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

Discover strange new universes that turn up at the core of Einstein’s General Relativity. Head to brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.
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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 - ve42.co/Euclid
Geodesy survey via ams - ve42.co/Geodesy
John Wheeler via NAS Online - 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. - ve42.co/LibofCongress
Euclid’s Elements, Wikipedia - ve42.co/Elements
The History of Non-Euclidean Geometry, Extra History via KZbin - ve42.co/ExtraHistory
We (could) live on a 4D Pringle - Physics for the Birds via KZbin - ve42.co/4DPringle
Parallel Postulate, Wikipedia - 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. - ve42.co/Bolyai
Bolyai, J. (1896). The Science Absolute of Space.. (Vol. 3). The Neomon.
Gauss, Wikipedia - ve42.co/Gauss
Singh, U. (2022). Gauss-Bolyai-Lobachevsky: The dawn of non-euclidean geometry. Medium. - ve42.co/CPNonEuclidean
Landvermessung, D. Z. (1929). Abhandlungen ueber Gauss' wissenschaftliche Taetigkeit auf den Gebieten der Geodaesie, Physik und Astronomie Bd. 11, Abt. - ve42.co/Landvermessung
Nikolai Lobachevsky, Wikipedia - 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 KZbin - ve42.co/NumberphileParallel
Riemann, B. (2016). On the hypotheses which lie at the bases of geometry. Birkhäuser. - ve42.co/Riemann
Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der physik, 17(10), 891-921. - ve42.co/Einstein1905
ESA/Hubble. (n.d.). Hubblecast 90: The final frontier of the Frontier Fields. ESA/Hubble. - 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. - ve42.co/NANOGrav
Secrets of the Cosmic Microwave Background, PBS Spacetime via KZbin - ve42.co/PBSCMB
Wood, C. (2020). How Ancient Light Reveals the Universe's Contents. Quanta Magazine. - ve42.co/AncientLight
Collaboration (2014). Planck 2013 results. XVI. Cosmological parameters. A&A, 571, A16. - ve42.co/Planck2013
WMAP Science Team, NASA. (2014). Matter in the Universe. WMAP, NASA. - ve42.co/WMAP2014
What Is The Shape of Space, minutephysics via KZbin - ve42.co/SpaceShape
Shape of the universe, Wikipedia - ve42.co/UniverseShape
Crocheting Hyperbolic Planes: Daina Taimina by Ted, via KZbin - ve42.co/Hyperbolic
Hyperbolic Crochet model - ve42.co/Crochet
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Special thanks to our Patreon supporters:
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Directed by Casper Mebius
Written by Casper Mebius, Petr Lebedev, Emily Zhang, Derek Muller, and Alex Kontorovich
Edited by Jack Saxon
Animated by Fabio Albertelli, Ivy Tello, and Mike Radjabov
Illustrations by Jakub Misiek and Celia Bode
Filmed by Derek Muller
Produced by Casper Mebius, Derek Muller, and Han Evans
Additional video/photos supplied by Getty Images, Pond5, and by courtesy of: NASA, NASA's Goddard Space Flight Center, NASA Goddard Flight Lab/ CI Lab, NASA’s WMAP science teams, ESO, and ESA/Hubble.
Music from Epidemic Sound
Thumbnail by Ren Hurley

[@birindersingh4146]

Imagine the greatness of the man who wrote 1 little paragraph and made mathematicians mad for 2000 years

[@IsaacGroff12]

Fr bros a menace

[@sarc143]

Haters pocket watching bro smh

[@m1k3vroom]

The level of genius involved is unparalleled.

[@dwdelve]

Greeks have been doing that for a long time

[@topherthe11th23]

@@dwdelve And so have many non-Greeks. There's nothing special about Greeks. There was something special about Euclid's achievement, and many special achievements of OTHER Greeks too, but that doesn't reflect glory on Greeks who DID NOT accomplish anything special with their lives. "Greekness" isn't some special trait that any Greek person should be allowed to feel good about.

[@Gene-ns2wk]

Let’s all appreciate Euclid’s effort he put into writing The Elements just so that Veritasium could make a video about it

[@TheOneAndOnlyCumGuzzler]

credit to veritasium for supporting small authors like euclid

[@Byzantia]

I

[@mr.fhizzy]

Yes

[@PrabhasPatil]

@@TheOneAndOnlyCumGuzzler yes 💯

[@JohnDoe-qz1ql]

As he said, it's exerted great influence, so

[@clarencejohncabahug5466]

To call Euclid just "Father of Geometry" is an understatement. The major branches of math are built from Axioms, and Euclid pioneered that. He might as well be called the Father of Pure Mathematics itself.

[@gilbertogarbi4479]

Euclid had a great merit in consolidating in the Elements most of the mathematical knowledge of his time. But he is by no means the "father of Geometry". At least 2 centuries before him, other pioneers like Thales of Miletus and Pytagoras of Samos had already devised the logic-deductive Method, on whitch rests all of Mathematics.

[@isaacpianos5208]

​@@gilbertogarbi4479 yes he is by father of geometry by some means, saying "by no means father of geometry" is incorrect

[@Mayank-tm2km]

He collected all the known math knowledge at that time, he didn’t create all of it by himself

[@patrikfloding7985]

@@Mayank-tm2km ok, so not Father of Geometry. More like Midwife of Modern Geometry?

[@linktv7979]

​@@gilbertogarbi4479Greeks are famous in history for just stealing all the other countries best ideas of cultures, inventions, math and making it seem like research was just like inventing. Greek alchemy is just middle eastern alchemy etc they've never come up with anything only debated other cultures creations they stole from and improved the original idea. Pythagore was a cult leader who stole sumerian math and has gone down in history as it's inventor.

[@mikehenson819]

How terribly tragic it is that one lives, studies and discovers the incredible and never knew the greatness of their accomplishment in life.

[@-astrangerontheinternet6687]

How incredibly wonderful to live one’s life and celebrate each moment. Likely he lived the exact life he preferred and would spit on your pity.

[@justaspalabras]

Are you talking about Bolyai?

