The Soap Film: An Analogue Computer
Soap films provide a simple method of obtaining analogue solutions to some mathematical problems
More Difficult Problems
The Steiner problem involves two dimensions, but it is also possible to solve three-dimensional problems using the minimum-area property of soap films. An important problem, which was solved analytically in the 18th century, is the determination of the minimum area contained by two separated coaxial rings. Each ring is arranged perpendicular to the axis. The surface is a catenary—the curve produced by a chain fixed at both ends—of revolution, providing that the distance between the rings is less than a certain value. As the distance between the rings is increased, the minimum of the catenary moves toward the axis of the rings and finally breaks up into two discs bounded by each of the rings—this is known as the Goldschmidt discontinuous solution. The catenary of revolution and the Goldschmidt solution can be demonstrated using a soap film bounded by the two rings. Mathematicians consider this problem to be one dimensional because the equation defining the surface depends on one independent variable.
Consider the truly three-dimensional problem of determining the minimum area contained by the six edges of a tetrahedral framework. The surface is not immediately obvious; we might guess that it would consist of three triangular plane surfaces covering three faces of the tetrahedron. The analogue solution can be obtained by dipping the tetrahedral framework into soap solution, shown diagramatically in Figure 6. The soap film forms a series of plane surfaces, each of which begins at an edge of the tetrahedron and meets all the other surfaces at the center of the tetrahedron. From each vertex there is a line of soap film formed by the intersection of three planes of film. The angle between adjacent planes is 120 degrees, as was the case for the two-dimensional problem, and the four lines meet at the center of the tetrahedron and intersect each other at an angle of 109 degrees 28 minutes. Figure 7a is a photograph of this minimum surface. The horizontal interference fringes in the surface of the soap film are due to the wedge-shaped cross-section of the film which is produced by the draining away of the water in the film. The thickness of the film is comparable to the wavelength of light and hence gives rise to interference fringes.
For a cubic framework we might conjecture that the minimum surface would be a set of planes, each beginning at an edge and meeting at a point at the center. However, if this did occur, the surfaces would not intersect at 120 degrees; Figure 7b shows the actual minimum surface. A “square”-shaped surface at the center of the cube ensures that the film surfaces always intersect at 120 degrees and the lines of soap film meet at 109 degrees 28 minutes. The central surface has curved edges that intersect at angles of 109 degrees 28 minutes, and the “square” can occur in a direction parallel to any of the faces of the cube. Perturbing the film, by blowing onto it, will cause the “square” to flip over to another direction. Formation of this surface at the center of the cube could have been conjectured from the two-dimensional result for the four-town problem.
The minimum surface solution bounded by the edges of an octahedron has many properties in common with the two-dimensional solution of the hexagon of points. One of several different minimum surfaces—the surface with the highest symmetry (Figure 7c)—contains four lines that meet at the center of the octahedron, as was the case for the tetrahedron. Figure 7d shows one of the other minima, which has a square in the center, similar to that in the cubic framework. (Several others are discussed in detail in Wolf 1968.)
More general mathematical problems can be investigated by including an air bubble inside the various frameworks. The free energy will contain additional terms that take account of the air in the bubble, but the minimization principle can be extended to take into account these additional terms. As the configuration of the surface is altered, the equilibrium configuration will correspond to a minimum value of the total free energy. The equilibrium configuration will, of course, depend on the quantity of air trapped in the bubble. The resulting configurations for the tetrahedron, cube and octahedron are shown in Figure 8. In each case the trapped bubble has the symmetry of the framework.
These three-dimensional solutions, with and without bubbles, were first obtained by the blind Belgian physicist Joseph Plateau (1801–1883). Many mathematicians have attempted to obtain analytic solutions to these “Plateau” problems, but no general analytic solution has been obtained, despite considerable effort. Mathematicians would like to be able to prove the existence of solutions to these three- dimensional minimization problems, in the most general case, before attempting to obtain analytic solutions. As far as the physical scientist is concerned, this is a mathematical detail— he can demonstrate the existence of a solution using soap films. An important step forward was made in the 1920s by the mathematician Jesse Douglas, for which she received the Fields medal.
More recently a number of mathematicians—Jean Taylor, Frederick J. Almgren, Jr., and Enrico Bombieri— have made important contributions to this subject; however, the end of the search for a general existence proof is not in sight. The state of mathematical knowledge of the subject has been summarized recently in the popular articles of J. C. C. Nitsche (1974) and L. A. Steen (1976). Nitsche has written that “mathematical existence proofs for surface systems” such as those depicted in Figure 7 “as well as the investigations of the nature and the regularity of the branch lines are still in their infancy.”
- Boys, C. V. 1959. Soap Bubbles: Their Colours and the Forces which Mould Them. New York: Dover.
- Courant, R., and H. Robbins. 1951. What Is Mathematics? pp. 354–361,385–397. New York: Oxford University Press.
- Newman, J. R. 1956. The World of Mathematics, vol. 2, pp. 882–910. New York: Simon and Schuster.
- Nitsche, Johannes C. C. 1974. Plateau problems and their modern ramifications. American Mathematics Monthly 81(9):945–68.
- Plateau, J. A. F. 1873. Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moleculaires, 2 vols. Belgium: Clemm.
- Steen, L. A. 1976. Solving the great bubble mystery. Science 198:186–87.
- Wolf, K. L. 1968. Tropfen, Blasen und Lamellen. New York: Springer-Verlag.