Preface Scientific principles are often reflected in geometry. Whether it is the curve made by a hanging wire or the path that light takes around the sun, shapes are often the manifestations of Nature's design. This book is about the mathematics which describes the geometric prop- erties of soap films. Using ideas from plane geometry, differential geometry, complex analysis and the calculus of variations, we can be- gin to understand why soap films take the shapes they do. But it isn't just soap which interests us as mathematicians. Rather, it is the mathematization of the study of soap film shapes which serves as a prime example of the place geometry has in mathematical modeling. As we shall see, the effects of surface tension lead a soap film to minimize its surface area. This well-defined mathematical conse- quence allows us to study soap film shapes from a purely mathemati- cal viewpoint. The mathematics involved ranges from the elementary to the very advanced, but in this book we will focus on a point some- where in the middle. That is, readers are expected to know calculus and have some familiarity with differential equations, but the relevant notions from differential geometry and complex variables needed in the book are all discussed in Chapter 2. In fact, in order to get to the point quickly, we try to use only the essential ingredients of each of these subjects to begin to tell the mathematical story of soap films. XI
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