- Euclid's Elements, one of the oldest math books, held the key to understanding the universe.
- Mathematicians were skeptical of a single line in the book for over 2,000 years.
- Euclid's method of using postulates and logic to prove theorems became the gold standard for rigorous mathematical proof.
- The fifth postulate, known as the parallel postulate, caused controversy and attempts to prove it failed.
- Janos Boliae imagined a world where there could be more than one parallel line through a point, leading to the discovery of hyperbolic geometry.
- The hyperbolic plane is a curved surface where straight lines appear bent.
- The Poincaré disk model is used to represent the hyperbolic plane.

- János Bolyai discovered a new geometry called hyperbolic geometry, which was consistent mathematically but different from Euclid's geometry.
- Bolyai was also skilled in playing the violin and dueling, winning all 13 duels against cavalry officers.
- Bolyai published his findings as an appendix to his father's textbook and sent it to Carl Friedrich Gauss, who had independently discovered non-Euclidean geometry.
- Gauss did not publish his findings due to fear of ridicule.
- Spherical geometry, which is based on the geometry of a sphere, was also considered a non-Euclidean geometry.
- Bolyai believed Gauss was trying to undermine him and never published again.
- Riemann expanded on non-Euclidean geometry by introducing the concept of varying curvature in different parts of space.
- Beltrami proved the consistency of hyperbolic and spherical geometries in 1868.

- In 1868, Eugenio Beltrami proved that hyperbolic and spherical geometry were consistent with Euclid's flat geometry.
- Einstein's special theory of relativity proposed that space and time are relative, creating a problem for Newtonian gravity.
- Einstein realized that a free-falling man would feel weightless and be an inertial observer, leading to the concept of curved space-time.
- General relativity has been successful in explaining phenomena such as gravitational lensing and gravitational waves.
- The shape of the universe can be determined by measuring the angles of a triangle, and current data suggests that the universe is flat.
- The mass-energy density of the universe is serendipitous, and the reasons for its value are still unknown.

- Brilliant offers a personalized learning path to help you reach your goals
- Their latest course, Measurement, focuses on geometry and spatial reasoning skills
- A strong foundation in geometry can have practical applications in various fields
- Brilliant connects what you learn to real-world examples and offers hands-on lessons
- You can try Brilliant for free for 30 days and get 20% off their annual premium subscription

- Euclid's Elements was a significant math book that provided insights into the universe.
- Mathematicians were skeptical of a particular line in the book for centuries.
- Euclid's method of using postulates and logic for proving theorems became the standard for rigorous mathematical proof.
- The fifth postulate, known as the parallel postulate, caused controversy and attempts to prove it failed.
- Janos Bolyai discovered hyperbolic geometry, a new consistent geometry that differed from Euclid's.
- Bolyai's findings were published as an appendix and sent to Carl Friedrich Gauss, who had independently discovered non-Euclidean geometry.
- Spherical geometry, based on the geometry of a sphere, was also considered non-Euclidean.
- Riemann expanded on non-Euclidean geometry by introducing varying curvature in different parts of space.
- Beltrami proved the consistency of hyperbolic and spherical geometries with Euclid's flat geometry.
- Einstein's theory of relativity introduced the concept of curved space-time.
- General relativity has successfully explained phenomena like gravitational lensing and gravitational waves.
- The shape of the universe can be determined by measuring triangle angles, suggesting it is flat.
- Brilliant offers personalized learning with a focus on geometry and spatial reasoning skills.
- A strong foundation in geometry has practical applications in various fields.
- Brilliant provides real-world examples and hands-on lessons.
- A free trial and discount on the annual premium subscription are available.

Discover strange new universes that turn up at the core of Einstein’s General Relativity. Head to https://brilliant.org/veritasium  to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

If you’re looking for a molecular modeling kit, try Snatoms – a kit I invented where the atoms snap together magnetically – https://ve42.co/SnatomsV  

If you want to play around in hyperbolic space then check out: https://3-dimensional.space/geometries/hyp/ 

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A massive thank you to Prof. Alex Kontorovich for all his help with this video.

