I’ll level with you in the first sentence—I really enjoy geometry. For many years, that admission was not easy to make. In middle school, for example, peer pressure made it very difficult to even admit liking mathematics. I did, but I kept it to myself and sometimes wondered if something was wrong with me because I seemed to be the only one who did. In high school, I still couldn’t tell anyone I liked math and especially not geometry. It was possible, however, to say things like “math is more interesting than history” or “at least math isn’t as bad as French.” As an undergraduate in college, admitting that I liked math was acceptable. There were even other students who would admit the same thing. There weren’t many geometry courses available to me at the time, so admitting I really liked geometry was never a problem. Graduate school, however, was a different story. There were many geometry courses available and many students who enjoyed them. It may have been the first time I could tell anyone “I really enjoy this stuff” while studying geometry.

How can anyone enjoy geometry? What’s the appeal? I can only speak for myself, but here’s why I enjoy the subject. First, geometry is a delightful combination of the very practical to the completely abstract. For example, no architect, plumber, astronomer, pilot, artist, carpenter, mason, or landscape designer could succeed without a good grasp of geometry. He or she may have learned practical geometry on the job rather than in school, but it had to be learned somewhere. Abstract geometry is used in many branches of science as well as by science fiction writers and artists. The works of Pablo Picasso and Salvador Dali, for example, often reflect the application of abstract geometry. The first chapter of Geometric Delight in which worlds of one, two, three, and four dimensions are explored will give you a taste of the abstract.

My second reason for enjoying geometry is the subject’s gaming aspect. Doing geometry can be seen as playing a 2000-year-old version of Dungeons and Dragons. When playing Dungeons and Dragons, you make up characters and rules that the characters must follow. Playing the game then involves seeing where your characters and rules lead you and doing the best that you can when given a certain set of conditions. That’s exactly what you do in geometry! The characters are points, lines, triangles, and so forth. The rules are the assumptions you make about these characters. You proceed by demonstrating that other statements about the characters must be true. These other statements are the theorems of geometry. Geometry really can be considered a game, and the second chapter of Geometric Delight highlights some of the gaming aspects. This chapter deals with making geometric constructions using tools that no longer work as intended.

A third reason for enjoying geometry is that it is a large, logical structure built on a small set of assumptions. A short list of assumptions leads to a long list of theorems, each of which can be demonstrated as valid based on the assumptions and previously demonstrated theorems. For most students, geometry is the first opportunity to see the logical structure behind a topic in mathematics. Algebra has a similar logical structure, but that structure is rarely seen before college mathematics due to the way algebra is presented in middle schools and high schools.

One of the practical applications of geometry that has become very important is computer graphics. Geometry is also critical to any application in which we want a computer to interact with the real world. Computer vision and robotics could not exist without geometry. The third chapter of Geometric Delight will give you a taste of some of the difficulties involved in getting a computer to recognize some very basic geometric relationships.

The final chapter in Geometric Delight is about conic sections and analytic geometry. Analytic geometry is the study of geometric properties using algebraic operations upon terms that are defined by a coordinate system. Analytic geometry is a rich intersection of algebra and geometry that demonstrates how two separate logical systems can remain consistent when put together. It also shows that an idea in one system can be shown to be valid by a demonstration in a different logical system. This chapter will also help you develop practical skills in algebraic manipulation.

I hope that your work in this book will help bring you the same feeling that I have. I hope your really enjoy geometry!