"I had never heard this before but I find it too delightful not to tell." This dude's love for teaching is *SO OBVIOUS* and deep and genuine. Every video is made with special care and I won't be surprised if he edits each lesson about 20 times before uploading to get it just right. The *delight* is *ours,* Sensei.
I've been working through the lectures he did for Khan Academy for multivariable calculus and he just has an amazing method of conveying the intuition of a concept visually before teaching the proof. It isn't as refined as his more recent work on TH-cam, but I really appreciate what Grant does.
I always thought spirals r underrated hemchandra nos (popularly known as fibonacci no) himself showed the unique characteristics of spirals in nature let it be galaxies or flowers thats why the cholas had temples arranged according to golden ratio and golden mean
It's funny how the things you enjoy change when they become your job. For me as a mathematician, proving a theorem only to find out it's already been proven is frustrating. It's not entirely bad, because at least the fact that it's been done already means your proof (probably) isn't wrong. You also walk out of it understanding things very well, so it's not a waste of time. It's just frustrating that you can't turn your work into a paper (unless your proof is very different, in which case it's sometimes still worth publishing).
I remenber when i make the area formula for the diagonal of a square based on its side ( diagonal = sqrt of 2 Side) when i was at high school learning sen and cos , i was so freaking happy that i made a formula that give the awnser for common problems. Only to discover a year (?)later that that formula already exists.
@@AwakeAgainAtLast This is true in American English, but the convention is different in other countries. It's not a mistake, it's just a regional difference.
Hello! I'm currently taking a Mathematics course in college, and I'm kind of questioning myself why did I even enter this course. This video made me realize why I love math, and why I entered a Math course in the first place. Thank you very much for these super high quality videos!
@@dsdsspp7130 ?! That's not math at all and college math isn't that either. Math at the university level is seldom about memorizing formulas but rather about finding the right solutions to diverse problems and showing how.
@@dsdsspp7130 In my case it's all about demonstrations as of now. Knowing things like integrating or multiplying matrices is taught very quickly and isn't given much importance in homeworks/exams (most times, at least) compared to knowing how to demonstrate stuff.
My favorite approximation for pi is 977/311 because both numbers are themselves prime and have analogous locations when typed out on a standard number pad.
@@guilhermeottoni1367 at that point, why not just write 3.1416? The rest of the sixes only take you further from the true constant while also being more key presses and a division.
OMG, mankind is so lucky to have these two things: someone who can clearly explain some of the most complex subjects in math; and a simple means of making that knowledge accessible (TH-cam). I don't mean to imply that producing these videos is "simple".... no, it takes A LOT of time and effort to produce a video this wonderfully clear. Who ever thought that when TH-cam started, we would get to this point... we are so lucky.
I was essentially thinking the same thing. Comparing TH-cam in its early years to now, I certainly never imagined it being a vehicle for the best class lectures I never could have imagined.
As someone who understand only a little maths it's very easy to see a diagram like that and think there's some deeper truth to it. The way you explained that there isn't was absolutely brilliant, thanks.
@@sunnylilacs it's very easy to get swept up in patterns and start making broken logic leaps, consequence of the brain liking them so much there's an entire mental condition based on an extreme version of this tendency to get stuck to patterns
To begin with, I just can't even image how you even managed to make these stunning animations at such a large scale. ABSOLUTELY FANTASTIC!! Easily one of my favorite channels on TH-cam.
@@katech6020 I knew he programmed these demonstrations, but I didn't know exactly what he used to do so, so thanks for that! I'll have to check it out at some point.
"If you effectively reinvent ... before you've seen it defined... then when you do learn those topics, you'll see them as familiar friends, not as arbitrary definitions." This is my favorite thing about messing around with math and numbers, finding patterns, testing different ways of measuring their properties and more. I didn't know what integrals were before high school, but I knew that if I added up all the space underneath a graphed line or curve, then that would be useful for say... adding up the total distance a car travels while only knowing it's speed over time. When I finally learned about integrals, it made the topic so much more exciting for me. Thank you for continuing to make math fun and interesting for everyone who sees your videos!
Same... That really hit me right in the face when I heard it. I'd been looking at a number pattern thingy (the description isn't clear whenever I try to explain it so feel free to skip to the next paragraph) where I try to see how soon a digit repeats itself when raising a number to an integer power which I increase, and I found various patterns which seemed almost arbitrary. In the end, I spoke about it with my brother, and he told me how it was related to this very totient function, and gave me a brief explanation. So once I saw it even in this video, I felt more familiar and certain with myself.
What you said toward the end about accidentally rediscovering things people learned in the past bringing an intrinsic value to them that simply being taught lacks was...completely true. It reminds me of this time once in which I tried to use multidimensional arrays to represent the possible results of a series of coinflips and accidentally discovered that the number of heads has pascal's triangle embedded into it.
Important error correction: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence. Incidentally, his tactics also show that these residue classes have the same "density", but for an alternate formulation of density than the one shown in the video. Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this. In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann! My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know.
Variety of Everything Our host spent a lot of time putting together his own rendering and animation platform. I hope he’ll give us a comprehensive tour one day.
an important erratum and I surprise to myself get it in the second read. It is too specific information that is hard to google it and find some Wikipedia about it, well I don't research enough is true too. cheers!
@@wizedivine First, let's make it clear between ourselves, that plane is a surface of a 2-sphere with an infinite radius. Secondly: S1 sphere is a boundary of a 2d disc, S2 sphere is a surface of a 3d ball and S3 sphere is a surface of a 4d ball (neither of the latter two you can see, or, to remain on a side of caution, most of us can't see them). This goes on. I think, then, that you wanted to see something where spiral is drawn in 3 d space and coordinates are (r, α, β), where α and β are angles from the x and y positive axes. Pity we don't enjoy true 3d vision, but only a binocular ("stereographic"? - where did Riemann got his idea from) projection of such onto a part of a sphere. I guess, you can go on from here on your own. I think it would be doable in GeoGebra. (I think 3Blue1Brown should use the standard terminology for spheres in his other videos. Moreover, geometric algebra and a proper torus are waiting.)
Hands down one of the best "math-y" videos I've seen. One of the best concept breakdowns as well. Everything is clearly described in an easy-to-understand way, yet you don't shy from all the "overly pretentious" (lol) jargon. Finally, the call to study and understand interesting concepts ("be playful") where you may connect the dots later down the road is the best. Thank you
It’s always been amazing to me that early mathematicians could find the time to focus so deeply (without computers) on these abstract topics in number theory. Life then was generally shorter and rougher so they must have been incredibly dedicated.
On the contrary, one had a lot fewer distractions to lure them away from the thing that interested them. In this day and age of internet, it's very hard to keep yourself dedicated to one thing, there's always something else that demands your attention, that makes you feel like you're missing out on something.
Yeah they usually had other people to do their cooking, cleaning, and errands for them. Life was shorter and rougher for their cooks, their maids, and other house staff, not them so much. ;)
@@AroundTheBlockAgain Yes, this is true. The folks who figured this stuff out tended to be men of leisure, for whom day-to-day finances weren't a concern. Aside from the lack of modern amenities like electricity and running water, their lives were probably _easier_ than most of ours, not harder.
"Life was shorter" is largely a myth, caused by interpreting the "average lifespan" too narrowly. If you exclude those who died before reaching five years of age, the figure jumps up. By a _lot_. The fact that infant and toddler mortality was high enough to have such a substantial impact on the average can be taken as an indicator that the second part of that statement is broadly correct, though.
Hi, I'm a mathematician, and have to say, WOW, I enjoy your videos a lot, have just recommended your channel to a friend of mine who teaches in high school to show your vids to his students, perhaps, with your help, more young talented students will be "lured" to study mathematics:) thank you very much for your work!!
What field are you? I was doing Representation Theory and Number Theory, with a dash of Hyperbolic Geom....I wish I could get back to that. Its just that there is no way for me to return...and when youre sitting on an important proof, it is maddening.
