The T and CLP families of triply periodic minimal surfaces. Part 2. The properties and computation of T surfaces

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The T and CLP families of triply periodic minimal surfaces. Part 2. The properties and computation of T

surfaces

Djurdje Cvijović, Jacek Klinowski

To cite this version:

Djurdje Cvijović, Jacek Klinowski. The T and CLP families of triply periodic minimal surfaces. Part

2. The properties and computation of T surfaces. Journal de Physique I, EDP Sciences, 1992, 2 (12),

pp.2191-2205. �10.1051/jp1:1992276�. �jpa-00246695�

Classification

Physics

Abstracts

02.40 61.30 68.00

The T and CLP families of triply periodic minimal surfaces.

Part 2. The properties and computation of T surfaces

Djurdje Cvijov16

and Jacek Klinowski

Department of

Chemistry, University

of

Cambridge,

Lensfield Road,

Cambridge

CB2 IEW, G-B-

(Received 7 May J992, accepted in

final form 4August

J992)

Abstract. The geometry ^of a tD minimal surface

depends

_on the ratio cla of

tetragonal

_axes, and

can be

fully

describd in terms of _a

single

free parameter. ^{We offer} a choice of three such parameters, ^{all related} to surface geometry, and derive analytical

expressions

for their

relationships

to the _axes ratio and the normalization factor. The latter is crucial for the

matching

of

specific

surfaces _to real structures. Parametric

equations

for normalized tD surfaces make it possible, for the first time, to find the surface

corresponding

to any

given

value of the cla ratio, and _to compare it with actual structural data.

Straightforward physical applications

of tD surfaces _are therefore

possible.

We discuss the

geometric

consequences of this result and show that the _z coordinate of any tD surface _can be

approximated using elementary

functions. A rational

approximation

for the

relationship

between the free parameter and the cla ratio is also found. We list exact coordinates of tD surfaces

corresponding

to several

prescribed

^{values of} the cla ratio.

Introduction.

Approximately

40

triply periodic

embedded minimal surfaces

(TPEMS)

derived

by

various methods

(now mainly by

_group

theory),

have been described

[1-3].

Over the last 20 years TPEMS have been

applied

in many areas of the

physical

and

biological

sciences

[4]. They

are a useful

crystallographic _concept

for the

description

of condensed matter, as advocated

by

Scriven

[5], Mackay [6-7], Hyde

and Andersson

[8], Mackay

and Klinowski

[9]

and Sadoc

and Charvolin

[10].

In

principle,

the translation

symmetry

of TPEMS makes it

possible

_to

match them _to actual structures, I,e, _to compare surface coordinates with

spatial

_pattems of atoms, ^and to establish

relationships

between surface

properties,

such _as curvature and the volume-to-surface

ratio,

and

relationships

between structural

properties, especially

those

provided by X-ray

diffraction and solid-state NMR.

Modelling

of _structures also suggests a

method of

quantifying

structural

changes by relating

them to transformations

(such

_as the Bonnet and Goursat

transformations)

of minimal surfaces.

Finally,

studies of

interpenetrating crystalline

structures with

large

unit cells and

complicated

networks of _cages and channels

exposed

the limitations of classical

crystallography, showing

the need for _more

appropriate

concepts. ^{Since TPEMS have}

labyrinthine

structures,

they

are

likely

to lead _to

simple

structural

descriptions

^of such systems. ^A

rigorous

classification of all known and

hypothetical

TPEMS will be of great

importance

_to materials science in

general.

Unfortunately,

mathematical arguments

linking

TPEMS and

physical

structures are often far from

rigorous.

The main obstacle _{to a} wider

application

of minimal surfaces in

experimental

science is that most of them have been described

empirically,

without the

precise

mathematical

specification.

It is therefore essential _to

quantify

all known

TPEMS,

to establish which

geometric properties

are related _to the

properties

of

physical systems,

^and to find _a reliable method of

computing

their coordinates. In

particular,

it is

imperative

that

approximate computation

of TPEMS to _a

prescribed degree

of accuracy be

systematically investigated.

A TPEMS is described

by giving

its

parametric representation

_or the

Enneper-Weierstrass representation,

_or

by specifying

the

boundary

conditions of _a

partial

differential

equation [I1- l3].

