Cabinet: What exactly is a hyperbolic plane?

David Henderson: There are many ways of describing the hyperbolic plane. In formal geometric terms, it is “a simply connected Riemannian manifold with negative Gaussian curvature.“ In higher-level mathematics courses it is often defined as the geometry that is described by the upper half-plane model. One way of understanding it is that it’s the geometric opposite of the sphere. On a sphere, the surface curves in on itself and is closed. A hyperbolic plane is a surface in which the space curves away from itself at every point. Like a Euclidean plane it is open and infinite, but it has a more complex and counterintuitive geometry.

The hyperbolic plane is sometimes described as a surface in which the space expands. Can you explain what that means?

DH: Actually that is true for many spaces, but it’s true for hyperbolic space in a particular way. Consider how circles on a surface behave. If you think of a series of circles around a point on a regular Euclidean plane, as you draw larger circles, the length of the circumference increases linearly. Now on a hyperbolic plane, the circumference of the circles doesn’t just increase linearly, but exponentially. The perimeter and also the area of the circles gets bigger much faster. On a sphere, the circles get larger at first, but then as you go further they actually begin to get smaller. On a sphere the circumference of a circle is always less than 2r, on a hyperbolic plane it is more. A similar thing happens with the area, which also increases much faster in hyperbolic space.

So as you move away from any point on a hyperbolic surface you get exponentially more space, so to speak.

DH: Yes, and you can get an idea of that with the hyperbolic soccer ball model that was discovered by my son, Keith Henderson. A normal soccer ball has spherical geometry and is made up of hexagons and pentagons. Each pentagon, which has five sides, is surrounded by hexagons, which have six. If you just stuck together a lot of hexagons you’d get a plane, but in the soccer ball the presence of the pentagons pulls the hexagons away from flatness and the surface closes up on itself to form a sphere. To make a hyperbolic surface, you replace the pentagons with heptagons, which have seven sides. Now you have to put in seven hexagons for every five you had before, so instead of closing up, the surface opens out and curves away from itself.