Hyperbolic Geometry

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Introduction

We have seen two different geometries so far: Euclidean and spherical geometry. The idea is that geometry is meant to describe the world around us, and the geometry then depends on some fundamental properties of the world we are describing. Objects that live in a flat world are described by Euclidean (or flat) geometry, while objects that live on a spherical world will need to be described by spherical geometry. The obvious question at this point should be if there are any other geometries out there. In two dimensions there is a third geometry. This geometry is called hyperbolic geometry. We just reminded ourselves that Euclidean geometry describes objects in a flat world or a plane, and spherical geometry describes objects on the sphere. So, what world does hyperbolic geometry describe? It is a fact that we can think of this hyperbolic world in several different ways, but if we use Escher’s work as a reference we should think of hyperbolic space as the distorted interior of a disk.

Circle-Limit-III.jpg

Think of all the fish as living inside this hyperbolic world. They all live in the interior of the disk, and it is a distorted picture because, believe it or not, all these fish are the same size.

Have you ever noticed how a stick looks bent if you partially submerge it in water? The same happens to a straw in a glass of water. Something like that happened here too. Notice the white curves that look like they are bent? That is the effect of hyperbolic space. These are actually straight. These curves are the geodesics of hyperbolic space. They measure shortest distance, and segments of them can be used to make polygons. Do you see the 4 sided polygon in the center? Do you notice other 4 sided polygons closed to the edge of our hyperbolic space? Notice that all the sides of these 4-gons are exactly the length of one fish. This means that all the 4-gons have exactly the same shape and size! This gives you an idea I hope of what I meant when I said that hyperbolic space could be thought of as the distorted interior of a disk.