## Mathematical Recreations

by Ian Stewart

### Double Bubble,Toil and Trouble

The dodecahedron has 20 vertices, 30 edges and 12 faces- each with five sides. But what solid has 22.9 vertices, 34.14 edges and 13.39 faces -each with 5.103 sides? Some kind of elaborate fractal, perhaps? No, this solid is an ordinary, familiar shape, one that you can probably find in your own home. Look out for it when you drink a glass of cola or beer, take a shower or wash the dishes.

I've cheated, of course. My bizarre solid can be found in the typical home in much the same manner that, say, 2.3 children can be found in the typical family. It exists only as an average. And it's not a solid; it's a bubble. Foam contains thousands of bubbles, crowded together like tiny, irregular polyhedra-and the average number of vertices, edges and faces in these polyhedra is 22.9, 34.14 and 13.39, respectively. If the average bubble did exist, it would be like a dodecahedron, only slightly more so.

 Coalescing Bubbles that enclose unequal volumes have shapes that remain a mathematical challenge.

Bubbles have fascinated people ever since the invention of soap. But the mathematics of bubbles and foam only really got going in the 1830s, when Belgian physicist Joseph A. Plateau began dipping wire frames into soap solution and was astounded by the results. Despite 170 years of research, we still have not arrived at complete mathematical explanations - or even descriptions of - several interesting phenomena that Plateau had observed.

A notorious case is the Double Bubble Conjecture, which states that the shape formed when two bubbles coalesce consists of three spherical surfaces. In 1995 Joel Hass of the University of California at Davis and Roger Schiafly of Real Software in Soquel, Calif., announced a proof of this conjecture in the special case when both bubbles enclose the same volume, but the case of unequal volumes remains open. Many other phenomena found by Plateau, however; are now well understood, and experiments with soap films have repeatedly helped mathematicians develop rigorous proofs of important geometric theorems.

In 1829 Plateau had carried out an optical experiment that involved looking at the sun for 25 seconds: this damaged his eyes, and eventually he became blind. Despite his loss of vision, he continued to make major contributions to that most intensely visual area of mathematics, three-dimensional geometry.

Soap bubbles and films are examples of an immensely important mathematical idea called a minimal surface. This is a surface whose area is the smallest possible, subject to certain additional constraints. Minimal surfaces relate to bubbles because the energy caused by surface tension in a soap film is proportional to its area. Nature likes to minimize energy-so bubbles minimize area. For example, the surface of smallest area that encloses a given volume is a sphere, and that's why isolated soap bubbles are spherical.

A soap film is so thin-about a millionth of a meter-that it closely resembles an infinitely thin mathematical surface. (Moving bubbles are another matter; because dynamical forces can make them wobble into all kinds of fantastic shapes.) Without some constraint, the area of a minimal surface would be zero. The most common constraints are that the surface should enclose some given volume or that its boundary should lie on some given surface or curve, or both. A bubble that forms against a flat tabletop, for example, is usually a hemisphere, and this is the smallest area surface that encloses a given volume and has a boundary lying in a plane (the top of the table).

 Surface of Least Area is always formed by a bubble. As a result,the soap film joining two parallel circles has the shape of a catenoid. A tetrahedron and a cube give rise to complicated arrangements of nearly flat surfaces that meet at characteristic angles.

Plateau was especially interested in surfaces whose boundary was some chosen curve. In his experiments the curve was represented by a length of wire bent into shape or several wires joined together in a frame. What, for instance, is the shape of a minimal surface whose boundary comprises two identical parallel circles? A first guess might well be that it is a cylinder. But we can do better. Leonhard Euler proved that the true minimal surface with such a boundary is a catenoid, formed by revolving a U- shaped curve known as a catenary about an axis running through the centers of the two circles.

The catenary is the shape formed by a heavy, uniform chain hanging between two hooks of the same height: it looks rather like a parabola but has a slightly fatter shape. (A hoary mathematical joke goes, "How do you make a catenoid?" Answer: "By pulling its tail.") Euler's theorem can be demonstrated by making two circular wire rings, with handle-like fishing net frames. Hold them together, dip them into a bowl of soap solution or detergent, and gently pull them apart to reveal the catenoid in all its glistening beauty.

One of the most famous descriptions of soap films can be found in the classic What is Mathematics? by Richard Courant and Herbert Robbins (Oxford University Press, updated in 1996). They relate some of Plateau's original experiments, in which he dipped wire frames shaped like regular polyhedra. The simplest case, which they don't discuss, arises when the frame is a tetrahedron, a shape with four triangular sides and six equal edges. Here the minimal spanning surface consists of six triangles, all meeting at the center of the tetrahedron.