In

general,

any surface in three-dimensional _space is

completely

described

by

the

parametric representation

^{of the form}

(X, _y,

Z)

" IX(U,

V), Y(U, V), Z(U, V)j

where the coordinates _are functions of two parameters, u and _v. However, in most cases

analytical expressions

for the coordinates of TPEMS are unknown. Sometimes _a

parametric representation

of such surfaces _cannot be

expressed

in terms of

elementary

^{functions alone.}

For

example, parametric representation

of surfaces of the T and CLP

family

involves

special

functions

[14].

Any

minimal surface is described

by

the

following

three

complex integrals

iw

x = Re

R(r) (I r~)

_dr

w~

y =

Re

i~ ^{iR(r) (I}

+

r~)

_dr

_(i)

w~

z = Re

j~

²

^R(r)

r dr

w~

known _as the

Enneper-Weierstrass representation.

The

problem

of

finding

_a minimal surface is thus reduced to

solving

these

integrals.

^For this to be

possible,

the

complex

_(« Weierstrass

»)

function R

(r )

must be known. Since _{a new} method for the construction of R

(r )

for _a

specific

TPEMS has been

developed [15, 30],

20 different surfaces have been _so described. In

theory,

it _seems

possible

to construct the function

R(r)

for any

TPEMS,

but this may involve

considerable

practical

difficulties

[16].

Coordinates of _any minimal surface _can be found

by solving partial

differential

equations

of the second order, known _as the minimal surface

equations,

but this method is not easy and is

rarely

used. As

rule, computational

difficulties increase in the order _:

parametric

_represen-

tation, Enneper-Weierstrass representation

and

partial

differential

equations.

Our

primary objective

is the

application

of

triply periodic

embedded minimal surfaces _to solid-state

chemistry.

We _are therefore interested in

developing straightforward procedures

to compare such surfaces with actual structures. We will demonstrate that

parametric equations

for normalized tD surfaces achieve this

objective.

The tD

family

of TPEMS.

Following

Koch and Fischer

[17]

_{we use} the

symbol

tD _to refer to the oldest known

family

of

triply periodic

embedded minimal surfaces. The first tD surface _was described

by Gergonne

_as

one his famous

problems conceming

a free

boundary [18],

which _was

subsequently

solved

by

Schwarz

[19].

The tD surfaces _are

variously

known _as T

surfaces,

_D~

surfaces,

_superman

surfaces and

Gergonne

surfaces. Because of their

tetragonal symmetry, they

_can be

thought

of

as a «

tetragonal

deformation of the D surface _{» ;} thus the abbreviation tD.

There _are four

types

of tD surfaces with different

symmetries

which could be

generated

in six different _ways

by using

various skew

straight-edged polygons [17]. Among

them is the skew

straight-edged 8-gon

obtained

by taking eight

of the twelve

edges

of _a

tetragonal parallelepiped (right tetragonal prism)

with

edges

_{a, a} and _c. The solution of the Plateau Problem for such

8-gon

is _a finite minimal surface

piece

of the tD

surface,

which will henceforth be referred to as the tD saddle surface

(see Fig.

I

).

It is

composed

of

eight

congruent parts ^related

by

four _axes of symmetry and mirror

planes.

The tD saddle surface contains _two

straight

line segments, ^which divide it into four congruent parts ^bounded

by straight

line segments

only.

A tD saddle surface _can be used _{as a}

building

block for

constructing

_an infinite minimal surface _: individual saddle surfaces assembled in such _a way are related _to their

neighbours by

twofold _axes.

Fig.

I. The tD saddle surface (a finite minimal surface

piece

of the tD surface

spanning eight

of the twelve

edges

of _a

right tetragonal prism) corresponding

_to ^~

=

~

a 2

Since there _are

denumerably

_many skew

straight-edged 8-gons which, prior

_to

scaling,

_can

differ for different cla

ratios,

tD surfaces _are one-parameter surfaces. This _means that the different surfaces of the tD

family _correspond

_to different values of the _same free parameter.

We offer _a choice of _one of three such parameters, all of them real and

closely

related _to the

eight

flat

points

of tD saddle surface. We shall

also,

for the first

time,

derive

analytical

expression

for their

relationships

to the cla ratio. The

parameter

^{E with 0} _~ ^E _~

l,

determines coordinates _on the unit

sphere

of the

images

of the flat

points

(± /$, _0,

±

E), (0,

_±

/$,

±

E)

which _are obtained

by

the Gauss map of _a

specific

tD saddle surface. The parameter

fl (0

~

fl

~ l _occurs in the standard

stereographic projection

of the

image points

onto the

complex plane.

