# Hyperbolic Geometry

## Contents

## 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:

His Circle Limit III (see top of the article) is more subtle. Looking at the white spines of the fish, it appears to be a tessellation of hyperbolic space by triangles and squares, with three triangles and three squares coming together at each vertex. This would mean that the corner angles of both polygons are 60° each, but this gives the triangle an angle sum of 180°, which cannot happen in hyperbolic space! And yet, Circle Limit III clearly has the flavor of a hyperbolic tessellation. H.S.M Coxeter, a mathematician and friend of Escher’s analyzed the print, and discovered that the white circular arcs along the spines meet the boundary of the disk at 80° each, rather than the 90° required to be geodesic. Since the white lines are not geodesics, neither the “triangle” or the “square” is really a polygon at all! Even Escher does not seem to have realized this, although most probably he would not have cared, as he was very satisfied with the print and the suggestion of infinity it presents. Ideal tessellations are tessellations by ideal polygons. Because ideal polygons have 0° angles, infinitely many come together at each vertex (but keep in mind that ideal polygons aren’t really polygons and don’t actually have vertices).

## Regular Tessellations in Three Geometries

A regular tessellation is described completely by a pair of numbers - the number of sides on each tile, and the number of tiles meeting at a vertex. The “Schlafli symbol” for a regular tessellation is just this pair of numbers, written {n,k}. For example, the regular tessellation of the plane by hexagons is written {6,3}, since three hexagons meet at each vertex. There is a regular tessellation for every Schlafli symbol {n,k} (with n and k at least 2). Some are spherical, some are Euclidean, and some are hyperbolic. To classify which {n,k} go with which geometry, we consider angle sums.

TO BE COMPLETED

## Another look at the Role of Axioms in Geometry

**The five axioms for Euclidean geometry are:**

- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

The fifth postulate is called the parallel postulate, which leads to the same geometry as either one of the following statements:

"If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough." or "The sum of the angles in a triangle is exactly 180 degrees."

The axioms are basic statements about lines, line segments, circles, angles and parallel lines. We need these statements to determine the nature of our geometry.

In our two other geometries, spherical geometry and hyperbolic geometry, we keep the first four axioms and the fifth axiom is the one that changes. It should be noted that even though we keep our statements of the first four axioms, their interpretation might change!

The five axioms for spherical geometry are:

1. Any two points can be joined by a straight line. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. There are NO parallel lines.

How do we interpret the first four axioms on the sphere? Lines: What would a “line” be on the sphere? In Euclidean geometry a line segment measures the shortest distance between two points. This is the characteristic we want to carry over to spherical geometry. The shortest distance between two points on a sphere always lies on a great circle. Longitude lines and the equator on a globe are examples of great circles. Note that we can always draw a great circle, which we will from now on call a geodesic, through any two points. We have to be careful here, because unlike in Euclidean geometry this geodesic (“line”) may not be unique. Take for instance the north and South Pole on the globe. There are infinitely many great circles through these two poles. In general, any two points that lie on opposite sides of the sphere, so called antipodal points, can be joined by infinitely many geodesics.

Line segments: We can extend any line segment, but at some point the line segment will then connect up with itself. A line of infinite length would go around the sphere an infinite amount of times.

Circles: As we have stated the circle axiom it is true on the sphere. Note that it does not make sense to say that given any center C and any radius R we can draw a circle of radius R centered at C. If we take a radius less than half the circumference of the sphere, then we can draw the circle. If the radius is exactly half the circumference of the sphere, then the circle degenerates into a point. If the radius were greater than half the circumference of the sphere, then we would repeat one of the circles described before. Note that great circles are both geodesics (“lines”) and circles.

Angles: Right angles are congruent. Think about the intersection of the equator with any longitude. These two geodesics will meet at a right angle.

No parallel lines: Any two geodesics will intersect in exactly two points. Note that the two intersection points will always be antipodal points.

