Foams and Honeycombs

A reprint from American Scientist the magazine of Sigma Xi, The Scientific Research Society This reprint is provided for personal and noncommercial use. For any other use, please send a request to Permissions, American Scientist, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A., or by electronic mail to [email protected]. ©Sigma Xi, The Scientific Research Society and other rightsholders Foams and Honeycombs For centuries, the precise architecture of soap foams has been a source of wonder to children and a challenge to mathematicians Erica G. Klarreich early all of us have been enchanted ble foam, one in which all the bubbles in the sciences that aesthetically pleas- Nat some time in our lives by the have the same volume. Kelvin realized ing solutions seem to have a higher fragile symmetry of soap bubbles and that this question could be boiled down chance of being correct than more com- soap films. But few have had the privi- to the following purely mathematical plicated, less symmetric ones (a topic lege of being introduced to them by a problem: What is the best way to divide elaborated in these pages by James W. great scientist, as Agnes Gardner King three-dimensional space into cells of McAllister in March–April 1998). As did. In the fall of 1887, the fledgling equal volume, if one wants to minimize Oxford University mathematician and painter paid a visit to her uncle, Lord the surface area of the cell walls? physicist Roger Penrose put it, Kelvin (then Sir William Thomson), a Over the course of two months, I have noticed on many occasions in physicist who was already renowned Kelvin searched for the shape that my own work where there might, for his work in thermodynamics. A day would form the best partition. By No- for example, be two guesses that after arriving, she recorded that: vember 4, 1887, he had found what he could be made as to the solution of a believed to be the answer, a polyhedron Uncle William and Aunt Fanny met problem and in the first case I would to which he gave the tongue-twisting me at the door, Uncle William armed think how nice it would be if it were name of “tetrakaidecahedron” (mean- with a vessel of soap and glycerine true; whereas in the second case I ing 14-sided polyhedron) but which is prepared for blowing soap bubbles, would not care very much about the more commonly known as the truncat- and a tray with a number of mathe- result even if it were true. So often, ed octahedron. Copies of this shape, matical figures made of wire. These in fact, it turns out that the more at- placed next to each other, fill all of he dips into the soap mixture and a tractive possibility is the true one— space, with a lower surface area than film forms and adheres to the wires or that while thinking about the more familiar partitions, such as a par- very beautifully and perfectly regu- problem in this way the true solu- tition into cubical “rooms.” Kelvin’s larly. With some scientific end in tion would finally emerge to reveal partition is highly symmetrical, like the view he is studying these films. itself as even more attractive than ei- cubical partition, and has a strong aes- ther contemplated earlier. Kelvin was drawn to soap films in thetic appeal. the course of his efforts to understand Although Kelvin did not prove that Weaire and Phelan’s structure, which the nature of light. By the late 19th cen- the truncated octahedron was the best is less symmetrical than Kelvin’s, is a re- tury, a multitude of experiments had possible division of space, for more than minder that our inclination toward beau- shown that light exhibits wavelike a century his solution was generally ac- tiful solutions must always be examined properties, and many scientists sought cepted as correct. It received the stamp critically—especially when nature itself to explain light by describing some sort of approval from such illustrious math- provides an alternative. On the other of medium, called the “ether,” through ematicians as Hermann Weyl, who hand, in the past two years Thomas which light waves would propagate. wrote of it in 1952 in his famous book, Hales, a colleague of mine at the Univer- Kelvin conceived of a foamlike model Symmetry. But in 1994, physicists Denis sity of Michigan, has struck a blow for for the ether, so he attacked the prob- Weaire of Trinity College Dublin and symmetry in two other problems relat- lem of understanding what would be Robert Phelan, now at the Shell Re- ed to foams: the Kepler conjecture, which the structure of the most perfect possi- search and Technology Center in Ams- states that the way to pack equal-volume terdam, astonished mathematicians and balls together with the least wasted space physicists by producing a partition of is the familiar