What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the productions of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth — the very experience out of which Love rises.

- — Plotinus,

This book is an expansion and revision of the book *Experiencing Geometry on Plane and Sphere*. The most important change is that I have included material on hyperbolic geometry that was missing in the first book. This has also necessitated more discussions of circles and their properties. In addition, there is added material on geometric manifolds and the shape of space. I decided to include hyperbolic geometry for two reasons: 1) the cosmologists say that our physical universe very likely has (at least in part) hyperbolic geometry, and 2) Daina Taimiða, a mathematician at the University of Latvia and now my wife, figured out how to crochet a hyperbolic plane, which allowed us to explore intuitively for the first time the geometry of the hyperbolic plane. In addition, Daina Taimiða has been responsible for including in this edition significantly more historical material. In this historical material we discuss and try to clear up many current misconceptions that are commonly held about some mathematical ideas.

This book is based on a junior/senior-level course I have been teaching since 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. Certain problems and sections in this book require from the reader a background more advanced than first-semester calculus. These sections are indicated with an asterisk (*) and the background required is indicated (usually at the beginning of the chapter).

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. I listen to and critique their thinking and use it to stimulate whole class discussions.

The formal expression of "straightness" is a very difficult formal area of mathematics. However, the concept of "straight" an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the (Euclidean) plane, on a sphere, or on a hyperbolic plane. We will study these ideas and questions, as much as is possible, from an intrinsic point-of-view — that is, the point-of-view of a 2-dimensional bug crawling around on the surface. This will lead to the question: "What is the shape of our physical three-dimensional universe?" Here we are like 3-dimensional bugs who can only view the universe intrinsically.

Most of the problems are approached both in the context of the plane and in the context of a sphere or hyperbolic plane (and sometimes a geometric manifold). I find that by exploring the geometry of a sphere and a hyperbolic plane my students gain a deeper understanding of the geometry of the (Euclidean) plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also 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 the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere or hyperbolic plane are not adequately described by the usual Parallel Postulate. I 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.)

**Useful Supplements**

A faculty member may obtain from the publisher the *Instructor's Manual* (containing possible solutions to each problem and discussions on how to use this book in a course) by sending a request via e-mail to George_Lobell@prenhall.com or calling 1-201-236-7407.

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase Lénárt Sphere^{®} sets — a transparent sphere, a spherical compass, and a spherical "straight edge" that doubles as a protractor. They work well for small group explorations in the classroom and are available from Key Curriculum Press. However, considerably less expensive alternatives are available: A beach ball or basketball will work for classroom demonstrations, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles. Also, many craft stores carry inexpensive plastic spheres that can be used successfully.

I strongly urge that you have a hyperbolic surface such as those described in Chapter 5. Unfortunately, such hyperbolic surfaces are not readily available commercially. However, directions for making such surfaces (out of paper or by crocheting) are contained in Chapter 5, and I will list patterns for making paper models and sources for crocheted hyperbolic surfaces at

www.math.cornell.edu/~dwh/books/eg00/supplements.html

as they become available. Most books that explore hyperbolic geometry do so by considering only one of the various "models" of hyperbolic geometry, which give representations of hyperbolic geometry in the same way that a map of a portion of the earth gives a representation of a portion of the earth. Each of these representations necessarily (see Chapter 16) distorts either straight lines or angles or both.

In addition, the use of dynamic geometry software such as *Geometers Sketchpad ^{®}*,

**My Teaching Background**

My teaching is a product of Western Civilization. My known ancestors lived in England, Scotland, Ireland, Germany, and Luxembourg and I am a descendent from a long line of academics stretching back (according to family traditions) to at least the seventeenth century. My mode of teaching also has deep Western roots that reach back to the Socratic dialogues recorded by Plato in ancient Greece. More directly, my teaching has been influenced by my experiences in high school, college, and graduate school. In Ames, Iowa, my high school world literature teacher, Mary McNally, coaxed deep creative thinking out of us through her many writing assignments which she read with great interest in our ideas. At Swarthmore College in Pennsylvania instead of classes I spent my last two years in student participation seminars and tutorials, where I learned to take charge of my own learning and become an academic scholar. In graduate school at the University of Wisconsin my mentor, R.H. Bing, taught without lectures or textbooks in a style which is often known as the Moore Method, named after Bing's graduate mentor R.L. Moore at the University of Texas. (See [**TG: **Traylor] for more information on the Moore Method.) R.L. Moore received his PhD at the University of Chicago before the turn of the century and was one of the very first Americans to receive a PhD in mathematics in this country. My teaching of the geometry course and the writing of this book evolved from this background.

**Acknowledgments for First Edition**

I acknowledge my debt to all the students and teachers who have attended my geometry courses. Most of these people have been students at Cornell or teachers in the surrounding area of upstate New York, but they also include students at Birzeit University in Palestine and teachers in the new South Africa. Without them this book would have been an impossibility.

Starting in 1986, Avery Solomon and I organized and taught a program of inservice courses for high school teachers under the financial support of Title IIA Grants administered by the New York State Department of Education. This is now called the Cornell/Schools Mathematics Resource Program (CSMRP). As a part of CSMRP we started recording classes and writing notes on the material. Some of the material in this book had its origins in those notes, but they never threatened to become a textbook. I thank Avery for his modeling of enthusiastic teaching, his sharp insights, and his insistence on preserving the teaching materials. In addition to Avery, my friends, Marwan Awartani, a professor at Birzeit University, and John Volmink, the director of the Centre for the Advancement of Science and Mathematics Education in Durban, South Africa, have for a long period of time consistently encouraged me to write this book.