A huge thank you to Prof. Geraint Lewis and Dr. Ashmeet Singh for helping us understand the applications of Non-Euclidean geometry in astronomy/cosmology.

Lastly, a big thank you to Dr. Henry Segerman and Dr. Rémi Coulon for helping us visualize what it’s like to be inside hyperbolic space and helping us understand hyperbolic geometry.

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Images:
Euclid via Science Museum Group - https://ve42.co/Euclid

Geodesy survey via ams - https://ve42.co/Geodesy

John Wheeler via NAS Online - https://ve42.co/Wheeler

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References:
Dunham, W. (1991). Journey through Genius: Great Theorems of Mathematics. John Wiley & Sons.

Bonola, R. (1955). Non-Euclidean geometry: A critical and historical study of its development. Courier Corporation.

Library of Congress. (n.d.). The Library of Congress. - https://ve42.co/LibofCongress
 
Euclid’s Elements, Wikipedia - https://ve42.co/Elements

The History of Non-Euclidean Geometry, Extra History via YouTube - https://ve42.co/ExtraHistory

We (could) live on a 4D Pringle - Physics for the Birds via YouTube - https://ve42.co/4DPringle

Parallel Postulate, Wikipedia - https://ve42.co/Parallel

Prékopa, A., & Molnár, E. (Eds.). (2006). Non-euclidean geometries: János Bolyai memorial volume (Vol. 581). Springer Science & Business Media. 

St Andrews, University of. (n.d.). Bolyai. MacTutor History of Mathematics.  - https://ve42.co/Bolyai

Bolyai, J. (1896). The Science Absolute of Space.. (Vol. 3). The Neomon.

Gauss, Wikipedia - https://ve42.co/Gauss
 
Landvermessung, D. Z. (1929). Abhandlungen ueber Gauss' wissenschaftliche Taetigkeit auf den Gebieten der Geodaesie, Physik und Astronomie Bd. 11, Abt.  - https://ve42.co/Landvermessung

Nikolai Lobachevsky, Wikipedia - https://ve42.co/Lobachevsky

Lobachevskiĭ, N. I. (1891). Geometrical researches on the theory of parallels. University of Texas. 

A Problem with the Parallel Postulate, Numberphile via YouTube - https://ve42.co/NumberphileParallel

Riemann, B. (2016). On the hypotheses which lie at the bases of geometry. Birkhäuser. - https://ve42.co/Riemann

Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der physik, 17(10), 891-921. - https://ve42.co/Einstein1905

Why Gravity is NOT a Force, Veritasium via YouTube - https://www.youtube.com/watch?v=XRr1kaXKBsU 

ESA/Hubble. (n.d.). Hubblecast 90: The final frontier of the Frontier Fields. ESA/Hubble.  - https://ve42.co/Einstein1905

Agazie, G., et al. (2023). The NANOGrav 15 yr data set: Constraints on supermassive black hole binaries from the gravitational-wave background. - https://ve42.co/NANOGrav

Secrets of the Cosmic Microwave Background, PBS Spacetime via YouTube - https://ve42.co/PBSCMB

Wood, C. (2020). How Ancient Light Reveals the Universe's Contents. Quanta Magazine. - https://ve42.co/AncientLight

Collaboration (2014). Planck 2013 results. XVI. Cosmological parameters. A&A, 571, A16. - https://ve42.co/Planck2013

WMAP Science Team, NASA. (2014). Matter in the Universe. WMAP, NASA.  - https://ve42.co/WMAP2014

Shape of the universe, Wikipedia - https://ve42.co/UniverseShape

Crocheting Hyperbolic Planes: Daina Taimina by Ted, via YouTube - https://ve42.co/Hyperbolic

Hyperbolic Crochet model - https://ve42.co/Crochet

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Written by Casper Mebius, Petr Lebedev, Emily Zhang and Derek Muller
Edited by Jack Saxon
Animated by Fabio Albertelli, Ivy Tello, and Mike Radjabov 
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A single sentence in 1 of the oldest math books held the key to understanding our universe.