[@divyasasidharan2960]

I think we should not have that awareness.. u just keep doing it like animals procreate

[@jinfin221]

Why tragic

[@benjaminj883]

@@-astrangerontheinternet6687might be true for him, definitily isn’t for others. People have been hanged or imprisonned for stuff they discovered. Or people like Kafka who has been depressed his whole life, thinking it amounted to nothing only to be successful after his death. It is tragic

[@vaibhav3955]

Veritasium's videos are generally great but the math ones are on another level

[@feynman_QED]

This is a fast-paced video. It doesn't give the opportunity to deepen and enjoy the beauty and the importance of these discoveries. Don't let yourself be derived by animations, a double-edged sword tool.

[@n1ppe]

​​​@@feynman_QED it's a perfectly paced video. It's not supposed to make you understand everything about the subject, just to introduce it in an interesting way and make people interested to learn more about it on their own. If it was longer and dived deeper, less people would care about it in the first place

[@feynman_QED]

​@@n1ppe First flaw of your comment: I have never said that a video should let you understand EVERYTHING. Second, the reason why people cannot catch this aspect is the same behind the many thumbs-ups received by an illogical comment: you don't want a more articulated video but you want "to dive deeper". Please, make a decision and select which one you wanna pursue because it's not possible to satisfy that requirement simultaneously. I love this guy and how he produces videos. But videos are very often aimed at knowledgeable audiences who can keep up with a shortage of details and the fast pace. And it is absolutely not true that one requires 3 hours to make a more articulated video. If you indeed are interested, you just find a few slots of time during your week and you watch it carefully. Finally, I don't believe you're going to search a book and then study the topic in-depth. You're simply in the "infancy" stage when you are impressed easily by animation and hype, but you haven't developed an internal and sincere urge and interest for learning something more deeply. It's what happens with children: they are excited by toys, but after playing for a while they get annoyed.

[@hattix6713]

Derek has good sources, and actually listens to them!

[@noway905]

Yeah, I wholeheartedly agree. I took freshman Algeria 4 years in a row and graduated with a cumulative grade of a D- . So believe when I tell you, of everything that I think I know, there is only one thing that I know for certain and that is that I don't know anything. 🤣

[@swarry3508]

The way this entire video beautifully transforms right from a single point in euclidean geometry to the shape of the entire observational universe itself is so fascinating

[@atinkapruwan6780]

fascinating indeed

[@matroxman11]

I’m convinced that math is the key to the secrets of the universe

[@eddielally2045]

​@@matroxman11 bruhhhhhh obviously

[@ukleth]

a book call Quran (1400 years old ) talk beautifully about the shape of the universe and what will happen to it the verse says : "" On the Day when We fold the heaven, like the folding of a book. Just as We began the first creation, We will repeat it-a promise binding on Us. We will act. ""

[@daemongamingtv]

(Comment deleted due to comment crybabies; enjoy the contextless whining below!)

[@meemdic8682]

Related note: if you’re interested in seeing more of and playing around with hyperbolic (and possibly others) geometry, I highly recommend the videogame HyperRogue. It’s a top-down (with poincare projection by default) roguelike, but also features many tools for building projections, tilings and pictures. It’s unique, offers an interesting point of view on a lot of these things and plays around with them in many different ways.

[@silviodc1309]

the game Hyperbolica brings it even into 3D/VR

[@meemdic8682]

@@silviodc1309 So does Hyperrogue!

[@Victor-lr2xr]

Being able to explain complex ideas simply indicates understanding. Well done.

[@mosgon]

As an astrophysics major, I love how a video on ancient math turns into a cosmology lesson

[@TNT-km2eg]

Mayor , no less

[@Tomico.]

That's how it all started.

[@no36963]

Your Standard Model of Cosmology is a dead and stinking. The BigBang-to-BlackHole sex-and-death cult has no scientific verification. It's all false assumptions for a foundation for a house of cards. The god of gravity is dead. Long live the Electric Universe Model. Good luck in your search for a better understanding of reality. Best wishes, Charles A Campbell III

[@truthhub7395]

Checkout The Greatest Lie on Earth by Edward Hendrie. I can promise you are being fed a bunch of garbage

[@Jakekelley-hs8bk]

Sounds like a religious massive psyop. @@truthhub7395

[@killmajaro1]

Being able to observe and predict a phenomenon as large scale as light bending around an entire galaxy to make a cosmic lense is insane. What a time to be alive.

[@tinobemellow]

Can't really do anything practical with it like funnel energy and hack into the quantum mainframe of reality, but it's cool.

[@yashsarda2263]

The act of observing such a phenomena is practical in itself no? It's not a theory that we could observe gravitational lensing, we have done it.

[@irg008]

wrong channel ;)

[@TheEpicProOfMinecraf]

​@@irg008 I was about to say this myself

[@vattghern7592]

People out here measuring galaxies while I’m struggling in trigonometry….

[@miketerrell9530]

What an amazing episode, thank you for putting that together and making the mathematical thinking accessible.

[@ausgoogtube01]

Brilliant work this video. There is no way I could have understood these concepts from a text book. This approach opens up access to even to relativity and quantum physics imho. Thanks to Veritasium I have some hope of comprehending even modern terminologies without completely understanding the math.

[@trunghungpham9414]

Veritasium’s math videos are so good. Just never gets bored watching them.

[@atinkapruwan6780]

me too

[@prerakmann1418]

Shortest 30 mins on KZbin

[@Ostinat0]

Man I used to think I hated math even though I was really good at it...every Veritasium video about math that I watch makes me feel more and more like they just trained me wrong as a joke

[@daddycationxx]

for real, learning has never been more exciting

[@Micke240]

yeah and you will always learn alot watching his videos.

[@adityavardhanjain]

This video combined 3 best kinds of videos you make: 1. History of science & math 2. Visualization of difficult concepts especially those of physics & mathematics 3. The current great curiousity of humanity

[@dinogt8477]

.

[@ryugo7713]

@@dinogt8477do you need a tampon?

[@aniket385]

So Flat Earth were way ahead of curve and Meant Flat Universe??? Also a side note... Euclid Book was one of the things demanded by Arabs ...after one of those Byzantine Arab wars...

[@jordanfrielingsdorf4761]

I agree it is very Cosmos-esque and I think Carl Sagan would be proud.

