Section 1 Preface
¶Euclidean geometry attempts to capture the geometric structure of the world around us in a few fundamental postulates, then use them to derive intuitively obvious results.
With the wisdom of hindsight, this effort was doomed to failure.
First of all, it took nearly 2000 years to get the postulates right, both in terms of providing clear statements of what was being assumed, and in terms of which postulates depended on which other postulates. The best-known difficulty was the unsuccessful attempt to show that the Euclidean parallel postulate is a consequence of the other postulates. It's not.
These efforts led to the realization that there were “almost Euclidean” geometries, in which most of the postulates hold, but not the Euclidean parallel postulate. It is these geometries that are normally meant when one uses the phrase non-Euclidean geometry.
However, there are other geometries that are not Euclidean. Euclidean distance is not very helpful when navigating city streets. More fundamentally, the universe around us turns out not to be Euclidean! Rather, its underlying structure is described by special and general relativity, perhaps the most geometric theories of physics ever developed.
Thus, Euclidean geometry as proposed by Euclid was not only incomplete, but also fails to capture the true structure of the world around us, however good an approximation it is on both counts. The study of non-Euclidean geometry is both fascinating in its own right, but also sheds light on both of these issues.
This book provides an informal guide to many of the models of non-Euclidean geometry. In most cases, interactive applets allow readers to explore these models for themselves. Little attempt is made to provide mathematically rigorous constructions, although the logical progression should be clear.
Let's begin.