I love your final point. I remember when I was at school, adding up the number of spots on a normal six-sided die, and then independently asking myself, and coming up with, the formula for "how many spots on a die on any number of sides?" - a question that was probably helped due to my D&D hobby making me familiar with the idea of dice with different numbers of sides." So I independently "invented" the formula for the triangle numbers, which is not a particularly great mathematical feat, but did allow me to stun a teacher who set the classic "Add up the numbers from 1 to 100" by answering it within a few seconds. Great video!
i loved the first point too. "How pretty but pointless patterns in polar plots of primes prompt pretty important ponderings on properties of those primes"
Hey guys, god here. I’m calling it. You’re digging too far into the source code and my gpu (godly processing unit) is struggling to keep up. I know this stuff is pretty and all, but it’s really just disguised spaghetti code (the Italians have yet to thank me for that). Another few million years of this and I’m going to get a 111 (angel numbers are just exit codes btw, don’t mess with that shit) error
"in case this is too clear for the reader" lmao Also, I absolutely love the ending. 3b1b, I wouldn't be half as enthusiastic about maths without your videos. Thanks so much!
@@henryg.8762 I would, and many others as well, but we sure do appreciate 3b1b for making amazing videos. It's people like him who make people that don't like math much at first, like it.
@@henryg.8762 i mean, i love math since i was 4 years old, and for the next 3 years i didn't even understand english enough to be able to understand any of this except that there's cool spirals and gaps between them. love for math is the kind of thing that you just need to start somehow, and then it grows on its own. it can grow faster, when you find things like googology or good math videos, or other stuff that is very enjoyable, but doesn't seem too useful at first, but even if you don't find these kinds of things, you can get just as far.
@@mironhunia300 Although it compares less in usefulness to e.g. calculus. I agree that every branch of mathematics which potentially has an application is very useful, I'm just doing a comparison. Personal opinions might differ, but eh.
@@GetIntoItDuhh applied math is not that fun. It needs you to focus on things you are not interested in. I'm a SWE, but hate programming, when I have to work on shit someone else wrote 15 years ago, probably drunk.
Aside from the astonishingly clear explanation of this problem, this is a great insight into why many people find maths difficult. Effective learning is about making connections between things. We often teach maths as "learn this set of rules", which has very few connections. Exploring patterns and then explaining them as this video does is much more powerful.
i just thought to myself: "wow this is fascinating. i cant believe i didnt know" but then saw that i actually already liked this video. it fucking sucks to be stupid
@3blue1brown What an amazing video would it be if you found out how the nature of these primes shown on screen interact with the compression algorithm meta-ly to the video causing the algo to glitch out like that
@@SARGAMESH The reason why the video “glitches” when he zooms is that the video has a certain maximum bitrate. TH-cam’s compression algorithm will not update pixels that don’t change. That saves bandwidth. However, when too much stuff is changing it has to reduce the resolution. There’s an amazing video on this. Google “tom scott youtube compression”. His video title is something about snow/confetti.
They should make the compression based on prime numbers, maybe that will specificly make this video better. But anyway, C++ is better than Rust. Just want to let you know aswell.
Euler was overcompensating for for the non-phonetic pronunciation of the spelling of his name. First class, every semester of math, teacher calls his name, "Leonard Yuler." "That's pronounced LeonHard Oiler." "Well, it looks like Yuler to me." "I can't help what it looks like to you...." and over time, decides to write 800-plus mathematical treatises just to make math teachers' lives everywhere miserable. And also, ours.
"Euler's totient function" sounds like something discovered not by Euler himself, but by his mother, during his toilet training, during the years he was studying enuresis. "He's got to get control of his totient function, or he'll never leave home"
When I was in sixth grade, I realized that the difference between any two consecutive squares was equal to the sum of their square roots. I was blown away by this fact, presented it to my teacher, and was ecstatic to learn that that tidbit generalized to the Difference of Two Squares. I then spent the next three years telling people I had discovered a theorem on my own, and I was so proud of what I had discovered by playing around and chasing patterns.
I didn't look at the time of the video before starting, and kind of assumed it was about 10 minutes. Then at the end, I thought...wow, that must have been only 5 minutes. LOL
I think this is an expression of the fact that math is typically motivated by the goal of explaining some particular phenomenon. For the Greeks, the were trying to explain and model the properties of geometry. For Newton, he was trying to explain motion. For Einstein, he was trying to explain weird things about gravity (although he took the math of general relativity from others). As math has grown increasingly complex over the centuries, it developed its own, non-physical phenomenons of interest. It's this sense of discovering patterns and relationships and being able to describe and explain them relatively simply that motivates us as humans to do math, and playing with problems on your own leads you to that sense in a way that memorizing and practicing a set of theorems can't.
This was awesome! Have you considered doing a follow-up on Dirichlet's theorem about Chebyshev's bias? For example, when you showed the histogram of primes 1, 3, 7, and 9 mod 10, there is a bias towards 3 and 7 mod 10 (because these are non-squares). Even though the categories all have 25% in the limit, there is quantitatively more primes 3 and 7 mod 10. The primes race is really compelling and not too hard to understand.
One implication he didn't go into: When plotting the numbers in whole number radians, each new number was 1/2pi rotations from the last one. So, the numbers made a spiral arm every time they encountered a number that was close to the denominator of a rational approximation of 1/2pi (that is to say, close to twice the numerator of a rational approximation of pi itself). But what if we didn't want to make spirals? What if we wanted all of our points to be as far away from other points as possible, *in every direction*? (Why we would need to do this is a point I'll come back to later.) If you're making spiral arms, there's a lot of space in between the arms that's wasted, and much less space between two neighbors on the same arm. Is there a way to avoid this? Well, if we want to find a number that gives us no spirals, we need it to have as few rational approximations as possible, (some of you might see where I'm going with this) we can look at continued fractions, since as explained in that Mathologer video, every time you encounter a large number in a number's continued fraction, you can truncate the sequence there and get a pretty good approximation. Thus, the ultimate not-close-to-any-rational-number number would have a continued fraction with numbers as low as possible. Ideally, made up of all 1's. This number happens to be (sqrt(5)+1)/2, known as the Golden Ratio. But getting back to why we would need to find points as far away from each other as possible: Well, what if we were a plant putting out seeds? We have a chemical process that rotates by a certain amount and then makes a seed. And we want those seeds to be spread out as efficiently as possible so that they don't have to compete for resources. If you've heard that the Golden Ratio shows up in nature, this is why.
I really appreciate your comment pointing towards the connections between math and nature and I think it would make another great video (I hope @3blue1brown reads this)! Do you maybe have a source for this that I can go to?
Yes, people should give the golden ratio more attention! It's got some crazy (cool) things too! Try this out: Find the line that connects the two inflection points of a quartic polynomial curve. Then, measure the distance between the outer intersections (the rightmost point and the leftmost point) and the inner intersections (the inflection points). It turns out that, provided that four distinct intersections exist, the ratio of the inner segment (the distance between the inflection points) to the outer segments (the distance between each of the outer intersections and the nearest inner intersection) is exactly the golden ratio. Furthermore, the two smaller areas enclosed (on the left and right) by the inflection line and the quartic curve are each exactly half the size of the larger area (in the middle). Why this happens probably comes down to a nasty algebraic nightmare with calculus, and things simplify to the golden ratio and whatnot. I'm sure it's possible to prove it. I tried to do it myself but got lost in the awfully complicated algebra (trust me, it's ridiculous). Maybe there's a neater and more elegant proof than that, though. 3Blue1Brown? Care to tackle this one?
@seba cea Andre Weil once said that understanding a problem that you have been working on endlessly can lead to a feeling of ecstasy for weeks at a time
The end of this video took me to such an emotional high. Its nice to see others who care so deeply about a subject you also care so deeply about. In a way it was like our spirits became one ... although philosophically I am not sold on 'spirits', that's the language I have to use to describe this feeling.
I cant wrap around how math can be so beatiful, it's like reading a really good novel that has many intresting characters and plots that are always more deep and connected that they lead you on at the start. Sometimes it requires more work to piece all the parts together but man the result is incredible
@@jamesr2936it's not as fancy as an entire novel filled with gestures... It's more like musiComposition. (It repeatStatic loops when charted, but cannot "break the mold" via willpower. It is the environmental medium, the reflecTensor allowing language). Language, on the other hand, is not exact or predictable, for having synonym varianTone stretching.
Max Planck is my favorite physicist, Wolfgang Paulie for chemist and the father of my own specialty, Mendelev for biology (more genetics, but close enough). I spent 25 odd years of my life treating mathematics as just a tool to an end, but every time I watch one of your videos, I'm reminded that you are my favorite mathematician... A sentence I never thought I'd say about anyone. Thank you for everything.
