Mathematics and Soap Films

John Oprea

Cleveland State University

Surface tension creates a “skin” on a liquid whose

molecules are polar.

Example: H2O is polar, so water has a “skin”.

Soap reduces surface tension by adding long polar

molecules with hydrocarbon tails.

Surface tension dominance, but with noticeable effects of gravity

1st Principle of Soap Films. A soap film minimizes

its surface area.

Surface tension pulls a soap film as tight as it can be.

But what is the exact geometry dictated by the

force due to surface tension?

To see this, let’s analyze a piece of a soap film

surface that is expanded outward by an applied

pressure p.

Take a piece of the film given by two

perpendicular (tangent) directions and

compute the work done to expand the surface

area under some pressure.

Also, if we take

p = pressure and S = surface area, then

The change in surface area is given by

A Physicist’s first words:

“Neglect the higher order Term!”

Laplace-Young Equation

Laplace-Young involves Mean Curvature!

Definition. A surface S is minimal if H = 0.

Theorem. A soap film is a minimal surface.

What property does a soap bubble have?

What happens when bubbles fuse?

Does the Laplace-Young equation have

medical consequences?

Questions

Alveoli Alveoli are modeled

by spheres which

expand when we

inhale and contract

when we exhale.

When is the pressure

difference the greatest?

So how can we ever inhale?

It was the development of artificial surfactant in

the 1960’s that was essential to the survival of

premature babies!

Now consider the “loop on a hoop” experiment.

What does it say?

Theorem. A closed curve which maximizes the

enclosed area subject to having a fixed

perimeter is a circle.

For other closed curves,

A

L

A

L1

A1

L1

Same A Same L1

by Theorem applied to 2nd and 3rd curves

For the circle, Consequences:

Theorem. For fixed area, the curve which

minimizes perimeter is the circle.

3-Dimensional Version. For fixed volume, the

closed surface which minimizes surface area is

a sphere!

Physical Consequence. Every soap bubble

is a sphere.

A soap bubble minimizes its surface area

subject to enclosing a fixed volume.

Plateau’s Rules

Weierstrass-Enneper Representations

Complex analysis may be used to obtain “formulas” for

minimal surfaces.

Theorem. (First Representation)

Theorem. (Second Representation)

The Catenoid

The Helicoid

Enneper’s Surface

La Chauve-Souris

The Trinoid

Appendix: One-celled Organisms

Minimizing surface area subject to fixed volume for

surfaces of revolution (without the extra requirement

of compactness) produces spheres, cylinders, nodoids

and unduloids.

One-celled creatures often take shapes (truncated by

cilia or flagella) similar to spheres, cylinders, nodoids

and unduloids. Here we present some drawings of

one-celled organisms taken from

On Growth and Form

By

D’Arcy Wentworth Thompson

Unduloids (surfaces of

constant mean curvature)

Other aspects of differential geometry make

themselves apparent in biology also.

Theorem. The only ruled minimal surface

is the helicoid.

Water Films in Space

Question: What is the exact shape of the “lens”?

Les faits mathématiques dignes d’être étudiés, ce

sont ceux qui, par leur analogie avec d’autres faits,

sont susceptibles de nous conduire à la

connaissance d’une loi mathématique, de la même

façon que les faits expérimentaux nous conduisent

à la connaissance d’une loi physique. Ce sont ceux

qui nous révèlent des parentés insoupçonnées

entre d’autres faits, connus depuis longtemps,

mais qu’on croyait à tort étrangers les uns aux

autres.

---------- Henri Poincaré

The mathematical facts worthy of being

studied are those which, by their analogy

with other facts, are capable of leading us

to the knowledge of a mathematical law

just as experimental facts lead us to the

knowledge of a physical law. They reveal

the kinship between other facts, long

known, but wrongly believed to be

strangers to one another.

----- Henri Poincaré