Hyperbolic Space
Introduction

We have created a world of rectilinearity. The rooms we inhabit, the skyscrapers we work in, the grid-like arrangement of our streets, the shelves on which we store our possessions, and the freeways we cruise on our daily commute speak to us in straight lines. But what exactly is a straight line? And how do such “objects” relate to one another?

This question, so seemingly trivial, lies at the heart of a conundrum that dates back to the dawn of the Western mathematical tradition. Though seemingly obvious, the property of “straightness” turns out to be a subtle and surprisingly fecund concept. Understanding this quality ultimately led mathematicians to discover a radical new kind of space that had hitherto seemed abhorrent and impossible.