The parameter ^A

(with

A _~

2)

_appears in the Weierstrass function of tD saddle surfaces

[15]

as a coefficient in the

polynomial

under square ^root with roots

given by (2).

Since all three parameters are related

(see

Tab.

I),

either of them _can be used. We will often refer to the tD surface with A

=

14

(corresponding

to E

=

l13

_{and fl}

=

fi)

and

limiting

_cases

A - ~x~

(E

-

I, fl

-

0)

and A

- 2

(E

-

0,

_fl

-

1).

Table I.

-Relationships

between the

free

_parameters which

fully

describe the tD

surfaces.

The parameter domains

of E, fl

and _are

(0, 1), (0, 1)

and

(-

_~x~, 2

), respectively.

E

p

A

I

p

² ~

l

p

^{2 2}

~'~

~/-

A ~

~

j~

~2

+

~~

~~~~

IA ^+fi

^{I -E}

^p8-1

2+A

fi _fl4

Two surfaces

corresponding

to the _same value of the free

parameter (I.e.

with the _same cla

ratio)

differ

only by

the

multiplication

constant, ^known as the _« normalization factor _» and

denoted

by

_K. For the _same value of the free parameter, ^{but different values of}

K, a

multiplicity

of surfaces _are

produced,

the coordinates of which differ

by

the factor

K. The normalization

factor,

_so far

unknown,

is crucial to

matching

of _a

specific

tD surface _to

a real structure, and enables _a

comparison

of different surfaces. We will show that

K is _a function of the free

parameter,

^{and derive} an

analytical expression

for it.

The

parametric representation

of the

family

of tD saddle surfaces is

[14]

x(u, v)

= Kgx Re

iF

_(q~x,

kx)i y(u, v)

= Kgx Re

iF

(q~x,

/fi)1 ₍₄₎

z(u, v)

= Kg~ ^Re

IF

_(q~~,

k~)j

where all components, except K, are defined in tables

IIa,

IIb and

w is

complex

w = u +

iv,

with

w « I

(see Fig. 2a),

^All components ^of

equations (4) depend

on the value

Table II.

a) Multiplication

_{constants g~} and g~ ^{and the} moduli k~ and k~ used in

parametric

the

equations for

the tD

su~fiaces expressed

in terms

of JFee

_parameters

E, p

and A

(see text).

b)

The

amplitudes

_q~~ and

q~~ in

parametric equations for

the tD

su~fiaces

can be

expressed

in various ways. g~ ^and g~ are

defined

in table IIa.

I-E~

₂

fl~ /~$~

gx

~~2 fi I+fl~

~

l-E

~

/

~ ~~~

~ ~/-A

+ 4

~2

E~ i~ p~ I_

~

l+E~

2

1+fl~

²

fi

~ ~~+~~~ ^~~ ~~/_A~ ^1~

a)

I _cos 2

q~~

cos~

q~~

sin~

q~~

w~ 6(w/g~)~

₊ l

w

~

2(w/g~)~

₊ l

4(w/g~)~

x

~ ~ ~ ~

w +

2(w/g~)

+ I

w~

+

2(w/g~)~

₊ l _w +

2(w/g~)

+ I

z 2

(

_{w ~/g~})~ l

(w

_{~/g~ )~}

(

_w

9g~

)~

b)

of the _same free real parameter, ^{and the}

relationship

between them and either of

E,

p and A, is

given

in tables

IIa,

b. Re in

(4)

stand for real part ^of

incomplete elliptic integrals

of the first kind F

(q~, k),

defined in its traditional

Legendre-Jacobi

normal form _as

~

~~~ ~~ i~ ~/(I ^~~~l

_k~

_t~)

~ ~

where _{we assume} that the variable

k,

the

modulus,

is real and lies in the interval

[0, 1]

and the variable _q~, the

amplitude,

is _a

complex

number. We will often _use various

properties

of

F(q~, k)

which _are

explained

in detail in standard texts

[20-21].

The

multiplication

factors g~ ^and g~ ^{and the} moduli k~ and k~ in

(4) depend only

_on the value of free

parameter (Tab. IIa),

and lie the interval

(0,1). Further, simple trigonometric

transformations allow _{us to} express the

amplitudes

_q~~ and

q~~ in number of

equivalent

_ways in terms of

complex

inverse

trigonometric

functions

(Tab. IIb).

