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The Mathematics of Soap Films

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The Mathematics of Soap Films 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 Surface tension pulls a soap film as tight as it can be. 1st Principle of Soap Films. A soap film minimizes its surface area. 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. Questions What property does a soap bubble have? What happens when bubbles fuse? Does the Laplace-Young equation have medical consequences? Alveoli are modeled Alveoli 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. Consequences: For the circle, For other closed curves, 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 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 .
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