= 2 Re

kj

and _s

= 2 Im

kj

₌ Im

k~,

^{this is the result announced in the} introduction.

Finally,

it follows that the

semi-regular dodecagon

A, which

corresponds

to a fundamental

piece

of all the three surfaces for their

respective

translation groups, also

corresponds

to a

fundamental

piece

of the surface in

C~. Again,

such _a property ^{cannot be inferred}

by

_mere

contemplation

of the

projections

alone because, if it is _true

that, by projection,

a symmetry ⁱⁿ the

higher

dimensional space

yields

a

symmetry

^{in the}

subspace,

the

reciprocal

^is not

always

true. The fact that the

quotients

do indeed reduce _to linear

projections

^{in the} abelianized group

L

implies

that fundamental

patches

lift _to fundamental

patches.

5. Summaries and

perspectives.

In summary, the

crystallography

of the Bonnet transformation is

entirely

contained in this six

(over R)

dimensional

representation.

When referred _to the semi-cubic basis

[r, is]fl~~

of

C~

m

R~,

this _one parameter

family

^is indexed

by

the

angle

of _a

hyper-rotation (or

a

screw)

which commands the direction of

projection.

In

doing

_{so, one}

immediately

^has to face _one of the

major

difficulties in minimal surfaces : when is the surface

regularly imbedded,

_{or even}

properly immersed,

in

R~

?

Imbedding

the surface in

C~

_poses less

problems (generically,

2-

surfaces do not intersect in R~ _{as soon as n m}

5).

As

readily checked,

the PDG

complex

surface is indeed

regularly

imhedded in

C~.

_{The accidents occur once the} surface is

projected

down

into the

physical 3-space

_:

taken _{as a}

whole,

the surface fills the space

R~ densely

when the direction is irrational.

Indeed, the result of

projecting

the 6D lattice is _a module with 6

generators.

When at least four

850 JOURNAL DE PHYSIQUE I N° 3

of them _are

rationally independent,

the module has accumulation

points filling densely

a set of lines,

planes

or even the whole

imbedding

_space. So the

surface,

which is the orbit of _a fundamental

piece

under this

module,

will accumulate

densely

in _some

regions

of space, with foils

arbitrarily

close to each other _;

most of the

time,

_even in rational _cases, there _are many self-intersections.

Only

_very

specific angles

related to low indices in L

yield regularly

imbedded surfaces in

R~.

_{In the}

example, exactly

3 orientations

(modulo

the semi-cubic

symmetries)

of the

projector yield regular global surfaces, namely

the base spaces Re and Im

plus

the first

diagonal:

(Arg ^{(kj )}

= 90° 51.985°

= 38.015°.

From _a

physical viewpoint,

the first aspect ^is

incompatible

with _mere conservation of matter and the hard _core of the molecules. The second one is _more

questionable.

On one

side,

the

occurrence of self-intersections does _not appear

fundamentally implausible

_{: at}

macroscopic

scale,

_soap bubbles accommodate self-intersections

perfectly

well. Closer to

crystals

^of

films, mesophases

such _as the

hexagonal

_one do not,

properly speaking,

form surfaces because of the 3~fold

junctions parallel

to the

hexagonal

axis.

And,

after

all,

self intersections

locally

look like

suitably equilibrated

4-fold

junctions.

On the other

side,

self-intersections _are

hardly compatible

with the fact that the surface

splits

_space into _two disconnected

labyrinths

_; this feature _seems to

play

_a role in the behavior of sponge

phases [18].

Further

investigations

in this direction

ought

to include _a finer

analysis

of the

bilayer

and _more

quantitative (energetic)

criteria.

To escape these

peculiarities,

one can suggest ^to

give

_up either the

isometry

condition _or,

eventually,

the

topological integrity

of the surface. In the first _case, of _course, the _set of

possible

motions is

tremendous,

but the surface will in

general

not stay ^minimal

(H

₌

0).

Deformations of all kinds _occur, for

example,

in thermal fluctuations, acoustical

modes etc..., but

they

do _not fulfill the nice and

simple predictions

of the Bonnet

transformation. The other solution _seems

physically

artificial _: the energy necessary ^{to tear} the surface is

expected

_to be _an order of

magnitude larger

than the

bending

_or tension

energies,

_so

that if there is

enough

available energy ^to break the

integrity

_(«

bicontinuity »)

of the

surface,

there is little chance for the features

specific

to the Bonnet transformation local

isometry,

H

= 0 to be of any relevance.

Even from _a

purely crystallographic point

of view

(making

abstraction of the surface and

keeping

in mind

only

the

operation

on the

lattices),

the

pieces

of the

puzzle

_are not

straightforward

to put

together.

