Surface Evolver by Kenneth A. Brakke, and: Computing Soap Films and Crystals (review)

mathematical research project would be a scientific paper carrying a proof or refutation ofa theorem. Now it can consist of some brightly coloured pictures and a joyous cry of 'Look what I found'." As an example of 'seeing is believing', several pictures from Not Knotwere published by Bown. Ifuntil now it has been unacceptable to mathematicians that "a series of pictures can be as convincing as a series of equations", it is true that they can be useful for mathematical research. The video is a guided tour into computer-animated hyperbolic space (Fig 1). It starts with some examples of knots and then moves to the hyperbolic structure of the complementary space of the Borromean rings, in order to show a special case ofThurston's theorem. The Borromean rings have been chosen because of their particular symmetry; therefore the pictures in the movie are particularly pleasing. Even the use of lighting in the movies is 'hyperbolic ', so that when the axes of the Borromean rings are pushed all the way to infinity, it is like flying through hyperbolic space. Reference 1. W. Bown, "New-Wave Mathematicians", New Scientist 131, No. 1780,33-37 (1991). SURFACE EVOLVER by Kenneth A. Brakke. Geometry Supercomputer Project, Univ. of Minnesota , Minneapolis, MN, U.S.A., 1990. Software and manual. COMPUTING SOAP FILMS AND CRYSTALS The Minimal Surface Team. Geometry Supercomputer Project, Univ. of Minnesota, MN, U.S.A., 1991. Videotape , 18 min. Reviewed lJy Michele Emmer, Via Santa Maria dellaSperanza 5, 00139 Rome, Italy. Mathematicians began to take an interest in thin films and soapy water around 1873, the date when Belgian physicistjoseph Plateau published the results of his experiments in his two-volume work Statiqueexpenmentale et theorique desliquides soumisaux seules forcesmoleculaires [1]. The study of the geometry of thin films and soap bubbles is one of the most interesting sectors of modern mathematics. Soap films and soap bubbles are used frequently as abstract minimal forms. "A collection of surfaces, interfaces or membranes is called minimal when it has assumed a geometrical configuration of least area among those configurations into which it can readily deform, subject to its constraints. A minimal surface form is the shape of such a least-area configuration" [2]. .So it is possible to use a movie to investigate and explain some minimal surfaces forms, as seen in SoapBubbles , filmed in 1979 [3]. Furthermore, new techniques such as computer graphics have provided us with a 'new soap' with which it is possible to construct 'soap-film-type' models otherwise unobtainable with real soapy water. Until 1982, only three complete , embedded minimal surfaces of finite topology were known: the plane, the cathenoid and the helicoid . David A. Hoffman and William H. Meeks III, by considering the equations obtained by the Brazilian mathematician Costa, were able to 'see' a new example of a surface of a different topological type on their video terminal , a surface impossible to obtain with soapy water. One of the projects of the Geometry Center for Supercomputing at the University of Minnesota in Minneapolis was to produce software in order to study minimalsurface forms. Kenneth A. Brakke has realized a software called Surface Evolver; it is an interactive program for the study of surfaces shaped by surface tension, as is the case with soap bubbles and soap films. Given an initial surface, Surface Evolver can evolve it toward minimal energy. This software makes it possible to handle arbitrary topology, volume constraints , boundary constraints, periodic surfaces, prescribed mean curvature, crystalline integrands and constraints expressed as surface integrals. The Minimal Surface Team (Fred Almgren, Kenneth Brakke,john Sullivan ,jean Taylor and others) has also realized a video in order to demonstrate how the Surface Evolver can be used. The video is a computer-graphics animation and has enough explanations so that it can also be understood by nonexperts in the area of calculus ofvariations. A personal observation: it was really exciting to see, 12 years after the real clusters of soap bubbles filmed in 1979, the computer -generated clusters of many soap bubbles impossible to obtain with real soapy water. References 1. J. Plateau, Statique exphimentale etthiorique des liquides soumisaux seulesforces moleculaires (Paris: Gauthier-Villars, 1873). 2. F.J. Almgren...