A cubic frame leads to a more complicated arrangement of 13 nearly flat surfaces. The tetrahedron case is fully understood, but a complete analysis for the cube remains elusive.
The tetrahedral frame illustrates two general features of soap films, observed by Plateau. Along the lines running from the vertices of the frame to its central point, soap films meet in threes, at angles of 120 degrees; at the central point, four edges meet at angles of 109 degrees 28 minutes. These two angles are fundamental to any problem in which several soap films abut one another. Angles of 120o between faces and 109o 28' between edges arise not just in the regular tetrahedron but in any arrangement of soap films -provided there is no trapped air or if there is, the pressures on the two sides of each film are equal (hence canceling each other out).

The films in a foam are slightly curved but can be approximated by plane faces: with this approximation, the two stated angles will be observed in the interior of a foam, though not for films near the foam's external surfaces. This fact is the basis of a curious calculation, which leads to the strange numbers with which I began this column. By pretending that foam is made from many identical polyhedra whose faces are regular polygons with angles of 109o 28' (which is impossible, but who cares?), we can estimate the average numbers of vertices, edges and faces in any foam.

 Plateau's Rule for the angle between four bubble edges was proved by considering the possible ways in which six faces meet. The vertices are enclosed in a sphere, on which the faces meet at angles of 120 degrees. As shown, only 10 shapes meet this criterion; of these, only the first , three are physically plausible, because they correspond to minimal areas.

Plateau's observation about the 120o angle was quickly established as a mathematical fact. The proof is often credited to the great geometer Jacob Steiner in 1837, but Steiner was beaten to the punch by Evangelista Torricelli and Francesco B. Cavalieri around 1640. All these mathematicians actually studied an analogous problem for triangles. Given a triangle and a point inside it, draw the three lines from that point to the triangle's vertices and add up their lengths. Which point makes this total distance smallest? Answer: the point that makes the three lines meet at angles of 120o . (Provided no angle of the triangle exceeds 120o , that is - otherwise the desired point is the corresponding vertex.) The problem for soap films can be reduced to that for triangles by intersecting the films with a suitable plane.

In 1976 Frederick J. Almgren, Jr., then at Princeton University and Jean E. Taylor; then at the Massachusetts Institute of Technology, proved Plateau's second rule about 109o 28' angles. They started by considering any vertex where six faces meet along four common edges. First, they showed that the slight curvature that occurs in most soap films can be ignored, so that the films can be taken as planar. They then considered the system of circular arcs formed by these planes when they intersect a small sphere centered on that vertex. Because the soap films are minimal surfaces, these arcs are "minimal curves": their total length is as small as possible. By the spherical analogue of the Torricelli-Cavalieri theorem, these arcs must always meet in threes at angles of 120o .

Almgren and Taylor proved that exactly 10 distinct configurations of arcs [see illustration on this page] can satisfy this criterion. For each case, they asked whether the total area of the films inside the sphere could be made smaller by deforming the surfaces slightly, perhaps introducing new bits of film. Any such cases could be discarded, because they could not correspond to a true minimal surface. Exactly three cases survived this treatment, the first three shown in the illustration above. The corresponding arrangements of film are a single film, three meeting along an edge at 120o , or six meeting at 109o 28'-just as Plateau observed.

 Radii of two coalescing bubbles (r and s) and their common surface (t) obey a simple relationship.

The detailed techniques required for the proof went beyond geometry into analysis - calculus and its more esoteric descendants. Almgren and Taylor used abstract concepts known as measures to contemplate bubble shapes far more complex than smooth surfaces.
The 120o rule leads to a beautiful property of two coalescing bubbles. It has long been assumed on empirical grounds that when two bubbles stick together; they form three spherical surfaces, arranged as in the illustration on the opposite page. This is the Double Bubble Conjecture. If it is true, the radii of the spherical surfaces must satisfy a simple relationship. Let the radii of the two bubbles be r and s and let the radius of the surface along which they meet be t. Then the relationship is

1/r = 1/s + 1/t

This fact is proved in Cyril Isenberg's delightful hook The Science of Soap Films and Soap Bubbles (Dover; 1992), using no more than elementary geometry and the 120o property. All that remains is to prove that the surfaces are parts of spheres, and it is this that Hass and Schlafly achieved in 1995-but only by making the additional assumption that the bubbles are of equal volume. Their proof required the assistance of a computer; which had to work out 200,260 integrals associated with competing possibilities-a task that took the machine a mere 20 minutes!

One curious fact that is known about the unequal volume case is that whatever the double-bubble minimal configuration is, it must he a surface of revolution. The problem thus reduces to one about a system of curves in the plane. Despite this simple feature, the answer remains as elusive as it was when near blind Plateau dipped his first wire frame into a bowl of sudsy water.

 Further Reading "The Double Bubble Conjecture," by Frank Morgan in Focus (Mathematical Association of America), Vol.15, No.6, pages 6-7; December 1995.

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