ABSTRACT
This chapter explores ways of visualizing space, and starts with the mathematically simplest case—the infinite space of Euclid's geometry. It looks at spherical space, a finite "round" space that Dante seems to describe in his Divine Comedy. The chapter describes it mathematically by analogy with the spherical surface. It also looks at hyperbolic space, an infinite space even more "spacious" than Euclidean space. The chapter contains hyperbolic planes— surfaces more spacious than Euclidean planes. In a hyperbolic plane there is more than one parallel to a given line through a given point. Spherical space and hyperbolic space have curvature analogous to the curvature of surfaces. It can be detected by the behavior of lines, and in particular by whether or not parallels exist and are unique. The Greeks believed the universe should reflect the geometric perfection of circles and spheres, and they imagined space structured by a system of spheres.