Gregor Mendel was the pioneer of genetics. Dimitri Mendeleev was a chemist. To err is human, to forgive divine, but please forgive a correction also. ❤
This is such a beautifully clear video. I've seen this prime spiral meme before and like you said thought it was due to some mysterious property of primes. Thank you for demystifying this and somehow leaving me even more amazed by the simplicity of the mathematics causing it and the more interesting topic that it brushes up against.
I am not a mathematical phenom or engineer, but ai do find immense beauty in the visualizations of mathematical concepts! Like this, visualizing Fourier transform, visualizing how to turn a circle inside out, mathematical proofs, etc!
"How pretty but pointless patterns on polar plots of primes prompt pretty important ponderings on properties of those primes." C'mon man, let me just watch a minute or two of the video before forcing me to like it.
It's more like showing a meme to your parents and they say "oh, cool" and then share a really deep story from their lives that relates to that meme, showing you worlds beyond and making you feel really good and loved.
Shatterdpixel And a random fourth whose parents didn't say anything at all... But the children still heard everything that needed to be said and eventually learned why primes form spirals. Clearly it's used in the flex capacitor to initialize time travel 🤓
2:25 that animation and change of music was utterly beautiful, that type of beauty you wouldn't expect to find, yet still it's there, waiting to be discovered.
I just randomly stumbled upon this, and it has me absolutely fascinated (from both the resulting math and the lucidity of the video/explanation itself). Amongst my other playlists for memes, drumming, etc., I now have one titled "Beautiful Math". I feel compelled to fill it with others and take the time to understand it all! Thank you so much for creating this incredible lesson! :)
Oh yes, waited such a long time for this! Quick Request: since you're doing Number Theory, can you prove Fermat's Last Theorem? I believe the proof is quite trivial, so it shouldn't be too bad :P
Yeah, Fermat's last theorem is an easy one.. definitely should be a video. In fact, I just found a nice proof for it, but I'm afraid it won't fit in this youtube comment.
@@FacultyofKhan This is what he said: "I'm not saying no, but let's just say this would be a very big project :) Certainly some special cases might be doable and interesting."
I really like how this visualization shows the principles of emergent properties. Given a set of rules, any system, complex or simple, will have properties emerge that are non-obvious from the inception of the system. This is one of my favorite observations of universal principles.
Math has a lot of subtle patterns, often too convoluted to see the whole picture and beauty all at once. But with each careful step, you can get closer to seeing how various things and ideas/concepts fit together, and that, in the end, can give you a deep appreciation for how it all works. It's really cool just how abundant patterns can seem around us.
This is where it started for me. A recommendation of this video is how I found your channel and it let to sth important, at least for me. I am going to start teaching theoretical computer science soon and, although it is not really the topic of your channel, I will try to use as many of your tips on conveying ideas visually as I can. Thanks and keep up the great work!
When you zoomed out from the initial set of prime numbers I got chills all over seeing that beautiful spiral come out of the numbers like that. Bravo! This is fascinating!
OMG THIS VIDEO IS AWESOME! All of this is contained in the natural constant. The prime series diverge into Euler's natural constant e. This is the universe!
I feel I am not wasting my time when I watch these videos, when in reality I am. Having said that, I am immensely amazed at the dedication, knowledge and teaching skill of the creators of these videos. A big thank you to you.
I really enjoy this kind of thing, even if I don't understand it. I tried, I dabbled up until AP calculus in high school. But that was the point my eyes started to glaze over. So I understand 'just' enough to get what's happening and why the spiral, and the gaps in the spiral, is honestly not that big a deal. Fun video!
Hey I’m going through a very tough and stressful times and I wanted to say that seeing your video in my feed just made me smile and actually really excited me. Thank you
I am writing a big school paper on RSA encryption, and wanted to watch some 3b1b videos so i just took one i hadnt watched. And boy does it feel satisfying when you started talking about eulers totient and coprime numbers because i have been learning so much about that stuff. Great video
I remember learning about reside classes in school. We had this little experiment going on where the computer science and math teachers tag teamed us for a couple lessons to teach us about RSA. Honestly, that was a cool idea and I wish more students got to experience something like it. Seeing how different subjects connect with each other is really special.
14:29 In Dirichlet's theorem on arithmetic progressions, he actually proves that each of the none disappearing "arms" has infinitely many primes. Great Video! It brought back great memories of my Number Theory days. I thoroughly enjoyed it. Thanks!
I once had the idea of plotting points like this when someone asked me how I might calculate prime numbers. I suggested that it might be possible to create an algorithm to calculate the position of every "prime" point with increasing levels of accuracy. I never bothered actually trying it, but it's nice the see that the idea is worth something.
As someone who's dyscalculiac, this was freaking WILD to see. I couldn't have fathomed something like this in my lifetime, but.. Y'all out here killin this maths stuff and I'm so proud of you, and happy to have seen this.
I mean, a million primes sounds like a lot, but with a half-decent computer it would only take a few minutes at most to calculate it's still pretty awesome, though
You can use the sieve of eratosthenes and it should be very fast. For Python: Def Sieve(): Primes = [2,3,5,7] a = 8 While len(primes) < n: For i in Primes: If a % i == 0: Break Else: Primes.append(a) a += 1 print(Primes) Sieve()
I'm a 56-year old engineer. I sometimes wonder how my life (maybe the world) would have been different if on Saturday mornings other kids like me had watched videos like these instead of cartoons.
This was one of the most greatest math videos I've ever seen in my whole life. And it kinda makes you think that if an irrational number like Pi was a little different in our universe, how things would actually look like...
Pi can't really be different. It's just a result of the axioms of Euclidean Geometry and the definition of a circle. It doesn't even have anything to do with our universe because space in our universe is non-Euclidean.
I'm thankful to be taught this now. I feel for all the people now gone (or still alive but nevertheless will never have the opportunity) who may have been completely enamored by this privilege. But yes, I do wish this was available during school. No doubt there are some lucky students out there with splendid teachers at this moment
Schools are forced to ensure kids can pass tests more than anything else. In the UK the benchmark for GCSE exams (sat when you're 15/16) is (well, was) a C grade. Getting a D-grade kid to a C meant a lot to the school, so way more effort was expended by teachers in that area. The high flying kids, those that could get an A without too much trouble, weren't pushed anywhere near as much. Not the fault of the teachers, I might add. The school doesn't care if an A* possible kid only gets an A, that won't really affect stuff like funding. It's all about getting kids to pass. I'm not saying the struggling kids shouldn't be helped, but it shouldn't be that schools have to prioritise them any more than kids with a high potential in that subject. They recently changed the grading from letters (A* to F) to numbers (9 to 1). Why? No doubt there are 'reasons' but it does not seem a priority to me. But that shows the nature of schools nowadays.
I was lucky enough to have a math teacher who showed us the beauty of math (in the 80’s, no youtube just chalk but he did it). This guy even gave up his pauses if anyone didn’t grasp anything and would explain again or explain more (out of scope for the exams) for those who were interested. My wife didn’t have this luck and always thought she was bad at everything math until I started explaining things I remembered after we got married. Long story short, she went back to university while working full time (didn’t attend most classes), did every year in half the time and has since become top of the field in her profession and gives guest colleges at several universities. She’s the perfect example of how the spark was not lit up because her teachers failed. This is why youtube is such an important tool right now where capable and driven people can enlighten the people who are interested and hopefully light many sparks!
do we find patterns beautiful because everything is in a pattern - or do we find patterns beautiful because we were "programmed" to like patterns, or both?
@@glaswasser Now silly as this question/joke might seem, the answer is quite worth it to look into. You see, pattern give us an important ability: to predict. Then of course, creatures that are programed to see patterns might predict things better, and be better at living at a whole. And what is a better reason to look for patterns, than its beauty?
Patterns exist. Humans are keen towards them because our brains allow us to recognize them. Patterns are caused by stimuli. We are intelligent enough to domesticate those stimuli if we comprehend them.
I want to thank you for this, because I think the ending had more of a philosophical spiritual answer that helped me look at life in another new light that I enjoyed.