Euclid's Elements has been published in more editions than any other book except the Bible.

It was the go-to math text for over 2,000 years.

But for all that time, mathematicians were skeptical of a single line, which seemed like a mistake.

Ultimately, some of the greatest math minds realized that Euclid wasn't wrong after all.

Slight tweaks to this line opened up strange new universes out of nothing.

Surprisingly, 80 years later, we found out those strange new universes are core to understanding our own universe.

Around 300 BC, the Greek mathematician Euclid takes on a massive project to summarize all mathematics known at the time.

To essentially create the 1 book that contains everything that everyone knows about mathematics.

See, before Euclid, there was a bit of a problem with math.

People would prove things, but they would just be going around in circles.

Because if you take 2 parallel lines, yeah, but why do parallel lines exist?

Oh, that's because you can make a square.

You have this infinite recursion of what the fundamental reason why something is true is.

It's kind of like in the dictionary, every word is defined in terms of other words.

Euclid used a solution that was pioneered by the Greeks.

Let's just accept a few of the most simple basic things as being true.

Then based upon these postulates, we can prove theorems 1 at a time, building up our math using logic.

So that as long as those first statements are true, then everything else that follows from them must definitely be true.

He had perfected the gold standard for rigorous mathematical proof that all modern math relies on.

Euclid used this method when he published his 13 book series called The Elements, in which he proved 465 theorems covering almost all of mathematics known at the time, including geometry and number theory.

And all these theorems depended on were some definitions, a few common notions, and 5 postulates.

We go right to book 1, and book 1 starts with definitions.

And the definition is a point is that which has no part.

A line is a breadthless length and the ends of lines are points.

By line he really means a curve and then it has some ends.

A straight line is a line that lies evenly within the points on itself, and then so on and so forth.

He's got 23 definitions, then he's got the 5 postulates.

1, if you have 2 points, you can draw a straight line between them.

2, if you have a straight line, you can extend it indefinitely.

3, given a center and a radius, you can draw a circle.

And the fourth is that all right angles are equal to each other.

Which is that if a straight line falling on 2 straight lines makes the interior angles on the same side less than 2 right angles, The 2 straight lines, if produced indefinitely, meet on that side, on which are the angles less than the 2 right angles.

Like all of these are, you know, half of a sentence, and they're all blatantly obvious.

And then comes 5, out of left field, and it's like an entire paragraph.

Greek philosopher Proclus thought postulate 5 ought even to be struck out of the postulates altogether, for it is a theorem.

But if it's a theorem, we should be able to prove it from the first 4 postulates.

Some, including Ptolemy and Proclus, believed they had succeeded.

In fact, all they'd managed to do was just restate postulate 5 in different words.

If you have a line and a point that is not on that line, then there is a single unique line which will be parallel to the first line.

For this reason, the fifth postulate is often called the parallel postulate.

When the method of direct proof failed, other mathematicians, including Al-Haytham and Omar Khayyam, tried a different approach, proof by contradiction.

You keep the first 4 postulates the same, but assume that the fifth postulate is false.

Then you use those new postulates to prove theorems.

And if that leads to a contradiction, for example, true equals false, well then it means your new fifth postulate must be wrong.

Therefore, the only remaining option would be that Euclid's version of the fifth postulate is correct and you would have proven the fifth postulate.

So what would it look like if the fifth postulate were false?

Well, according to Euclid, through a point not on a line, there could only be 1 line that is parallel to the first.

1 alternative is that there are no parallel lines that you could draw through that point.

Well, people tried that and they realized that then lines had to be finite in length.

It contradicted the second postulate, which states that lines can be extended indefinitely.

The other alternative is that you can draw more than 1 parallel line through a point not on the first line.

In total, mathematicians spent more than 2,000 years trying to prove the fifth postulate, but everyone who tried failed.