[@cherniaktamir612]

And 4. An inspiring story about not listening to seniors about not pursuing your intuition, thinking from first principles

[@petsandpaws8906]

I truly enjoy watching your videos about math, geometry in particular. As a student in Belgium, I excelled in maths but was never motivated by the school or parents to pursue a degree in it. I switched from high maths to computer science before turning 17. Currently I work as an architect drawing in 3D and automating the software (Revit) with custom packages and dynamo scripts to assist collegues. I feel like ive wasted my potential in maths, especially since I was teaching my class at 13 in spherical geometry because the teacher himself didnt get his point across clearly. Seeing someonelike you, makes me realize what I wished I pursuit,even though I understand the chance I couldve gotten as far is slim. Keep making these videos please, thank you so much.

[@unvergebeneid]

I love how some things went unsolved for millennia and then multiple people have the same idea at the same time. This has happened over and over in the history of science and mathematics.

[@catfishingfornitro3416]

its cause they all get new info to work with. Some day some new proof may come out that allows everyone to figure something else out at once

[@culwin]

Often this also happen when and where institutions have been set up to publicize (or at least preserve) those findings in some way.

[@Tinil0]

It's because individual geniuses are utterly meaningless to the history of progress. Humanity has always had plenty of smart people, what matters is the opportunity. If one famous historical person didn't discover something, someone else would've, and for the same reason that person did: Not specific individual intelligence, but individual intelligence applied to the sum of human knowledge at that point in time.

[@johnlucas6683]

These are just the ones that have been on record, or at least have surviving records.

[@patu8010]

Makes you wonder if there was something about the time period of the 19th century (more mathematical geniuses?), or if older versions of the idea are lost to time. Possibly during 2000 years countless mathematicians came up with non-Euclidean geometry but never published it because they feared ridicule.

[@RealGhoda]

I find it so wild that mathematicians can do crazy things like predicting one supernova appearing 5 times spaced 1 year apart, but do things like spending 2000 years arguing about 1 sentence Edit - How did this start a war. I just exagerated some stuff to make a point

[@itzhexen0]

None of them spent 2000 years arguing over one sentence because they died.

[@NNOTM]

one is not possible without the other

[@capitano3483]

Physicists were the ones predicting the supernova appearing one year later.... the pragmatic mathematicians.

[@RealGhoda]

Mathematicians in general@@itzhexen0

[@pickle380]

I apologize for my actions

[@fun_at_work]

I really enjoy these videos. I really appreciate how you present modern theory in an Accessible way, and yet you do it with a humility that is so often lacking in how journalism covers these things. I think that the electric Universe criticisms of the cosmic microwave background research deserves to be answered, but I also believe that the work you're doing here is important no matter how science and Discovery changes what we're seeing.

[@alexchristakis4539]

You are gifted with a unique, original, very clear manner of explaining rather difficult-to-grasp or purely theoretical problems in a perspicuous way, easily understandable by a layman with no true mathematical background like myself. Only a person with deep knowledge, assiduousness and experience in math and physics can relay such information to non-experts. I cannot congratulate or thank you enough for the exceptional uploads of your channel. Please keep up the top-notch work. Many, many thanks!

[@TimeBucks]

Gauss never ceases to amaze me.

[@walifortune-hc4fe]

Wow lovely

[@user-nj3dp5sj9n]

Nice one

[@IsmailMansir-qq4hl]

Vnvb

[@dsoli807]

Sweet 🎉🎉🎉

[@dilshadjmati]

Good jobs gauss never ceases to amaze me.

[@myoky]

I took a geometry course in college where we started from Euclid and went on to derive essentially everything that you covered in this video to end with the shape and dimensionality of the universe using relativity. It was the best class that I ever took and this video was an amazing refresher on it.

[@happysailor315]

Whoa sounds great, what was it called?

[@black_crest]

​@@happysailor315Probably differential geometry

[@emmanuellawal2694]

What was it called, and please can you share resources that might help

[@AissamElkirafi]

Bump

[@matthewnoyce9089]

iI loved my geometry course in uni and we did almost the same up to spherical and hyberbolic geometry, but your ending on the dimensionality of the universe sounds brilliant

[@nicholasrogillio4280]

I’m only a minute into this video, and already the production feels like old school discovery or history channel and I’m here for it

[@baomao7243]

I thoroughly enjoyed this video. Reminded me of learning electromagnetics - it starts in simple plane geometry, then is extended to solid geometry, which then considers “infinite wires” which pull you into cylindrical coordinates, then “electrically small elements” which takes you into Spherical coordinates. Then you hit relativity and the curvature of space takes you into hyperbolic/elliptical coordinates, … I find these analyses both vexing and wonderful to contemplate.

[@alex-nb3lh]

the buildup from first principles to the payoff via einstein’s relativity is phenomenal. one of my favorite videos by you so far.

[@opticalreticle]

ti is

[@user-fx5ii1kt8i]

Yeah this was amazing

[@0RbIt1000]

Agreed

[@huyxiun2085]

Haven't seen the video yet, just the introduction (0:55 now). And I must say: it sounds extremely fishy for now. Don't get me wrong, I very much like Veritasium and I appreciate is work. His means to get the public interested in sciences, even the fishy ones, I see them as a good ideas. Doesn't change the fact that suggesting Euclide already knew about "Hidden Universes" (modern science ones) is, at best, sketchy. I know a bit of history of sciences and Euclide most certainly is one of this genius among geniuses. A mind like that would probably (hard to prove) make discoveries ahead of its time in any era. But still. It's very confusing to suggest that, since it leads to a very inaccurate understanding of what was the state of science (philosophy) back then. It's a great way to introduce the concept. But I don't think you should repeat this idea at lunch time, you'll propagate a misunderstanding or pass for a fool. Well, maybe not, I have to watch the rest of the video to now ;-)

[@canderson9167]

@@huyxiun2085 Watch the video.. I don't think you should repeat this idea at lunch time, you'll propagate a misunderstanding or pass for a fool.

[@omkarpawar2620]

Euclid: Makes 5th postulate bit long Other mathematicians: * thinks intensly for 2000 years*

[@adrian9098]

😅👍

[@ivanleon6164]

they took it personal, lmao.

