Space filling minimal surfaces and sphere packings

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Space filling minimal surfaces and sphere packings

Veit Elser

To cite this version:

Veit Elser. Space filling minimal surfaces and sphere packings. Journal de Physique I, EDP Sciences,

1994, 4 (5), pp.731-735. �10.1051/jp1:1994172�. �jpa-00246943�

J. Phys. I Fianc.e 4 (1994) 731-735 MAY 1994, PAGE 731

Classification Physics Absn.ac<s

61.50E

Space filling minimal surfaces and sphere packings

Veit Elser

Laboratory

of Atomic and Sobd State

Physics,

Cornell

University,

Ithaca, New York 14853-2501, U-S-A-

(Received 25 Oc<ober J993, a<.c.epted in

final

form 20

Januaiy

J994)

Abstract. A space

filling

minimal surface is defined _tu be any embedded minimal surface without boundary with the property ^{(haï the} area and genus enclosed by any

large

spherical region scales in proportion tu the volume of the region. ^The

tfiply periodic

minimal surfaces _{are une} realization, but _net necessarily the only une.

By usiqg

^the genus per unit volume of the surface, _a

meartirtgful

_companson of surface _{areas can} be made _evert in _cases where there _{is no} unit cell. Of the krtowrt

periodic

minimal surfaces this _measure of the surface _{area is} smallest for Schoert's F- RD surface. This surface _{is une} of several that _is

closely

related tu

packirtgs

of

spheres.

Its low _area

is

largely

due tu the fact that the

corresponding sphere packing

(fcc) has the maximal kissirtg number.

The

triply periodic

minimal surfaces have found

expression

in _a diverse

variety

of

physical

structures

[1].

The basic

length

scale of these structures is set

by

^the dimensions of their

primitive

unit cells. On much

larger length

scales

they

_{appear as a} foam which fills space

homogeneously.

Considered _{as an}

example

of _a

therrnodynamic phase

in

equilibrium,

the

macroscopic homogeneity

of _a foam _{is more}

significant

than _its

microscopic penodicity.

This fact _is

nicely

illustrated

by

the existence of

quasicrystals

in

metallurgy.

Are there

analogous quasicrystalline

minimal surface foams ? In

anticipation

that the _{answer to} this

question

_is

« yes », I consider below the _use of

topology

as

opposed

to

penodicity

_m

normalizmg

measurements of _a surface. Of the known

triply penodic

minimal

surfaces,

the one with the

smallest value of the

topologically

norrnalized surface _{area is} Schoen's F-RD surface

[2].

In trying to understand this fact I _am lead to consider _a

family

of minimal surfaces

closely

related to

sphere packings.

The surface _areas of such surfaces _are

reproduced

to a remarkable

degree

of accuracy

by

a formula

involving

the

packing

fraction

f, kissing

number

k,

and

kissmg angle

0 of the

corresponding sphere packing.

Essentially

the same scale mvariant

quantity

defined below _was mtroduced

by

Anderson e<

ai.

[3]

and

by Hyde [4]

_in his defimtion of the _«

packing

^index _» ^of a minimal surface. In addition _{to its}

geometrical significance,

considered here, this index _is beheved _to be correlated

732 jOURNAL DE PHYSIQUE I N° 5

with the

phase stability

of _a sequence of cubic

phases

in the

DDAB-water-styrene

system

[5],

and

measurable,

_m

principle, by

NMR

techniques [6].

Consider _an embedded minimal surface without

boundary

and any sequence of

spherical

volumes V

(R

with

increasing

radius R. Let A

(R

be the _area and G

(R

the genus of the surface inside

V(R).

The surface will be called _« space

filling

_» if the

following

two limits exist

a~~j =

lim A

(R )IV (R

,

(1)

R _{- «}

g~~j =

lim G

(R )IV (R (2)

R

- «

The limits a~~j and g~~, are

respectwely

the _area and genus « per unit volume

». Confusion about the value of g~~, for

periodic

surfaces _can be avoided

by using

the

asymptotic

relation

[7]

~

~~~ fi

~ ~ ~~~~ ~~~

2 ^'

where _X

(R

is the Euler charactefistic of the surface inside V

(R) (').

The Euler charactefistic has the

advantage

over the genus in

being

additive for _two surfaces

being jomed along

_a dosed

curve

(such

_as considered below j. In _a

crystalline

surface the situation is

simplest

when the unit

cell _cuts the surface _so that each

boundary

_curve lies

entirely

within _one of the cell s faces. The Euler charactefistic of the

crystal

is then

just

_Xceii, the Euler charactefistic inside _one primitive cell, limes the number of _unit cells.

Using (3)

_{one may} define the genus « pet

primitive

cell _»

as well _as the genus « pet ^unit volume _»

gcell ~

@

" gvol ~cell,

(4)

where v~~jj is the volume inside _one

primitive

cell. With this definition g~~,j is one less the

« genus »

conventionally

associated with the

compactified

surface

[7] (e.g.

_g~~,j

=

2 for the P

surface).

A dimensionless combination of a~~j and g~~,

(and proportional

to a~~j) ^{defines the}

topologically

normalized surface _area of _a space

filling

minimal surface

a =

~()(. ⁽⁵⁾

gvoi

Table I.

Sphere

~

~

o _r

a(k, f, 0)

a