Then, around 1820, Janos Voliae, a 17-year-old student, started spending his days and nights working on the mystery.

His father became worried and he wrote to his son, "'You must not attempt this approach to parallels.

"'I have traversed this bottomless night "'which extinguished all light and joy in my life.

"'I entreat you, leave the science of parallels alone.

Learn from my example." But the young Boliai didn't listen to his father.

He could not leave the science of parallels alone.

After years of work, he realized that maybe the fifth postulate can't be proven from the other 4.

See, according to Euclid you could have only 1 parallel line through a point.

But Bolle I imagined a world where there could be more than 1 parallel line through that point.

Well, who said you needed to have a flat surface?

On a surface that is curved like this, you can draw more than 1 line that is parallel to the original line.

But wait a second, those lines don't look straight.

Well, what makes straight lines special is that they're the shortest paths between 2 points.

On this surface, those shortest paths just look bent because the surface is curved.

Airplanes always try to fly the shortest path between 2 cities.

They're basically flying in a straight line.

But that line doesn't look straight on a map because the surface is curved.

These shortest paths on curved surfaces are called geodesics.

So all these lines are straight, they just don't look it because the world Bolleae had imagined turned out to be curved.

You know, when I used to think of the hyperbolic plane, I would just imagine it as 1 giant saddle.

The hyperbolic plane is much more like this piece of crocheting.

So it starts out pretty flat and even in the middle, but as you move outwards, more and more fabric is created, and that would push parallel lines apart and the further and further out you go the amount of fabric grows Exponentially and that ends up causing this crumpling effect So if you really want to think about the hyperbolic plane, I think you've got to think about saddles on saddles on saddles, like it's a infinite crumpling mess.

But that little piece of crochet isn't the full hyperbolic plane.

To show that, we need to make a map, 1 that fits the entire plane into a disc.

To show how this works, we're gonna fill the entire plane with these triangles.

Starting in the middle, just as with the crochet, things look pretty normal.

But as you go farther out from the center, you get all this extra space, so you can fit more and more triangles.

So they appear smaller, but they are actually the same size.

Now, since the hyperbolic plane is infinite, you can keep adding triangles forever, and they all need to fit on the disc So as you get closer to the edge, the triangles will appear smaller and smaller and smaller Infinitely smaller never ending and you can never quite reach the edge This is known as the Poincaré disk model.

Here, straight lines are arcs of circles that intersect the disk at 90 degrees.

And just like on our original shape, a straight line down the middle appears straight, while straight lines next to it appear to curve away.

What's remarkable is that Baliai didn't have a model of hyperbolic geometry yet.

He was just drawing Euclidean triangles with the assumption that Euclid's fifth postulate didn't hold.

And while Bollei found that the behavior in hyperbolic geometry is very different than Euclid's, mathematically it seemed just as consistent.

In 1823, the 20-year-old Janos wrote to his dad, I have discovered such wonderful things that I was amazed.

Out of nothing, I have created a strange new universe." But Bolleae had been doing more than just tackling ancient math mysteries.

In his 20s, he joined the army, where he continued developing 2 of his other passions, playing the violin and dueling.

He had mastered both, but with a sword in particular he was unmatched.

Perhaps because of his many talents, Boliai grew arrogant and found it difficult to accept authority from his superiors.

This reached a peak when during 1 of his deployments, 13 cavalry officers from his garrison challenged him to a duel.

Boliai accepted their challenge on the condition that after every 2 duels he could play for a little while on his violin.

Bollei fought each of them in succession, winning all 13 duels and leaving behind all his adversaries on the square.

While Bollei loved dueling, his first love was still mathematics.

In 1832, 9 years after he discovered his strange new universe, he published his findings as a 24-page appendix to his father's textbook.

Extremely proud and excited about his son's work, Farkas Bolliai sent it to perhaps the greatest mathematician of all time, Carl Friedrich Gauss.