[@shivamanand8998]

I love you videos especially the entropy video changed my way of thinking

[@DaveCalx]

Quite incredible how brilliant your content is. Thank you.

[@tomvesely4008]

If someone had told me this when I was in highschool (I was fascinated with astronomy as a kid, so maybe even earlier), my relationship with math would be completely different. This is fascinating

[@firstnamelastname9215]

That’s what I’m saying. Now I wish I cared for it because it would make life easier. But no they wanted to torture us instead of nurture.

[@alecmartin8543]

​@@firstnamelastname9215they didn't necessarily want to torture you, there aren't many teachers as good as this channel but there are a lot of honest people trying their best

[@milferdjones2573]

The system wanted to torture them is the point. And in a system like the best Finland lot of those honest people would be unable to qualify to be a teacher which is harder to do then get in med school. @@alecmartin8543

[@LumaSloth]

What change do you believe it caused you?

[@d_lta]

@@alecmartin8543 I love math and had a good teacher, but there are a lot of teachers who hate/mistreat children or aren't good at their job. I've definitely had more teachers that made me dread going to school than good ones

[@timothyshapka1309]

You just summed up my entire university foundations of geometry course in 30 minutes. I admire your ability to educate so concisely immensely.

[@briondalion3696]

It was a very great explanation, and hopefully many get inspiration from! I am actually thinking about gravity again, time travel, entropy, antimatter, gravitational waves, and antigravity.

[@grim_reaper977]

My question, was the hardship worth it, (the long nights to prove one theorem problem, while miscalculating several times)??

[@KhuongTuan-ef7hi]

zzz

[@maythesciencebewithyou]

So you are saying that your university course was only superficial and just taught you some history and only explained concepts, but didn`t teach you how to calculate the stuff?

[@tacticalidiots2340]

i like how the most replayed part of the video is the first mentioning of the postulate like none of us understood it all and went back to hear it again

[@sakshamconsul1389]

You explain really well. Really enjoyed this and learnt quite a bit from this video!

[@ChristopherCrimi]

Your coverage is fantastic!

[@rhov-anion]

I had a math teacher with Elements sitting on his desk. I have dyscalculia, numbers are extremely confusing (if not downright nightmarish) and math was always my worst subject. Straight A's with a D- in algebra. However, glancing through that book a little bit every class period, I found that the THEORY of math fascinated me. Sadly, schools only cared on if you could find answers to questions, and they didn't give a damn on if you knew WHY math worked. In college, after struggling for 3 years with "self paced math" meant for people with learning disabilities, my campus came up with an experimental "Algebra for Liberal Arts Majors" class, where we had the option of doing 30 math problems, creating art projects to illustrate the math theory, or writing essays on the theorems, how mathematicians came about discovering and proving them, or how this particular type of math applies to the real world. THAT I could do with ease, and it was my very first A in a math class. Numbers are still a mystery, something I just cannot sort out with my weird brain, but I love the history of math and what went into the geniuses who came up with these ideas.

[@ATF2099]

check out "fractals" that's probably how your brain is trying to operate. like mine, in reality, not fictional numbers. this thousand year old stuff is cool, but new thinkers might benefit more from realizing 3d math exists but is just way too complicated for most to even fathom. i figured it out on my own but then learned somebody already did in the 70s, thank god, cuz i wasn't trying to write that book myself. the way i understand it is every atom is 3 parts. and reality is base 3 number system. everything in 3s, based on that simple postulate. expanding out in all directions, continuously. interesting rabbit hole, but that's the real infinite realm. base 10 makes sense on paper, and counting on our fingers, but that's not how god designed the universe.

[@Sorrowdusk]

I've never heard of a course like that!

[@rhov-anion]

@@Sorrowdusk It was brand new when I took it back in 2004, a test program for students who don't need advanced math in their given major, especially folks like me with a learning disability that nearly prevented me from getting a degree. It was humiliating during matriculation when my counselor exclaimed, "How can you be in the top 1% in English and Logic but the bottom 1% in Mathematics?" Dyscalculia. That's how. It's because my brain cannot process numbers, so it compensated by using logic. (Also part of being autistic.) Yet the California education system refused to give me a pass on the math requirement, even though it had NOTHING to do with my major. Struggling with numbers doesn't mean that I can't have a successful career; in fact, the way my brain works is THE IDEAL for my job.... it just means I have to hire someone else to do my taxes, because my brain can't process numbers in the right order. If this sort of program didn't catch on at other colleges, that's a real shame. We need an education style that evolves with our growing understanding of neurodiversity. I've begun to work with Umbrella US, a NPO trying to change this sort of thing and get society to realize that not all brains work in "typical" ways. There are talented people out there being held back because they can't pass a course designed for neurotypical students. Look at all the scientists over the generations who struggled in school, because their brain danced with numbers but struggled with language, or danced to the music of etymology but fell flat with numbers. Imagine if they lived today, where they had to pass all these frivolous classes mandated by the federal government, but have nothing to due with their interests, or else their ideas and theories are disregarded due to not graduating from the "right school." One day, hopefully, programs like what I took become the norm. Give students the option to show the teacher that they truly understand a subject, even if the way they do that is not neurotypical.

[@suzuki8951]

Me too i hated math growing up but now that i am learning it's history and creative applications, I am really starting to appreciate it more and more. I'd say it's just a beautiful subject taught in an ugly manner

[@aviilokinkshi]

Brother/sister; math is just about seeing coherences and expressing these coherences with symbols we call numbers. Art and music are ALSO about seeing coherences and expressing these, but NOT in numeral symbols but having the coherences expressed in form and colour; or in the case of music: sound. You too are a mathematician, you just use a different language to describe the coherences you see.

[@orngjce223]

Hyperbolic geometry is more than just a giant cosmological thing. You know how the outside edges of some types of lettuce and kale go all crinkly, just like with the crochet model? The cells in them are effectively living on a hyperbolic plane, and studying the shapes and sizes of the cells there - how they fit together and exchange resources and such - requires some of this geometry.