~~~~~~~

packing

D diamond 4 0.340 54.7° 1.016 1.898 1.919

P _sc 6 0.524 45° 1.148 1.859 1.861

1-WP bcc 8 0.680 35.3° 1,199 1.923 1.908

F-RD fcc 12 0.740 30° 1.195 1.770 1.758

(1) Equation (3) _is aise valid for rtortpefiodic surfaces provided the _{errer term,} proporiional tu the number of bourtdary _curves gerterated at the surface of V(R), remains subdominant.

N° 5 SPACE FILLING MINIMAL SURFACES AND SPHERE PACKINGS 733

The _«

packing

index _» of _a minimal

surface,

_as defined

by Hyde [4],

_is

just ,/

the index defined

by

Anderson et ai.

[3]

_is

a~lar.

Values of _a for four

penodic

minimal surfaces _are

given

in the last column of table I. The small

dispersion

ⁱⁿ these numbers is remarkable.

Obtaining

bounds _{on a} appears ^to be _a difficult

problem.

^Of the known

periodic

surfaces for which surface _areas are known, the F-RD surface has the minimum value of _a

[3, 4].

The F-RD surface and the three others in table I each

belong

to a one

parameter family

^of

constant _mean curvature surfaces which terminates in _a

packing

of identical

spheres [7].

This

greatly

facilitates the visuahzation of these surfaces since one may

begin

with the

correspond-

mg

sphere packing

and then

replace

the

regions

around the contact

points by

catenoids _as shown _in

figure

1. The limit of the F-RD

family

of _{constant mean} curvature surfaces _is the fcc

packing

of

spheres

the lattice

packing

which achieves both the maximum

packing

fraction

~f

=

0.740 and

kissing

number

(k

=

12

).

An

approximate

calculation below shows that it _is

the latter which _is

responsible

^for

givmg

the F-RD surface the smallest value of

a.

--_

',~

/ ~

J ',

;' '

'

/

L /

' /

_"

,.' ,,

/

/

' ,'

j

i i

A

/i, /

' /

.'

'.~ ~,' .,

~

~

',

_.~ ~,

_.'

/

Fig.

l

Consider _a

general packing

of identical

spheres

where each

sphere

has radius

~ l and

makes contact with

exactly

k other

spheres. Figure

shows _one

sphere

centered at A

making

contact with two other

spheres

_; the separation ^between «

kissmg

_»

spheres

_is

^W

= 2 _r. The

arrangement ^of

kissing spheres

_{may vary} from

sphere

to

sphere

_; even so, there _{is a minimum}

angle

2 0 subtended

by

_{any pair} of

kissing spheres.

For the

particular packing

considered in

figure1

_we will _assume this minimum

angular

_{separation is} realized

by

the two

spheres making

contact with

sphere

A. Smaller

spheres,

concentnc with the

original spheres

of the

packing,

_can be

joined together smoothly by

catenoids. The transformation of each small

sphere (see Fig, l) corresponds

to the

replacement

of k

sphencal

_caps

by

k half-catenoids which

smoothly jour

the surface of the small

sphere.

At each circular

junction

between

sphere

and catenoid, the first denvatives _{are contmuous.} This cannot be extended _to the second denvatives _smce the _two surfaces

clearly

have different _constant values of _{mean curvature.} In order to have the

largest possible

fraction of catenoid (zero mean

curvature)

surface, the small

spheres

_are made _as small _as

possible without,

of _course,

allowmg

catenoids _to intersect each

734 JOURNAL DE PHYSIQUE ^{I N° 5}

other. The

optimal

construction is shown in

figure1

where two catenoids _at the minimum

angular separation just

touch each other _at C.

By rescaling

the small

sphere

radius AC _to 1, the radius of

spheres

in the

packing

is

given by

i = cos 0 ₊

(sin

0)~ arcsinh

(col

0

). (6)

The _areas of _one ehminated

spherical

_cap and _one

replacement

half-catenoid _are

given

respectively by

a~~~ ⁼ 2 _ar (1 Cos 0

), (7)

a~~, = wr

sin~

₀

₍₈₎

Viewing

each of the small

punctured spheres

with its k half-catenoids _{as a} node of _an

approximate

^{minimal surface,} we obtam for the _area per node the expression

~node "

4 _{" +} k(~cal

~cap) (9)

The _area per unit volume of the

approximate

minimal surface is obtained

by dividing

the _area per node

by

the volume associated with each

node,

_v~~~~. Since the latter is

just

the volume per

sphere

^of the

sphere packing,

it is

simply

related _to the

packing

fraction

by

f

₌ ^~

ri~~)

/v~~~~. (10)

3

Similarly,

the genus per unit volume _is obtained

by dividing

the genus per

node,

g~~~~,

by

_v~~~~,

Agam using (3),

but this _time

applied

to _a node rather than _a

primitive cell,

_we have g~~~~ =

Xnode/2

smce the entire surface is constructed

by joining together

a

large

^number

of nodes,

always along

circular

boundary

curves.

