Last time we've seen a topological classification of all closed surfaces as connected sums of tori or connected sums of projective planes. Now let's shift our attention to the different geometries surfaces can have. You will probably be surprised to find out that there are only three different types of surface geometries where each point looks exactly the same as any other. These three are spherical geometry, which we've encountered on the sphere and projective plane; planar or flat geometry, which we've seen on the torus and Klein bottle; and finally hyperbolic geometry, the most important and mysterious of the three, and one which we haven't encountered yet.
Before we plunge into the depths of hyperbolic geometry, I want to emphasize the criteria of homogeneity—of having the same geometry around every point. Consider the closed surface on the right. It inherits a geometry from three dimensional Euclidean space it's in. Your avatar the cyclops, by observing the surrounding and possibly a walking friend can easily tell between different points on the surface. The view of at point p, for instance is very different from point q. On the other hand, all points on a torus or a Klein bottle or the sphere are exactly the same to the cyclops (remember, there are no seams). Mathematicians are most often interested in understanding precisely those geometries that are everywhere the same.
Now back to the topic at hand. You may well ask which surfaces from the ones we've classified last time have hyperbolic geometry. And the answer is: all of the ones we haven't explicitly talked about. That is, all the surfaces except the sphere, torus, Klein bottle, and the projective plane have hyperbolic geometry. Let's try to understand this starting with the double torus, which is just torus#torus.
As you've seen in the preceding activity, an octagon has a large angle surplus. All the vertices get glued to a single vertex and so the total angle around that vertex becomes 8x135=1080 degrees, while there are only 360 degrees around a point. Note that this is the opposite of what we've seen happen in the sphere and the projective plane, where we had an angle deficit. Let's investigate this phenomenon further.
Thus on a sphere triangles, and polygons in general, have larger angle sums than on a plane. So if a fundamental polygon is too angle-skinny, we can place this polygon on a sphere and increase its angles so that they'll add up to 360 upon identification of the appropriate vertices. This is the case for the projective plane, and of course for the sphere itself; in this case we say the surface has a spherical geometry.
The torus and Klein bottle have neither angle surpluses nor deficits, since 4x90 is 360 on the nose. Thus both have Euclidean or flat geometry.
On the other hand, the remaining infinite chain of surfaces we've seen in the classification, like the double torus in activity 1, have an angle surplus in their fundamental polygons; they are too angle-fat. Thus we need to find a surface in which polygons have smaller angle sums than in the plane, so that when we transfer these fundamental polygons to it, they will have smaller angles that will add up to 360 around each identified vertex. This surface is called the hyperbolic plane and correspondingly, the surfaces which have an angle surplus in their fundamental polygons have a hyperbolic geometry.
It would not be an exaggeration to say that the hyperbolic plane is one of the most important objects in modern geometry. It is, however, not as easy to picture or describe as the spaces we've studied up to now. One way to look at it is through the so called Poincaré disk model. In it the hyperbolic plane is represented as the interior of a disk in the plane in which distances get exponentially larger as you approach the boundary of the disk. One of the easiest ways to see this is by looking at the art of M.C. Escher, like the woodcut Circle Limit III below. Note that while the fish appear smaller the closer you get to the boundary of the disk, they are the same size in the geometry of the hyperbolic plane. Similarly, all the white segments going from the fishes' tails to their noses are the same length.
Just like in the Euclidean plane and the sphere, polygons in the hyperbolic space are composed of segments of hyperbolic lines. One way to explore the hyperbolic plane is using a computer program, like the Java application NonEuclid which can be found at http://www.cs.unm.edu/~joel/NonEuclid/NonEuclid.html.
Here's the pictorial summary of how we go about giving the double torus, as a side-identified octagon, a hyperbolic geometry.
Just like in the case of the torus or Klein bottle and the Euclidean
plane, we can tile the whole hyperbolic plane by reflecting this
fundamental octagon across each of its sides, then reflecting again, and
again, and again. What you get is a tiling similar to the one in the
Java applet below. You can use p to increment the number of sides
and P decrement it, while q and Q increment/decrement the number of
polygons adjacent to each vertex. Dragging with the mouse moves the
plane in the corresponding direction. The starting values, for the
octagonal tiling, are p=8 and q=8. For complete instructions go to the
author's, Don Hatch's page.
Note that all octagons you see are regular: all their sides are
equal in length in the hyperbolic geometry, and all their angles are 45