How to Use This Book

This quote expresses the philosophy on which this book is based. 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 one's own experience of the meanings. 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.

This book will present you with a series of problems. You should explore each question and write out your thinking in a way that can be shared with others. By doing this you will be able to actively develop ideas prior to passively reading or listening to comments of others. When working on the problems, you should be open-minded and flexible and let your thinking wander. Some problems will have short, fairly definitive answers, and others will lead into deep areas of meaning that can be probed almost indefinitely. You should not accept anything just because you remember it from school or because some authority says it's good. Insist on understanding (or seeing) why it is true or what it means for you. Pay attention to your deep experience of these meanings.

You should think about each problem and express your thinking even when you know you cannot complete it. This is important because

An important thing to keep in mind is that there is no one correct solution. There are many different ways of solving the problems — as many as there are ways of understanding the problems. Insist on understanding (or seeing) why it is true or what it means to you. Everyone understands things in a different way, and one person's "obvious" solution may not work for you. However, it is helpful to talk with others — listen to their ideas and confusions and then share your ideas and confusions with them. In the experience of those using this book and the earlier edition, it is very important to be able to talk with others in small groups whether inside or outside of class. In fact, small groups have successfully gone through this book as self-study without a teacher.

Also, some of the problems are difficult to visualize in your head. Make models, draw pictures, use rubber bands on a ball, use scissors and paper — play!

For exploring properties on a sphere it is important that you have a model of a sphere that you can use. You can draw on worn tennis balls ("worn" because the fuzz can get in the way) and they are the right size for ordinary rubber bands to represent great circles. You may find useful clear plastic spheres from craft stores. Most any ball you have around will workyou can even use an orange and then eat it when you get hungry!

For exploring the geometric properties of a hyperbolic plane it is very important to have a hyperbolic surface in your hands. Instructions on how you can make (either out of paper or by crocheting) hyperbolic surfaces are contained in the beginning of Chapter 5. It will be very helpful to your understanding of the hyperbolic plane for you to actually make one of these hyperbolic surfaces yourself.

How I Use This Book in a Course

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.

Some problems are investigated by my students in small cooperative learning groups (with no written work) and the groups report back to the class, or not, depending on how it goes.

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. For more discussion of this, see the "Message to the Reader" on pages xxviiixxxi.

For a final project I usually ask each student to pick a chapter that we have not covered in class and to investigate it on his or her own (with my assistance as necessary in office hours). I have found from the experiences of my students that all the chapters work for such projects.

But, Do It Your Own Way

From feedback I have received from instructors using the first edition of this book and preliminary versions of this edition, I have learned that many of the most successful uses of this book in the classroom are when the instructors does not follow this book! In many of these successful courses, the instructors have used some of the problems from this book and then added their own favorite problems. The chapters in the book have been used in a variety of different orders and at many different speeds. Some instructors have, with careful organization and direction, successfully covered the whole of the first edition of this book in a semester. Other instructors have let their students explore and wander in such a way that they covered less than half of the book. Some instructors use the book in a course with traditional preliminary and final exams. Others have had no exams, but instead relied on portfolios, student journals, and/or projects. Some have used small cooperative learning groups regularly throughout the semester and others have never divided the students into small groups.

Also, the book has been used in courses for sophomores who are prospective teachers either elementary or secondary. Or, in courses for senior mathematics majors. Or, in courses for Masters students in Education. Many of the problems in this book (but not the book itself) have been used successfully with freshman-level courses for students weak in mathematics ("liberal arts mathematics"). Portions of the book have also been used successfully in a freshman writing course.

Chapter Sequences

I have attempted to make the chapters more independent of each other than was the case in the first edition. But there is a minimum core that is necessary before exploring the other chapters.

Minimum core: Cover, at a minimum, Chapters 13, 6, 8, and either 7 or 10.

All chapters after Chapter 10 are independent: After exploring the minimum core, all of the other chapters are essentially independent. Each chapter refers to some results from other chapters but these references can be looked up without destroying the experience of the chapter you are reading.

Starred problems and sections: 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).

Here are some chapter sequences that put emphasis on different aspects of geometry:

To use this book the same as the first edition: Leave out Chapters 5, 14, 16, 17, and 22; and ignore the references to hyperbolic geometry and the hyperbolic plane in the remaining chapters.

To emphasize traditional topics in Euclidean plane geometry: Explore Chapters 13, (4), 6, 814, 18, 21. This includes much geometry on spheres (and cones and cylinders), which is needed to bring out the meanings of Euclidean geometry.

To emphasize spherical geometry: Explore Chapters 13, 6–9, 11, 12, 15, 19.

To emphasize hyperbolic geometry: Explore hyperbolic plane parts of Chapters 13, 512, 14, and 16.

To get to Chapter 22 on the shape of space as quickly as possible: Explore Chapters 18, 17, and Problems 20.1 and 20.6.

For chapters using similar triangles (Chapters 1419): Explore Problem 12.1 and 13.3 or assume the two Criteria for Similar Triangles (Problem 13.3).

For three-dimensional investigations in Chapters 21 and 22: At least explore Problem 20.1 and assume the results of Problem 20.6.

These chapter sequences are summarized in the following table:

Goal Chapters
Minimum Core 1–3, 6, 8, and either 7 or 10
Use book similar to

first edition

4   7 9 10 11 12 13   15     18 19 20 21  
To emphasize

Euclidean geometry

-4     9 10 11 12 13 14       18     21  
To emphasize

spherical geometry

    7 9   11 12     15       19      
To emphasize

hyperbolic geometry

  5 7 9 10 11 12   14   16            
To get to Chapter 22

"Shape of Space"

as soon as possible

4 5 7                 17     20.1



To prove and apply similar triangle criteria (13.3)       9 10   12 13 14 15 16 17 18 19      

1 From an unpublished lecture in London, April 1984. Used here by permission.