A few years ago my colleague Maria Terrell suggested that five of us at Cornell who have been teaching non-traditional geometry courses (Avery Solomon, Bob Connelly, Tom Rishel, Maria, and I) submit a proposal to the National Science Foundation for a grant to write up materials on our courses. The fact that we were awarded the grant (in 1992) is largely due to Maria's persistence, clear thinking, and encouragement. It is this grant that gave me the necessary support to start the writing of this book. I thank the NSF's Program on Course and Curriculum Development for its support.

The major portions of this book were written during the 1992–93 academic year, in which I taught the course both semesters. Eduarda Moura was my teaching assistant for these courses. She was supported by the NSF grant to assist me by describing the classroom discussion and the student homework on which the content of this book is based. Much of this book (and especially the instructor's manual) are derived from her efforts. In addition to Eduarda, Kelly Gaddis, Beth Porter, Hal Schnee, and Justin Collins were also supported by the grant and made significant contributions to the writing of this book. I thank them all for their excellent contributions, their support of my work, and their friendship. The final writing and the decisions as to what to include and what not to include have all been mine, but they have been based on the foundation started with Avery and the CSMRP materials and continued with Eduarda, Justin, Kelly, Beth, and Hal during 1992–93.

Since the spring of 1992, the early drafts of the book have been used by me and others at Cornell and 13 other institutions. Various other individuals have worked through the book outside a classroom setting. From these students, instructors, and others I have received encouragement and much valuable feedback that has resulted in what I consider to be a better book. In particular, I want to thank the following persons for giving me feedback and ideas I have used in this final version: David Bray, Douglas Cashing, Helen Doerr, Jay Graening, Christine Kinsey, István Lénárt, Julie Lubell, Richard Pryor, Amanda Cramer and her students, Erica Flapan and her students, Linda Hill and her students, Tim Kurtz and his students, Judy Roitman and her students, Bob Strichartz and his students, and Walter Whitely and his students. Susan Alida spent many hours proofreading and refining the text, and was my consultant on matters of aesthetics.

**Acknowledgments for This Edition**

I wish to thank the instructors and students (all over the world) who have used the first edition: *Experiencing Geometry on Plane and Sphere*. Their responses were the first encouragement to expand and revise the book.

I wish to thank Jeffrey Weeks for introducing me to the current issues and upcoming experiments about the shape of our physical universe. He is the first to have informed me about the observations that are due to be performed in 2000–2001 that may allow a group of mathematicians and physicists (including Weeks) to determine the global geometry of our physical universe. It is my hope that this book provides the necessary background to understand these observations and determinations.

Without Daina Taimiða's crocheted hyperbolic planes I would not have had the intuitive experiences that encouraged me to write this expansion and revision. The ideas for the expansion and revisions were discussed between us and much of the rewriting and expansion was completed with her able assistance.

In addition, the following persons gave me special comments that were incorporated into the expansion and revision: sarah-marie belcastro (University of Northern Iowa), David Bellamy (University of Delaware), Gian Mario Besana (Eastern Michigan University), Alexander Bogomolny (CTK Software, East Brunswick, NJ), Katherine Borgen (University of British Columbia), Sean Bradley (Clarke College), David Dennis (University of Texas at El Paso), Kelly Gaddis (Lewis and Clarke University), Paul J. Gies (University of Maine at Farmington), Chaim Goodman-Strauss (University of Arkansas), Alice Guckin (College of Saint Scholastica), Cathy Hayes, Keith Henderson (Thomas Jefferson School, St. Louis, MO), George H. Litman (National-Louis University), Jane-Jane Lo (Ithaca College and Cornell University), Alan Macdonald (Luther College), John McCleary (Vassar College), Nathaniel Miller (Cornell University), David Mond (University of Warwick), Colm Mulcahy (Spelman College), Jodie Novak (University of Northern Colorado), David A. Olson (MTU), Mary Platt (Salem State College, MA), John Poland (Carleton College), Nancy Rodgers (Hanover College), Thomas Sibley (St. John's University), Judith Roitman (University of Kansas), Frances Rosamond (National University), Avery Solomon (Cornell University), Daniel H. Steinberg (Case Western Reserve University), Robert Stolz (University of the Virgin Islands), John Sullivan (University of Illinois – Urbana), Margaret Symington (University of Texas), George Tintera (Texas A&M University – Corpus Christi), Susan Tolman (University of Illinois – Urbana), Andy Vidan (Cornell University), Tad Watanabe (Towson State University), Walter Whiteley (York University), Jeffrey Weeks (Canton, NY), Steve Weissburg (Ithaca High School, Ithaca, NY), Cindy Wyels (California Lutheran University), and Michelle Zandieh (Arizona State University). I may have inadvertently left out a few names; if so, I apologize.

I produced the entire manuscript (typing, formatting, drawings, and final layout) using the integrated word processing software WordPro. Finally, I wish to thank George Lobell, Senior Editor at Prentice Hall, for his encouragement and support and for the vision and enthusiasm with which he shepherded both editions through the publication process. I also thank Betsy Williams, Production Editor, for thoughtfully improving the style of this book, which has made the book much more pleasing and readable.

* David W. Henderson*

* Ithaca, NY*