Finally,

since the Euler characteristic of _a

sphere

^{with k} hales _is 2 k, _we obtam

g~~~~ =

~

l Il

Dividing (9)

and

(11) by

_v~~~~ to form a~~j and g~~j and

using

these in

(5),

_we arrive at the

topologically

norrnalized surface _area of _Dur approximate ^{minimal surface}

~~~'~'

~

Î~

^{~~~ ~}

^~~j

2

~~~ ~

~

~

~Î~

^{~ ~~~~}

We _note that the _r in

(12)

is net an

independent

_parameter but

given explicitly by (6).

Formula

(12)

is

compared

with the surface _areas of the _true minimal surfaces _in table I. The agreement

is

quite good considenng

the fact that pieces of the

original spheres havmg

posit<ve Gaussian curvature remam in the approximate ^{minimal surface. To} the extent that the _mean

curvature of the approximate ^surface is net constant, its surface _area should be greater ^{than that}

of _a constant _mean curvature surface

enclosmg

the _same volume.

However,

the endosed

volume of the approximate ^{surface does} net

correspond exactly

to that of the _{true zero mean}

curvature surface. This leads to an underestimate of the surface _area since ail four surfaces

considered here have _{maximum area} at the true volume fraction

[7].

The _{two sources} of _errer

(non-constant

_{mean curvature, wrong}

volume)

have opposite signs and may

partially

compensate ^{each other.}

Nevertheless,

the _correct identification of the smallest _area minimal surface

(F-RD)

is

probably

net fortuitous and formula

(12)

_grues some

explanation

of this

outcome. We _see

immediately

that it is _net the

packing

fraction but the rather

large kissing

number that _is

responsible

for the small _area.

N° 5 SPACE FILLING MINIMAL SURFACES AND SPHERE PACKINGS 735

A definite

shortcoming

of the

sphere packing representation

of minimal surfaces _is the

asymetrical

treatment of the inside and outside volumes. Surfaces which possess this symmetry, ^such as P and D,

actually

lead _{to two} different

approximations, depending

_on what

one considers _to be the

« inside _» and

« outside

». This

discrepancy

_can be illustrated for the

case of the P surface

by computing

the location of the _« flat

points

_» which in fact _are net flat _at ail _m the

approximation

and _are better defined _as the

points

of intersection with _axes of

3-fold symmetry. ^In a x x unit oeil the flat

point

coordinates _are

given by (±

_{>., ± x, ± x}

)

where _x

=

1/4 in the _exact surface. In the approximate

surface,

with the choice made of

putting

the

spheres

at the

integer coordinates,

_we find instead _>.

=

1/(2 ,1r)

=

0.2515. Additional

shortcommgs

of the

approximation

considered here _can be

expected

in the _case of

surfaces,

such _as

C(P) [7],

where the

corresponding family

of _{constant mean curvature} surfaces

terminales in

self-intersecting spheres.

The close

correspondence

between the F-RD minimal surface and the fcc

sphere packing

suggests some

possibilities

for other minimal

surfaces,

in

particular,

surfaces which _{are net}

periodic. Viewing

the fcc

packing

_as a

particular periodic

_sequence

(...ABCABC...)

of close-

packed planes

of

spheres,

I

speculate

that other sequences

quasiperiodic,

random also

correspond

to minimal surfaces. The

parameters k, f,

and 0 would be the _same in each _{case so} that their

topologically

normahzed _areas would ail be close _to that of the F-RD surface. In this

regard

there _{are two} interestmg

problems

_:

(1) Among

these

possibilities,

does the F-RD surface remain the surface with the smallest

topologically

normahzed _area ?

(2)

Can the

corrections _to formula

(12)

for this class of surfaces be understood

directly

_m terras of the

layering

_sequence ?

Acknowledgments.

This work _was

supported by

_a Presidential

Young Investigator

grant administered

by

the National Science Foundation, DMR-1234567.

References

Ii For _a survey the reader _is directed _tu the

Proceedings

of _{a recent} conference, J. Phys. Colloq. Fianc.e Si (1990) rt° C7.

[2] Schoert A., ^Irtfirtite

pefiodic

minimal surfaces without self-intersections, ^{NASA Tech.} Rep, _no. D- 554l (1970).

[3] Artdersort D. M., Wertrterstrüm H., Olsson U., J. Phys. Chem. 93 (1989) 4243-4253.

[4] Hyde S. T., Ref. Il, _pp. 209-228.

[5] Strüm P., Artdersort D. M., Langmmr 8

(1992)

691-709.

[6] Halle B.,

Ljurtggrert

S., Lidirt S., J. Chem.

Phys.

97 (1992) 1401-1415.

[7] Artdersort D. M., Davis H. T., Nitsche J. C. C., Scfivert L. E., Adv. Chem. Phys. 77 (1990) 337-396.