Introduction

Geometry is full of beautiful theorems, and its logical structure can be inspiring. As the poet Edna St. Vincent Millay wrote, "Euclid alone has looked on beauty bare." But beyond beauty and logic, geometry also contains important tools for applied mathematics. This should be no surprise, since the word geometry means "earth measurement" in Greek. As just one example, we will illustrate the appropriateness of this name by showing how geometry was applied by the ancient Greeks to measure the circumference of the earth without actually going around it. But the story of geometric applications is modern as well as ancient. The upsurge in science and technology in the last few decades has brought with it an outpouring ofnew questions for geometers. In this introduction we provide a sampler of the big ideas and important applications that will be discussed in this book.1

Individuals often have preferences, either for applications in contrast to theory or vice versa. This is unavoidable and understandable. But the premise of this book is that, whatever our preferences may be, it is good to be aware of how the two faces of geometry enrich each other. Applications can’t proceed without an underlying theory. And theoretical ideas, although they can stand alone, often surprise us with unexpected applications. Throughout the history of mathematics, theory and applications have carried out an intricate dance, sometimes dancing far apart, sometimes close. My hope is that this book gives a balanced picture of the dance at this time, in the early years of a new millennium.

Axiomatic Geometry

Of all the marvelous abilities we human beings possess, nothing is more impressive than our visual systems. We have no trouble telling circles apart from squares, estimating sizes, noticing when triangles appear congruent, and so on. Despite this, the earliest big idea in geometry was to achieve truth by proof and not by eye. Was that really necessary or useful? These ideas are explored in Chapters 1 and 2.

Creating a geometry based on proof required some basic truths — which are called axioms in geometry. Axioms are supposed to be uncontroversial and obviously true, but Euclid seemed nervous about his parallel axiom. Other geometers caught this whiff of uncertainty and, about 2000 years later, some were bold enough to deny the parallel axiom. In doing this they denied the evidence of their own eyes and the weight of 2000 years of tradition. In addition, they created a challenge for students of this so-called "non-Euclidean geometry," which asks them to accept axioms and theorems that seem to contradict our everyday visual experience. According to our visual experience, these non-Euclidean geometers are cranks and crackpots. But eventually they were promoted to visionaries when physicists discovered that the faraway behavior of light rays (physical examples of straight lines) is different from the close-to-home behavior our eyes observe. Astronomers are working to make use of this non-Euclidean behavior of light rays to search for "dark matter" and to foretell the fate of the universe. These revolutionary ideas are explored in Chapter 3.

Rigidity and Architecture

If you are reading this indoors, the building you are in undoubtedly has a skeleton of either wooden or steel beams, and your safety depends partly on the rigidity of this skeleton (see Figure I.1b). Neither a single rectangle (Figure I.1b), nor a grid of them, would be rigid if it had hinges where the beams meet. Therefore, when we build frameworks for buildings, we certainly don’t put hinges at the corners — in fact, we make these corners as strong as we can. But it is hard to make a corner perfectly rigid, so we will think of the corners as a little bit like hinges that need to be kept from flexing. If a rectangle is built of four sticks hinged together in such a way that any flexing produces a four-sided figure in a plane, and if we add a diagonal brace, then the rectangle can’t be flexed at all. Perhaps surprisingly, if we have a grid of many rectangles, it is not necessary to brace every rectangle. The braced grid in Figure I.1b turns out to be rigid even if every corner is hinged. In Section 2.3, we work out a procedure for determining when a set of braces makes a grid of rectangles rigid even though all corners are hinged.

Computer Graphics

The impressionist painter Paul Signac (1863–1935) painted "The Dining Room" by putting lots of tiny dots on a canvas (Figure I.2a). If you stand back, the tiny dots blend together to make a picture. This painting technique, called pointillism, was a sensation at the time, and it foreshadows modern image technology. For example, if you take a close look at your TV screen, you’ll see that the picture is composed of tiny dots of light. Likewise, a computer screen creates a picture by "turning on" little patches of color called pixels. Think of them as forming an array of very tiny light bulbs, arranged in hundreds of rows and columns in the x-y plane so that each point with integer coordinates is the center of a pixel (Figure I.2b).

When a graphics program shows a picture, how does it calculate which pixels to turn on and what colors they should be? Here is a simple version of the problem: If we are given two pixels (shaded in Figure I.2b) and want to connect them with a set of pixels to give the impression of a blue straight line, which "in between" pixels should be turned blue? Neither Signac nor you would have any problem with this, painting by eye, but how can the computer do it by calculations based on the coordinates of the centers of the start and end pixels? In this computer version of the problem, the desired answer is a list of pixel centers, each center specified by x and y coordinates. Chapter 1 gives you an idea of how to create such a list.

Symmetries in Anthropology

In studying vanished cultures, anthropologists often learn a great deal from the artistic patterns these cultures produced. Figure I.3 shows two patterns you might find on a cloth or circling around a clay pot. Each of these patterns has some kind of symmetry. But what kind? What do we mean by the word symmetry? Some would say the bottom pattern has more symmetry than the top pattern. How can symmetry be measured? The questions we pose here for patterns are similar to ones that arise, in three-dimensional form, in the study of crystallography. We study these questions in Chapter 4, with particular attention paid to the pottery of the San Ildefonso pueblo in the southwestern United States.

Robotics

Figure I.4 shows a robot about to drill a hole. Perhaps the hole is being drilled into someone’s skull in preparation for brain surgery, or maybe it’s part of the manufacture of an automobile. Whatever the purpose, the drill tip has to be in just the right place and pointing just the right way. The robot moves the drill about by changing its joint angles. If we specify the x,y,z coordinates of the drill tip, how do we calculate the values of θ1 θ2, and θ3 needed to bring the drill tip to the desired point? Can we also specify the direction of the drill? These are questions in robot kinematics. In Section 6.5 we study the basics of robot kinematics for a two-dimensional robot. This is the same kind of mathematics used for three-dimensional robot kinematics.

Molecular Shapes

As chemistry advances, it pays more attention to the geometric shapes of molecules. In 1985 a new molecular shape was discovered, called the buckyball,2 that reminded chemists of the pattern on a soccer ball. The molecule consisted of 60 carbon atoms distributed in a roughly spherical shape — like the corners of the pattern on the soccer ball. Coincidentally, this pattern had been discovered not only by soccer ball manufacturers but many centuries ago by mathematicians. Figure I.5 shows the pattern in a Renaissance drawing of a truncated icosahedron, by Leonardo da Vinci. To understand the structure of the buckyball, think of the corners of the truncated icosahedron as being occupied by carbon atoms and think of the connecting links as representing chemical bonds betweeen certain carbon atoms. Each carbon atom is connected to each of three other nearby carbon atoms with a chemical bond. This pattern contains 12 pentagons and 20 hexagons.

Once chemists discovered this molecule, they looked for other molecules involving just carbons, where each carbon is connected to exactly...