After careful examination, Gauss replied a few months later, "...to praise it would amount to praising myself, for the entire content of the work coincides almost exactly with my own meditations which have occupied my mind for the past 30 or 35 years.

Years earlier, Gauss had wandered a similar path.

In 1824, he wrote a private letter to 1 of his friends in which he describes discovering a curious geometry, 1 with paradoxical and, to the uninitiated, absurd theorems.

For example, Gauss writes, the 3 angles of a triangle become as small as 1 wishes if only the sides are taken large enough.

Yet the area of the triangle can never exceed a definite limit.

In other words, you can have a triangle that's infinitely long, but the area is finite.

You can see why by using the Poincaré disk model.

A small triangle looks pretty ordinary, but as you make it bigger the angles start to become smaller and smaller.

Eventually all those angles go to 0 because all these lines intersect the disk at 90 degrees.

Now, these lines are infinitely long, but because of the geometry, the area is finite.

In the same private letter, Gauss wrote, all my efforts to discover a contradiction and inconsistency in this non-Euclidean geometry have been without success.

Just like Bolyai, Gauss had found that this geometry seemed thoroughly consistent.

He named it non-Euclidean geometry, a name that stuck.

It describes geometries where Euclid's first 4 postulates hold, but the fifth doesn't.

But Gauss decided not to publish his findings for fear of ridicule.

This aversion to a different kind of geometry should at least be a little surprising, because there is 1 other geometry that we should be very familiar with, spherical geometry, since we all live on a sphere.

On a sphere, straight lines are parts of great circles.

These are the circles with the largest possible circumference.

On Earth, the equator and circles of longitude are examples of great circles.

And we can use this to see how straight lines behave.

These lines seem to go in the same direction, but as you keep extending them, you find that they intersect once and then again on the other side of the earth.

And this will always happen for any 2 great circles because they must each have the largest possible circumference.

So on a sphere, there are no parallel lines.

Gauss had long been fascinated by spherical geometry.

He was also a geodesist and frequently took measurements of the earth.

In the 1820s, he was tasked with surveying the kingdom of Hanover to help make a map.

As part of this survey, he climbed the mountains near Göttingen.

With the help of people situated on other landmarks, they were able to carefully measure the angles of several triangles, which would then be used to determine the position of 1 place relative to another.

As a reference for the survey and to help determine the roundness of the earth, they also precisely measured the angles of a large triangle formed by 3 mountains.

But for all of Gauss's romantic notions of taking measurements on top of mountains, he was not the kindest correspondent.

When Bolyai received his hero's response, he was devastated, believing that Gauss was trying to undermine him and steal his ideas.

He was so embittered by Gauss's response that he never published again.

In 1848, Bolliai had to endure another hardship when he found out that Russian mathematician Nikolai Lobachevsky had independently discovered non-Euclidean geometry, several years before Bolliai had published his 24-page-long appendix.

When Bolliai died in 1860, he left behind 20,000 pages of unpublished mathematical manuscripts.

He would not know that Gauss did independently discover non-Euclidean geometry, nor would he know that upon receiving the appendix Gauss had written to a friend, I regard this young geometer, Bollei, as a genius of the first order.

While Bollei was embittered, non-Euclidean geometries continued to develop.

Until 1854, spherical geometry wasn't actually considered a non-Euclidean geometry.

That's because on a sphere, lines can't be extended indefinitely.

This is what earlier mathematicians had stumbled upon and therefore dismissed this geometry, since Euclid's second postulate wouldn't hold.

But in 1854, Riemann changed the second postulate from an infinite extension to something that is quote, unbounded so that the second postulate still holds on a sphere.

With this change, spherical geometry became another valid non-Euclidean geometry.

By using the generalized 4 postulates and taking the fifth as there being no parallel lines, you could now derive spherical or elliptic geometry.

Would you consider the fifth postulate a mistake?

Would it have been better if he just never wrote that down?

If he never wrote that down, he would have stunted his geometry because he wouldn't be able to prove a lot of the things that he claimed to have.