^The

large degeneracy

of the

projection along

^{directions of} low indices

(in Z~) immediately implies

_a drastic

proliferation

of nodes as soon as we

slightly change

the Bonnet

angle.

To get ^{rid of this}

proliferation,

_one could think of

performing

_a

selection in the superspace

(a preliminary tentative,

in the _context of

surfaces,

_was

proposed

in

j19]),

ⁱⁿ a way

analogous

to what is done in

deriving quasicrystals

and Penrose

tilings by

cut and

project

and related methods

[20].

In that respect, ^{the Bonnet} transformation looks similar to the various tilts of

projectors

sometimes made to

get approximants,

_or other _structures with

pattems

^close to those of the

quasiperiodic phases.

If _we

roughly

decide _{to cut out} all the nodes

but those which _are contained in _a

cylinder

around the

physical subspace

of _some

given

periodic

^surface

(say

the P

one),

then the Bonnet transformation

by

^90° would map all these

points

^into a compact

region

of the

complementary

_space

(the

F

space)

_; this is not what _we expect to occur in martensitic transformations for

example

Even

choosing

another

cylinder

does not

help

^much in

situations,

_as

here,

^{where the} co-dimension of the

physical

_space in the

larger

_one exceeds _one. Nevertheless, ^solutions to the

problem

^of

global displacive

transformations for lattices and

quasilattices

_were

proposed

ⁱⁿ

[21].

As stated several

times,

the Bonnet transformation is

locally

isometric _; the

isometry

extends at least to the

simply

connected fundamental

patch

shown in

figure 6,

but it cannot be

Fig. 6. Bonnet transformation of the fundamental

patch

A. The Bonnet

angle

takes the values 0°, 30°, 51.985° and 90°. The second _one is

generic

and does _not lead to a

regularly

imbedded surface in R~ when extended _to

larger

_parts. As

Arg

(a) varies, _every

point

in the surface follows _an

elliptic

trajectory

and the unit normal

keeps

_a constant direction

throughout.

a

global isometry

for closed

loops

in _one surface _are

mapped by

Bonnet to infinite helical _or

straight

lines in the associates. In the

particular

_case of the P and F

surfaces,

there is another way to transform _one into the

other, namely by

rhombohedral distortion. The rPD

family

is

indeed

self-adjoint.

However, this

transformation,

which is continuous and

global,

is not an

isometry.

Finally,

let _us sketch how the elements of the

analysis presented

in the

body

of this article

would appear when

dealing

with _more

general periodic

minimal surfaces.

Specific

to the

example

treated here and to its continuous

deformations,

the

tetragonal

tD and tP

[10, 22]

surfaces,

_etc. is that the fundamental group of the

complex

surface

(PDG)

coincides with

852 JOURNAL DE

PHYSIQUE

I N° 3

the commutator

subgroup

of the discrete

hyperbolic

_group. This _means that the

only loops

in

the surface

correspond

to closed

paths

built with

edges

of the lattice.

Somehow,

the surface has the minimal genus

compatible

with full

periodicity

in 6 dimensions _or, what is

equivalent,

to

wind around and in the torus

T~

_in

a non retractible _manner. In

general,

the fundamental group

ar j has _more than 6

generators

so that the surface in C~ is

expected

to have a richer

topology,

a

genus

(per

unit

cell) larger

than 3

(examples

in Ref.

[I]

and

analysis

in Refs.

[8, 9])

there will be closed

loops

not accessible _to motions

through only

^lattice vectors.

Denoting, again,

the

fundamental group of the

periodic

surface M

by T~,

^the

automorphism (symmetry

for

physicists)

_groups

LM

^of intermediate surfaces

(in

the _sense of

coverings

the lowest

lying

surface is the _one,

M~, wrapped

around the torus obtained

by taking

the real _or

complex

surface modulo its translation lattice the other _extreme is the universal

covering ^H~

_gotten

by unfolding

all the

loops) correspond

to

quotients

of the

hyperbolic

_{group arj}

by TM.

In other

words,

_{if p is} a linear

projector,

_we have

H~~WI(M~)~TpjM)~TM~Q.

If M is imbedded in _a linear space, as

C~,

then

TM

^contains the commutator

Q

of

arj ^{so as to}

give

_a commutative group

~Z~

_{in the}

quotient but,

in

general,

TM

^is

strictly larger

than

Q.

This does not solve all the

problems,

ⁱⁿ

geometry

or

physics,

but it reduces the

study

to compact ^{surfaces in} compact _spaces

(tori).

From the mathematical

point

of

view,

it _seems more

interesting

_{to go} to still

higher

dimensions: indeed,

according

to Abel's

theorem,

a Riemann surface of genus g is

conformally

imbedded in _a

complex

torus of dimension g then it

corresponds

to a nice, self- intersection

free, peRodic

surface in C~. This is the basis of the

analysis

of Meeks

[8]

and

Nagano

and

Smyth [9],

and has been

used,

here, as a

background,

ⁱⁿ

particular

for the _case g = 3. What has

pleased

_us is that

significant

information _can be

gained by purely algebraic

methods

(discontinuous

_groups and linear

algebra).