Even during desinging of simple DSP for my radio amateur transceiver, I've been taking pi to about 9 decimal places to have enough frequency accuracy (nearly 10 Hz) in my "narrow" working band (30 MHz). And I wouldn't be surprised, if serious engineers take pi much more accurate. E.g. in automatic control theory, while estimaing safety margin of some closed-loop control system.
Even if I don't understand all technical aspects of your videos I really appreciate the visualization that give me an deeper understanding of mathematical problems. Thanks!
he didn't do any of this by hand, he codes the animations in a custom python library. so it would probably be something like he creates a list of primes and then feeds that list into a function that sorts it into the categories for the histogram.
@@tylerbreisacher5841 Oh goddamn thank you. Every time I watch his videos, I find myself really wanting to animate other formulae/functions. You just made my week.
@@tylerbreisacher5841 does he actually upload the code of his videos on manim repository? I doubt so? because he once mentioned that it is more like a work in progress. please give me a heads up if you find something he did in the videos on the repository, I'll be super motivated to recreate or create new animations
@@sudheernaidu6646 at least some of it. For example here's some of the code for the colliding blocks. github.com/3b1b/manim/tree/master/old_projects/clacks
One of your best videos. I remember how Harry Potter books made some kids into reading books. Well this kind of videos can make kids into liking maths. Bravo!
I took number theory the previous semester in uni and now i can see the point of it. Brilliant job! The visualization of such theoretical problems is so helpful
Indeed. Curiosity may have killed the cat, but it also made us consider the possibility that the cat may in fact also be alive too until one opens the box and the possibilities collapse.
@@shandy_2001 but then they should pick another video. the easy to understand visual preparation is teaching in its highest art. those videos pose a reference for how to spread the ideas of Mathematics to the general public/audience. there's nothing to dislike about that, imho.
What you said toward the end about accidentally rediscovering things people learned in the past bringing an intrinsic value to them that simply being taught lacks was...completely true. It reminds me of this time once in which I tried to use multidimensional arrays to represent the possible results of a series of coinflips and accidentally discovered that the number of heads has pascal's triangle embedded into it.
i know this is gonna sound cheesy but, your explanations and the amazing visuals on this video are so outstandingly clear!! i've never found a channel on youtube so far as great as conveying knowledge and information as you are, the way you approach the content and explain it is truly marvelous. so thanks for making these awesome high quality videos :)
@@hbushnell Dirichlet (despite the Pierre and Lejeune in P.G.L. Dirichlet) was a German mathematician. So the usual tendency is to pronounce the 'ch' closer to a 'k' sound than an 'sh' sound. I myself as a native English speaker say it more like Dearie Clay (emphasis on first and third syllables). I consider the narrator's pronunciation somewhat unusual.
This video has had me thinking about this for a long, long time. I love how it is so easy for the mind to "see" the rays. But the "spirals" are just discontinuous dots that happen to be near each other, and the human brain is compelled to connect the dots. I wonder how often we get fooled by things like sub atomic particle data or astrophysics data that is really discontinuous, and assign some kind of relation that might not really be there, but "kind of" explains the data so we stop looking for other explanations. Fine job on the whole idea, and the presentation. Many thanks.
yes, i do learn more effectively when it's something i voluntarily click on to learn. I like learning when it's interesting, not when i'm stuck doing an hour of homework to "review" things i can already do perfectly well.
![[TH] 2024 PMGC League | Survival Stage วันที่ 1 | PUBG MOBILE Global Championship](http://i.ytimg.com/vi/FVFT-PFwU80/mqdefault.jpg)
At this point, the word “beautiful” isn’t even enough to describe the sheer elegance and clarity of these videos. Amazing as always.
Yep!!!
i was just about to say beautiful
Couldn't agree more!!!
also Amazingk!!
How about bootyful?
"I had never heard this before but I find it too delightful not to tell." This dude's love for teaching is *SO OBVIOUS* and deep and genuine. Every video is made with special care and I won't be surprised if he edits each lesson about 20 times before uploading to get it just right. The *delight* is *ours,* Sensei.
Your reply is so apt and true.
I want to teach like him.
I've been working through the lectures he did for Khan Academy for multivariable calculus and he just has an amazing method of conveying the intuition of a concept visually before teaching the proof. It isn't as refined as his more recent work on TH-cam, but I really appreciate what Grant does.
When I find something to delightful not to tell, some people around me just say "does it sell?".
I always thought spirals r underrated hemchandra nos (popularly known as fibonacci no) himself showed the unique characteristics of spirals in nature let it be galaxies or flowers thats why the cholas had temples arranged according to golden ratio and golden mean
totally agree... subscribed right a way
As a maths lover, proving a theorem before you knew it existed is undeniably the best feeling I would ever experience
It's funny how the things you enjoy change when they become your job. For me as a mathematician, proving a theorem only to find out it's already been proven is frustrating. It's not entirely bad, because at least the fact that it's been done already means your proof (probably) isn't wrong. You also walk out of it understanding things very well, so it's not a waste of time. It's just frustrating that you can't turn your work into a paper (unless your proof is very different, in which case it's sometimes still worth publishing).
I remenber when i make the area formula for the diagonal of a square based on its side ( diagonal = sqrt of 2 Side) when i was at high school learning sen and cos , i was so freaking happy that i made a formula that give the awnser for common problems. Only to discover a year (?)later that that formula already exists.
Math is already plural. You don't need to add an s to say "maths", it's redundant.
@@AwakeAgainAtLast This is true in American English, but the convention is different in other countries. It's not a mistake, it's just a regional difference.
@@AwakeAgainAtLast woke up on the wrong side of the bed?
3:22 "If you patiently went through each ray"
I can hear it in your voice, thank you 3Blue1Brown for your meticulous work in counting each ray
Well he can just use the theorem to see that theres 280
it's 3:15 but nice comment
It's clearly not pointless, I mean, look at all of those dots!
I feel this comment is underrated.
@@juhonuorala3512 That's a nice line.
lmfao
But even a dot has an inside and an outside.
Of course, that is not the point of zero dimensions.
you now have 0 likes in 8 bit systems.
Hello! I'm currently taking a Mathematics course in college, and I'm kind of questioning myself why did I even enter this course. This video made me realize why I love math, and why I entered a Math course in the first place. Thank you very much for these super high quality videos!
@@dsdsspp7130 ?! That's not math at all and college math isn't that either. Math at the university level is seldom about memorizing formulas but rather about finding the right solutions to diverse problems and showing how.
@@_cytosine Depends on the country.
@@dsdsspp7130 In my case it's all about demonstrations as of now. Knowing things like integrating or multiplying matrices is taught very quickly and isn't given much importance in homeworks/exams (most times, at least) compared to knowing how to demonstrate stuff.
unite perry at least take a proofs class. That's where math gets fun
Keep going!! Math is pretty cool.
My favorite approximation for pi is 977/311 because both numbers are themselves prime and have analogous locations when typed out on a standard number pad.
That's actually really cool
Mine is always when you calculate (355+22)/(113+7) = 377/120 = 3.14166666... The repeating part has only one digit.
977-311=666
@@guilhermeottoni1367 at that point, why not just write 3.1416? The rest of the sixes only take you further from the true constant while also being more key presses and a division.
That's the most nerdy thing i ever heard anyone say, and i like it.
The 977-311=666 makes it even better xD
OMG, mankind is so lucky to have these two things: someone who can clearly explain some of the most complex subjects in math; and a simple means of making that knowledge accessible (TH-cam). I don't mean to imply that producing these videos is "simple".... no, it takes A LOT of time and effort to produce a video this wonderfully clear. Who ever thought that when TH-cam started, we would get to this point... we are so lucky.
I was essentially thinking the same thing. Comparing TH-cam in its early years to now, I certainly never imagined it being a vehicle for the best class lectures I never could have imagined.
As someone who understand only a little maths it's very easy to see a diagram like that and think there's some deeper truth to it. The way you explained that there isn't was absolutely brilliant, thanks.
Wobblycogs Workshop Why does learning the explanation behind it make you think there isn’t deeper truth?
@@sunnylilacs Same reason nobody believes in unicorns - they don't _need_ to exist because there is no evidence requiring unicorns as an explanation.
@@sunnylilacs it's very easy to get swept up in patterns and start making broken logic leaps, consequence of the brain liking them so much
there's an entire mental condition based on an extreme version of this tendency to get stuck to patterns
Dopamine Cloud so why are there patterns at all?