Sum of the angles in a triangle: On the sphere the sum of the angles in a triangle is always strictly greater than 180 degrees.

These basic facts really turn the properties of this geometry on its head. We will have to rethink all of our theorems and facts! Here are some examples of the difference between Euclidean and spherical geometry.

In Euclidean geometry an equilateral triangle must be a 60-60-60 triangle. In spherical geometry you can create equilateral triangles with many different angle measures. Take for instance two longitudes that meet at 90 and intersect them with the equator. This gives ride to a 90-90-90 equilateral triangle! If you shrink this triangle just a little bit, you can make an 80-80-80 triangle. If you expand it a bit, you can make a 100-100-100 triangle. As a matter of fact you can make a X-X-X triangle as long as 60 < X < 300.

Note that not having any parallel lines means that parallelograms do not exist. Recall that a parallelogram is a 4-gon that has the property that opposite sides are parallel. In Euclidean geometry this definition is equivalent to the definition that states that a parallelogram is a 4-gon where opposite angles are equal. In spherical geometry these two definitions are not equivalent. There are quadrilaterals of the second type on the sphere.

The five axioms for hyperbolic geometry are:

1. Any two points can be joined by a straight line. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Through a point not on a given straight line, infinitely many lines can be drawn that never meet the given line.

How do we interpret the first four axioms in hyperbolic space? We first have to agree on a model of hyperbolic space. We will choose the Poincare Disk Model. We will think of all of hyperbolic space as living inside a disk. Putting an entire infinite world inside a disk will lead to some distortion as you might expect. We think of the center of the disk as being close to Euclidean geometry, but the closer we get to the edge of the disk, the more distorted the picture will become. We have to think of the boundary of the disk as being infinitely far away from the center of the disk. This means that anything we see close to the edge of hyperbolic space will appear much, much smaller than it actually is.

Lines: In hyperbolic geometry a geodesic line segment measures the shortest distance between two points. There are two types of geodesics in the Poincare Disk Model (PDM). Geodesics will be Euclidean line segments passing through the center of the disk, or semi-circles, which meet the boundary of the disk in right angles.

Line segments: Any finite piece of a geodesic.

Circles: Given any center C and any radius R we can draw a circle of radius R centered at C. Hyperbolic circles look just like Euclidean circles, but the center is not located where a Euclidean center would be. The center of the circle will be slightly closed to the boundary of the PDM than it’s Euclidean counterpart.

Angles: Right angles are congruent.

Infinitely many parallel lines: Given a line and a point not on the line, we can always draw infinitely many parallel lines through the point. Remember that two lines are parallel if they never meet. Because the geodesics in hyperbolic space include semi-circles, we have a bit more freedom in our choice of geodesic. The easiest way to see this is to choose a geodesic that is a fairly small semi-circle near the boundary of the PDM. Now think of all the geodesics passing through the center of the PDM. You can draw infinitely many of these straight looking geodesics that never meet the semi-circle, so all of those are parallel to the small semi-circle.

Sum of the angles in a triangle: On the sphere the sum of the angles in a triangle is always strictly less than 180 degrees.

These basic facts also turn the properties of this geometry on its head. We will have to rethink all of our theorems and facts for hyperbolic geometry too. Here are some examples of the difference between Euclidean and spherical geometry.

In Euclidean geometry an equilateral triangle must be a 60-60-60 triangle. In hyperbolic geometry you can create equilateral triangles with many different angle measures. Take for instance three ideal points on the boundary of the PDM. If we connect these three ideal points by geodesics we create a 0-0-0 equilateral triangle. Moving the vertices into the interior of hyperbolic space will result in equiangular triangles with small angle measures. We will be able to create X-X-X triangles with 0 ≤ X < 60.

Having infinitely many parallel lines means that parallelograms will look different than you expect!

Note that we cannot have squares or rectangles in hyperbolic space, because the sum of the angles of a quadrilateral has to be strictly less than 360.