The formal expression of "straightness" is a part of differential geometry and 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. In addition, we will explore the geometry on geometric manifolds which are spaces that locally have the same geometry are one of these three spaces but which globally are different — for example, a cylinder (extending indefinitely and without top or bottom) locally has the geometry of the plane and is a geometric Euclidean manifold. 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?", where we are like 3-dimensional bugs who can only view the universe intrinsically — presumably our physical universe is a 3-dimensional geometric manifold.
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.
This book is based on a junior/senior level course I have been teaching for twenty five years 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, are encourage to write and speak their reasonings and understandings; and then I listen to and critique their thinking and use it to stimulate the whole class discussions.
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.
In my course the distinction between learning activities and assessment activities is blurred. I present a sequence of problems (together with motivation, discussion of contexts, and connections of the problems with other areas of mathematics and life). I tell the students:
The students then work on the problems either individually or in small groups and report their thinking back to me and the class. This cycle of writing, comments, discussion continues on each problem until both the students and I are satisfied, unless external constraints of time and resources intervene.
What I have discovered is that in this process not only have the students learned from the course, but also I have learned much about geometry from them. At first I was surprised; how could I, the teacher, learn mathematics from the students? But this learning has continued for 25 years and I now expect its occurrence. In fact, as I expect it more and more and learn to listen more effectively to them, I find that a greater portion of my students show something new to me about geometry. I have also discovered that I am learning more (percentage-wise) from those students who differ from me in terms of gender and race. For more discussion of this, see the "Message to the Reader" on pages xx-xxiv and my article, "I learn mathematics from my students -- multiculturalism in action", For the Learning of Mathematics, 16, 34-40, 1996.
Useful Supplements
A faculty member using this book in a course can contact me at dwh2@cornell.edu for information on obtaining an Instructor's Manual.
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 plastic sphere sets which include a transparent sphere, a spherical compass, and a spherical "straight edge" which doubles as a protractor. These sets should be available in your bookstore or from Key Curriculum Press, Berkeley, CA. They work well for small group explorations in the classroom. The instructor should also have a large "black board" sphere that can be written on with chalk — these spheres are often common in chemistry classrooms. However, a beach ball or basketball will also work, 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.
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 at
http://math.cornell.edu/~dwh/books/eg99/title.html/supplements
sources for the hyperbolic surfaces as they become available. Most books which 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 the a portion of the earth. Each of these representations necessarily (see Chapter 16) distort either straight lines or angles or both.
In addition, the use of dynamic geometry software such as Geometers Sketchpad® or Capri® will enhance any geometry course. These software packages were originally written for exploring Euclidean plane geometry, but recent versions allow one to also dynamically explore spherical and hyperbolic geometries. I will maintain at
http://math.cornell.edu/~dwh/books/eg99/title.html/supplements
links to information about these software packages and to web pages which give examples on how to use them for self learning or in a classroom.
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 which 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 upon 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 that was started with Avery and the CSMRP materials and was 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 which 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 proof-reading 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 preformed 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.
And, 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: George H. Litman (National-Louis University), David A. Olson (MTU), Colm Mulcahy (Spelman College), Sean Bradley (Clarke College), George Tintera (Texas A&M University-Corpus Christi), Chaim Goodman-Strauss (University of Arkansas), Walter Whiteley (York University), John Sullivan (UIUC), Nathaniel Miller (Cornell University), Nancy Rodgers (Hanover College), Robert Stolz (University of the Virgin Islands), Gian Mario Besana (Eastern Michigan University), Michelle Zandieh (ASU), Paul J Gies (University of Maine at Farmington), David Mond (Warwick University), Daniel H. Steinberg (Case Western Reserve University), Alice Guckin (College of Saint Scholastica), Keith Henderson (Thomas Jefferson School), Margaret Symington (University of Texas), Mary Platt (University of Massachusetts), Jane-Jane Lo (Ithaca College and Cornell University, and Kelly Gaddis (Lewis and Clarke 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 this book through the publication process.
Ithaca, NY, June 1999
David W. Henderson