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# The Soap Film: An Analogue Computer

Soap films provide a simple method of obtaining analogue solutions to some mathematical problems

Editors’ Note: This Classic article was first published in the September–October 1976 issue, and has been reprinted as part of American Scientist’s centennial-year celebrations. The author based his article on a popular lecture-demonstration given during his visit to American universities. Dr. Isenberg, who remains at the University of Kent since the writing of this article, still gives his soap-film demonstrations about 50 times a year, to audiences that range from young people to members of learned societies. In 2008, Queen Elizabeth II made him a member of the Most Excellent Order of the British Empire for his contributions to physics. Reprints of letters in response to the original printing of this article are available in the links section to the right. Other articles on soap films and mathematics have appeared in American Scientist in May–June 1986, September–October 1996 and March–April 2000. The latter, "Foams and Honeycombs," is also available in the links section.

The advent of digital computers in the 1950s and their rapid growth in the 1960s and 1970s has resulted in the neglect of analogue computers and analogue methods. Most analogue computers are designed to solve specific problems and are usually only capable of two-figure accuracy. Digital computers do not have these limitations, but analogue computers usually have one advantage: they provide a speedy visual solution to a problem. One of the simplest and most impressive analogue methods is based on a physical property of soap films, and in this article I will show how these methods can be used to solve mathematical minimization problems.

A well-known principle in thermodynamics states that, at constant temperature, the free energy of a system will be minimized when it is in thermodynamic equilibrium. The system may consist of a fluid, a solid, a gas, or combinations of different substances in these three phases. In a system consisting of a soap film, this free energy, F, is proportional to the area of the film, A. Thus F = σA, where σ is the surface tension and is constant at a fixed temperature. Consequently it is possible to obtain analogue solutions to mathematical problems that require the minimization of surface area.

Consider the simple problem of finding the minimum area contained by a circular ring: it is well known that the area is a circular disc contained by the ring. An analogue solution to this problem can be obtained by dipping a ring into a bath of soap solution; when the ring is withdrawn, a soap film surface will form bounded by the ring, and when it has come to thermodynamic equilibrium it will have the shape of the disc. In practice, thermodynamic equilibrium of the surface configuration is reached as soon as the film has come to rest, in only a few seconds.