Non-Euclidian Geometry

Podcast Transcript

If anyone has taken some basic mathematics, you are probably familiar with Euclidian Geometry. Euclidean geometry is what most people just call geometry.

It is the study of shapes like triangles and circles in a simple plane. This type of geometry was developed over 2000 years ago, and it is based on certain set axioms.

However, later mathematicians challenged one of those axioms, and it completely changed how we thought of geometry.

Learn more about non-Euclidian geometry and what it means on this episode of Everything Everywhere Daily.

Many of you might find the phrase non-Euclidian geometry to be daunting. However, it really isn’t.

Euclidian geometry is just the geometry that you were taught in school. For example, the fact that all of the angles in a triangle add up to 180 degrees is part of Euclidian geometry.

The word Euclidian comes from the ancient Greek mathematician Euclid and the system that he developed.

So, before we can get into what non-Euclidian geometry is, we first need to understand exactly what Euclidian geometry is and who this guy Euclid was.

Euclid, and that is the only name we know him by, was born about 2,300 years ago. We know almost nothing about his early life, but we do know that he did most of his work in the City of Alexandra in Egypt, which was very much a culturally Greek city at the time.

Euclid might have attended the Platonic Academy, which Plato established, and he may have taught at the Musaeum, which was part of the Library of Alexandria.

Euclid was a prodigious writer who wrote on many subjects relating to science and mathematics, although many of his works have been lost to history.

What Euclid is known best for, however, was his work on geometry. Euclid’s book titled Elements is his best-known work and is considered the foundational work in the subject of geometry. Elements was so important for the development of geometry that Euclid is known as the father of geometry.

Here, I should note that mathematics 2300 years ago was nothing like it is today. There were no equations. There were no pencils or paper. They didn’t even have base-10 numbers or mathematical symbols like we have today.

Mathematics, which was largely geometry back then, was done using two primary tools: a compass, with which you could draw circles and arcs, and a straight edge, with which you could draw straight lines.

What Euclid did in Elements that was so special is that all of the geometric theorems he developed were logically based on a few simple axioms.

An axiom is a simple statement or proposition that is accepted as true without proof. An axiom is usually considered to be self-evident and is the starting point for all the logical conclusions that can then be drawn from it, which are called theorems.

Euclid laid out five axioms, which were the foundation for geometry. Even if you aren’t well-versed in mathematics, you will still find most of these axioms to be extremely easy to comprehend.

They are as follows:

Axiom 1: A straight line segment can be drawn, joining any two points.

Axiom 2: Any straight line segment can be extended indefinitely in a straight line.

Axiom 3: Given any straight line segment, a circle can be drawn, having the segment as the radius and one endpoint as the center.

Axiom 4: All right angles are equal to one another.

Having heard these, your reaction might just be…..duh. But that is sort of the point of axioms. They are simple, self-evident statements that are the logical foundation of everything else.

Before I mentioned that, there were five axioms, and I only read four. That is because the fifth axiom is really what this entire episode is all about. You will notice immediately that it is different from all the others.

Axiom 5: If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Axiom 5 is known as the Parallel Postulate. You will notice that it is much more complicated than the other four axioms and not nearly as self-evident.

The thing that bothered many mathematicians about that fifth axiom is many of them thought that it shouldn’t have been an axiom. They felt that it was something that could have been derived from the previous four axioms.

For centuries, there were attempts to derive the fifth axiom as a theorem from the first four axioms, and for centuries, mathematicians failed.

It seemed like old Euclid actually knew what he was doing and put in the fifth axiom for a good reason.

There is one assumption that Euclid never explicitly stated, but it is actually really important, and it turns out to have something to do with that fifth axiom. That everything Euclid was talking about was taking place on a flat plane.

That assumption was never really challenged, and that is why the term “Euclidian Geometry” didn’t exist for 2000 years. It was just called…..geometry.

That began to change in the 18th century

This restatement of the fifth axiom was developed by the Scottish mathematician John Playfair. His version, known as Playfairs Axiom, says,

In a plane, given a line and a point not on it, at most, one line parallel to the given line can be drawn through the point.

As mathematicians tried to prove the fifth axiom, they discovered a bunch of axioms that could be used in place of the fifth axiom that didn’t at first seem to have anything to do with parallel lines but served the same function.

One such axiom states that the sum of the angles in every triangle is 180°

The sum of the angles is the same for every triangle.

After centuries of trying to prove the fifth axiom, some mathematicians just began to wonder what would happen if the fifth axiom wasn’t true. The axioms were the logical basis for geometry.

What would happen if you used the same logical basis in the first four axioms but didn’t assume the fifth axiom? By not assuming the fifth axiom, it meant that if you had a line and a point, what if there were either zero or an infinite number of parallel lines that went through the point?