We shall _use

complex

inverse sine

functions,

which differ for different values of the free parameter.

Quantitative

characteristics of the tD

family.

In order _to match actual structural data

describing

a

tetragonal

_system to a tD

surface,

it is necessary ^that the

following problem

be solved

given

the _a and _c dimensions of _a

right

tetragonal prism,

find the coordinates of the

appropriate

tD saddle surface. To do that _{we must}

examine closer the normalization factor _K, and establish the

relationship

between the

cla ratio and other

properties

of the surface.

Our

parametric equations (4)

^show

(Fig. 2b) that,

for any value of free parameter, ^the

point

A _on the real axis of the

complex plane

with coordinates

(p, 0) (where p

^{is defined} in Tab.

IIa)

is

mapped

into

point

A with coordinates

(a/2, 0, c/2)

^{of the} tD saddle surface. This the basis of the derivation of _an

analytical expression

for the ratio of

tetragonal

axes

cla

(which

_we

designate by x),

which _cannot be derived from the

Enneper-Weierstrass representation (Eqs. (I)

and

(3)

alone.

Thus,

from

(4)

_we obtain

where

ll'~(fl

and

ll'~(fl )

_are

~ 2

V'z

(P )

=

z

~ ~

~

j

⁴

^(w/g~)2

^~~~

~~~

W

~ + 2

(w/g~)2

₊

~ ~

Table IIa shows that

expressions (6) give ll'~(p

=

I and

ll'~(fl

=

I for any value of the free parameter. ^{In terms of the}

complete elliptic integral

of the first

kind, K(k)

=

F

(w/2, k), expressions (5)

are

a ₌ Kg~

K(k~)

c = 2 Kg~

K(k~)

^~~~

ant the

tetragonal

axis ratio is therefore

g~

K(k~)

x = 2

(8)

gx

K(k~)

where g~, g~, k~ and k~ ^are defined in table

IIa,

and

depend

_on the value of the free

parameter

alone.

Using

table

IIa,

the ratio

(8)

can be

easily expressed

in terms of either of the parameters

E, p

_or

A,

_so that

expressions

for _x

= x

(E),

_x

= x

(p )

and _x

= x

(A

_are

readily

available.

We _see therefore that the

tetragonal

axis ratio _x is

independent

^{of the} normalization factor, and

only depends (in

a continuous

manner)

_on the free parameter. ^{It follows that} x is

uniquely

characteristic

(an invariant)

of _a

specific

surface

belonging

to the tD

family.

We

verify (8) by

_ap

fling

^{it to the tD surface with A = 14 which has the}

^only

^{known value}

of the cla ratio

(x

=

2/2

) [22]

and

limiting

_cases

(A

_{- ~x~} and A

-

2).

It is easy ^to show that for A

=

14

relationship

^between _k~ and k~ is

~

l

/~

~ l ₊

/~

I.e, the moduli _are related

by

^Landen's

descending

transformation which

gives [21]

K(k~)

I ₊

/~

K(k~)

²

(a)

^{~' 1°~~ z} y

Re (to) x

fi

~~

I _m (w~

C A

B

,

~, A

' Re(to)

C

, '

D

B

m (to)

(c)

B A

,

A

,

Re (to)

D

~ ~

ifii

Fig.

2. (a) Construction of _a tD surface

piece spanning

_an

8-gon by projecting

a closed unit disc in the

complex plane using

parametric

equations

(I I).

Shading

indicates that

integration

(I I) is carried out over

the entire unit disc

including

the

boundary.

(b) and (c) construction of characteristic parts ^{of the tD} surface

by projecting

thick lines _on the unit disc.

Since k~ = 1/2 and

g~/g~

₌

/(2 _/),

the

required

ratio

(8)

is _x

=

l12.

_The

_limiting

behaviour of _x

(Tab. III)

_can be

explained by using only

the concept ^of

tetragonal

distortion _as follows. Normalized tD surfaces span the

edges

of

right tetragonal prisms

with the _same base

edge

_a

(which

is

conveniently

used _{as a}

unit)

but different

edges

_c. In other

words,

the surfaces

Table III.

Limiting

values

of

the various _constants

for

E

- 0

(corresponding

to

p

- I and

A -

2)

and

for

E

- I

(corresponding

to

fl

_- ⁰ and A

- ~x~).

gx gz k~ k~ x K

E-o I 0 _~x~ 2/w

E-1 0 0

/

0 0 _~x~

2

are

tetragonally

distorted. When

approaching

a limit of the free parameter

(A

_- ~x~

)

the

length

of the

edge

c decreases

(hence

_x

-

0),

and with

increasing _c(A

-

2),

the ratio x increases towards

infinity.

It is

