Do not just pay attention to the words;
Instead pay attention to meanings behind the words.
But, do not just pay attention to meanings behind the words;
Instead pay attention to your deep experience of those meanings.
— Tenzin Gyatso, The Fourteenth Dalai Lama1
This quote expresses the philosophy upon 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 ones 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 which 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 the problems and express your thinking about them even when you know you cannot do them completely. This is important because:
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 in craft stores. Most any ball you have around will work — you 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.
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.
1
From an unpublished lecture in London, April, 1984. Used here by permission.