@@hermanubis96 Because pattern recognition was adaptive and beneficial, therefore it was selected for during human evolution.
To begin with, I just can't even image how you even managed to make these stunning animations at such a large scale. ABSOLUTELY FANTASTIC!! Easily one of my favorite channels on TH-cam.
He is using manim, which is a python library that he created to make this video. you can check it out in Github
@@katech6020 I knew he programmed these demonstrations, but I didn't know exactly what he used to do so, so thanks for that! I'll have to check it out at some point.
"If you effectively reinvent ... before you've seen it defined... then when you do learn those topics, you'll see them as familiar friends, not as arbitrary definitions."
This is my favorite thing about messing around with math and numbers, finding patterns, testing different ways of measuring their properties and more.
I didn't know what integrals were before high school, but I knew that if I added up all the space underneath a graphed line or curve, then that would be useful for say... adding up the total distance a car travels while only knowing it's speed over time. When I finally learned about integrals, it made the topic so much more exciting for me.
Thank you for continuing to make math fun and interesting for everyone who sees your videos!
Same... That really hit me right in the face when I heard it. I'd been looking at a number pattern thingy (the description isn't clear whenever I try to explain it so feel free to skip to the next paragraph) where I try to see how soon a digit repeats itself when raising a number to an integer power which I increase, and I found various patterns which seemed almost arbitrary. In the end, I spoke about it with my brother, and he told me how it was related to this very totient function, and gave me a brief explanation. So once I saw it even in this video, I felt more familiar and certain with myself.
What you said toward the end about accidentally rediscovering things people learned in the past bringing an intrinsic value to them that simply being taught lacks was...completely true.
It reminds me of this time once in which I tried to use multidimensional arrays to represent the possible results of a series of coinflips and accidentally discovered that the number of heads has pascal's triangle embedded into it.
Important error correction: In the video, I say that Dirichlet showed that the primes are equally distributed among allowable residue classes, but this is not historically accurate. (By "allowable", here, I mean a residue class whose elements are coprime to the modulus, as described in the video). What he actually showed is that the sum of the reciprocals of all primes in a given allowable residue class diverges, which proves that there are infinitely many primes in such a sequence. Incidentally, his tactics also show that these residue classes have the same "density", but for an alternate formulation of density than the one shown in the video.
Dirichlet observed this equal distribution numerically and noted this in his paper, but it wasn't until decades later that this fact was properly proved, as it required building on some of the work of Riemann in his famous 1859 paper. If I'm not mistaken, I think it wasn't until Vallée Poussin in (1899), with a version of the prime number theorem for residue classes like this.
In many ways, this was a very silly error for me to have let through. It is true that this result was proven with heavy use of complex analysis, and in fact, it's in a complex analysis lecture that I remember first learning about it. But of course, this would have to have happened after Dirichlet because it would have to have happened after Riemann!
My apologies for the mistake. If you notice factual errors in videos that are not already mentioned in the video's description or pinned comment, don't hesitate to let me know.
(first) How do you even make the visuals and graphs on your computer? Probably some programming or something :P
Variety of Everything Our host spent a lot of time putting together his own rendering and animation platform. I hope he’ll give us a comprehensive tour one day.
an important erratum and I surprise to myself get it in the second read. It is too specific information that is hard to google it and find some Wikipedia about it, well I don't research enough is true too. cheers!
@@wizedivine First, let's make it clear between ourselves, that plane is a surface of a 2-sphere with an infinite radius. Secondly: S1 sphere is a boundary of a 2d disc, S2 sphere is a surface of a 3d ball and S3 sphere is a surface of a 4d ball (neither of the latter two you can see, or, to remain on a side of caution, most of us can't see them). This goes on. I think, then, that you wanted to see something where spiral is drawn in 3 d space and coordinates are (r, α, β), where α and β are angles from the x and y positive axes. Pity we don't enjoy true 3d vision, but only a binocular ("stereographic"? - where did Riemann got his idea from) projection of such onto a part of a sphere. I guess, you can go on from here on your own. I think it would be doable in GeoGebra. (I think 3Blue1Brown should use the standard terminology for spheres in his other videos. Moreover, geometric algebra and a proper torus are waiting.)
3Blue1Brown I’m a German ninth grader and I like maths and ur vid but now my brain makes weird noises and smokes.
Oh, truly a piece of art. I’ve never seen a movie which expresses the cliché that “math is beautiful” better than this video!!! I love this!
I wasn't ready for how beautiful the "zoom out" was going to be
it loses so much after first viewing but is still brilliant
When he zoomed out all I could do was to stare at it, fascinated with my mouth open.
Check the original answer from the link above. Once zoom out will shock you more. Because the beams are actually spiral again.
@@TheHwiwonKim do they turen abck into beams?
Time Stamp?
Hands down one of the best "math-y" videos I've seen. One of the best concept breakdowns as well. Everything is clearly described in an easy-to-understand way, yet you don't shy from all the "overly pretentious" (lol) jargon. Finally, the call to study and understand interesting concepts ("be playful") where you may connect the dots later down the road is the best. Thank you
It’s always been amazing to me that early mathematicians could find the time to focus so deeply (without computers) on these abstract topics in number theory. Life then was generally shorter and rougher so they must have been incredibly dedicated.
On the contrary, one had a lot fewer distractions to lure them away from the thing that interested them. In this day and age of internet, it's very hard to keep yourself dedicated to one thing, there's always something else that demands your attention, that makes you feel like you're missing out on something.
Yeah they usually had other people to do their cooking, cleaning, and errands for them. Life was shorter and rougher for their cooks, their maids, and other house staff, not them so much. ;)
@@AroundTheBlockAgain Yes, this is true. The folks who figured this stuff out tended to be men of leisure, for whom day-to-day finances weren't a concern. Aside from the lack of modern amenities like electricity and running water, their lives were probably _easier_ than most of ours, not harder.
@@WhyBhanshu That's exactly what I was thinking last evening.
"Life was shorter" is largely a myth, caused by interpreting the "average lifespan" too narrowly. If you exclude those who died before reaching five years of age, the figure jumps up. By a _lot_.
The fact that infant and toddler mortality was high enough to have such a substantial impact on the average can be taken as an indicator that the second part of that statement is broadly correct, though.
Hi, I'm a mathematician, and have to say, WOW, I enjoy your videos a lot, have just recommended your channel to a friend of mine who teaches in high school to show your vids to his students, perhaps, with your help, more young talented students will be "lured" to study mathematics:) thank you very much for your work!!
What field are you? I was doing Representation Theory and Number Theory, with a dash of Hyperbolic Geom....I wish I could get back to that. Its just that there is no way for me to return...and when youre sitting on an important proof, it is maddening.
I'm 43 and it made me pick up the books!
True! Also I wonder if perfect numbers would do something...
@@RipRoaringGarage😂 anyone can claim to be a mathetician online.
I love your final point. I remember when I was at school, adding up the number of spots on a normal six-sided die, and then independently asking myself, and coming up with, the formula for "how many spots on a die on any number of sides?" - a question that was probably helped due to my D&D hobby making me familiar with the idea of dice with different numbers of sides." So I independently "invented" the formula for the triangle numbers, which is not a particularly great mathematical feat, but did allow me to stun a teacher who set the classic "Add up the numbers from 1 to 100" by answering it within a few seconds. Great video!
You gaussed them!
same
i loved the first point too. "How pretty but pointless patterns in polar plots of primes prompt pretty important ponderings on properties of those primes"
Yeah exactly 💯
@@Adventurin_hobbit yo sir give more formulaes
Hey guys, god here. I’m calling it. You’re digging too far into the source code and my gpu (godly processing unit) is struggling to keep up. I know this stuff is pretty and all, but it’s really just disguised spaghetti code (the Italians have yet to thank me for that). Another few million years of this and I’m going to get a 111 (angel numbers are just exit codes btw, don’t mess with that shit) error
gimme more ram - sincerely, a program.
"in case this is too clear for the reader" lmao
Also, I absolutely love the ending. 3b1b, I wouldn't be half as enthusiastic about maths without your videos. Thanks so much!
who would
@@henryg.8762 I would, and many others as well, but we sure do appreciate 3b1b for making amazing videos. It's people like him who make people that don't like math much at first, like it.
