On Beyond Euclid

This curious little book is basically a hitchhiker's guide to Book I of Euclid's Elements. We travel through each of the forty-eight Propositions-more or less in order-and see how each one generalizes-or does not generalize-to hyperbolic and other non-Euclidean spaces. Few people seem to realize that Einstein's special theory of relativity is a model of hyperbolic geometry. The connection between Minkowski geometry and special relativity is well-known, while the connection between hyperbolic geometry and special relativity is, rather, known of. But this book makes the hyperbolic connection explicit; Poincare´ disks rather than the traditional Minkowski diagrams are used to illustrate concepts of special relativity. As we progress through Euclid's propositions, it becomes increasingly clear that every theorem in neutral and hyperbolic geometry can be translated into a true statement in special relativity.