important

_to be able to determine _a tD surface for any

prescribed

value of xo. The

equation

_{x = xo,} where _x is

given by (8)

is transcendental, and can be solved

only numerically.

We have

adopted

the

following procedure.

To shorten the

root-searching

interval,

we express x in _terms of either E _or fl. Since there is

only

one real _root and the interval is

known,

it is not necessary to

employ root-finding

methods which involve derivatives. Brent's

algorithm [23]

is

probably

the best numerical method of

solving

the

equation

_x

= xo.

However,

in _most

applications,

it is sufficient _to

approximate

_x

by

_a rational function. We _use

a rational

approximation

^{of the form}

4

p~ E' X #

' 4

(9a)

I

_{~, ~'}

i 0

where E is the free parameter ^{with 0} _~ ^E _~ l, and the coefficients p~, q,

(I

=

0,

1,

2,

3 and

4)

need _to be determined.

By employing

the standard method of rational

interpolation [23]

_we have found rational

approximations

for _x

(see

Tab.

IV)

with _a numerical _accuracy better than 7

significant figures.

Thus, for _a

given

_xo, the determination of free parameter ^{is reduced} to a

numerical solution of the

quartic

£

4 _{~Pi xo qi} E~

= o

(9b)

, o

with coefficients

given

in table IV.

Table IV. Three rational

approximations of

the

form (9) for

ratio _x

(E) for different

intervals

of

the _parameter E.

0.05£ E £ 0,13 £L73 ) 0.13£E£0.50 0.50 £E£0.89

I

0 4.65nWWM2642 1 3.80339n64598255 2.7l697576312VM

1 95.016254824041 45.213715340127 1.051796760637337 5.37555S39100W09

2 %15.10224088421 4445.36501919f0& 1485%2176933356 213.073S84195259

3 15023.24273163% 10.97%3834344544

4 85.7994121752395 1.95526487007l338

We note that A

=

14 has been

wrongly

attributed to the tD surface

spanning

an

8-gon

formed

by eight

of the twelve

edges

of _a cube

(so

that xo =

I). By solving

the above

equation numerically

_we find that the _correct value is A

= 5.3485782.

The

expressions (7)

and

(8) give

~

x

~

2

g~K(k~)

so it is clear that the normalization factor _K is related to the size of the

right tetragonal prism

for

given

_a and _c and if

parametric representation

of surface is considered _as dimensional

equations,

then that _K should be

given

units of inverse

length.

Further

simplification gives

the

following expression

for the normalization factor _K in the

parametric equations (4)

for tD saddle surfaces

K = a

( IO)

~x

K(kx)

where g~ and k~ are defined in table

IIa,

and both

depend

_on the value of free

parameter

^alone.

Equation (10)

and

expressions

in table IIa

give

an

analytical expression

for _K in _terms of the free parameter.

Therefore,

for _a

specific

value of the

length

of the

edge

_{a, K}

depends only

on

the value of free parameter ^and

changes continuously

with

it,

with

limiting

values listed in table III. Until _now

only

for tD surface with A

=

14 has been known normalization factor

(K

₌ 0.8389223 in Ref.

[24])

which _can be

easily

obtained from

(lo)

_as

K =

~~~~

^a =

0.8389223 _a

Parametric

equations

for normalized tD surfaces.

For the

parametric equations (4)

with the

amplitudes

_q~~ and

q~~

given

in _terms of

complex

inverse sine functions to be

practically useful,

it is necessary to examine the

complex

_square

root function of the

complex

variable

w = u + iv with

w w

I,

which appears in

equations

for the coordinates _x and y

(see

Tab.

IIb).

It is

clearly

desirable _to choose the _same branch of the

complex

_square root for both coordinates _on the entire unit disc. In what follows _we

consider

only

that branch of the square ^root which is I _at I. In this way, the

equation (4)

^{for the}

x coordinate

correctly _represents

the tD

tetragonal

surface for _{u m}

0,

I.e. _on

imaginary

axis and

right-hand

half of the unit disc. In order _to

represent correctly

the surface _on the left-hand part of the unit

disc,

it is sufficient to

change

the

sign

of the

expression

for _x.