@@henryg.8762 i mean, i love math since i was 4 years old, and for the next 3 years i didn't even understand english enough to be able to understand any of this except that there's cool spirals and gaps between them. love for math is the kind of thing that you just need to start somehow, and then it grows on its own. it can grow faster, when you find things like googology or good math videos, or other stuff that is very enjoyable, but doesn't seem too useful at first, but even if you don't find these kinds of things, you can get just as far.
And I almost cried after reading "Be playful". Really amazing conclusions you gave us here!
I feel yah, me too man.
th-cam.com/video/iFuR97YcSLM/w-d-xo.html - slightly similar concept, but not. Conclusions are totally different.
The number theory ones are always so interesting!
I feel like number theory is more useless than other branches but it poses some interesting and often difficult problems.
@@erikkonstas Number theory is the basis for cryptography, so it's pretty much one of the most useful branches of mathematics right now.
@@mironhunia300 Although it compares less in usefulness to e.g. calculus. I agree that every branch of mathematics which potentially has an application is very useful, I'm just doing a comparison. Personal opinions might differ, but eh.
All the ones are always so interesting!
number theory is bs
I don't even LIKE math, but this was amazing.... and I wasn't completely lost for most of the video! You're a brilliant communicator.
Math is like beer. You won't like it to begin with, but drink some good... And you are lost to it :)
@@shardator ive worked in a math-focused field for almost a decade; still hate it.
@@GetIntoItDuhh applied math is not that fun. It needs you to focus on things you are not interested in. I'm a SWE, but hate programming, when I have to work on shit someone else wrote 15 years ago, probably drunk.
Me: *a math disliker* (20-1 kicked my ass cus my teacher sucked)
Also me: PATTERNSSS PATTERNS PATTERNS!!
Aside from the astonishingly clear explanation of this problem, this is a great insight into why many people find maths difficult. Effective learning is about making connections between things. We often teach maths as "learn this set of rules", which has very few connections. Exploring patterns and then explaining them as this video does is much more powerful.
i just thought to myself: "wow this is fascinating. i cant believe i didnt know"
but then saw that i actually already liked this video. it fucking sucks to be stupid
LMFAO
LMAO
RIP.
@ゴゴ Joji Joestar ゴゴ heroin or hero in? x)
@@aeiouaeiouaeiou hero(br)in(e)?
@@aeiouaeiouaeiou clever junkie
3Blue1Brown: Zooming out
TH-cam Compression: Dies
@3blue1brown What an amazing video would it be if you found out how the nature of these primes shown on screen interact with the compression algorithm meta-ly to the video causing the algo to glitch out like that
@@SARGAMESH The reason why the video “glitches” when he zooms is that the video has a certain maximum bitrate. TH-cam’s compression algorithm will not update pixels that don’t change. That saves bandwidth. However, when too much stuff is changing it has to reduce the resolution. There’s an amazing video on this. Google “tom scott youtube compression”. His video title is something about snow/confetti.
Arpan Dhatt mkbhd proved it with his 1000 upload test
@@arpandhatt6011 Not to mention aliasing, which you'll unavoidably have at that kind of graphics
They should make the compression based on prime numbers, maybe that will specificly make this video better. But anyway, C++ is better than Rust. Just want to let you know aswell.
You, sir, are one of the most effective teachers on the planet.
"Euler's totient function." I swear, Euler had a hand in everything.
Euler, the madlad of math.
euler’s hoarding problem
Soo... you're telling me I need to look up Euler
Euler was overcompensating for for the non-phonetic pronunciation of the spelling of his name.
First class, every semester of math, teacher calls his name, "Leonard Yuler."
"That's pronounced LeonHard Oiler."
"Well, it looks like Yuler to me."
"I can't help what it looks like to you...." and over time, decides to write 800-plus mathematical treatises just to make math teachers' lives everywhere miserable. And also, ours.
"Euler's totient function" sounds like something discovered not by Euler himself, but by his mother, during his toilet training, during the years he was studying enuresis. "He's got to get control of his totient function, or he'll never leave home"
When I was in sixth grade, I realized that the difference between any two consecutive squares was equal to the sum of their square roots. I was blown away by this fact, presented it to my teacher, and was ecstatic to learn that that tidbit generalized to the Difference of Two Squares. I then spent the next three years telling people I had discovered a theorem on my own, and I was so proud of what I had discovered by playing around and chasing patterns.
This is the exact same thing I discovered, litterally exact same story. mind blowing
In 6th grade?
Yep! I got bored a lot in school. Haha.
OMG I DID THE EXACT SAME THING WHEN I WAS A KID TOO
This was so entertaining I didn't even realize that was 22 minutes long, I love this♥️
I didn't look at the time of the video before starting, and kind of assumed it was about 10 minutes. Then at the end, I thought...wow, that must have been only 5 minutes. LOL
Same, you were the one that made me look at the time stamp for the first time
The excitement in your voice reflects the love you've got for mathematics. Hence, your videos are truly labour of love. KEEP IT UP!
When you discover math before you learn the math theorem, then the theorem becomes your friend instead of an arbitrary inconvenience.
Well said
You're a fucking genius.
I think this is an expression of the fact that math is typically motivated by the goal of explaining some particular phenomenon. For the Greeks, the were trying to explain and model the properties of geometry. For Newton, he was trying to explain motion. For Einstein, he was trying to explain weird things about gravity (although he took the math of general relativity from others). As math has grown increasingly complex over the centuries, it developed its own, non-physical phenomenons of interest. It's this sense of discovering patterns and relationships and being able to describe and explain them relatively simply that motivates us as humans to do math, and playing with problems on your own leads you to that sense in a way that memorizing and practicing a set of theorems can't.
@@BladeOfLight16 waffle
@@heroricspiritfreinen38 not really
I'm smiling before even the video is started :)
Same :)
Samea
First you click on the video, then thumbsup, then it loads :D
Why? Something is wrong with you guys? Maybe you need professional help?
(.❛ ᴗ ❛.)
*The ultimate connect the dots game*
300 likes comment-less?? Don't think so.
*takes a deep breath*
Let me take the time to talk to you about category theory.
Indeed, and on so many levels. ☺
IMPOSTER
I AM THE TRUE POTATO
This was awesome! Have you considered doing a follow-up on Dirichlet's theorem about Chebyshev's bias? For example, when you showed the histogram of primes 1, 3, 7, and 9 mod 10, there is a bias towards 3 and 7 mod 10 (because these are non-squares). Even though the categories all have 25% in the limit, there is quantitatively more primes 3 and 7 mod 10. The primes race is really compelling and not too hard to understand.
One implication he didn't go into:
When plotting the numbers in whole number radians, each new number was 1/2pi rotations from the last one. So, the numbers made a spiral arm every time they encountered a number that was close to the denominator of a rational approximation of 1/2pi (that is to say, close to twice the numerator of a rational approximation of pi itself). But what if we didn't want to make spirals? What if we wanted all of our points to be as far away from other points as possible, *in every direction*? (Why we would need to do this is a point I'll come back to later.) If you're making spiral arms, there's a lot of space in between the arms that's wasted, and much less space between two neighbors on the same arm. Is there a way to avoid this?
Well, if we want to find a number that gives us no spirals, we need it to have as few rational approximations as possible, (some of you might see where I'm going with this) we can look at continued fractions, since as explained in that Mathologer video, every time you encounter a large number in a number's continued fraction, you can truncate the sequence there and get a pretty good approximation. Thus, the ultimate not-close-to-any-rational-number number would have a continued fraction with numbers as low as possible. Ideally, made up of all 1's. This number happens to be (sqrt(5)+1)/2, known as the Golden Ratio.
But getting back to why we would need to find points as far away from each other as possible: Well, what if we were a plant putting out seeds? We have a chemical process that rotates by a certain amount and then makes a seed. And we want those seeds to be spread out as efficiently as possible so that they don't have to compete for resources.
If you've heard that the Golden Ratio shows up in nature, this is why.
I really appreciate your comment pointing towards the connections between math and nature and I think it would make another great video (I hope @3blue1brown reads this)! Do you maybe have a source for this that I can go to?
Awesome!
Yes, people should give the golden ratio more attention! It's got some crazy (cool) things too!
