Geometries of Surfaces

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.

Too Much Angle!

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.

Activity 1: Gluing octagons for fun and profit

1. On the right you see an octagon with marked side identification. When you glue respective sides, what surface do you get? (Hint: carefully draw a series of pictures, gluing one pair of arrows at a time.)
2. What is the (inside) angle between two consecutive sides in a regular octagon?
3. Does this octagon tile the plane? Why not? (Hint: use part 2.)

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.

Activity 2: A triangle here, a triangle there

1. What is the sum of the angles of a triangle in a plane? Can you prove it? (Hint: see this helpful picture.)
2. What is the minimum and maximum sum of the angles of a triangle on the sphere with one vertex on the North pole and two on the equator? (Hint: the sum depends on where the last two vertices are on the equator.)
3. Recall the square identification scheme for the projective plane, illustrated on the right. Which vertices of the square are identified? That is, when you do the gluing, which vertices are glued together?
4. What are the angle deficits of the resulting vertices? (Hint: remember how this question was analyzed in case of the sphere.)
5. Can you find a square on the sphere in which each of the four angles measures 180 degrees? (Hint: think equator.)
6. Let this square be the fundamental polygon for the projective plane. Looks familiar?

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.

The Hyperbolic Plane

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.

Activity 3: Exponentially vast expanse

1. Note that the white segments in Escher's woodcut form equilateral triangles and squares. Can we tile the plane with equilateral triangles and squares in such a manner? (Hint: if they were triangles and squares in the plane, what would the angle sum around a vertex be?)
2. While distances in the Poincaré disk are distorted, angles are not. Assuming all angles between adjacent white segments in picture are the same (which is not quite true, but almost!), what does each measure?
3. Lines which look straight to someone living in the hyperbolic plane look like circle arcs which intersect the disk boundary at right angles in the Poincaré model (see illustration below) and diameter lines going through the center of the disk. Given such a line L in the hyperbolic plane and a point p not on the line, how many lines parallel to L go through that point? (Hint: in the normal Euclidean plane the answer is 1; it's an axiom of Euclidean geometry. Recall that two lines are parallel if they do not intersect.)

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 degrees.