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Mathematics and Films

John Oprea Cleveland State University tension creates a “skin” on a whose molecules are polar.

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

Soap reduces by adding long polar molecules with hydrocarbon tails.

Surface tension dominance, but with noticeable effects of gravity Surface tension pulls a as tight as it can be.

1st Principle of Soap Films. A soap film minimizes its surface .

But what is the exact 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 p. Take a piece of the film given by two perpendicular (tangent) directions and compute the work done to expand the 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 Laplace-Young involves Mean ! Definition. A surface S is minimal if H = 0.

Theorem. A soap film is a .

Questions What property does a soap have?

What happens when bubbles fuse?

Does the Laplace-Young equation have medical consequences? Alveoli are modeled Alveoli by 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 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 which maximizes the enclosed area subject to having a fixed is a circle. Consequences: For the circle,

For other closed ,

Same A Same L1 A A A1

L L1 L1

by Theorem applied to 2nd and 3rd curves 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 ! Physical Consequence. Every 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 . 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, , nodoids and unduloids. One-celled creatures often take (truncated by cilia or flagella) similar to spheres, cylinders, nodoids and unduloids. Here we present some drawings of one-celled organisms taken from By D’Arcy Wentworth Thompson

Unduloids (surfaces of constant ) Other aspects of make themselves apparent in also. Theorem. The only ruled minimal surface is the helicoid. Water Films in Space

Question: What is the exact 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