If you've got a surface in your hand, you can find a homeomorphic version of the surface on which to construct hyperbolic geometry, elliptic geometry, or Euclidean geometry. And the choice of geometry...If you've got a surface in your hand, you can find a homeomorphic version of the surface on which to construct hyperbolic geometry, elliptic geometry, or Euclidean geometry. And the choice of geometry is unique: No surface admits more than one of these geometries. As we shall see, of the infinitely many surfaces, all but four admit hyperbolic geometry (two admit Euclidean geometry and two admit elliptic geometry).