Symmetry

considerations

require

that for the calculation of the y coordinate the surface be

correctly represented

_on the upper

(I

and II

quadrant)

or lower

(III

and IV

quadrant) parts

of the unit disc. Were it the

only requirement,

it could be satisfied

by

a

simple change

of

sign, just

_as

was done for the _x coordinate.

However,

the

expression (4)

for the _y coordinate

gives

a correct

representation

in the I and the IV

quadrant

of the

complex plane,

instead of the I and II

quadrant.

To resolve this

problem,

we must «rotate» the solution function counter-

clockwise

by

w/2. The

required

function for y is found

using

the

procedure

described in

[14].

The _new

equation

for y, obtained from

integral

231.00 in reference

[20],

which satisfies the convention

conceming

the branch of the square root is

y

(u,

v

)

=

KgxRe iF

_(q~y,

kx)i

where the

expression

for

q~~ is derived from that for _q~~ in the first row of table IIb

by changing signs

of all

quadratic

terms. In other

words,

the

relationship

between

q~~ and

q~~ is

~y(W ⁾

" ~x

(I

W

).

Further,

it is convenient _{to use}

polar

coordinates instead of the Cartesian. Hence, for

given lengths

_a and _c of the

edges

of _a

right tetragonal prism

it is necessary ^to find the value of the free parameter

(conveniently

chosen from among

E, fl

and

A),

from

(8). According (4)

and

( lo),

^the coordinates of the

appropriate

tD saddle surface

spanning eight

of the twelve

edges

of

a

right tetragonal prism

are

x(r,

0

= ±

~ Re

[F (arcsin ll'~(w ), k~)]

2

K(k~)

y

(r,

0

= ±

£

^' _Re

IF (arcsin

_{~l~~ (< w}

), k~) I1)

2

K(k~)

z(r,

0

=

£

^' _Re

iF (arcsin

_~l~~(w

), k~)j

2 K(k~

where

~/

4(w/g~)~

'~'~~~°

w~

₊

2(w/g~)~

₊ l

~2 ll'~(w )

₌

gz

with w = r cos 0 + ir sin 0 and 0 _{« r «} I and _w

~ 0 _{w w,} while k~, k~, _g~ ^and g~ are

defined in table IIa. In the

equation

for _x, the

plus sign

relates to

right

half and the minus

sign

to left half of the

complex plane.

In the

expression

_{for y, the}

plus sign

relates to upper half and the minus

sign

to lower half. In this way, we arrive at

parametric equations

of the normalized tD

family.

Unfortunately,

there is _no true addition formula for F

(q~, k)

^with a

complex amplitude,

_so

that F

(u

₊ iv,

k)

cannot be

expressed

in _terms of F

(u, k)

and F

(iv, k) only.

It is therefore

impossible

to separate ^{the real and}

imaginary

_parts, and

parametric equations (11)

cannot be

simplified

further.

Consider the

limiting

behaviour of the

parametric equations

when the free parameter

approaches

its limits

(see

^Tab.

III).

Close to one limit

(A

- ~x~ we find that the

appropriate

F (q~,

k) exist,

but _x

= y = z ₌

0,

because g~ = g~ = 0. For the other limit (A =

2)

the

moduli k~ ^and k~ in

equations (4)

are 0 and I

respectively,

and therefore the

appropriate

functions F (q~,

k)

reduce _to

elementary

functions. Since the

multiplication

constants are finite, the

equations

_are

expressed

ⁱⁿ terms of

elementary

functions

only,

and describe surface with the cla ratio

approaching infinity.

It is easy ^to show that for A

=

2 _we obtain

parametric equations

of Scherk's surface _z

=

In

(cos y/cos x) [25],

_so that

equations (4)

derived with the

assumption

A

~ 2 _can be extended to the

limiting

case of A

= 2.

The

elliptic integrals

in II are finite in the entire

complex plane,

_even at

points

defined

by

(2).

This _means that F

(q~, k)

is finite _at

singular points

of the Weierstrass function

(3).

This is

particularly important,

in view of the fact that numerical estimation of

integrals

fails close to the

singularities.

