# The Three Geometries

M.C. Escher, Three Worlds (1955)

## The Classification of Regular Tessellations

The simplest tessellations are the regular tessellations. They are simple, because each involves only a single shape of tile, and that tile has all sides the same length and all angles the same measure. We have studied regular tessellations in three different geometries: Euclidean, spherical, and hyperbolic. In each geometry, the key step to forming regular tessellations was to choose the corner angles of the tile so that multiple tiles could fit together around a vertex. That is, we needed the corner angle to evenly divide 360°.

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 [itex]\{n,k\}[/itex]. 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 [itex]\{n,k\}[/itex] (with [itex]n[/itex] and [itex]k[/itex] at least 2). Some are spherical, some are Euclidean, and some are hyperbolic. To classify which [itex]\{n,k\}[/itex] go with which geometry, we consider angle sums.

Regular Tessellations
[itex]n \backslash k[/itex] 2 3 4 5 6 7
2 S S S S S S
3 S S
Tet
S
Oct
S
Ico
E H
4 S S
Cube
E H H H
5 S S
Dod
H H H H
6 S E H H H H
7 S H H H H H

## Axioms and the History of Non-Euclidean Geometry

In about 300 BCE, Euclid penned the Elements, the basic treatise on geometry for almost two thousand years. Euclid starts of the Elements by giving some 23 definitions. After giving the basic definitions he gives us five “postulates”. The postulates (or axioms) are the assumptions used to define what we now call Euclidean geometry. [1]

The five axioms for Euclidean geometry are:

1. Any two points can be joined by a straight line. (This line is unique given that the points are distinct)
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, 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." or “given a line L, and a point P not on that line, there is exactly one line through P which is parallel to L”.

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.

The 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.

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.

## Notes

This section is unfinished.