Try this out: Find the line that connects the two inflection points of a quartic polynomial curve. Then, measure the distance between the outer intersections (the rightmost point and the leftmost point) and the inner intersections (the inflection points). It turns out that, provided that four distinct intersections exist, the ratio of the inner segment (the distance between the inflection points) to the outer segments (the distance between each of the outer intersections and the nearest inner intersection) is exactly the golden ratio. Furthermore, the two smaller areas enclosed (on the left and right) by the inflection line and the quartic curve are each exactly half the size of the larger area (in the middle). Why this happens probably comes down to a nasty algebraic nightmare with calculus, and things simplify to the golden ratio and whatnot. I'm sure it's possible to prove it. I tried to do it myself but got lost in the awfully complicated algebra (trust me, it's ridiculous). Maybe there's a neater and more elegant proof than that, though. 3Blue1Brown? Care to tackle this one?
Pheww!!!!! So long thst i couldn't help liking!!
m.th-cam.com/video/sj8Sg8qnjOg/w-d-xo.html
This video made me... feel emotions that I can't quite put into words.
Forty Two.
try to put it in numbers instead :)
i know what u mean. me too...
@seba cea Andre Weil once said that understanding a problem that you have been working on endlessly can lead to a feeling of ecstasy for weeks at a time
The end of this video took me to such an emotional high. Its nice to see others who care so deeply about a subject you also care so deeply about. In a way it was like our spirits became one ... although philosophically I am not sold on 'spirits', that's the language I have to use to describe this feeling.
I cant wrap around how math can be so beatiful, it's like reading a really good novel that has many intresting characters and plots that are always more deep and connected that they lead you on at the start.
Sometimes it requires more work to piece all the parts together but man the result is incredible
Nice analogy well said! And in the case of our universe, math is the language in which the novel is written. As Kepler said :)
@@jamesr2936it's not as fancy as an entire novel filled with gestures... It's more like musiComposition. (It repeatStatic loops when charted, but cannot "break the mold" via willpower. It is the environmental medium, the reflecTensor allowing language). Language, on the other hand, is not exact or predictable, for having synonym varianTone stretching.
Max Planck is my favorite physicist, Wolfgang Paulie for chemist and the father of my own specialty, Mendelev for biology (more genetics, but close enough). I spent 25 odd years of my life treating mathematics as just a tool to an end, but every time I watch one of your videos, I'm reminded that you are my favorite mathematician... A sentence I never thought I'd say about anyone. Thank you for everything.
Gregor Mendel was the pioneer of genetics. Dimitri Mendeleev was a chemist. To err is human, to forgive divine, but please forgive a correction also.
❤
The last minute of your talk was profound, enlightening and valuable: the connections of deep math concepts to many manifestations of reality. Thanks.
This is such a beautifully clear video. I've seen this prime spiral meme before and like you said thought it was due to some mysterious property of primes. Thank you for demystifying this and somehow leaving me even more amazed by the simplicity of the mathematics causing it and the more interesting topic that it brushes up against.
Note that you might be thinking of the "Ulam spiral", which is a different spiral- and prime-related picture...!
Absolutely stunning. I am a part-time mathematics teacher myself and the epilogue was truly inspirational. Thank you.
I am not a mathematical phenom or engineer, but ai do find immense beauty in the visualizations of mathematical concepts! Like this, visualizing Fourier transform, visualizing how to turn a circle inside out, mathematical proofs, etc!
"How pretty but pointless patterns on polar plots of primes prompt pretty important ponderings on properties of those primes."
C'mon man, let me just watch a minute or two of the video before forcing me to like it.
P'shaw.
Plliteration
important-->purposeful
Also the patterns cannot be pointless when they are clearly coming from a bunch of points
There is beauty in innocence
This is analogous to showing a meme to your parent and instead of saying “oh cool”, they give you a piece of life advice
._.
It's more like showing a meme to your parents and they say "oh, cool" and then share a really deep story from their lives that relates to that meme, showing you worlds beyond and making you feel really good and loved.
@@johnnyswatts There are two types of people, those who listened to their parents' stories and those who rolled their eyes.
NortheastGamer Or a mystical third kind where their parents just yelled at them
Shatterdpixel And a random fourth whose parents didn't say anything at all... But the children still heard everything that needed to be said and eventually learned why primes form spirals. Clearly it's used in the flex capacitor to initialize time travel 🤓
2:25 that animation and change of music was utterly beautiful, that type of beauty you wouldn't expect to find, yet still it's there, waiting to be discovered.
The way you seamlessly explained the jargon for modulo was perfect
I just randomly stumbled upon this, and it has me absolutely fascinated (from both the resulting math and the lucidity of the video/explanation itself). Amongst my other playlists for memes, drumming, etc., I now have one titled "Beautiful Math". I feel compelled to fill it with others and take the time to understand it all!
Thank you so much for creating this incredible lesson! :)
1am: i have to sleep
*3b1b uploads*
Same.
Exactly same. 😂
Same.
For me currently 12 am but same
Same
Oh yes, waited such a long time for this!
Quick Request: since you're doing Number Theory, can you prove Fermat's Last Theorem? I believe the proof is quite trivial, so it shouldn't be too bad :P
Yeah, Fermat's last theorem is an easy one.. definitely should be a video. In fact, I just found a nice proof for it, but I'm afraid it won't fit in this youtube comment.
He's addressed this: www.reddit.com/r/3Blue1Brown/comments/7aubxv/fermats_last_theorem/
@@chumbucket6989 Aww nooo, is the project too big to fit on the margin of his paper?
@@runningcrocodile8051 lol nice one.
@@FacultyofKhan This is what he said: "I'm not saying no, but let's just say this would be a very big project :) Certainly some special cases might be doable and interesting."
I really like how this visualization shows the principles of emergent properties. Given a set of rules, any system, complex or simple, will have properties emerge that are non-obvious from the inception of the system. This is one of my favorite observations of universal principles.
I’ll be real, seeing the switch of the spiral from clockwise to counter clockwise when we move from mod 6 to mod 44 is super satisfying.
I literally finished a number theory course in my degree two weeks ago, and was tested on almost everything you brought up in the video!
Seriouly , i never realized there is so much beauty hidden in math before watching your videos..thank you 3blue1brown❤️❤️❤️
Math has a lot of subtle patterns, often too convoluted to see the whole picture and beauty all at once. But with each careful step, you can get closer to seeing how various things and ideas/concepts fit together, and that, in the end, can give you a deep appreciation for how it all works. It's really cool just how abundant patterns can seem around us.
This is where it started for me. A recommendation of this video is how I found your channel and it let to sth important, at least for me. I am going to start teaching theoretical computer science soon and, although it is not really the topic of your channel, I will try to use as many of your tips on conveying ideas visually as I can. Thanks and keep up the great work!
Awesome !!
I love the way you take the time to teach math jargon and other tidbits in these videos. So well done. I wish every single lecture was like this
🕊️ *The beauty of mathematics only shows itself to more patient followers.* 🕊️
Math teachers be takin notes on this channel. Superb
Or to those follower of 3B1B even if they are less patient.
When you zoomed out from the initial set of prime numbers I got chills all over seeing that beautiful spiral come out of the numbers like that. Bravo! This is fascinating!
But the spiral is coming from the whole numbers also.
OMG THIS VIDEO IS AWESOME! All of this is contained in the natural constant. The prime series diverge into Euler's natural constant e. This is the universe!
I feel I am not wasting my time when I watch these videos, when in reality I am. Having said that, I am immensely amazed at the dedication, knowledge and teaching skill of the creators of these videos. A big thank you to you.
It’s not wasting of time instead it’s a enjoying beauty
I really enjoy this kind of thing, even if I don't understand it. I tried, I dabbled up until AP calculus in high school. But that was the point my eyes started to glaze over. So I understand 'just' enough to get what's happening and why the spiral, and the gaps in the spiral, is honestly not that big a deal. Fun video!
Hey I’m going through a very tough and stressful times and I wanted to say that seeing your video in my feed just made me smile and actually really excited me.
Thank you
Hope the times are doing you better my friend
I am writing a big school paper on RSA encryption, and wanted to watch some 3b1b videos so i just took one i hadnt watched. And boy does it feel satisfying when you started talking about eulers totient and coprime numbers because i have been learning so much about that stuff. Great video
I remember learning about reside classes in school. We had this little experiment going on where the computer science and math teachers tag teamed us for a couple lessons to teach us about RSA. Honestly, that was a cool idea and I wish more students got to experience something like it. Seeing how different subjects connect with each other is really special.