No

expressions

for the

approximate computations

of tD surfaces _are known. It is clear that

they

_can be deduced from

equations (11) by

_an

appropriate approximation

of the

incomplete elliptic integrals

of the first kind

by elementary

functions. This

procedure

deserves additional

analysis,

but _an

approximation

for _z

immediately

suggests ^{itself. Thus for A} _~ ^{2 the} values of the modulus k~ in the

expression

for _{z are} between 0 and I, but for A

~ 8 the value of

k~ is less than 0, I. This makes it

possible

to express the _z coordinate in _terms of

elementary

functions with _a

high

numerical accuracy. Since

F(q~, k)

m q~ for small

k,

we have

z m

~ Re arcsin I

2

K(k~) g/

With w = r cos 0 + ir sin 0 the real

part

^is

explicitly

found _as

z m

)

_{arcsin &}

K(k~)

where

&

=

~

[~/(r~

_{+ k~}

^cos~

2 0)~ +

kj ^sin~

2 0 ₊

~/(r~

_k~

^cos~

2 0)~ +

k) ^sin~

2 0 2 k~

For A _« 50 this

approximation gives

numerical accuracy ^to 7

significant figures

on the entire unit

disc,

and for A _« 000 _{to as} many as 15

significant figures.

It is _even

possible

to

approximate

the coordinate, albeit less

accurately, using only

the

quadratic

function

c ~

w~

~ 2

K(k~)

^~

fi

and hence

~

~K~k~)fi~~~~~~~

Computation

of tD surfaces.

Since sketches

[26]

and

photographs

of models

[27]

^of tD surfaces cannot be

regarded

_as

quantitative, drawings [15]

obtained

using

the

Enneper-Weierstrass representation

are the

only

known _case of their

computation.

Reference

[15] gives

the Cartesian coordinates for

A

=

2.01, 3,

14 and 50 obtained

by taking

the real

parts

^of

complex integrals (our Eqs. (I)

and

(3))

^defined on the unit disc.

Computation

of _a tD surface

using

the

Enneper-Weierstrass representation requires

numerical

integration

to evaluate

hyperelliptic integrals,

and _cannot be carried _out without

prior analysis

of the behaviour of the

integrands

and the choice of _an

appropriate

numerical

method from among a number which _are available. For direct

comparison,

_we have

performed

such

computations

for tD surfaces described below. These

computations

_are

quite

cumbersome and

require

a careful

interpretation

of data in the

vicinity

of

singularities

of the Weierstrass

function,

since

large portions

of the surfaces _are

generated by

small

regions surrounding

these

singularities. However,

the main weakness of

using

the

Enneper-Weierstrass representation directly

is that surfaces

corresponding

to a

specific

value of A

computed

from them _cannot be

applied,

In other

words,

without the

knowledge

of

K and _x

(neither

of which _can be deduced from

it),

the

Enneper-Weierstrass representation

does not tell _us how _to

generate

a surface to match

prescribed

data.

Also,

it is

impossible

to compare different tD surfaces without

normalization.

By

contrast,

parametric equations (11) require

the evaluation of functions F

(q~, k)

and

K(k)

which is much

easier,

and

yield

all information

required

for

matching,

thus

making straightforward physical applications possible.

We _use them to

compute

^{tD saddle} surfaces for several

prescribed

values of the _axes ratio.

The evaluation of the functions

F(q~, k)

and

K(k),

^is now

usually

^based on the Gauss

algorithm

of the

arithmetic-geometric

mean and Landen's transformation

[21, 23, 28]

and Carlson's

algorithm [29].

The latter is

implemented

in

NAG,

SLATEC and IMSL numerical

libraries.

However,

_{we use} the Gauss

algorithm

because its _use for the evaluation of

F(q~,

k) for

complex

values of

amplitude

is well documented. In order _to demonstrate the

simplicity

of this

algorithm,

we outline it below. The

arithmetic-geometric

_mean of _two

numbers ao and

bo (where bo

is any

complex

number different from 0 and from

JOURNAL DEPHYSIQUE i T 2, N' 12, DECEMBER <992 ⁷⁹

ao),

is denoted

by AG(ao,bo),

and defined _as the _common limit of the arithmetic

(a,)

and

geometric (b;)

_sequences

~< ~

~' _'

j

^~~ '

~i

~

~i

with I

=

1, 2, 3,

Both sequences are convergent ^for any choice of ao and

bo.

^After

reaching

the

prescribed

_accuracy at the n-th iteration step

(I,e,

difference between a~ and b~ ^{is less than} the

required value), AG(ao, bo)

_can be

approximated by

_{a~ m}

b~.