The first person to seem to have taken this step was the man many people regard as the greatest mathematician in history, Carl Friedrich Gauss.

Gauss spent decades pondering this question and, strangely enough, never published any of his conclusions on the subject. All we have are his references to his ideas in letters. However, he was also the person who coined the term non-Euclidian geometry.

In the early 19th century, two mathematicians independently began working on this problem. The Russian mathematician Nikolai Ivanovich Lobachevsky and the Hungarian mathematician János Bolyai.

What they found is that you could create a logically consistent geometry that held to the first four Euclidian axioms but violated the fifth.

It turned out that there wasn’t simply geometry as had been thought for 2000 years…. there were actually geometries.

The way things are now roughly categorized is that there are two general types of non-Euclidian geometry. Elliptic geometry and Hyperbolic geometry.

Again, fancy-sounding words, but not that difficult to understand.

Elliptic geometry is when space has a positive curvature. The easiest example of this to comprehend is spherical geometry, which is the surface of a sphere or, in a more real-world example, the surface of the Earth.

In spherical geometry, you can still have the four original axioms but not the fifth.

Example: for any two points on the sphere, there is a great circle that will go through the points that would divide the sphere into two equal hemispheres.

Assume some point that is not on that great circle. Every great circle that can be created through that point will intersect with the given great circle. In other words, there are no parallel lines.

If you make a triangle out of three points on a sphere, the angles of the triangle will always be greater than 180 degrees, which sort of shows how the angles of a triangle and the parallel postulate are tied together.

Now consider another version of the parallel postulate, but this time, instead of there being one parallel line as in Euclidian geometry or zero parallel lines as in elliptic geometry, there are actually an infinite number of parallel lines.

It turns out this is a perfectly valid geometry as well. It is known as hyperbolic geometry.

Instead of a sphere that has a positive curvature, in hyperbolic geometry, space has a negative curvature, sort of like being in the middle of a horse saddle.

It turns out that if you take a line in hyperbolic space or on a Saddle surface and then run another line through a point not on that line, there are no lines that will ever intersect with the original line. In other words, there are an infinite number of lines that meet the definition of a parallel line.

Also, in hyperbolic geometry, the angles in a triangle add up to less than 180 degrees.

So, it turns out that Euclidian geometry is just geometry in a space with zero curvature, a.k .a. a flat plane. If the space curves, you can get totally different geometries that violate the fifth axiom but still preserve the first four.

You might be thinking that this is all just a bunch of theoretical nonsense. Parallel lines are parallel are parallel lines. The angles in a triangle are 180 degrees, and the rest is just a fantasy.

However, that most certainly isn’t true.

In the case of spherical geometry, the applications are rather obvious.

When an airplane flies from one place to another, the route it will usually fly is a great circle route across the surface of a sphere. If you have ever been on a long flight, you can usually track the progress of the flight on the entertainment system.

When doing surveying for projects that span long distances, the curvature of the Earth needs to be taken into consideration

It turns out there are real-world uses for hyperbolic geometry as well. Perhaps the best-known use of hyperbolic geometry is the theory of Special Relativity. According to Special Relativity, space and time are bound up into something known as spacetime, which can have a curvature.

If spacetime has positive curvature, aka elliptic, it is known as a de Sitter space, and if it is negative, aka hyperbolic, it is known as a Minkowski space.

So, these non-Euclidian geometries have very real applications indeed.

I realize some of the things I’ve talked about in this episode might have gone over the heads of some people. However, I assure you the concepts behind them are rather easy to understand.

For centuries, mathematicians tried to disprove that Euclid’s fifth axiom was, in fact, an axiom, and they failed miserably.

It wasn’t until later that mathematicians took a different approach and tried to envision what things might look like if the fifth axiom were false….and fr that they were able to develop entirely new geometries.

It turned out that Euclid’s decision to add that fifth axiom was, in fact, a stroke of genius. A decision taken over 2000 years ago that has led to new fields of mathematics today.

The Executive Producer of Everything Everywhere Daily is Charles Daniel.

The associate producers are Peter Bennett and Cameron Kieffer.

Today’s review comes from listener JSD078-14 over on Apple Podcasts in the United States. They write…

Finally a Completionist Club Member

Finally, I slogged down all the episodes! That was an epic ride. I Hope Gary has episodes covering the Philippines soon.

By the way, where do I get my badge and my fez? Do we all have a secret handshake before each meeting?

Thanks, JSD! You can get your membership card at the front desk. The concierge will also show you the facilities.

…and yes, I do have some episodes planned about the Philippines. At this point it is just a matter of what order I want to do the episodes.

Remember, if you leave a review or send me a boostagram, you too can have it read on the show.