The Three Geometries

M.C. Escher, Three Worlds (1955)

Axioms and the History of Non-Euclidean Geometry

Euclidean Geometry and 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. Euclid used a different version of the parallel postulate, and there are several ways one can write the 5th postulate. They are all equivalent and lead to the same geometry.

• "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." (Euclid's version)
• "The sum of the angles in a triangle is exactly 180 degrees."
• “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.

The Greeks already studied spherical trigonometry. Hipparchus (190 BC-120 BC) was a Greek astronemer. hipparchus was known for his work in trigonometry and he may have known some results about spherical triangles. [2] Menelaus of Alexandria (ca 100 AD) worked on spherical geometry and was the first known to publish a treatise on the subject. Menelaus' work was called Sphaerica (3 volumes) and included material on the properties of spherical triangles. [3] Ptolemy (ca 90 - 168 AD) also included some studies of spherical triangles in his work. [4] Hyperbolic geometry, in comparison, took a lot longer to develop.

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 (plus the assumption that lines are infinite) 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?[5]

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!

Spherical Geometry

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.

How can we formulate the 5th postulate?

• 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.

Hyperbolic Geometry

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.

How do we interpret the 5th postulate?

• 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.

The Classification of Regular Tessellations

Two topics we studied - regular tessellations and two dimensional geometries - are connected in a special way. This is not something that is immediately obvious and you may not have expected such a connection at all, but surprising connections such as this are what make mathematics 'beautiful'.

This section is a summary of our work on regular tessellations. It provides all the desirable features of a classification: It is a complete list of all possibilities, the possibilities are organized in a way that reveals their structure, and it is mathematically complete. Not only this, but once the classification is started, it extends to 'degenerate' tessellations that may have been ignored, and reveals a simple interpretation of duality in the form of a symmetry of the classification itself.

Recall that 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 $\{n,k\}$. 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.

Regular Tessellations
$n\backslash k$ 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

The table can be extended indefinitely, and this symmetric diagram still has the property that the tessellations along the top and left edge will exist in spherical geometry, and the rest will exist in hyperbolic geometry. The three Euclidean regular tessellations are the only ones possible (as we showed in one of the explorations).

For example $\{n,k\}=\{2,8\}$ and $\{n,k\}=\{2,10\}$ will appear on the top right side of the diagram and correspond to tessellations by 2-gons, with 8 (resp 10) 2-gons meeting at a vertex. These tessellations can only appear on the sphere, so these are spherical tessellations. Similarly $\{n,k\}=\{8,2\}$, $\{n,k\}=\{9,2\}$ and $\{n,k\}=\{10,2\}$ will appear in the left hand column and correspond to tessellation by 8-gons (resp 9-gons and 10-gons) and have 2 polygons meet at a vertex. These tessellations can only be realized on the sphere and will there fore be spherical geometries.

The Shape of the Universe

This section is unfinished.

In two dimensions there are 3 geometries: Euclidean, spherical, and hyperbolic. These are the only geometries possible for 2-dimensional objects, although a proof of this is beyond the scope of this book.

What about in three dimensions, which corresponds to the space we actually live in? It has been shown that in three dimensions there are eight possible geometries. There is a 3-dimensional version of Euclidean geometry, a 3-dimensional version of spherical geometry and a 3-dimensional version of Hyperbolic geometry. There is also a geometry which is a combination of spherical and Euclidean, and a geometry which is a combination of hyperbolic and Euclidean. The three other geometries are a bit more exotic and are harder to describe.

Since all these geometries look the same at small scales, we cannot tell the shape of the space that we live in without studying difficult questions about the universe itself. In particular, there are still two very fundamental questions about the universe that remain unknown:

• Is the universe finite or infinite?
• Do we know which of the geometries describes the shape of the universe?

When we ask if the universe is finite, we are really asking if it closes up like a sphere, or extends infinitely like the plane or hyperbolic space. Another way to ask this question is to think about a rocket traveling through space in a straight line: If the universe is finite, it will eventually wrap around and return. On a 2 dimensional surface, if we travel in a straight path and never return we would be on something like an infinite plane. If we did manage to return, even though we travel in a straight line, then we would have been on something like a sphere. Some scientists believe our universe is more like the 3-dimensional version of the sphere. Our rocket would eventually return to Earth (after an impossibly long time).

What about the geometry of the universe? Euclidean, spherical and hyperbolic geometry are different on small scales. The sum of the angles in a triangle is different, for example. However, for really small triangles in spherical and hyperbolic geometry, the triangles begin to look a lot like their Euclidean cousins. One would have to be able to do very precise measurement to measure the angle defects. We run into a similar problem when trying to measure the geometry of the universe. So far, measurements are not accurate enough or large enough to decide the issue. The universe could be what we call flat (which corresponds to Euclidean) or it could have some small amount of curvature (which could make it have some other geometry).

How do we picture possible 3-dimensional spaces? Think about some computer games where our screen in a square or a rectangle, but if we leave the screen on the right hand side, we re-appear on the left. Similarly if we were to leave the screen at the top, we would show up again at the bottom. This really means that the left is connected to the right and the top to the bottom. I little more thought would show that we were actually playing the game on a torus (a doughnut like shape).

The 3-torus is obtained from the cube by gluing the left side to the right side, top to bottom, and front to back
A dodecahedral space is created by gluing the sides of the dodecahedron in pairs

We can do similar things in 3-space. A so called 3-torus would be a 3-dimensional space made up of a cube but with the understanding that top and bottom, left and right and front and back are connected. Imagine a computer game with a spaceship. If the spaceship exits on the left it will re-appear on the right! This space is Euclidean in nature, and is finite.

The dodecahedron can similarly be used to create 3-dimensional spaces. They will be finite, but in some cases they will have a hyperbolic geometry.

See Geometry Center's page on the Shape of Space for more detail.

Related Sites

Geometry Games Jeff Weeks’ Topology and Geometry Software. This site includes the torus games, Kali and Kaleidotile.

Notes

1. A History of Greek Mathematics by Thomas Little Heath, 2nd Ed. 1981, retrieved from google books