[@user-xi7lr6oe6q]

The crinkly bits on kale and lettuce, remind me of fractals'....they are everywhere....gotta love em

[@user-tn4rx8pc3p]

Are you a biologist or something? Where would i look to learn more about this stuff?

[@jevilsugoma1743]

Ok

[@chetsenior7253]

No. Geometry requires real life things to give it value. Geometry is the explanation, not the reality.

[@niviamaeva]

Wait. What? 😮

[@BIM40K]

I love mathematics like this. It's totally fascinating.

[@nct948]

Stunning presentation sharing the excitement of mathematical challenges and their resolution relative to the actual reality of our universe. Thank you so much for guiding us through the thought processes leading to our present understanding of a flat universe. I am neither a physicist nor a mathematician and I can't help wondering why the cosmic background radiation is the same in any direction if the universe is flat!

[@hirakbandyopadhyay699]

I am a physicist, and this is one of the best explanations of curvature of spacetime I have seen on youtube, starting from absolute basics! Thank you so much, and keep up the good work. 🙂

[@cheoa1473]

ok

[@Steve-si8hx]

How does anybody know that postulates are true ?

[@archit1048]

😂@@Steve-si8hx

[@ForkGenesis]

@@Steve-si8hx we all just assume that they are true, and develop from them

[@efhi]

@@Steve-si8hx empirical observation of reality

[@TheRealQuickSilver]

Gauss never ceases to amaze me. There isn't a single math or science class I've taken where his name hasn't come up. Someday I'd love to spend some time learning about all of his greatest discoveries and trying to connect all of the dots of the contributions he's made.

[@ivanleon6164]

he was a monster, what a gigachad.

[@squibbelsmcjohnson]

​@@ivanleon6164😂😂😂

[@ImKinoNichtSabbeln]

The German author Daniel Kehlmann wrote the superb novel "Measuring the World" on C.F. Gauss' and Alexander von Humboldt's (yes, THAT Humboldt) lives. It was an sensation, depicting two crude geniuses in a absolutely entertaining, readable and intelligent way. The moment you open the book, you'll read it in one go, accompanying both Gauss and von Humboldt getting old, and even more strange. Gauss did not publish all his findings. He published only if he decided that he treated a subject in it's entirety. E.g. he did not publish his vast and deep findings in mathematical knot theory for he just wanted to complete some details, as we learned from his duaries.. Decades after Gauss' death, other mathematicians began to devolop the very same ideas, that Gauss already had knew.

[@XxZeldaxXXxLinkxX]

Don't sleep on my boy Euler. Lots of things are named after the person who discovered them after Euler, otherwise almost everything would be named after Euler

[@SylveonSimp]

He was on the 10 DM bank note in Germany. Now we have soulless windows on our Euro notes. What a pity.

[@judithhume9047]

I love listening to this...even though I don't really follow much of it. I especially love how the gentlemen smile and are so enthusiastic about it. Crikey...maths is fun!

[@AT-27182]

This is one of the greatest stories of human history. Thank you for explaining it for a large audience.

[@cereal-killer4455]

It really is. On a different note, flat earth confirmed /s Seriously: flat universe confirmed

[@LeonEdwardsFitnessOfficial]

@@cereal-killer4455 how

[@ninebreaker274]

ahaha normie

[@fearlessjoebanzai]

​@@cereal-killer4455, isn't that the bizarrest thing - he claims the universe to be essentially flat after just describing the reason why mathematically any sphere can be seen as essentially flat when zoomed in enough!

[@PinguinKeks]

It's funny how quick people can be to misunderstand things though, Gauss never said anything negative about Bolyai in his letter

[@caschque7242]

This video surpasses your usual Veritasium content in a unique way. The layered storytelling, which included extra details beyond just non-Euclidean geometry, enriched my experience. It felt like a 3D exploration rather than a linear journey, giving me a deeper and more nuanced understanding. I think this is one of the best ways to explain a topic yet. Good job!

[@em-agoo-481]

Great observation. In the 2nd episode of Cosmos, Sagan opens with a recounting of a 12th century Japanese battle only to segue into a discussion of selection/evolution: there's a magical quality to that scene that I've rarely felt elsewhere. This has a similar feeling, and "layered" captures it perfectly.

[@minhvan1216]

ok

[@dieSpinnt]

Yeah, you are absolutely right:) I really love the butchering of Greek and Arabic names, the misuse of the Latin Roman spelling for Greeks and those fantasy images of some material, where we actually have real historic content available. Stock Photos ... it's just a blessing ... Also getting Εὐκλείδης Axioms ignored and the purpose of Postulates confused is rather embarrassing. Well ... he had ONE JOB to do. I guess nobody's perfect?:) (Talking about "Journalists" here ... better consult a historian next time)

[@johnfkennedy1019]

22:00 -23:00 blew my mind. Thank you for finally explaining this concept so simply

[@guillermogutierrez710]

Exactly, hearing this is when all that bent space stuff finally made total sense to me!

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video.

[@chanceroberson7517]

This is I think the 3rd or 4th time having professor Alex on the channel and I’ve loved it everytime. He seems so enthusiastic with his explanation to the point where it’s infectious.

[@Cameron-ls3qt]

I find his argument about definitions ridiculous back then that language most likely lead to a specific understanding.. otherwise, why does everybody else agree and well mathematics as well. Seems Alex was the only one who didn't understand.

[@maxinator2002]

@@Cameron-ls3qtNo. At first, I too wasn’t buying it. However, I heard him out (and digested his whole argument), and I can now see it. It’s actually pretty brilliant. If you write a definition, the definition is made up of things that also could be defined, of which all are composed of even more things to define, and so on. It’s either a cyclic (circular logic, flawed) or never-ending (no useful definition) problem. Instead of establishing definitions, the professor suggests to establish relationships. As in, given this thing (regardless of how it could be defined), here’s how it relates to other things (regardless of how they could be defined). It all of a sudden makes this never-ending definition problem into a finite relationship problem, and it has much more rigor this way (as the postulates now only need to establish relationships, which then could be used to demonstrate other relationships, proving theorems). Yes, one could certainly understand Euclid’s definitions (and they don’t invalidate his results). However, his definitions are hand-wavey and non-rigorous… and most importantly (as the professor explains), unnecessary.

