The Hyperbolic Plane

The hyperbolic plane—illustrated by the poncho hyperbolic poncho in Figure 10.1—will complete the set of three homogeneous two-dimensional geometries. The first is the geometry of a ordinary sphere sphere!two-dimensional (Figure 10.2(a) ), which is called spherical geometry spherical geometry geometry!spherical and has positive curvature positive curvature curvature!positive . The second is the familiar geometry of the Euclidean plane Euclidean plane (Figure 10.2(b) ), which is called Euclidean geometry Euclidean geometry geometry!Euclidean and has zero curvature zero curvature curvature!zero . The third geometry shown is, loosely speaking, the geometry of a piece of leaf lettuce (Figure 10.2(c) ). It is less familiar than spherical or Euclidean geometry, but no less beautiful. It is called hyperbolic geometry hyperbolic geometry geometry!hyperbolic and has negative curvature negative curvature curvature!negative . The hyperbolic plane hyperbolic plane —abbreviated H ²—is an infinite plane that has hyperbolic geometry (= constant negative curvature) everywhere, just as the Euclidean plane E ² is an infinite plane that has Euclidean geometry (= zero curvature) everywhere. It is a fact of nature, though, that there can be no infinite plane with spherical geometry (= constant positive curvature) everywhere—it will, without fail, close back onto itself to form either a sphere or a projective plane.