The Soap Film: An Analogue Computer
Soap films provide a simple method of obtaining analogue solutions to some mathematical problems
Some Practical Problems
A problem that has not been solved analytically is that of linking a number of points, in a plane, by the shortest path. This problem has practical applications—for example, the linking of a number of towns by the shortest road, water pipe, or cable. The soap film provides a simple analogue mechanism for solving these minimum-path problems. They have become known as Steiner problems, after the 19th-century mathematician Jacob Steiner, who made a study of them, but it was the mathematician Richard Courant, in the 1940s, who popularized the analogue approach to the solution of these problems.
Consider the particular problem of linking four towns by the shortest road. For convenience let the towns, say A, B, C, and D, be arranged at the corners of a square. We might conjecture, on the grounds of symmetry, that the shortest roadway system joining all four towns is the X-shape given in Figure 1a, which has a length 2(2)1/2 for a square with sides of unit length. The verification or nullification of this conjecture is best examined using the analogue approach.
Before attempting to solve the four-town problem using the minimum-area property of soap films, we should first look at the one minimization problem for which the solution is well known—finding the shortest path linking two points. The solution, of course, is the straight line joining the two points. Consider two parallel Lucite plates connected by two pins perpendicular to the plates (Figure 1b).
After this system is dipped into soap solution a soap film will form between the plates, beginning on one pin and ending on the other. By symmetry it will be perpendicular to the plates and bounded, from above and below, by the plates. It will have the form of a tape of constant width, and its area will be proportional to the length of the film. When it reaches equilibrium, its area, and consequently its length, will be minimized; hence the final shape of the film joining the pins will be a straight line. Figure 1b shows a curved film while it is still not in equilibrium and the straight-line film after reaching equilibrium.
Similarly, in order to solve the problem of the four towns, we must construct two parallel Lucite plates joined by four pins perpendicular to the plates. The pins should represent, to scale, the four towns. When the plates are dipped into the soap solution, the soap film will take up the minimum area and will be in the form of a tape, of constant width, connected to all four pins (towns). The area of the film will be proportional to its length, and the minimum area will correspond to the minimum length of the film (Figure lc). At the two places where three soap films intersect, the angle between adjacent films is 120 degrees. The total length of roads is thus [1 + (3)1/2]—shorter than the X-system length of 2(2)1/2. Because the towns are at the corners of a square, another possible solution can be obtained by rotating the first analogue solution through 90 degrees (Figure 1d).
The method of solution of the general problem of linking n towns by the shortest path is now clear. Two parallel plates are constructed and joined by pins perpendicular to the plates so that they represent, to scale, the positions of the n towns. The equilibrium soap film formed between the plates will have the minimum length, and the solution will always consist of a number (greater than or equal to zero) of intersections with three films meeting at 120 degrees, as with the four-town problem. Physically this is a consequence of three equal forces, caused by surface tension at the intersection, being in equilibrium. The soap film system will always be connected to every pin, and only one or two films can be joined to any one pin. In the latter case, the angle between the films will be greater than or equal to 120 degrees.
A problem may well have more than one mathematical minimum, and the soap film will always take up one of these minima. All the minimum configurations must be determined in order to pick out the shortest one. Each minimum will correspond to an equilibrium configuration of the soap film. The different configurations can be obtained by first producing one of the equilibrium configurations and then blowing onto the soap film to perturb it so that it jumps into another equilibrium configuration. In this way all the minimum configurations can be determined. Figure 2 shows the possible variation of length of the film, L, as the shape of the configuration is altered, and the length passes through all the minimum configuration as some parameter, C, is altered. It is possible to change the shape of the film by passing through nonequilibrium configurations to equilibrium configurations, which correspond to C1, C2, C3, etc.
There is no way of determining the total number of minimum configurations. By perturbing the soap film or redipping the plates in many different ways we hope to obtain all the equilibrium configurations, and during this process we often obtain the same configuration several times. The only information available for the n-town problem is the knowledge that there can be at most (n – 2) intersections, and there may well be a smaller number.
An example of a problem with several minima is the determination of the smallest minimum length of path linking six points at the corners of a regular hexagon. There are three minimum configurations (Figures 3a, b, c). Figures 3b and 3c each contain four intersections in which three films meet at angles of 120 degrees; the smallest minimum (Figure 3a) can be determined by calculation using the 120-degree property, or by measurement. Figure 3a is an example of a case in which two films meet, at a single pin, with the limiting angle of 120 degrees.
If the hexagon of pins were distorted slightly, so that it is no longer regular, maxima similar to Figures 3b and 3c would be obtained with four intersections. Figure 3a, however, would alter. Those vertices with angles that are less than 120 degrees, and previously had two films, would form intersections near the pins and have only one film joined to the pin. Figure 3a would remain the shortest minimum configuration.
We have assumed throughout that the pins have zero diameter. In practice they will have a small diameter which can produce errors: A pin of finite diameter can cause two films to meet at the pin inclined at less than 120 degrees. This is not possible for pins of zero diameter. To obtain the same configuration as in the zero-diameter case it is necessary to perturb the films by blowing onto them to make them coalesce into one film. The only difference now between the finite and zero-diameter case is the exclusion of the soap film from the region of the pin. For calculations and measurements one can extend the lengths to the center of the pins and hence obtain the same results as those for a system with pins of zero thickness.
It is possible to take the curvature of the Earth into account in solving this problem by constructing two concentric spherical shells connected by pins perpendicular to the surfaces. Again, the pins represent the towns. The lengths of the curved segments of soap film that form between the spherical shells will be proportional to the area of the soap film. The length of the film is measured along the surface of one of the spherical shells, and consequently the minimum area of the soap film will correspond to the minimum length of path connecting the “towns.”
A more convenient analogue system consists of only one shell, as shown in Figure 4. From the center of the sphere, of which the shell forms part of the surface, wires radiate to points on the surface of the shell at which the “towns” are situated. The soap film that forms between the wires and the shell consists of segments whose area is proportional to their length measured along the surface of the spherical shell. Once again the minimum area will correspond to minimum length.
We can also take account of constraints that exclude the roadway from a particular region. For example, for a three-town problem which requires that the roadway system exclude the region of a lake at the center, a circular hole is drilled, to scale, in the Lucite plates to represent the lake, and the three pins that represent the towns are inserted. The soap film can be made to avoid the circular lake by physically breaking any soap film that forms in the hole. In the resulting shape, the roadways go around the lake and form a three-way intersection (Figure 5). An obstacle of any shape can be dealt with in this way.