[@opticalreticle]

@@Cameron-ls3qt it is language

[@briondalion3696]

@@maxinator2002 It also forces people to actually do the math and confirm that it is correct.

[@UNr34]

The most impressive of all is how far ahead Euclid was that it took mathematicians thousands of years and forced them to invent a whole new field of mathematics and non-Euclidian geometry.

[@vidal9747]

"The consequences of the fifth postulate are left as an exercise for the reader"

[@AkiraDemi]

the entire math community: oh for fu-

[@TheSkystrider]

😂

[@localcragdirtbag8049]

Veritasium's have helped me with science, math, spirituality, and mental wellnes; as well as illness depending on the density of info!

[@successmeditations110]

This is a brilliant explanation of non-euclidian geometry. Brilliant.

[@successmeditations110]

I didn't mean to make a joke, sorry.

[@feliperiquelme8504]

As a Mathematician that loves hyperbolic geometry, I'm grateful to you for making this video. I'm going to share it with all my students for sure!

[@chaosopher23]

Psst... Euclid is describing parallax. Postulate 5 is a really stretched out triangle.

[@Roxor128]

Take a look at the game Hyperbolica. It's set in a world with hyperbolic geometry. It takes a bit of adjusting to go back to moving around the real world after playing it for a while. Appropriately, the first tag to show up on Steam is "surreal".

[@rocketscience4516]

Just call them your nerds, it's three less letters to type.

[@BisexualPlagueDoctor]

@@rocketscience4516he’s a mathematician, who loves hyperbolic geometry, why do you expect him to care about the length of words Lmao

[@rocketscience4516]

@@BisexualPlagueDoctor Aw, someone lacked the perspicacity to realise it must have been said in jest. Think a bit before you type.

[@sundhar5229]

His father not only understood his scientific work, he was so proud and sent it to his peers.. What's more wholesome than that..!

[@XmarkedSpot]

Taking the literal five seconds to check how his name is pronounced. Nails on the chalk board, I tell you.

[@zwojack7285]

Isnt that like every stereotypical Asian parent?

[@GudiUTube]

@@zwojack7285 When Farakas cautioned his son Jonas Bolyai with a father's concern that he should come to nothing as he himself failed there being a teacher in the field. But when he later realized his son's originality and brilliance he accepted his mistake, when 1) he made his son's work as an addendum to his own and 2) he sent it to his old friend Gauss. So Farakas did the best thing and was not responsible for what happened next, viz. Bolyai's frustration due to Gauss's fear of "Bootier".

[@rockjano]

@@zwojack7285 The were Hungarians not asian....

[@sexmansex4776]

​@@rockjanohungarians are steppe people anyway

[@res00xua]

Fascinating. Great job!!!

[@JustDiscipline]

I've watched this 3 times.. great video, what a complicated subject!

[@18th_King]

I am sad for Bolyai as he thought his hero Guass would appreciate his work. Still, when Guass replied that he didn't do anything except repredicting his work Bolyai would have been crushed deeply. I definitely see Bolyai as a legend doing such great work at the age of 23 in just 5 years while Guass had spent nearly decades finding out about it. "Appreciation can make someone's day, even change lives. So even if the work is small or the best it is incomplete without a 'Good Job'." - A Wise Man.

[@Sharkie1717]

Nice 👌

[@alexc4924]

What's even sadder is HE DID APPRECIATE IT VERY MUCH AND PRAISED BOLYAI TO ANOTHER MATHEMATICIAN but Bolyai never knew it!

[@SamageetDutta]

If I were Gauss, I'd tell Bolyai why I couldn't publish the previous works, and would invite him to come and work with me

[@kolyashinkarev7366]

He didn't really say that the work was unimpressive or bad, the sentence is taken out of contex, we don't really know if he followed it up with praising

[@apokalypthoapokalypsys9573]

Ah yes, Guass, my favourite character from Code Guass.

[@cbwavy]

15:57. That is devastating. Imagine if Bolyai and Gauss actually collaborated.

[@thisispiero2807]

I learn more from one of your videos than I do in a week of high school. insane work.

[@thecapone45]

Man this was such a great video. Appreciate your effort in this mini-doc. Anyways, how'd you get your teeth so white?

[@jacopolibera6974]

This has to be one of the better animated veritasium videos ever. A pleasure to watch as usual

[@jacobshirley3457]

Everybody says that after every history-based Veritasium video, lol. Which is saying something.

[@benmastar]

Hey Derek, you probably won't read this but I just wanted to say that I've been watching your videos casually for many years when you first started. I just need to say that your production quality and ability to keep audiences engaged has evolved so beautifully. I watched this 30min video and it felt like only 5 minutes had gone by... and I'm not even that interested in mathematics let alone history. All I'm trying to say is well done. Thank you for all your years of educational content and thank you for continuing to impart the same level of passion in every project. I hope you continue for many years to come.

[@j_m3102]

12:29 12:34 12:36

[@WREFMAN]

Your

[@CaptainPeterRMiller]

Well said Monsieur.

[@karupt422]

Your right , he didn't read it😂

[@fearsomefoursome4]

HAVE ERIC WEINSTEIN AND GIVE US GEOMETRIC UNITY

[@mikecarter4258]

This is a classic case of the "more you know, you realize the more you don't know." Aristotle

[@chrisn8349]

18:57 Geometry as a game with three different worlds is a really cool way to explain this.

[@erictaylor5462]

16:00 The amount of human suffering that is born of misunderstanding, or assumptions that are in error is astonishing.

[@borisborcic]

ambiguities are like microbes, the pathogenic ones steal the attention...

[@shininio]

I find the quality of the content in this channel to be in a completely different league than most of the things you find in KZbin. Kudos Derek

[@opticalreticle]

good

[@KcyL0709]

True

[@MysticPing]

Apart from all the misleading startup ads

[@opticalreticle]

@@MysticPing what startup ads?

[@ACuriousChild]

@@MysticPing .... and much more ... THE BLIND LEADING THE BLIND!

[@Joel-es5le]

I think this video just made me start understanding some things I never did before. Really wild.

[@N0Xa880iUL]

Mind-blowing. Such great minds.

