Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces

Geometry has always come as a difficult subject for me. For some reason, it's hard for me to "see" the answers and proofs like some people can. But, that's where a book like this came in handy.Basically, this book is good for two things: first-principle arguments and model building. The problems in here almost require you to make models and draw pictures to see for sure what's going on. So, for a person raised on the calculator like myself, having to make models is a good thing (although very challenging!). Also, the first principle arguments are good for thought and frustration: Almost every problem in this book takes a very long time to solve, with combined thinking time, study time with your notes, and actually writing out your solution. Only on a couple of the problems that my class worked on was I able to get away with using less than one sheet of paper to write out my answer. Basically, you start from the very beginning and look at concepts you know already, like straight lines, and try to explain them as clearly and concisely as you can, which isn't always easy. But, Henderson gives hints for almost every problem, gives background on each problem, and sometimes provides solutions to the problems (but rarely!); he also provides motivation for studying the concepts rather than to pass a math course, which is very good. He also does an excellent job of introducing the concept of hyperbolic geometry by first discussing what properties stay the same or differ between the sphere and plane, which is excellent because hyperbolic geometry is not as easy to grasp as the other two; this instructional method is actually carried out through the book, an excellent way to introduce two types of geometry that many people (including myself) may have never seen before.Honestly, my only gripe is that the book is SO based on first principle arguments that, while it provides excellent framework for prospective elementary and high-school teachers, it won't do much for the applied mathematician who wants to see some computational examples because there are barely any! It forces a student to be held accountable for abstract thought and proof, but there are many aspects of geometry that can be handled with computations like arclength, areas of polygons, etc., which are not discussed in the book.Basically, here's the short version: It's a great book for the student who wants to work through arguments based on first principles, but be prepared to work hard! Because almost none of the exercises come with answers, I don't find this book suitable for independent study, so I hope you have a good teacher! Also, get a ball, some rubberbands, and plenty of paper with some good erasers; you'll need them!

I found this book caused me to rethink much of how I approach Geometry personally and in my classroom. From the first problem, "What is Straight?" which had me thinking about my own assumptions straight lines, I have been thinking of ways to approach Geometry differently with my students. I enjoy the problem-centered exposition, but at times, I wish I had a little more direction. The emphasis on INTUITIVELY understanding what is going on in these different spaces, and on working with physical models (the hyperbolic models are cool), is a refreshing change from an algebraic/matrix approach. This book is all about DOING geometry, and formulating convincing arguements to your "Why?" questions.