Foams and Honeycombs
For centuries, the precise architecture of soap foams has been a source of wonder to children and a challenge to mathematicians
This article appeared in the March-April 2000 issue of American Scientist.
Nearly all of us have been enchanted at some time in our lives by the fragile symmetry of soap bubbles and soap films. But few have had the privilege of being introduced to them by a great scientist, as Agnes Gardner King did. In the fall of 1887, the fledgling painter paid a visit to her uncle, Lord Kelvin (then Sir William Thomson), a physicist who was already renowned for his work in thermodynamics. A day after arriving, she recorded that:
Uncle William and Aunt Fanny met me at the door, Uncle William armed with a vessel of soap and glycerine prepared for blowing soap bubbles, and a tray with a number of mathematical figures made of wire. These he dips into the soap mixture and a film forms and adheres to the wires very beautifully and perfectly regularly. With some scientific end in view he is studying these films.
Kelvin was drawn to soap films in the course of his efforts to understand the nature of light. By the late 19th century, a multitude of experiments had shown that light exhibits wavelike properties, and many scientists sought to explain light by describing some sort of medium, called the "ether," through which light waves would propagate. Kelvin conceived of a foamlike model for the ether, so he attacked the problem of understanding what would be the structure of the most perfect possible foam, one in which all the bubbles have the same volume. Kelvin realized that this question could be boiled down to the following purely mathematical problem: What is the best way to divide three-dimensional space into cells of equal volume, if one wants to minimize the surface area of the cell walls?
Over the course of two months, Kelvin searched for the shape that would form the best partition. By November 4, 1887, he had found what he believed to be the answer, a polyhedron to which he gave the tongue-twisting name of "tetrakaidecahedron" (meaning 14-sided polyhedron) but which is more commonly known as the truncated octahedron. Copies of this shape, placed next to each other, fill all of space, with a lower surface area than more familiar partitions, such as a partition into cubical "rooms." Kelvin's partition is highly symmetrical, like the cubical partition, and has a strong aesthetic appeal.
Although Kelvin did not prove that the truncated octahedron was the best possible division of space, for more than a century his solution was generally accepted as correct. It received the stamp of approval from such illustrious mathematicians as Hermann Weyl, who wrote of it in 1952 in his famous book, Symmetry. But in 1994, physicists Denis Weaire of Trinity College Dublin and Robert Phelan, now at the Shell Research and Technology Center in Amsterdam, astonished mathematicians and physicists by producing a partition of space with lower surface area. Weaire and Phelan's partition, like Kelvin's, was inspired by naturally occurring structures, this time from chemistry.
One of the reasons that scientists expected Kelvin's solution to hold up was that his partition is one of the most appealing, most symmetrical divisions of space. It is an intriguing phenomenon in the sciences that aesthetically pleasing solutions seem to have a higher chance of being correct than more complicated, less symmetric ones (a topic elaborated in these pages by James W. McAllister in March–April 1998). As Oxford University mathematician and physicist Roger Penrose put it,
I have noticed on many occasions in my own work where there might, for example, be two guesses that could be made as to the solution of a problem and in the first case I would think how nice it would be if it were true; whereas in the second case I would not care very much about the result even if it were true. So often, in fact, it turns out that the more attractive possibility is the true one—or that while thinking about the problem in this way the true solution would finally emerge to reveal itself as even more attractive than either contemplated earlier.
Weaire and Phelan's structure, which is less symmetrical than Kelvin's, is a reminder that our inclination toward beautiful solutions must always be examined critically—especially when nature itself provides an alternative. On the other hand, in the past two years Thomas Hales, a colleague of mine at the University of Michigan, has struck a blow for symmetry in two other problems related to foams: the Kepler conjecture, which states that the way to pack equal-volume balls together with the least wasted space is the familiar pyramid packing used to stack fruit in groceries (discussed in Science Observer, November–December 1998); and the honeycomb problem, a two-dimensional version of the Kelvin question. The study of foams, then, seems to be a subject poised very delicately between complexity and symmetry, and one in which the final word has not yet been said.