We believe that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone. Much of mathematics is not accessible through formal approaches except to those with specialized learning. However, through the use of nonformal experience and geometric imagery, many levels of meaning in mathematics can be opened up in a way that most humans can experience and find intellectually challenging and stimulating.
Formalism contains the power of the meaning but not the meaning. It is necessary to bring the power back to the meaning.
A formal proof as we normally conceive of it is not the goal of mathematics—it is a tool—a means to an end. The goal is understanding. Without understanding we will never be satisfied—with understanding we want to expand that understanding and to communicate it to others. This book is based on a view of proof as a convincing communication that answers—Why?
Many formal aspects of mathematics have now been mechanized and this mechanization is widely available on personal computers or even handheld calculators, but the experience of meaning in mathematics is still a human enterprise that is necessary for creative work.
In this book we invite the reader to explore the basic ideas of geometry from a more mature standpoint. We will suggest some of the deeper meanings, larger contexts, and interrelations of the ideas. We are interested in conveying a different approach to mathematics, stimulating the reader to take a broader and deeper view of mathematics and to experience for herself/himself a sense of mathematizing. Through an active participation with these ideas, including exploring and writing about them, people can gain a broader context and experience. This active participation is vital for anyone who wishes to understand mathematics at a deeper level, or anyone wishing to understand something in their experience through the vehicle of mathematics.
This is particularly true for teachers or prospective teachers who are approaching related topics in the school curriculum. All too often we convey to students that mathematics is a closed system, with a single answer or approach to every problem, and often without a larger context. We believe that even where there are strict curricular constraints, there is room to change the meaning and the experience of mathematics in the classroom.
This book is based on a junior/senior-level course that David started teaching in 1974 at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication.
The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems and are encouraged to write and speak their reasonings and understandings.
Most of the problems are placed in an appropriate history perspective and approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). We find that by exploring the geometry of a sphere and a hyperbolic plane, our students gain a deeper understanding of the geometry of the (Euclidean) plane.
We introduce the modern notion of "parallel transport along a geodesic," which is a notion of parallelism that makes sense on the plane but also on a sphere or hyperbolic plane (in fact, on any surface). While exploring parallel transport on a sphere, students are able to appreciate more fully that the similarities and differences between the Euclidean geometry of the plane and the nonEuclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. We find that the early interplay between the plane and spheres and hyperbolic planes enriches all the later topics whether on the plane or on spheres and hyperbolic planes. All of these benefits will also exist by only studying the plane and spheres for those instructors that choose to do so.
CHANGES IN THIS EDITION
This book is an expansion and revision of the book Experiencing Geometry on Plane and Sphere (1996) and the book Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces (2001). There are several important changes: First, there are now coauthors—Daina was a "contributor" to the second edition. She brings considerable experience with and knowledge of the history of mathematics. We start in Chapter 0 with an introduction to four strands in the history of geometry and use the framework of these strands to infuse history into (almost) every chapter in the book in ways to enhance the students understanding and to clear up many misconceptions. There are two new chapters—the old Chapter 14 (Circles in the Plane) has been split into two new chapters: Chapter 15 (on circles with added results on spheres and hyperbolic plane and about trisecting angles and other constructions) and Chapter 16 (on inversions with added material on applications). There is also a new Chapter 21, on the geometry of mechanisms that includes historical machines and results in modern mathematics.
We have included discussions of four new geometric results announced' in 2003-2004: In Chapter 12 we describe the discovery and solution of Archimedes' Stomacion Problem. Problem 15.2 is based on the 2003 generalization of the notion of power of a point to spheres. In Chapter 16 we talk about applications of a problem of Apollonius to modern pharmacology. In Chapter 18 we discuss the newly announced solution of the Poincare Conjecture. In Chapter 22 we bring in a new result about unfolding linkages. In Chapter 24 we discuss the latest updates on the shape of space, including the possibility that the shape of the universe is based on a dodecahedron. In addition, we have rearranged and clarified other chapters from the earlier editions.
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