LETTERS TO THE EDITORS

# Soap Film Letters

Editors’ Note: These letters appeared in the January–February 1977 issue of *American Scientist.* in response to the 1976 feature "The Soap Film: An Analogue Computer." The feature has been reprinted as part of *American Scientist*'s centennial celebrations and is linked at the right.

To the Editors:

Certain impressions left in the article “The Soap Film: An Analogue Computer,” by Cyril Isenberg (*Am. Sci.* 64:514-18, Sept. 1976), should be corrected. First, Dr. Isenberg’s comments on the Steiner problem (linking a number of points in a plane by the shortest path) are misleading. He says that the “problem has not been solved analytically, and that “there is no way of determining the total number of [locally] minimum configurations.” But in fact there is an explicitly computed maximum for the number of such solutions (Gilbert and Pollack, *SIAM J. Ap. Math. *16:11,1968), they can all be listed, and the major computing problem is simply to find out which of these possible local minima is the smallest (this, however, is an NP complete problem, and thus the difficulty of doing this comparison may rise exponentially with the number of points). A good reference is L. Steen, *Science News *109:298-301,1976. More important, Dr. Isenberg’s comments on the mathematical aspects of the general soap film problem are several years out of date. In fact, the existence of solutions to the general soap film and bubble problem was proved by F. J. Almgren, Jr., and the nature and smoothness of their branch lines was proved by me; these results were published in mathematical journals early this year and were discussed in the July 1976 issue of *Scientific American; *they have been known informally by the mathematical community since about the time of publication of Nitsche’s article quoted by Dr. Isenberg (1974) and were outlined in the article by Steen cited by Dr. Isenberg (the correct reference is *Science News* 108:186-7, 1975). Also, note that the bubble in Figure 8c does not have the symmetry of its boundary.

Jean E. Taylor

Department of Mathematics

Rutgers University

New Brunswick, NJ

Dr. Isenberg replies:

In my article I implied that the general case of the Steiner problem, with* n* points, could not be written down explicitly with pen and paper. I was comparing the analogue method based on soap film with results that can be obtained with pen and ink. I excluded the use of high-speed digital computers. At the time I visited the United States in January of last year Dr. Taylor’s publication was not known to me, and I could not have been aware of her results prior to publication. I will look forward to seeing the explicit solutions to the Plateau surfaces and bubbles contained by frameworks.

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