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.
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