In two dimensions, Euclidean geometry is the geometry of an infinite sheet of paper. The postulates now exist in several different forms, but all address the basic properties of lines and angles. Key among them is the parallel postulate, which says that, given any line and a point not on that line, the exists a unique line through the given point which is parallel to the given line.

It was thought by Euclid (and many after him) that the parallel postulate was so obvious that it should follow from the other postulates. In the centuries following Euclid, many incorrect proofs were proposed, purporting to show just that, but the claim is false; the parallel postulate turns out to be independent of the others. These attempted proofs ultimately gave rise in the early 19th century to hyperbolic geometry, the geometry in which the parallel postulate fails due to the existence of more than one parallel line.

The existence of such non-Euclidean geometries was ultimately established by considering particular models of hyperbolic geometry, several of which we discuss below. However, just as the sphere provides the most intuitive model for elliptic geometry, in which there are no parallel lines, it is the hyperboloid in Minkowski space that provides the most intuitive model of hyperbolic geometry. But Minkowski space and special relativity did not exist until the early 20th century, and were therefore not available when these other models were being developed.

In this chapter, we run history backward, first describing the hyperboloid model of hyperbolic geometry, then discussing briefly how to derive the most common traditional models of hyperbolic geometry from the hyperboloid.