# Hyperbolic Geometry

## Preliminaries

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.

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.

## Introduction

In about 300BC, Euclid penned the Elements, the basic treatise on geometry for almost two thousand years. He clearly states his assumptions in five “postulates”. Euclid’s fifth postulate concerns parallel lines, and in a more modern form says that “given a line L, and a point P not on that line, there is exactly one line through P which is parallel to L”. This fifth postulate, the “parallel postulate” seemed more complicated and less obvious than the other four, so for many hundreds of years mathematicians attempted to prove it using only the first four postulates as assumptions. We saw that the parallel postulate is false for spherical geometry (since there are no parallel geodesics), but this is not helpful since some of the first four are false, too. For example there are many geodesics through a pair of antipodal points. In fact, the first four postulates imply that given a line and a point not on that line, there is a parallel line as required. The subtle question is: can there be more than one? In 1733, the Jesuit priest Giovanni Saccheri began by assuming the fifth postulate was false, and attempted (at great length) to derive a statement contradicting the other four. In doing so, he nearly produced the theory of hyperbolic geometry. However, his goal was not to discover new kinds of geometry, but to rule them out, so he concluded his treatise with a rant about the absurdity of everything he had just written. The great German mathematician Carl Freidrich Gauss apparently believed that a geometry did exist which satisfied Euclid’s first four postulates but not the fifth. However, Gauss never published or discussed this work because he felt his reputation would suffer if he admitted he believed in non-Euclidean geometry. In the early 1800’s, the idea was preposterous. Generally, Nikolai Ivanovich Lobachevsky is credited with the discovery of the non-Euclidean geometry now known as hyperbolic space. He presented his work in the 1820’s, but even it was not formally published until the 20th century, when Felix Klein and Henri Poincaré put the subject on firm footing.

## Models of Hyperbolic Space

The non-Euclidean geometry which satisfies Euclid’s first four postulates but not the parallel postulate is called hyperbolic geometry. Like spherical geometry, which takes place on a sphere, hyperbolic geometry takes place on a curved two dimensional surface called hyperbolic space. On a sphere, a small neighborhood of a point looks like a cap. In hyperbolic space, every point looks like a saddle. Unfortunately, while you can piece caps together to make a sphere, piecing saddles together quickly runs out of space.

These two images represent a bit of sphere (on the left) and a bit of hyperbolic space (on the right).

To get the flavor of hyperbolic space, make some hyperbolic paper. You could make Euclidean space by gluing equilateral triangles together so that six touch at each vertex (this is just the usual tessellation by triangles). You can make an icosahedron by gluing equilateral triangles together so that five touch at each vertex, which corresponds to a tessellation of the sphere and makes a pretty good model for an actual curved sphere. Gluing four, three, or two triangles also makes a sphere, of sorts. To make a model of hyperbolic space, cut out equilateral triangles and paste together so that seven come together at one point. This will be a floppy, saddle like object. Now continue gluing on triangles so that each vertex has seven of them. Probably you will run out of patience before you run out of room, but eventually you’ll need to put too many triangles in too small a space to continue. Because globes are unwieldy, navigators use flat maps of the spherical earth. Maps of the Earth are necessarily distorted, for example Greenland appears extremely large on the standard Mercator map of the Earth. Because models of hyperbolic space are unwieldy (not to mention infinite), we will do all of our work with a map of hyperbolic space called the Poincaré disk. The Poincaré disk is the inside of a circle (although the circle is not included) and is badly distorted near its edge. Objects near the edge of the Poincaré disk are larger than they appear.

Hyperbolic man takes a walk

The picture shows a stick man as he walks towards the edge of the disk. He appears to shrink, as does the distance he moves with each step. But this disk is a distorted map, and in the actual hyperbolic space his steps are all the same length and he stays the same size. The man will never reach the edge of the disk, because it is infinitely far away. The edge is drawn dashed because it is not actually part of hyperbolic space. The geodesics in hyperbolic space play the role of straight lines. Geodesics appear straight to an inhabitant of hyperbolic space, and they are the shortest paths between points. In the Poincaré disk model, geodesics appear curved. They are arcs of circles. Specifically:

Geodesics are arcs of circles which meet the edge of the disk at 90°. Geodesics which pass through the center of the disk appear straight.

## Polygons and Defect

Definition A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the sides of the polygon.

A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in Euclidean geometry. Here are some triangles in hyperbolic space:

From these pictures, you can see that: The sum of the angles in any hyperbolic triangle is less than 180°.

Definition The defect of a hyperbolic triangle is 180° – (angle sum of the triangle).

By cutting other polygons into triangles, we see that a hyperbolic polygon has angle sum less than that of the corresponding Euclidean polygon.

Define the defect for a hyperbolic polygon with n sides to be (n – 2)×180° – (angle sum of the polygon). Putting this together with the defect in spherical geometry: The defect of a polygon is the difference between it’s angle sum and the angle sum for a Euclidean polygon with the same number of sides. This statement works in spherical and hyperbolic geometry, for polygons with any number of sides. It even works for biangles, because a biangle in Euclidean geometry must have two 0° angles. The area of a hyperbolic polygon is still proportional to its defect:

Area of a hyperbolic polygon = π/180° × defect

The proof of this equality is usually referred to as the Gauss-Bonnet theorem. In spherical geometry, we had a formula for the fraction of the sphere covered. If the sphere has radius 1, the total area is 4π, and so the area of a polygon is 4π ×defect720° which (after a little simplification) is exactly the same formula as in hyperbolic space! In an extreme case, we can draw three geodesics that touch at the boundary of the disk. The boundary isn’t part of hyperbolic space, so these lines never actually meet, they just get closer and closer to meeting. Although this isn’t a triangle (because it has no vertices), we call such a figure an ideal triangle. You can create other ideal polygons in a similar manner:

Ideal triangles and an ideal hexagon

The area formula implies that any ideal triangle has area π, because the angle sum is zero and its defect is 180°. Both ideal triangles shown above have the same area even though the distortion of the Poincaré disk makes one look much smaller than the other. The ideal hexagon shown has angle sum zero, so it’s defect is 720° and its area is 4π.

## Hyperbolic Tessellations

A tessellation of hyperbolic space is a covering of the sphere by tiles, with no overlapping tiles and no gaps. Escher’s Circle Limit series are examples of hyperbolic tessellations. Like his other tessellations, Escher began with a geometric tessellation by polygons and worked from there. In hyperbolic space, there are infinitely many regular tessellations, which is in sharp contrast to Euclidean space, which has only three, and to spherical geometry, where there are only five non-degenerate possibilities. Hyperbolic space is easy to tessellate because the corner angles of polygons want to be small, and small angles fit nicely around a vertex. Here are a few regular tessellations of hyperbolic space:

In the first, the triangles have 360°/7 ≈51.4° angles and angle sums of approximately 3×51.4°=154.3°. This gives a defect of approximately 180–154.3° = 25.7° and an area of approximately 0.45. In the second tessellation, the pentagons have five 90° angles each, so their angle sums are 450° giving a defect of 540°-450° = 90° and an area of approximately 1.57. The third tessellation is left as an exercise. Escher’s Circle Limit I is based on a tessellation by quadrilaterals with angles 90°-60°-90°-60° (which is not regular). You can see the underlying grid by focusing on the spines of the fish: File:Circel-LimitI.svg