[@apolotorresart]

My goodness, man, I think that quite often but I have to say this is one of your best videos to date. The level of storytelling and the way you managed to tie ancient knowledge all the way to the very edge of our current understanding of the universe was absolutely mesmerizing. And poetic, even, because just as in culture, arts, music and so forth, the new doesn't necessarily destroys the old, but rather builds higher grounds over earlier foundations. Chapeau!

[@opticalreticle]

because God made it that way

[@taisa8776]

I honestly believe that learning to sew could have given insight for those mathematicians that spent 2000 years trying to understand the 5th postulate. All fabrics are essentially planes, but our body isn't and, depending on culture and time, it needs to closely follow our body's format - that is definitively non-euclidian.

[@Grace-ms7un]

You are my new hero for using sewing to describe math ❤

[@wawaweewa9159]

Thsi is why it's so important to make men way more feminine, toxic masculinity has stunted those great men of the past

[@Qay]

@@wawaweewa9159 This has nothing to do with the comment. Maybe concern troll under a different bridge?

[@wawaweewa9159]

how dare you? @@Qay

[@StevenBranz]

The best understanding I could come up with is like a huge mosaic of Sedimentary rock separated by space time!

[@paulgrobar7098]

Regarding circular/spherical geomitry: it is not possible to form a right angle with a curved line. Such an intersection will always be acute because the circle curves toward the line.

[@user-zk4lv4hz6r]

If you have a straight line you can't have a sphere, no matter the scale. You can't square the circle.

[@Nanbread-bw7nq]

what if you have the line be straight at the intersection point and then curve off afterwards? like a line intersecting a circle that passes through its centre (in euclidean geometry)?

[@MrAulic]

I feel like I finally understand a bunch of stuff that I've always heard, in relation to math and physics, but was never able to fully grasp and connect together, for example what people meant exactly by the curvature of space time caused by gravity and such. This video's narrative is absurdly good and really helps tie it all together in a very comprehensible way. Just wow... this channel never ceases to amaze me.

[@usadefcon1]

If you want more checkout the ScienceClic English KZbin channel. It's the best for visualizing the math.

[@joerionis5902]

Finally understood what physics channels meant when they say that "the universe is flat."

[@Oscaragious]

@@joerionis5902 Unless you draw REALLY big triangles.

[@T_Dot94]

I love these history of science videos. It makes math and science feel like a lively art form rather than something mechanical and dead.

[@nineveh17]

real

[@storiesshubham4145]

Rightly said

[@tamasjenovari2248]

these pronanciations are gold

[@colinbyerly5212]

Thank you great achievement in presentation.❤️👍🏼

[@vladbobe26]

Leaving aside the educational aspect, the production value of this was incredible. Congratulations Veritasium team!

[@clionekimura9604]

Leonard Mlodinow's Euclid's Window is a good book for those who want to dive deep into the history of geometry. Thank you, Veritasium, for animating the history and concepts. I am a math teacher and have always wanted to incorporate historical development of mathematics into math curriculum. I want to dilute the goal of math education from only problem solving to explaining math concepts. I want to tell stories like Galois' duel, Zeno's paradox, Newton & Leibniz's letters/rivalry, the centuries of proving Fermat's theorem (and Langland's program), al-Khwarizmi's completing the square, the discovery of the formula for cubic & higher (and the affairs) ....etc. Veritasiim and 3blue1brown are my favorite youtubers who have great influence to our online math community. Now we find THOUSANDS of math explainers on youtube who actually teach better than professors and grade school teachers (myself included).

[@sharadsemilo]

Start a channel. Please

[@albertandearthie7138]

I read it all 2 years ago lol

[@dlevi67]

More power to you!

[@Adroit1911]

The kind of mind you must have in order to actually want to teach math to the kids..... I commend you, and appreciate your drive to teach difficult things to difficult peoples.

[@kwaherikwasasa]

I read his Drunkard's Walk book, and it is spectacular! I will look into Euclids Window...😂

[@vvoodoocrafts8511]

I learned a lot from this. Thank you

[@anthonygordon9483]

Amazing how Vertasium can take a topic that anyone can explain in 5 minutes and stretch it out to 30 minutes. Everytime I see one of his fascinating videos. It makes me want to search it on youtube.

[@Sweet9964]

I used to hate math class in school, but the topics that Veritasium covers are so interesting I always find myself going down a mathematics rabbit hole after i watch one of his videos.

[@zwan1886]

because this isn't math, it's a history lesson and a story. You just hate making an effort which is what math requires and listening to a story does not.

[@IIT24Aspirant]

yeah and I bet this guy didn't understand the last 10mins of the video lol@@zwan1886

[@StarfireReborn]

​@@zwan1886 Funny Thing, I Hate Math But Love The Effort It Takes To Solve. 🤷🏽‍♀️

[@MsMiDC]

Learning is fun, but its different when its out of your own curiosity, rather than being forced upon you.

[@allrequiredfields]

I know. I'm realizing that my brain is no less inclined towards understanding complicated maths but at least I enjoy TRYING to understand it now.

[@kolakoala6702]

Hi Derek! I send you greetings from Göttingen. Right now, while I'm watching this, I am sitting at the Gauss Tower in the cities forest east of Göttingen and I am deeply touched by the emotional gravitas and the history behind the place and it's meaning for human society in conection with the 2000 years of history of the topic of your video. This is truly an intense moment for me it will probably stay in my mind for many, many years to come. Thank you very much ❤

[@moonman8450]

As a Geismaraner I can see the tower every day but as far as I can remember the last time I visited the tower was over 15 years ago 😅

[@tarmorboi5307]

As soon as he started to talk about Gauss and Göttingen, I thought about going to the Gaussturm to admire the stars and get high on my curiosity about this universe we live in :)

[@tarmorboi5307]

Göttingers unite!!🚀

[@sjsomething4936]

A new place to add to my bucket list! Enjoy your contemplations ☮️

[@thitran6105]

ok

[@DihelsonMendonca]

I love these episodes on the "History of science". It's great to know the names of those who created our time, and the world we live. 🎉❤❤❤

[@thomasguidon3763]

Excellent overview!

