mathematicians, including undergraduates, look at soap bubbles

Mathematicians, Including Undergraduates, Look at Soap BubblesAuthor(s): Frank MorganSource: The American Mathematical Monthly, Vol. 101, No. 4 (Apr., 1994), pp. 343-351Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2975627 .

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Mathematicians, Including Undergraduates, Look at Soap Bubbles

Frank Morgan

Why are soap bubbles so beautifully round? A soap bubble tries to minimize surface energy or area, and the round sphere has the least surface area for the fixed volume of air trapped inside. See FIGURE 1. A soap bubble very quickly succeeds in finding this mathematically optimal shape. Thus the underlying principle is area minimization.

Figure 1. A spherical soap bubble has found the least-area way to enclose a given volume of air. Jim Bredt [M4].

Similarly bubble clusters try to minimize the total surface area enclosing and separating several volumes. Whether the number of enclosed volumes is one or one thousand, it is the same principle of area minimization at work. See FIGURES

2, 3. This principle alone has sufficed to produce computer simulations of bubble clusters, as in the frame in FIGURE 4 from the video "Computing soap films and crystals" by the Minimal Surface Team at the Geometry Center (formerly the Minnesota Geometry Supercomputer Project).

Do soap bubble clusters always find the absolute least-area shape? Not always: FIGURE 5 illustrates two clusters enclosing and separating the same five volumes. In the first, the tiny fifth volume is comfortably nestled deep in the crevice between the largest bubbles. In the second, the tiny fifth volume less comfortably sits between the medium size bubbles. The first cluster has less surface area, although I do not know' for sure that there is not a third possibility of still less area.

1994] MATHEMATICIANS LOOK AT SOAP BUBBLES 343



Figure 2a and 2b. Double and triple bubbles seek the least-area way to enclose and separate two or three given volumes of air.

As a matter of fact, it is an open question whether the standard triple bubble of FIGURE 2 is the least-area way to enclose three given volumes. Even for the double bubble, the proof seems incomplete, as realized over the course of a recent undergraduate thesis [Fl] by Joel Foisy, Williams '91.

In fact, for area-minimizing clusters, it is an open question whether each separate region is connected, or whether it might conceivably help to subdivide the regions of prescribed volume, with perhaps half the volume nestled in one crevice here, and the other half in another crevice there.

Similarly, it is an open question whether an area-minimizing cluster may incidentally trap inside "empty chambers," which do not contribute to the prescribed volumes. FIGURE 6 shows a 12-bubble with a dodecahedral-like empty chamber on the inside, obtained by Tyler Jarvis of Princeton University using a program of Ken Brakke [B] of the Geometry Center. The computation postulated the empty chamber; without such a restriction, empty chambers probably never occur.

344 MATHEMATICIANS LOOK AT SOAP BUBBLES [April



* .'''.. .'t ' 'g'.?a.8-'ewis-? _ _~~~~~wl tfts0 ... . . ... ...

Figure 3. Configurations of thousands of bubbles are governed by the same principle of area minimization. Photograph from Science '84 [S] courtesy of Gordon Graham/Prism.

.. : .. ... ..........;

Figure 4. A computer simulation of a bubble cluster. "Computing soap films and crystals" video by the Minimal Surface Team, The Geometry Center. Photo provided by John Sullivan.




Figure 5. Soap bubble clusters are sometimes only relative minima for area. These two clusters enclose and separate the same five volumes, but the first has less surface area than the second.

Figure 6. It is an open question whether area-minimizing clusters may have empty chambers, such as the dodecahedral-like chamber at the center of this 12-bubble. Tyler Jarvis, Princeton University.




You must be wondering what is known! It is known that soap bubble clusters consist of constant-mean-curvature surfaces (such as pieces of spheres) meeting in threes at 1200 angles along seams, which in turn meet in fours at about 1090 angles at points. Four such seams and two such points are visible already in the triple bubble of FIGURE 2. Nothing more complicated ever happens, even in such complicated clusters as that of FIGURE 3.

