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A systematic method for parametrising periodic minimal surfaces: the F-RD surface A. Fogden

To cite this version: A. Fogden. A systematic method for parametrising periodic minimal surfaces: the F-RD surface. Journal de Physique I, EDP Sciences, 1992, 2 (3), pp.233-239. .

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J.

Phys.

J

France

(1992)

2

233-239

1992,

MARCH

PAGE

233

Classification

Physics

Abstracts

02.40

68.00

A systematic surfaces : the

parametrising

for surface

method F.RD

periodic

minimal

Fogden

A.

Department

(Received

Mathematics, University, PO

National

July 1991,

23

Abstract.

which

accepted

algorithm

An

developed by discovered by from

Applied

of

Australian

the Alan

all

for is

author

Schoen.

quantitative

in

the

exact

features

2

with

expression of

Physical Australia

December

1991)

of

construction

simply

exact

of

Canberra

final form

illustrated An

4,

2601,

School

Research Box

this

surface

the for

Sciences

triply periodic minimal parametrisation of

first the

may

F-RD be

Weierstrass

readily

and

Engineering,

surfaces the

function

recently

F-RD

surface

is

derived,

established.

Introduction.

key participant in all stages of the development of the physical microstructures increasing diversity of bicontinuous observed in condensed Iii provides a challenge to conventional geometric intuition. Recently, matter infinite periodic minimal surfaces (IPMS) have been adopted as the basis of a vocabulary for this specific class of this evolving language of It is an open question as to whether structure. periodic surfaces of the physical however context systems, possess any real significance in the their utility as a of classification is undeniable. The cubic TAMS of genus three the means D, P and G surfaces frequently invoked, in modelling the underlying a most structures are self-assemblies variety of molecular [2]. It has been postulated that these particular IPMS are favoured basis surface homogeneity uniform the of [3], that is, they represent the most on interracial scenario attainable globally. Bicontinuous phases related to higher genus TAMS with in the phase diagram increasing with also the region of their existence may occur, decreasing sample purity. This expectation has been bome out by recent experiments [4]. With the of sophisticated techniques, the principal limitation advent measurement to the observation of topologically possible and recognition complex remains the structures incomplete knowledge regarding accompanying IPMS descriptions. This problem has been partially redressed by the discovery of a multitude of new IPMS by Schoen [5] and, lately, by While

sciences,

geometry

has

explanation

been

a

of the

and Fischer [6]. These TAMS empirically, without the precise mathematical isolated were specification of the surface for quantitative comparison with observed systems. necessary Many of the Schoen examples have now been parametrised by Karcher [7]. A systematic of a general IPMS has recently been procedure for the parametrisation simplified developed by the author [8]. There the verified with greatly method a was description of the C(P) surface, then utilised in deriving the first rederivation of Neovius'

Koch

234

JOURNAL

PHYSIQUE

DE

I

N° 3

parametrisations of the hexagonal HS2 and orthorhombic surfaces PT of Koch and Fischer, together with those of the new orthorhombic IPMS and surface (denoted pentagonal new VAL and pCLP, respectively) arising from earlier work with Hyde [9, 10]. While the general formulation of the procedure is mathematically involved, the fundamentals simple and its are seeking application should accessible experimentalists classification of observed be to with With view this aim illustrate the method here the first parametrisation structures. to we a of the F-RD surface of Schoen. The choice of this example is physically motivated the Fsurface is a cubic IPMS of relatively simple topology (in fact, the lowest cubic IPMS RD genus of Schoen media. remaining undescribed) and thus may be expected to occur in bicontinuous The The

surface.

F-RD

topology

summarise lational

unit

absence two

by

the

of

(or

3 ~

m

m

).

and

RD).

The

in

the

Accordingly rotational

two-fold

partitioned by

Schoen's briefly is specified in report [5] we parametrisation here. The fundamental transfigure I, has a genus of six and the space group

surface

to

displayed

~

on-surface

labyrinths F

F

F-RD

relevant

IPMS,

the

of any

types

of the

symmetry information

Fm3m

symmetry

the

and the

the

interface

translational

unit

IPMS

the

axes

(as is the

are

geometrically

is

for

case

the

from

a

surface

I-WP

(and

distinct

constructed

planes

mirror

contains

basic

in II

the

])

so

are

characterised

surface

element

or

(identified in Fig. I ) delimited by four mirror planes of the surface. Propagation element by repeated reflection in the bounding mirror planes, until the full set of of this basic symmetries are exhausted, translational unit, thus comprising 48 such elements. generates the intersection surface is referred The of a mirror plane with the to as a plane line of curvature. Flhchenstfick

perpendicularly. The remaining defining the basic element intersect angles of w/3 and MN at points of 3-fold and 4-fold perpendicular respectively. These are degenerate points of the surface, rotational symmetry of the surface, is zero. called flat points, at which the Gaussian The degree of these flat points curvature surface of normal this point traced the number revolutions about the vector at out by the surface enclosing point is respectively. normal circuit the the 2 and 3, along any vector on Two

two

Fig.

pairs pairs

I.

of

such

curves

intersect

The

Parametrisation

at

fundamental

of

unit

of

F-RD

surface into the plane, such Mapping a minimal normal of of its surface point intersection the to in tum stereographically projected to a complex Weierstrass [12] to have the general form

(x,

y,

z

Re =

comprising

surface

48

of

the

surface

elements

shown.

surface.