[@edsmith2562]

By far this is my favorite in the millions of explanitions on the subject of math.

[@valinhorn42]

As an EE and ham radio enthusiast, the Poincaré disk looked very familiar -- this transformation from 2D euclidian space to a finite-sized disk is exactly what we do to visualize complex impedances more easily. It's called a Smith diagram, it's an invaluable tool, and using it feels like doing black magic.

[@JBNemeth]

If only my teachers in high school had had this ability to present math in this way.... I'm grateful to finally be able to understand/grasp these concepts.

[@tannerman46]

Unfortunately making a KZbin video is completely different from teaching a class, and teaching is not scripted

[@jmc8076]

Teaching is live vs having an editing team for a content creator on social media w/millions of views. First relies on a fraction of the income and does their own prep. Not saying this channel didn’t take effort to build just not a fair comparison.

[@tylerhaslam2083]

I have had this conversation with my class. About 5% of students are riveted-- having in every word. The other 95% are zoned out.

[@rev0live752]

you didnt learn anything from this video its just an illusion

[@rizasid]

These math videos are fun to watch but go way over my head.

[@okeemchristie5847]

I can barely follow this guy but everything is so fascinating ❤

[@flintsparks8406]

The amount of work Veritasium puts in his videos is amazing

[@barnabas.csermely]

Literally doesn't know how to pronounce the MAIN guy's name. Did a 30 minute video and didn't bother to press play on Wikipedia next to the name.

[@iateuranium-235forbreakfas7]

he definitly has a production team working on the videos with him, but that doesnt take away from the passion he puts into explaining the topics he covers

[@kevinmahaley4916]

He's good but he's been wrong on some things

[@Nyxyz999]

​@@kevinmahaley4916maybe because he's human?

[@ekksoku]

@@kevinmahaley4916 wish I knew what being wrong was like, luckily I'm infallible

[@iamleonidus1742]

So what I've gathered from this video is that flat earthers just aren't thinking large enough. The earth may not be flat the universe is

[@Bynk333]

Wait flat universe? We are in Petri dish? :D

[@joebeastyg5686]

Not going to lie, this video has me on the edge. One thing I believe can still remain true.... Just because what we see above our heads my have a certain shape doesn't mean that the ground below us has to have the same.

[@charlesherman286]

No no no, the flat earthers are right. The earth IS flat, it just exists as a large mass in a spherical universe

[@WTfire10]

Problem is these guys deny the existence of the universe too.

[@joebeastyg5686]

@@WTfire10 Who are "these guys"?

[@danieledwards4305]

Hey, this is really good thank you very much!

[@PoopusLoopus]

I cannot stress how much this makes me wanna binge Euclid's Elements on a 5 hour road trip I have tomorrow

[@Kyanzes]

Just a side note: Bolyai is pronounced as Boyai or Bo-ya-e. The "ly" is an old form of "j" in Hungarian which is pronounced similarly to the English "y". Cool video.

[@sujoms]

Haha I just wrote the same.

[@csokasg]

Yes, he should have asked/looked it up how to pronunciate his name :)

[@melybuvar]

It was really bugging me during the video. Not a hard task to check the pronouncing on google translate... On the other hand great video

[@majdnemkocka]

Yeah, the very first line in the Wikipedia article on Bolyai gives the pronunciation as [ˈjaːnoʃ ˈboːjɒi], it wouldn't have been that hard to look up.

[@akosbakonyi5749]

I am Hungarian I didn’t realise he was talking about Bolyai until the middle of the video when it was written on the screen 😅

[@ClutchCps]

It's always nice to see Professor Kontorovich in a Veritasium video. You can see that he really loves what he is talking about.

[@ccost]

bro how many videos do u comment on for me to keep seeing you

[@ginalley]

@@Joe-sg9ll You'd be happy to know that mathematics proved that mathematics has true statements that cannot be proven

[@M2ProMBP]

All of you liking bot comments 😂😂😂

[@carnap355]

wot

[@carnap355]

wat

[@jackbuff_I]

just wanna say a sincere thank you to the whole Veratisium team. I hope you know you're creating future geniuses, by the bucket load! If my generation (~90's) had these resources to educate ourselves, I might have fulfilled my potential!

[@fishhuntadventure]

My advice: maximize your potential NOW; one can do nothing about the past yet the future is still in reach.

[@brienmacgearailt7801]

This just blew my mind

[@spunktasticjismmonkey8569]

I've always had some kind of mental block when it comes to taking in mathematics, like a minor learning disability, but this video was made in a way where I could follow and understand what was being explained. Thank you for that. 😊

[@tka3]

That's the magic of a professional science communicator

[@LordGino]

21:17 The man is falling from a skyscraper. Everyone: "Call an ambulance!". Einstein: *Is joyful*💀 Of course I know is just for the animation.

[@Trev0r98]

I still like the integral of the reciprocal function, which, when graphed and and then rotated in 3 dimensions about the x-axis, yields a conical object that has the following strange property: infinite surface area, but finite volume. In other words, one can quickly fill this conical object with paint, but there is not enough paint in the entire universe to cover the outer surface of this conical object.

[@dustin8420]

As a kid I worked on a client-side mod for a video game and ended up hearing a bit about Euclid when I needed an algorithm to boil down everyone's kill to death ratio.

[@deepakpottavatri7613]

10:33 how can someone be good at mathematics, violin, and dueling 🥶

[@assarlannerborn9342]

That’s what I am saying. In Ancient times everyone was a main character

[@theewzrrd-dc6dh]

😂😂😂😂

[@weshbesh2635]

It’s because they didn’t have TikTok

[@3548374]

I know, I know... Hungarian is a pretty difficult language, but if you just tap listen in the Google translator, you will be amazed, how big the difference pronouncing Bolyai János in English and in Hungarian. As a proud Hungarian, I thank you for this video!

[@TheTinytwirl]

I fricking love this show. Channel. Knowledge. This Guy

[@Iluhajjot]

Great video, but a bit weird that Lobachevsky is mentioned so briefly - he developed non-euclidian geometry earlier than Bolyai and in more detail. Lobachevsky definitely deserves more than just 3 seconds in a video about non-euclidian geometry which he pioneered.