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

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