These laws were observed and recorded by the Belgian physicist J. A. F. Plateau [P1 over a century ago, but it was over 100 years until a complete explanation was proved by Jean Taylor, now a professor at Rutgers University. Her demonstration required no physics or chemistry, just a single mathematical hypothesis: area-minimization. Many pages of complicated mathematics later came the conclusion: Plateau's laws, 1200 angles, 1090 angles, and all.

51 52 perimeter 9.33 perimeter 9.3 area 2.5 area 2.15 p2 p2

6.96n 8.05 na na

61 6 perimeter 9.99 62 area 2.47 perimeter 9.64 p2 2area 2.27

6.73 p na 6.82

na

71 72 perimeter 11.21 perimeter 11.0 area 2.77 area 2.59 p2 p2 - 6.48 - 6.67 na na

Figure 7a. Comparison of candidates for the least-perimeter way to enclose and separate n = 5, 6, 7 unit planar areas. Note that only sometimes are the most symmetric candidates the winners.




Taylor's work used geometric measure theory, a relatively new kind of geometry that allows singularities such as the seams in soap bubble clusters (cf. [M2]). Classically, geometry studied mainly smooth surfaces.

Although Taylor's mathematical proof [T] is hard to read, a beautiful expository account [AT] appeared in Scientific American in July, 1976.

Joel Foisy, whose undergraduate thesis work on the double bubble I mentioned earlier, got started the previous summer with an undergraduate research Geometry Group at Williams studying the

Planar soap bubble problem. Find the least-perimeter way to enclose and separate n given areas (say unit areas) in the plane.

FIGURES 7ab show the results of their experiments with soap bubbles between plexiglass plates [A]. Since it is hard to keep all the areas exactly 1, it is better to minimize the normalized quantity p2/na, where p is the total perimeter (counting all inside and outside walls once) and a is the total area. Could you have guessed the winners? I find it hard to see any pattern.

There was also one notable theoretical result:


6.33 - 6.52 na na


- 6.52 - 8.2 na na

Figure 7b. Comparison of four candidates for the least-perimeter way to enclose and separate eight unit areas. For the fourth candidate, the eight enclosed areas incidentally surround and enclose an extraneous "empty chamber."




81 Stanton Road Brooklinie, MA 02146 January 2, 1983

Professor Frank Morgan Rice University Department of Mathematics Houston, Texas 77001

Dear Professor Morgan:

Thank you again for sending me the 18.01-18.02 MIT Calculus materials. I recently took my first 18.02 test and made no errors on it.

I believe that I have discovered a solution to the soap film problem which was the topic of your speech at the 1982 Massachusetts State Science Fair, namely, 57Z'd a 4ZmptL, 4mooth, cto4ed cmrve. which can bournd 4nfZnkt.4t# ma,u7 mZ^kmaZt 4urfae.

Start with the shape in Figure 1, a shape that has two minimal surfaces,

cFigure 1. and elongate it at points A, B, C, and D to create the shape in Figure 2, A

Figure 2. and coat both of the two minimal surfaces with bubble suds. Introduce crosspieces AB, BC, CD, and DA and coat the resultant square with bubble suds too. By popping the proper bubble films you will get the configuration portrayed in Figure 3,

Figure 3.

ho Remove the crosspieces with care socas to not pop any more of the

bubble films, and the shape will settle into the saddle of Figure 4.

Figure 4. This saddle can be shifted along the entire length of the shape, creating infinitely many minimal surfaces.

I built a model of Figure 4 and found that the saddle could be slid along the length of the figure by tilting one end towards the floor. (The weight of the bubble suds made it slide - I wonder what would happen in a weightless condition.)

I have been unable to form a minimal surface on a mobius strip -

I think that this happens because there is only one edge or surface for the bubble suds to adhere to. (Am I right?)

Sincerely yours,

Mark Kantrowitz

Figure 8. As a high-school sophomore, Mark Kantrowitz addresses an open question on minimal surfaces.




Theorem ([A], [F1], [F2]). The standard double bubble is uniquely perimeter minimizing.

The hard part is showing that both areas are connected and that there are no empty chambers. For more bubbles or in space, such questions remain open.

Undergraduate research. Undergraduates do mathematics, prove theorems, write papers for publication, and give talks at mathematics meetings. In the Williams College SMALL undergraduate research project, each student belongs to two of seven research groups, each with a student leader and a faculty advisor.

