I have this book called The Shape of Space which is sort of a topology book for non-mathematicians. Remember when we looked at geometry done on the surface of a sphere, and how triangles had interior angles which added up to more than 180 degrees, as well as other weird properties? That was a geometry with positive curvature. Hyperbolic geometry is geometry that has a negative curvature. Escher has some drawings that illustrate it really well. If I'm not mistaken, saddle shapes exhibit this sort of geometry.