Finally,

the

operation

that

brings

_us back _to the

physical

3D space is

simple

in this context :

it is _a linear

projection

^which can be cast into the real part ^{of the}

complex

argument ^times a

phase

a. If the interface _or the

bilayer

is _to generate a

properly

immersed

periodic

surface, with

a translation group ar

i/T~~Mj isomorphic

to

Z~,

then the kemel _must be _a lattice

plane, meaning

that the

slope

associated _{to a} is rational

(in

the lattice

coordinates).

Appendix

A.

Q«TpandosT~.

The generators of

Tp

are

"1 ~ t3 ~l t2 _{"2 ~} tl _~2 t3 _{"3 ~} t2 ~3 tl

"4 ~ t3 _~l t2 _{"5 ~} tl _~2 t3 _{"6 ~} t2 _~3 t1

and all the

conjugates by

_arj.

Straightforward

calculations

yield

t3 "1t3 "4 ~ ~l t2 ~l t2 _"

C(Tl> t2)

"6 tl t2 "3 t2 tl ~ C(t2>

tl)

tl "2 "1t2 tl _{"5 "4} tl t2 tl _~ C(T2>

VI) C(t2

^>

tl)

>

where _c

(, )

^denotes commutators. To get c

(r~,

_vi

),

_{use c}

(tj ',

_ti

_)~

'

"

tj

^' c

(t~,

tj

)

_t~. The _rest

follows

by permutation

of the indices and

taking products.

Similarly, T~

is

generated by

the

following

elements and their

conjugates

_:

fll

~ T3 tl _~2

fl2

_{~ ~l t2 ~3}

fl3

~ ~2 t3 Tl

fl4

~ T3

tl

T2

fl5

_{~ Tl t2 T3}

fl6

_{~ T2 t3 Tl}

and _we

get

fl2

_T3

fl5

_T3

~

C(Tl> t2) fl3

_{Tl T2}

fl6

_{T2 Tl}

~ C(T2>

Tl)

Tl

fl2 iii

_{T2 Tl}

fl5 fl4

_{Tl T2 Tl}

~ C(t2>

tl)

C(T2> _T1 ~) etc..

We

join

_to this

appendix

_an

equivalent expression

of the LHS of

(6),

C(t3, T~) C(T~

t3, tj T~ t~

Tj) C(tj

_{~, T(} ~)

C(T~ t/

~, t~

Tj) C(t~, Tj),

which is _a

product

of commutators. Still another way ^to cast

(6)

is

C(t3>

T2) C(T2

_t3> tl ~)

C(tj

_T~ t3, T3 t~

Tj) C(T3

_~, t~

Tj) C(t~, Tj)

=

Appendix

B.

Let

A,

B be _two

subgroups

of _a group G and C _a normal

subgroup

of G

satisfying

Cm An B. Then

(A

n B

)/C

=

(A/C)

n

(A/C).

Indeed,

_any element of the LHS is _a class aC with _a

belonging

to An B. Therefore

aC is in

A/C

and in

B/C.

To show the _converse

inclusion,

let _y be _a class in

(A/C)

n

(A/C).

This _means that _y

=

aC

=

bC with _a in A and b in B. In other words,

a~ ^' b

belongs

_to

C,

which is

subgroup

of A. Therefore b

= a

(a~

^' b also

belongs

to

A,

_so that indeed _y

=

bC with b in A n B.

Appendix

C.

Lattices.

Suppose

we have _an n-dimensional lattice L in

R~,

and _a

projector

_p. Let K

=

Ker

~p)

n L and R

=

p(L).

Then

(m

means

isomorphic)

~

~~~~~~' ~mf(I).

Here _s is a section in L _over

R,

that is _a I I

mapping

of R _{onto a} sublattice of L such that

p(s(R))

= R. The lattice

s(R)

is maximal. To construct such _a

section,

take _a basis of R _; every ^vector of R is the

projection

of

(eventually

_more

than)

_one vector of L Choose _one for each element of the basis. Then _s

(R)

^{is the lattice}

generated by

^{this lifted basis.}

(In particular,

when _p is

irrational,

K

=

(0)

_so that _{p :} L

- R is I I and the section _s

(R

is

unique, being

L

itself~.

Example

_: L

=

BCC,

_p

=

projector

onto the horizontal

plane

: K

= Z and R is _a square lattice with _a

=

2~'/~

_A

possible

section is

s(R)

=

[(1,

0,

0), (1,1,1)/2].

Then indeed