F-RD

the

the

l~

(I

that

w'~, I(I

a

number

+

point (x, y, z) unit sphere

surface with

vector

w,

w'~),

the the

2

is

Gauss

centred

inverse

function

w') R (w')

dw'

was

mapped there, shown

and

by

(1)



3

PARAMETRISATION

EXACT

OF

THE

235

SURFACE

F-RD

complex function R (w ). The image of the basic element of the F-RD surface in the some complex plane under this composite map is the shaded polygon in figure 2. For an IPMS the corresponding Weierstrass function R (w ) is multivalued, specifically, each generic and more surface image w is normal shared by the (finite) number of points vector same (x, y, z) on the translational unit. Gauss of gives the multiple Hence the this unit map covering of the unit sphere (and the complex plane on projection) comprising this number of sheets. This is a topological and is equal to the less for an number constant, genus one orientable TAMS [9]. For the genus six F-RD surface Gauss the translational unit map image from figure 2 then comprise five sheets. This is apparent just as successive reflections must of the basic in its bounding planes yields the set of 48 such surface element elements forming unit, in the Gauss map image repeated the translational reflection of the spherical geodesic polygon in its edges generates a set of 48 such polygons tessellating a five-sheeted spherical covering. for

Fig.

Projected

2.

Gauss

map

image

of

the

surface

F-RD

element.

surface normal images w of flat points, the number of points on the the special vector with the possessing this normal is only equal to the generic value if each is counted multiplicity of its flat point degree. Accordingly, above the image w on the multiple covering, identified numbering this degree, corresponding to the flat point are the set of points, as point, which is associated pinned at this equivalent. We regard the sheets set of as common referred branch point of order the degree less one. The projected Gauss map image to as a of the Weierstrass with imposed by this identification is the Riemann surface structure Riemann surface of the is function R (w ). For the F-RD surface the branch point structure generated by the propagation of the first and second order branch points (labelled in Fig. 2, and corresponding to the flat points of degree two and three, respectively, on the periphery of For

unit

the

basic

the

five As

and

The

the

is

surface

element)

at

the

2w/3

and

3 MN

vertices

of

the

polygon

in

tessellation

the

of

sheets. Riemann

hence

surface

specified by

Weierstrass

function

is

finite-sheeted,

polynomial

a

of

the

F-RD

Weierstrass

the

equation surface

of

is

function

degree equal then

the

to

solution

of the

of

an

IPMS

number the

is of

algebraic, sheets

[13].

quintic equation

a5(w)R~+a~(w)R~+a~(w)R~+a2(w)R~+aj(w)R+ao(w)

0 =

(2)

236

JOURNAL

particular completed specified.

for

As

the

cubic

on

Gauss

polygon

this

must

image of the surface triangulated by one

map be

symmetry

derived

are

images

reflection

polynomials a~(w ), combining equations (I)

of

from

geodesic

the

~~"'~)

*

/

((/

and

TAMS

an

figure tiling

the

of

the

sphere).

2.

For

[14]. The TAMS of sphere by the 48 w/3, w/2) (equivalent to unit

underlying

This

positioned

are

is

been

spherical covering,

a

orientation

this

surface have

tiles

angles (MN,

concentric

in

Schwarz

tiling

4

vertex

a

tessellates

basic

case

with

at

l) e'~"'~), respectively

I) e~"'~,

±

of

Schwarz

triangle

parametrisation of this these polynomials

once

fifteen

3



The

(2)

and

element

the

I

,

of the

projection of the hexakis octalJedron onto tiling is shown in stereographic projection vertices of angle Mm, w/3 and w/2 in

~

ao(w ).

of six

set

a

then

PHYSIQUE

DE

(0,

(where

triangular

the 6, cc, m

is

8

12

and

e~"'~e~~"'~), integer).

an

of each coefficient polynomial a~(w) are necessarily symmetric with underlying Schwarz tiling. Thus they must comprise the 6, 8 and/or 12 tiling vertices and/or the 24 (respectively 48) images of a general edge (respectively face) point. The polynomials with roots at the Mm, w/3 and w/2 vertices are The

respect

set

of

to

the

roots

5

pi

fl

=

(w

w~)

=

w

=

w

(w~ +1)

(3a)

1=1

~

p2

fl

=

(w

w~)

(w

w~)

14 w~ +

~

(3b)

j=1 12

p~

fl

=

~~ =

+

w

33 w~

w~-

33

(3c)

,=1

(where the Mm vertex tessellated edges of the the degree of a~(w facts

These

where

n~,

ao

,

surface

4M

be

must

sufficient

equation (2)

particular

to

becomes

now

numbers,

real

are

complex plane is suppressed in pi). Further, for the IPMS, curvature to represent plane lines of on (again counting with the suppressed zero at infinity) [8]. specify the form of each coefficient polynomial in closed

in the

Riemann

then

are

infinity

at

and

the

indicate

braces

the

possibilities

two

for

the

cubic

term.

information

Additional known

that,

point

branch local

pinned image

to

a

the

at

the

wo,

results

demanding

from

Riemann

of the

structure

On

tessellation

the 2

three

unbranched.

