Preface Why study geometry? Those who have progressed far enough in their mathematical edu- cation to read this book can probably come up with lots of answers to that question: Geometry is useful. It’s hard to find a branch of mathematics that has more practi- cal applications than geometry. A precise understanding of geometric relationships is prerequisite for making progress in architecture, astronomy, computer graphics, engineering, mapmaking, medical imaging, physics, robotics, sewing, or surveying, among many other fields. Geometry is beautiful. Because geometry is primarily about spatial relationships, the subject comes with plenty of illustrations, many of which have an austere beauty in their own right. On a deeper level, the study of geometry uncovers surprising and unexpected relationships among shapes, the contemplation of which can inspire an exquisitely satisfying sense of beauty. And, of course, to some degree, geometric relationships underlie almost all visual arts. Geometry comes naturally. Along with counting and arithmetic, geometry is one of the earliest areas of intellectual inquiry to have been systematically pursued by human societies. Similarly, children start to learn about geometry (naming shapes) as early as two years old, about the same time they start learning about numbers. Almost every culture has developed some detailed understanding of geometrical relationships. Geometry is logical. As will be explored in some detail in this book, very early in Western history geometry became the paradigm for logical thought and analysis, and students have learned the rudiments of logic and proof in geometry courses for more than two millennia. All of these are excellent reasons to devote serious study to geometry. But there is a more profound consideration that animates this book: the story of geometry is the story of mathematics itself. There is no better way to understand what modern mathematics is, how it is done, and why it is the way it is than by undertaking a thorough study of the roots of geometry. xi
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