pyramid packing used to Erica G. Klarreich is an assistant professor of space with lower surface area. Weaire stack fruit in groceries (discussed in Sci- mathematics at the University of Michigan. She and Phelan’s partition, like Kelvin’s, ence Observer, November–December received her bachelor’s degree from Brooklyn was inspired by naturally occurring 1998); and the honeycomb problem, a College in 1992 and her Ph.D. in mathematics structures, this time from chemistry. two-dimensional version of the Kelvin from the State University of New York at Stony Brook in 1997. Her research interests include One of the reasons that scientists ex- question. The study of foams, then, hyperbolic geometry, Kleinian groups and pected Kelvin’s solution to hold up was seems to be a subject poised very deli- Teichmüller theory. Address: Department of that his partition is one of the most ap- cately between complexity and symme- Mathematics, University of Michigan, Ann Arbor, pealing, most symmetrical divisions of try, and one in which the final word has MI 48109. Internet: [email protected]. space. It is an intriguing phenomenon not yet been said. 152 American Scientist, Volume 88 © 2000 Sigma Xi, The Scientific Research Society. Reproduction with permission only. Contact [email protected]. Films and Foams Kelvin opened his article on partition- ing space with the words, “this problem is solved in foam.” It may not be obvious that mathematics applies to foams, but indeed physics and mathematics are intimately intertwined in soap films. The force that enables soap bubbles to exist is known as surface tension and is caused by the attraction between the molecules of a soap film. Since the molecules tend to pull together, main- taining a large surface area is energy- expensive; the soap film naturally de- forms toward the surface of least area and lowest energy. Consider what happens when you dip a circular wire into a soapy liquid. When you draw it out, you see a beau- tiful iridescent soap film shaped like a flat disk. In this simple case, it is possible to confirm mathematically that the flat disk is the lowest-area surface that the wire can support. What if I dip a more complicated wire loop into the soap solution? The surface that forms when I draw out the wire is no longer as familiar, but there are still some things that mathematicians can say Figure 1. Computer rendition of a soap foam, filling up all of space with equal- sized bubbles, suggests the symmetry and infinite complexity of mathematical foams. In the last decade, mathematicians have solved problems concerning both “dry” foams with infinitely thin cell walls—such as the idealized soap foam pictured here— and “wet” foams, in which walls can be thicker in places. However, one of the most fundamental questions about dry foams, called the Kelvin problem, remains unre- solved: What is the way to partition all of space into equal-sized bubbles, using the smallest amount of surface area? In this image, John M. Sullivan of the University of Illinois simulates a dry foam representing a new candidate solution to this problem, the Weaire-Phelan foam. © 2000 Sigma Xi, The Scientific Research Society. Reproduction with permission only. Contact [email protected]. Figure 2. After extensive experiments with wire models such as the “Kelvin bedspring” (photograph), Lord Kelvin arrived at a conjecture that lasted for more than a century. He proposed that, in the ideal foam, the cells or bubbles are shaped like a very gently curved version of a truncated octahedron. These cells do fit together to fill all of space, as suggested in the computer drawing above. Moreover, it is possible to force a small number of soap bubbles into this configuration by dipping the “Kelvin bedspring” into a soap solution. But Dublin physicists Denis Weaire and Robert Phelan startled the mathematical world in 1994 by showing that Kelvin’s foam uses more surface area per bubble than an alternative, somewhat less symmetrical arrangement. (Photograph by Mike Jewkes, Hunterian Museum, Uni- versity of Glasgow; illustration adapted from Surface Evolver image generated by Kenneth Brakke, Susquehanna University.) about it. When a wire loop supports a the only forces that can disturb the bal- forces that the molecules exert on one soap film, the air pressures on both ance of the film are the attractive forces another must exactly cancel one anoth- sides of the film are equal. Setting aside of the film’s own molecules. er. This happens, for example, at the gravity, which is much weaker than Therefore, in order for the soap film central point (or “saddle point”) of a surface tension for a small soap film, to be in equilibrium, the attractive saddle surface.

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