Here

between utilise

we

the

equation (4) and the general observation

surface flat point of degree bo + I, on the bo + I sheets of the Riemann corresponding branch point (of order bo) above the flat point normal Weierstrass function branches asymptotic form [8] have the

R~°~ ~(w )

figure

consistency

surface.

of

sheets The

are

five

R~(w )

the

five

pinned branches

y~ (w

sheets, at

a

are

w~

wo)~~°

yo(w

then

each

over

second

order

given

)~~ (once),

Mm

w

-

vertex

point asymptotically branch

R(w )

surface vector

(5)

wo.

w~

of

with via

yj (twice)

:

the

underlying tiling

remaining equation (5) by the

w

-

w~

two

in

sheets

(6)



PARAMETRISATION

EXACT

3

Substituting terms

be

applies.

requirement that the quintinc and quadratic form into equation (4), the (and equal) asymptotic order in (w w~) implies that the quartic term 0) and that the first of the two options for the cubic (that is, a~ term

present Hence

which,

terms

=

form

the

of

consistent

pinned

with

at

order

R~(w ) The

final

a~, a~, another

a~

and

may of

be

choice

a

for

units

w~ of the

vertices

and

of

loss

coordinate

the

~

vertex

wj of the

Schwarz

the

three

unbranched

sheets,

surface

Riemann

(~)

°

"0

w/3

R

yj (three

(w

w

scaling of

the

on

figure 2,

each

five

R be

to

is

coefficients

equation (7) while (w (amounting to specified. This is

unbranched

point

such

(8)

real

the of

out

parameters

two

now

w~

-

of

divided

suitable

a

structure

From

be

may

constant

is

tiling the pair of sheets display the asymptotics

times

determination

the

is

generality by axes), leaving

underlying tiling.

be

now

equation (6), the quadratic and required. Similarly, equation (7)

as

unknown

such

one

must

"2P2~~+ of

form

(once),

'

+

zero),

parametrisation

the

the

each

wj)~

Clearly ao. without chosen

by analysing

achieved

w/2

of

stage

second

point,

yj(w

"3P~~~

+

(namely

over

branch

~

the

order

that,

fact

the

first

a

of

equal leading

equation

polynomial

the

substitution

on

of

are

237

SURFACE

F-RD

leading

"5P~P2~~ for

THE

first

the

of

are

cannot

OF

sheets

covered

above

the

by four

once

w/2 vertex interior point of the polygons meeting with there, and four times as an common reflection latter tour coverings in the polygon. In the the related by two case are perpendicular underlying tiling edges intersecting at the point, and accordingly the four function For example, corresponding Weierstrass values at this point occur as a repeated pair. 2 and evaluating equation (7) at w (where, equations (3a) and (3b), pj(1) I from -12) exist real conjugate pair p~(1) there and and complex must constants a a a =

=

=

(c, n)

that

such

-48a~R~+4a~R~-12a~R~+ao= Equating

yields

coefficients

pair

the

16 a~ a~

removing

thus a~

5

and

ao

=

final

the

25 =

degrees

two

48, the

of

Weierstrass

a(R-a)(R-c)~(R-n)~.

homogeneous ao

135

a~,

of

relations

al

of

al

a~

the

say surface is

F-RD

=

p)p~R~-

(10)

=

Choosing,

freedom.

function

(9)

lsp)R~+5p~R~-48

a~= then

I,

a~=

-15,

given by (II)

0 =

where

and

pi

equations

(3a)

p~ and

are

the

Weierstrass

polynomials

of

the

I-WP

and

D

surfaces,

given

in

(3b).

parametrisation

attained by substituting the representation (I). Although the quintic polynomial equation cannot be solved analytically, all required surface information (such as local coordinates and the global properties of surface-to-volume ratio, surface area and integral squared Gaussian curvature) be extracted numerically in a simple and can economical The of the path integration in equation (I) may be restricted extent to manner. the polygonal region of the complex plane in figure 2, as the complete TAMS is then shaded reflection in the bounding planar generated by mirror surface of the basic element. The curves corresponding polygonal region of the Riemann surface (that is, the corresponding branch of the function) is Weierstrass ascertained by determining the five root of equation (I I) for a The

exact

Weierstrass

function

solutions

of

of

the

F-RD

equation

surface

(11)

into

is

the

thus

JOURNAL

238

PHYSIQUE

DE



I

3

~ point

particular

comprise

real

a

the

on

w

real

number

edge

axis

and

0

w