Everything Everywhere Daily: History, Science, Geography & More - Non-Euclidean Geometry

Episode Date: December 21, 2023

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 in this episode of Everything Everywhere Daily. Sponsors BetterHelp Visit BetterHelp.com/everywhere today to get 10% off your first month ButcherBox Sign up today at butcherbox.com/daily and use code daily to choose your free steak for a year and get $20 off."  Subscribe to the podcast!  https://link.chtbl.com/EverythingEverywhere?sid=ShowNotes -------------------------------- Executive Producer: Charles Daniel Associate Producers: Peter Bennett & Cameron Kieffer   Become a supporter on Patreon: https://www.patreon.com/everythingeverywhere Update your podcast app at newpodcastapps.com Discord Server: https://discord.gg/UkRUJFh Instagram: https://www.instagram.com/everythingeverywhere/ Facebook Group: https://www.facebook.com/groups/everythingeverywheredaily Twitter: https://twitter.com/everywheretrip Website: https://everything-everywhere.com/ Learn more about your ad choices. Visit megaphone.fm/adchoices

If anyone has taken some basic mathematics, you're probably familiar with Euclidean geometry. Euclidean geometry is just what most people call geometry. It's a study of shapes like triangles and circles in a simple plane. This type of geometry was developed over 2,000 years ago, and it's based on a certain set of axioms. However, later mathematicians challenged one of those axioms, and it completely changed how we thought of geometry. Learn more about non-Euclidean geometry and, what it means on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong?

ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. Many of you might find the phrase, non-Euclid. geometry to be daunting. However, it really isn't. Euclidean 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 a part

of Euclidean geometry. The word Euclidean comes from the ancient Greek mathematician, Euclid, and the system that he developed. So before we can get into what non-Euclidean geometry is, we first need to understand exactly what Euclidean geometry is and who this guy Euclid was. Euclid, and that's 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 Alexandria and Egypt, which was very much a culturally Greek city at that time. Euclid may have attended the Platonic Academy, which Plato established, and he may have taught at the museum, which was,

was part of the Library of Alexandria, which I covered in a previous episode. 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 best known 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. And here I should note that mathematics, 2,300 years ago, was nothing like it is today. There were no equations.

There were no pencils or paper. They didn't even have a base 10 numbering system or mathematical symbols like we have today. Mathematics, which was largely geometry back then, was done using two primary tools, a compass from which you could draw circles and arcs, and a straight edge with which you could draw straight lines. What Euclid did in the elements that was so special is that all of the geometric theorems he developed were logically based on just 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 be drawn from it, which are called theorems. Euclid laid out five axioms, which were the foundation for geometry. Even if you aren't very well versed in mathematics, you'll still find most of these axioms to be extremely easy to comprehend.

They are as follows. Axiom one, a straight line segment can be drawn joining any two points. Axiom two, any straight line segment can be extended indefinitely in a straight line. Axiom three. given any straight line segment, a circle can be drawn having the segment as the radius of the circle and one end point as the center. Axiom four, all right angles are equal to one another. Now having heard those, your reaction might be, duh, but that is sort of the point of axioms. They're simple self-evidence statements that are the logical foundation of everything else.

but before I mentioned that there were five axioms and I only read four and that's because the fifth axiom is really what this entire episode is about and you'll notice immediately that it is different from all the others axiom five 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 angle sum to less than two right angles. Axiom five is known as the parallel postulate, and you'll notice that it's much more complicated than the other four axioms and not nearly as self-evident.

The thing that bothered many mathematicians about the fifth axiom is that many of them thought that it shouldn't have been an axiom at all. They felt that it was a theorem. that could be 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 had put the fifth axiom in for a good reason. There's one assumption that Euclid never explicitly stated, however,

but it's actually really important, and it turns out to have something to do with the fifth axiom. that everything Euclid was talking about was taking place on a flat plane. That assumption was never really challenged, and it's why the term Euclidean geometry didn't even exist for 2,000 years. That's because it was just called geometry. That, however, began to change in the 18th century. A restatement of the fifth axiom was developed by the Scottish mathematician John Playfair. His version, known as Playfair's axiom, says,

In a plane, given a line and a point knot 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 have been 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. For example, one axiom states that the sum of the angles in every triangle is 180 degrees. or 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. So what would happen if you use the same logical basis in the first four axioms but didn't assume the fifth was true? By not assuming the fifth axiom, it meant that if you had a line and then another point, what if there were either zero or an infinite number of parallel lines that went through that point? The first person to seem to have taken this step was the man that 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-Euclidean geometry.

In the early 19th century, two mathematicians independently began working on the problem, the Russian mathematician Nikolai Loewiczewski and the Hungarian mathematician Janos Bollai. What they found is that you could create a logically consistent geometry that held to the first four Euclidean axioms, but violated the fifth. So it turned out there wasn't simply geometry, as had been thought for 2,000 years, there were actually geometries. The way things are now roughly categorized is that there are two general types of non-Euclidean geometry, elliptic geometry and hyperbolic geometry. Again, these are fancy-sounding words, but not that difficult to understand. Ecliptic geometry is when space has a positive curvature.

The easiest example of this to comprehend is, 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. For example, for any two points on a sphere, there is a great circle that will go through the two points that would divide the sphere into two equal hemispheres.

Assume that there is some point that is not on that great circle. Every great circle that can be created through that point will intersect with the first 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 Euclidean geometry, or zero parallel lines as in elliptic geometry, there are actually an infinite number of parallel lines. It turns out that this is a perfectly valid geometry

as well, and it's 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 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 Euclidean geometry is just geometry in a space with zero curvature, a.k.a. a. a. A.k.a. A.k.a. If the space curves, you can get totally

different geometries that violate the fifth axiom, but still preserve the first four. Now, at this point, you might be thinking that this is all just a bunch of theoretical nonsense. Parallel lines are parallel lines. The angles in a triangle are 180 degrees, and all the rest is just fantasy. However, that is most certainly not true. In the case of spherical geometry, the applications are 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 the sphere. If you've ever been on a long flight, you can usually track the progress of the flight on the flight entertainment system. When doing surveying for projects that span long distances, the curvature of the Earth needs

to be taken into consideration. And it turns out there are real-world uses for hyperbolic geometry as well. Perhaps the best-known use of hyperbolic geometry is in the theory of special relativity. According to special relativity, space and time are bound up into something known as space-time, which can have a curvature. If spacetime has positive curvature, aka elliptic, it's known as a decider space, and if it's negative, aka hyperbolic, then it's known as a Minkowski space. So these non-Euclidean geometries do indeed have very, real-world applications. I realize some of the things that I've talked about in this episode may have gone over

the heads of some people. However, I assure you that the concepts behind them are actually 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 centuries later that mathematicians took a different approach and tried to envision what things might look like if the fifth axiom were false. And it's from that that they were able to develop entirely new geometries. It turned out that Euclid's decision to add a fifth axiom was in fact a stroke of genius.

A decision taken over 2,000 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 Kiefer. Today's review comes from listener JSD-078-14 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, JST. You can get your membership card at the front desk. The concierge will show you the facilities and all the other secret things you need to know. And yes, I do have some episodes planned about the Philippines.

At this point, it's really just a matter of what order I want. want to do the episodes in. Remember, if you leave a review or send me a boostagram, you two can have it right on the show.