When learning how Euclidean geometry works, it is necessary to understand the axioms/postulates Euclid provides. All of the theorems within Euclidean geometry are proven using those “simple” axioms, but the fifth postulate in actuality is not so simple. In fact, great mathematicians have been scratching their heads about this one for over 2000 years wondering how in the world to prove it to be true always. The problem mathematicians have with this postulate is that it states two lines will be parallel to each other out to infinity, and it is impossible to prove this postulate to be true for that length.

What is an axiom and what is a postulate? An axiom is a statement that is accepted as being true without any proof. A postulate is the same thing as an axiom, but it is usually specific to geometry. Both of these terms, axioms and postulates, serve as a basis for proving other important theorems within Euclidean geometry. It is important to note that with these axioms, we are also able to deduce 48 different propositions given to us by Euclid in his books The Elements.

Introduction to Euclid's Five Postulates

The following postulates are related to Euclidean geometry

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and distance.

4. That all right angles are equal to one another.

5. That if a straight line falling on two straight lines makes the interior angles on the same side less that two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles.

Let’s take a closer look at these 5 postulates. The first four postulates seem to be quite self-explanatory. Unfortunately, once we reach the fifth postulate it becomes a bit more complicated. This postulate is one that not only other mathematicians questioned, but Euclid himself also questioned, and he was the one who came up with the postulate! Luckily, we now have a simpler term for this which helps us understand what it means; this postulate is referred to as the parallel postulate.

Let’s think more about this idea. If you have a line, and there are two other straight lines intersecting the first line, the two intersecting lines are both going to create angles with the first line. If those two angles created added together are less than 180 degrees, when extended, the two lines will eventually cross. This also works in reverse though. If the two lines create angles which are greater than 180 degrees when added together, when extended in the opposite direction as before, the lines will also eventually intersect each other. If the line intersect one another, they cannot be parallel. It sounds complicated. Euclid questioned this postulate because he was not entirely sure how to prove it. Some people even believe the fifth postulate should be a part of the propositions rather than the postulates. Use the geogebra applet below to help better understand the idea of the parallel postulate.

Now that we have the axioms, let’s talk about what Euclidean Geometry even is. Euclidean geometry encompasses multiple different things that relate to geometry: points, angles, triangles, squares, lines, circles, etc. This type of geometry is what we refer to as “plane geometry” because we are using visual objects to help us prove propositions in 2 dimensional spaces. Using this type of geometry we are able to prove theorems as complicated as the Pythagorean theorem as well as prove something as simple as why two triangles are congruent when reflected over a line.

Most people probably would not know that Euclidean geometry also encompasses what is referred to as the Mobius strip. This strip is a two dimensional object, one that only has two faces, but when put together in a certain way tries to act like it is in a 3 dimensional space. Follow the link below to play on the surface of a Mobius Strip.

Mobius Strip Applet

You can also try this yourself. Take a strip of paper, twist it halfway and glue the ends together. Once you have done this you can take a pencil and draw it along the piece of paper, it does not matter what side you start to write on and draw the line until you end up where you started. What happened? If you did this correctly, you would have noticed that the pencil line drawn would have gone around the entire strip and ended up back where the line began. We will talk more about the importance of this idea later on.

This type of geometry is what is commonly taught in elementary school all the way through high school. Euclidean geometry is also taught in college, except, the difference being that you focus on re-creating the proofs more than just using the already proven proofs to create graphs and other figures.

Since you now know a little more about what Euclidean geometry is, we can talk about non-Euclidean geometry! This type of geometry relates to objects that are not flat, which means they are not in 2 dimensional spaces, and we cannot study them using plane geometry.

Following Euclidean geometry was the beginning of non-Euclidean geometry. Through a string of great mathematicians, non-Euclidean geometry was created by removing Euclid’s fifth postulate about parallel lines. Hyperbolic geometry was created by Russian mathematician Nikolai Lobachevsky, born in 1792, and Janos Bolyai, from Hungary, explained what we know as hyperbolic geometry. This type of geometry satisfies Euclid’s first four propositions, but not the fifth. Later on, about 1830 published works began appearing about non-Euclidean geometries, and then during 1871 Felix Klein came out with an explanation of the difference between the elliptical space within hyperbolic geometry and double-elliptical space. Within non-Euclidean geometry there are two different types: spherical and hyperbolic geometry. Before we move any further though, we should note the main difference between Euclidean, spherical, and hyperbolic geometry lies within the Euclid’s fifth postulate. As mentioned above, the fifth postulate within Euclidean geometry, also known as the parallel postulate, tells us if we have a line and then a point, there will only be one line that goes through the point that is parallel to the first line. Spherical geometry explains that there are no such lines through the point that are parallel to the first line. But hyperbolic geometry says there are multiple lines which go through a point and are parallel to the first line.

Non-Euclidean geometry allows us to explore objects in 4 dimensional spaces because Euclid's fifth postulate does not exist. For example, a Klein bottle similar to the one pictured below would be considered to be a part of hyperbolic geometry.

Within plane geometry there are such things as similar triangles. Similar triangles are triangles where their corresponding angles are congruent, and the corresponding sides are proportional, to each other. The triangles themselves do not have to be the same size. Within hyperbolic geometry, similar triangles do not exist. This is due to the fact that triangles in hyperbolic geometry do not have the same sum of angles.

As mentioned above, spherical geometry does not contain any parallel lines. Instead, straight lines on a sphere are great circles, and great circles cannot be parallel. When we talk about great circles though we are not talking about circles that are awesome or fantastic, we are talking about the largest circles, circles with the largest diameter, that can be drawn on the surface of a sphere. To help you understand what a great circle is let’s image the earth, specifically the equator. We know that the equator is the line that is drawn on maps going from east to west that splits the earth into the northern hemisphere and the southern hemisphere. If we were to cut the earth right across the equator, it would split the earth directly in half. This also means that all of the longitude lines on the earth – the vertical lines – are also great circles because if we cut the earth down any one of those lines we would also end with two exact halves. It is important to note that there are infinite great cirles on a sphere.