To calculate

K(k)

for any

particular

value of k we start from

ao =

I

bo

=

/$.

After _n iterations _we have

~~~~

2 AG

ao, bo)

~j~

To calculate F

(q~, k)

for any

particular

values of _q~ and k _we start from

ao =

I

bo

₌

k(

=

/$

q~o = q~

and

by using

^Landen's

descending

_sequence

k)

j

~ ~ _' ~ "~~~~

~~

' ~~' ~

'~

~~~~ _~~

l +

k)

_,

where I

=

1, 2,

..., n,

together

with the

arithmetic-geometric

_mean

algorithm.

After

prescribed

accuracy is reached at n iterations _we have

~

~~' ~~

2~

AG(ao, bo)

_{2~ a~}

The

algorithm

is

simple

and very fast

(it

_converges

quadratically)

_so that numerical accuracy to 7 and 15

figures

is

usually

reached at the

4th,

and 8th iteration

step, respectively.

Its

only

known

unsatisfactory

behaviour is for k

- I and _q~

- w/2, but these _cases do not arise here.

We have

computed

^the coordinates of tD saddle surfaces for _x

=

1,

0.5 and 0.25. The

length

of

edge

a is assumed _to be I and all

quantities required

in the calculation _are

given

in table V.

Table V. Constants used in the

computation of

the various saddle tD minimal

su~fiaces for different

values

of

^the axis ratio _x.

=1 =o.5

k 5.3485782 43.91463368 569.960o6523

K 0.71579938 1.054279364 1.899815152

0.85894265 0.543284378 0.289183541

0.44045702 0.15094129 0.041886923

0.36208944 0.593650606 0.676894704

0.19400238 0.022783273 0.00175451429

Tables VIa-c list

only

the coordinates of _one

eighth

of each saddle surface. These minimal surface

pieces

are

asymmetric units,

and for each the

(r,

b

) computational

domain is _{a sector} of the unit disc with 0 _{« r w} I and 0 _w b _« mm. The _sector is divided

using

_a 8 _x 5

grid

where the coordinates of

grid points

_are

given by

w

(n~,

no

)

= n~ ro cos

(no Ho)

+

in~

_ro ^sin

(no

_Ho

)

with ro =

1/8 and Ho = w/16

(n~

₌

1,

...,

8 and no ₌

0, 1,

...,

4).

It is clear that the coordinates of _a

complete

tD saddle surface _can be obtained

by _symmetry

considerations alone. Coordinates of surfaces

corresponding

to the _same value of _x but different values of

a can b£ obtained

by multiplying

the coordinates in tables VIa-c

by

_a.

Table VI. Cartesian

coordi~ates _of

_three

_different

_tD

_su~fiaces.

_{The columns}

_differ

_in

n~ and the _rows in no,

a)

_{x =} I,

b)

_x

=

0.5,

c)

_{x =} 0.25.

I Z 3 4 5 6 7 8

o

1

2

3

4

a)

1 2 3 4 S 6 7 8

o

i

z

3

4

b)

Table VI

(continued).

1 2 3 4 5 6 7 8

o

i

z

3

4

C)

Acknowledgements.

We _are

grateful

to Dr. C.

Briggs

of the British Council for support, ^and to Professor B. C.

Carlson of Iowa State

University

for discussions

conceming

the

properties

and

computation

of

elliptic integrals.

References

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Abhandlungen (Verlag

Julius

Springer,

Berlin, 1890) vol.1.

[2] SCHOEN A. H., Infinite Periodic Minimal Surfaces Without Self-intersections, NASA Technical

Report

No. TN D-05541 (1970).

[3] FISCHER W. and KOCH E., J.

Phys.

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[4] ANDERSSON S., HYDE S. T., LARSSON K, and LIDIN S., Chem. Rev. 88 (1988) 221-242.

[5] SCRIVEN L. E., Nature 266 (1976) 123.

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Physica

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168 (1984) 221-254 170 (1985) 225-239.

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Geometry

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Engelwood

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[16] FOGDEN A., Ph. D. Thesis,

Department

of

Applied

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Engineers

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[22] Reference [13], _p. 234 and

figure

25.

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University

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[25] Reference [13], _p. 128 and

equation

(80).

[26] Reference [I], _p. 126 and

figure

4 reference [I I], _p. 430 and

figure

21.

[27] Reference [2],

figure

6 reference [4],

figure

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