It's beautiful that people spent 2,000 years trying to refute him, only to discover that in fact he was right in writing this in the first place.

So while Euclid was right to write down the fifth postulate, he did make a different mistake.

So here's the problem with what Euclid was doing.

Definition 1, a point is that which has no part.

We read this 2 minutes ago and we were nodding along like, yeah, it completely makes sense what he's saying.

Don't give me a definition that's going to have an infinite recursion.

If you give me a definition in terms of other things then you have to tell me what those things are.

If you tell me what that is you have to tell me what the thing before it is.

All I'm gonna tell you is what the postulates are that they're assumed to satisfy.

It's the relationships between the objects that's important, not the definitions of the objects themselves.

And once you free your mind to that possibility, all of a sudden you realize that there's a perfectly good geometric world in which by line you mean great circle, and by plane you mean sphere, and by point you mean a point on a sphere, and then 4 of those axioms are satisfied, just not the fifth.

And similarly, there's another model, something called the disk model for hyperbolic space, which the disk is the plane.

What I mean by straight lines is arcs of circles that are orthogonal to the disc, and then points are points inside the disc, and the disc is the plane.

See, you can think of geometry as a game.

The first 4 postulates are like the minimum rules required to play that game.

And then the fifth postulate selects the world that you'll play in.

If you pick that there are no parallel lines, you're playing in spherical geometry.

If you choose 1 parallel line, you're playing in flat geometry.

And if you go for more than 1 parallel line then you're playing in hyperbolic geometry.

But Riemann decided to take it 1 step further.

Instead of selecting just 1 world to play in, why not combine them all into 1?

During his inaugural speech in 1854 he laid out the groundwork for a geometry where the curvature could differ from place to place.

1 part might be flat, another part might be slightly curved, and yet another part might have a very strong curvature.

And this geometry wouldn't be limited to two-dimensional planes either.

It could be extended to 3 or more dimensions.

Another breakthrough came in 1868 when Eugenio Beltrami unequivocally proved that hyperbolic and spherical geometry were just as consistent as Euclid's flat geometry.

That is, if there were any inconsistencies in hyperbolic or spherical geometry, then they must also be present in Euclid's flat geometry.

The prospects for these new geometries were looking great and it turns out this was just the beginning.

In 1905 Einstein proposed the special theory of relativity which is based on just 2 postulates.

1, the laws of physics are the same in all inertial frames of reference.

And 2, the speed of light in a vacuum is the same for all inertial observers.

So as a result, space and time must be relative.

But that created a problem for Newtonian gravity, because according to Newton, the force of gravity is inversely proportional to the distance between the 2 objects squared.

But in Einstein's special relativity, that distance is no longer well-defined.

In whose reference frame are we measuring?

And so Einstein had to find a way to reconcile relativity and gravity.

2 years later, in 1907, Einstein had the happiest thought of his life.

He imagined a man falling off the roof of a house.

And what made Einstein so joyful is that he realized that while the man is falling, he would feel absolutely weightless.

And if he let go of an object, it would just remain in uniform motion relative to him.

It would be just like being in space, not near any masses, floating around in a spaceship at constant velocity.

Einstein realized that they're not just similar, they are identical.

Because there is no experiment you could do to determine whether you're in free fall in a uniform gravitational field or whether you're in deep space, not near any massive objects.

And so the free-falling man too must be an inertial observer, meaning he is not accelerating and he's not experiencing any force of gravity.

But if gravity is not a force, then how do you explain things like the space station orbiting Earth?

Shouldn't it just fly off in a straight line?

Well, Astronauts in the space station also feel weightless.

It feels just as if they're traveling at constant velocity in a straight line.

It feels like that because that's precisely what they're doing.

How then could that straight line appear curved to a distant observer?

The answer is because the space-time that straight line is on is curved.

And objects moving through curved space-time will follow the shortest path through that curved geometry, the geodesic.

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How One Line in the Oldest Math Text Hinted at Hidden Universes