The student groups work largely independently, although the faculty advisor usually provides the problem and some guidance. The students generally are quite busy and do not like to be interrupted by the faculty! The less experienced students learn from the others, and the most advanced like to bounce ideas off the rest.

One woman, who had no intentions of attending graduate school, changed her mind during the summer project and is now a graduate student in computer science at Wisconsin. A nonmajor, who used to consider mathematics just an avocation, is now a graduate student in Mathematics at UC Berkeley. A Hispanic student, who never seemed to settle down in his coursework, succeeded outstand- ingly in the SMALL project his freshman year. In his junior year he wrote an Honors thesis, which has been accepted for publication in the Pacific Journal of Mathematics. He now has two jobs, teaching mathematics in his old high school and at a community college, including one course taught in Spanish.

The name SMALL, in case you were wondering, is an acronym for the faculty on the original proposal.

It is probably good to start research early on. One time I got a letter from Mark Kantrowitz, a high-school sophomore, that announced, "I believe that I have discovered a solution to the [open] soap film problem...: Find a simple, smooth, closed curve which can bound infinitely many minimal surfaces." See FIGURE 8. Although Mark's stated proposal used gravity, the basic idea was one I had used in at least one published example. Mark's continuing correspondence amazed me. My collaborators and I barely managed to keep ahead of him. The problem is still open.

REFERENCES

[A] Manuel Alfaro, Jeffrey Brock, Joel Foisy, Nickelous Hodges, and Jason Zimba, Compound soap bubbles in the plane, SMALL Geometry Group report, Williams College, 1990.

[AT] F. J. Almgren, Jr., and Jean E. Taylor, The geometry of soap films and soap bubbles, Scientific American, July, 1976, pp. 82-93.

[B] Kenneth A. Brakke, The surface evolver, Experimental Math. 1 (1992), 141-165. [CR] R. Courant and H. Robbins, What is Mathematics?, Oxford Univ. Press, 1941. [F1] Joel Foisy, Soap bubble clusters in R2 and R3, Honors thesis, Williams College, 1991. [F2] Joel Foisy, Manuel Alfaro, Jeffrey Brock, Nickelous Hodges, and Jason Zimba, The standard

double soap bubble in R2 uniquely minimizes perimeter, Pacific J. Math., 159 (1993), 47-59. [Ml] Frank Morgan, Compound soap bubbles, shortest networks, and minimal surfaces, AMS video,

1992. [M2] Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Academic Press, 1988. [M3] Frank Morgan, Minimal surfaces, crystals, shortest networks, and undergraduate research,

Math. Intel., Vol. 14, Summer, 1992, 37-44. [M4] Frank Morgan, Soap bubbles and soap films, in Joseph Malkevitch and Donald McCarthy, ed.,

Mathematical Vistas: New and Recent Publications in Mathematics from the New York Academy of Sciences, Vol. 607, 1990.




[P] J. A. F. Plateau, Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moleculaires, Paris, Gauthier-Villars, 1873.

[S] Bruce Schecter, Bubbles that bend the mind, Science 84, March, 1984. [T]. Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal

surfaces, Ann. of Math. 103 (1976), 489-539.

This article is based on an AMS-MAA address in San Francisco, 1991, available on video [MI]. The second half appears as [M3]. The work was partially supported by the National Science Foundation and the Institute for Advanced Study.

Added in proof There has been progress on the planar triple bubble by the 1992 Williams NSF SMALL undergraduate research Geometry Group, Chris Cox, group leader, Lisa Harrison, Michael Hutchings, Susan Kim, Janette Light, Andrew Mauer, Meg Tilton, "The shortest enclosure of three connected areas in R2" (preprint), and on the double bubble in space by Michael Hutchings. See the new chapter on "Soap bubble clusters," in the second edition of [M2], to appear this year.

Department of Mathematics Williams College Williamstown, MA 01267 [email protected]

"The advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of thc pencil. It is a field which has often been compared with chess, but differs from the latter in that it is'only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation."

Excerpt from Ex-Prodigy: My Childhood and Youth by Norbert Wiener, p. 21




mathematicians, including undergraduates, look at soap bubbles

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