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Hyperbolic Surfaces in Nature

A table-top is an example of an ordinary Euclidean or flat space. Like any two-dimensional flat surface, it has zero curvature. What about non-zero curvature? There are two kinds (naturally): positive and negative. A familiar example of a positively curved surface is a sphere. But what about negatively curved surfaces? Are there any familiar examples of those? One is the saddle of a horse. If you start at the front end of a saddle and move backwards, the saddle curves down and then up. If you move from side to side, however, it curves up and then down. This is the definition of negative curvature -- opposite curvature going in different directions. On a sphere, no matter which way you go, it curves the same way, so we call it positive curvature.

Another example you might have seen is this seedpod from a particular kind of tree, which has many saddle-shaped regions instead of just one:

Make your own Hyperbolic Surface

You can make a hyperbolic surface yourself. The key to doing this is a remarkable geometric fact about surfaces and curvature. On a flat plane, the three angles of a triangle add up to 180 degrees. On a sphere, the angles add up to more than 180, and on a hyperbolic surface, they add up to less. In a sense, there's an excess of area around a point on a hyperbolic surface. Spheres, on the other hand, have the minimum surface area for a given volume. That's why bubbles are spherical. Hyperbolic space, which is three-dimensional, has more volume than ordinary Euclidean space! (Just as a hyperbolic surface has more area.) For more technical details, check out this link.

Tiling the Plane

So, to make a hyperbolic plane surface, we need to arrange for there to be, in some sense, more surface around a point than usual. We do this by tiling the plane. You can tile a flat plane with hexagons -- this is the pattern of the honeycomb in a beehive. But what if you try to tile a plane with heptagons (seven sided instead of six)? You can't tile a flat plane this way, but you can tile a hyperbolic plane with heptagons (or octagons, etc.).

First we need to make some heptagons. Print out this Postscript File to make a page of heptagons, each one inch on a side. Print it out several times, and cut out all the heptagons. Then start constructing your surface by taping the edges of the heptagons together. Three heptagons come together at each vertex, like in the honeycomb, only the angle is more than 120 degrees, so your paper model won't be flat. The hyperbolic surface curves up and around like the seedpod above, until you have a model that looks something like this:

Links

The Geometry Center at the University of Minnesota has a number of exhibits devoted to Hyperbolic Geometry, including a Java Applet for drawing Hyperbolic Triangles and some pictures from a Computer Generated Fly-through of Hyperbolic Space.