14:29 In Dirichlet's theorem on arithmetic progressions, he actually proves that each of the none disappearing "arms" has infinitely many primes.
Great Video! It brought back great memories of my Number Theory days. I thoroughly enjoyed it. Thanks!
LOL! I guess I should have finished watching the video. :)
I once had the idea of plotting points like this when someone asked me how I might calculate prime numbers. I suggested that it might be possible to create an algorithm to calculate the position of every "prime" point with increasing levels of accuracy. I never bothered actually trying it, but it's nice the see that the idea is worth something.
If you think you could do it, definitely worth the try. If you can prove it by finding a new prime, wouldn't that be something!
That spiraled out of control quickly..
Underated comment
@@mikedamacenos why is it out of control? Since when has infinity been out of control? Just out of reach, out of sight, out of mind.
As someone who's dyscalculiac, this was freaking WILD to see.
I couldn't have fathomed something like this in my lifetime, but.. Y'all out here killin this maths stuff and I'm so proud of you, and happy to have seen this.
3:00
Can we just note that he had to calculate like the first million prime numbers to get that visual?
Edit: 13:00 was even better holy h e c c
I mean, a million primes sounds like a lot, but with a half-decent computer it would only take a few minutes at most to calculate
it's still pretty awesome, though
And here’s me like “I wish he would just zoom out further”
@@TheAechBomb Took 33.28 seconds in Matlab... Can probably be way faster though if you do it in a compiled language
@@repsajderapper I was only basing it off how fast I could do it in BASIC on a modded DSi (using Petit Computer), but I knew it wouldn't take long
You can use the sieve of eratosthenes and it should be very fast. For Python:
Def Sieve():
Primes = [2,3,5,7]
a = 8
While len(primes) < n:
For i in Primes:
If a % i == 0:
Break
Else:
Primes.append(a)
a += 1
print(Primes)
Sieve()
I'm a 56-year old engineer. I sometimes wonder how my life (maybe the world) would have been different if on Saturday mornings other kids like me had watched videos like these instead of cartoons.
I'm 15 and everyone is talking about how it would've been if they had those videos at that age, I guess I should be very glad :)
Life enriches in different ways. Laughter brings its own universal rewards.
I am 15 and i watch university math videos but animes too lol.
This was one of the most greatest math videos I've ever seen in my whole life. And it kinda makes you think that if an irrational number like Pi was a little different in our universe, how things would actually look like...
Pi can't really be different. It's just a result of the axioms of Euclidean Geometry and the definition of a circle.
It doesn't even have anything to do with our universe because space in our universe is non-Euclidean.
Whenever I feel discouraged by humanity, I come to this channel and get courage from knowing this video still can amass millions of views
Pfp (Profile Picture) and / or Banner Sauce (Source [Artist])? 🗿
I wish someone could have taught me like this in my school
TUSHAR SHAILY as far as I can see, all schools everywhere rush to the ‘answer’ when really it is the question that is really interesting.
I'm thankful to be taught this now. I feel for all the people now gone (or still alive but nevertheless will never have the opportunity) who may have been completely enamored by this privilege. But yes, I do wish this was available during school. No doubt there are some lucky students out there with splendid teachers at this moment
Schools are forced to ensure kids can pass tests more than anything else. In the UK the benchmark for GCSE exams (sat when you're 15/16) is (well, was) a C grade. Getting a D-grade kid to a C meant a lot to the school, so way more effort was expended by teachers in that area. The high flying kids, those that could get an A without too much trouble, weren't pushed anywhere near as much. Not the fault of the teachers, I might add. The school doesn't care if an A* possible kid only gets an A, that won't really affect stuff like funding. It's all about getting kids to pass. I'm not saying the struggling kids shouldn't be helped, but it shouldn't be that schools have to prioritise them any more than kids with a high potential in that subject.
They recently changed the grading from letters (A* to F) to numbers (9 to 1). Why? No doubt there are 'reasons' but it does not seem a priority to me. But that shows the nature of schools nowadays.
I was lucky enough to have a math teacher who showed us the beauty of math (in the 80’s, no youtube just chalk but he did it). This guy even gave up his pauses if anyone didn’t grasp anything and would explain again or explain more (out of scope for the exams) for those who were interested. My wife didn’t have this luck and always thought she was bad at everything math until I started explaining things I remembered after we got married. Long story short, she went back to university while working full time (didn’t attend most classes), did every year in half the time and has since become top of the field in her profession and gives guest colleges at several universities. She’s the perfect example of how the spark was not lit up because her teachers failed. This is why youtube is such an important tool right now where capable and driven people can enlighten the people who are interested and hopefully light many sparks!
I think i may have learned more maths from youtubers at this point then i did in college. and i minored in mathematics.
Humans love to find patterns so they can figure out why a pattern exists.
Patterns love to find humans. Oops.
do we find patterns beautiful because everything is in a pattern - or do we find patterns beautiful because we were "programmed" to like patterns, or both?
@@glaswasser Now silly as this question/joke might seem, the answer is quite worth it to look into. You see, pattern give us an important ability: to predict. Then of course, creatures that are programed to see patterns might predict things better, and be better at living at a whole. And what is a better reason to look for patterns, than its beauty?
Patterns exist. Humans are keen towards them because our brains allow us to recognize them. Patterns are caused by stimuli. We are intelligent enough to domesticate those stimuli if we comprehend them.
Patterns are sequence to follow for direction and routing and assessment of speed and vectors.
I want to thank you for this, because I think the ending had more of a philosophical spiritual answer that helped me look at life in another new light that I enjoyed.
i didn’t know you could edit a video with pauses long enough for people to be able to comprehend the math that was described. It’s very satisfying
I commend all the mathematicians who made these discoveries before computers were invented. 👏👏
Precisely! Discovering patterns on one's own greatly enriches one's understanding of Maths and one's appreciation for its intrinsic beauty.
"Why do prime numbers make these spirals?"
me before the video: how tf should i know that
me after the video: *what are prime numbers*
They are portals into and out of our minds simultaneously....yea pretty nuts i know.
@@FeedEgg god, is that you?
@@foooooooont4679 one of them...shh
@@FeedEgg ok i will keep my mouth shut
@@foooooooont4679 lol do not be afraid, they exist just not in the way you think, agnostic, all i know is that i know nothing at all.
If there was ever a vote for best TH-cam channel of the year, I would vote for you over the 500+ channels I'm subscribed to.
This video is really really really hard to make. Not many people can do this.
"3 is slightly less than Pi"
You have angered the engineers.
"Pi is 1." /a physicist/
Engineers would be the first to simplify pi to 3. You're thinking about mathematicians. Or school teachers or lawyers.
The engineers aren't the ones you have to watch out for on this one. It's the slapstick comedians. (lemon-meringue pi)
@@SuperPol1981 The (running) joke is that an engineer would say that pi = 3, while the statement here is pi > 3.
Even during desinging of simple DSP for my radio amateur transceiver, I've been taking pi to about 9 decimal places to have enough frequency accuracy (nearly 10 Hz) in my "narrow" working band (30 MHz).
And I wouldn't be surprised, if serious engineers take pi much more accurate.
E.g. in automatic control theory, while estimaing safety margin of some closed-loop control system.
Only three minutes in and already loving this. Haven’t had this much fun with math since your pi in clicking blocks series.
Even if I don't understand all technical aspects of your videos I really appreciate the visualization that give me an deeper understanding of mathematical problems. Thanks!
dude amazing video. one question though: how on earth did you animate those histograms??? that seems like it would've taken forever!! amazing work
he didn't do any of this by hand, he codes the animations in a custom python library. so it would probably be something like he creates a list of primes and then feeds that list into a function that sorts it into the categories for the histogram.
The code for those histograms in particular will probably be up on github.com/3b1b/manim soon.
@@tylerbreisacher5841 Oh goddamn thank you. Every time I watch his videos, I find myself really wanting to animate other formulae/functions. You just made my week.
@@tylerbreisacher5841 does he actually upload the code of his videos on manim repository? I doubt so? because he once mentioned that it is more like a work in progress. please give me a heads up if you find something he did in the videos on the repository, I'll be super motivated to recreate or create new animations
@@sudheernaidu6646 at least some of it. For example here's some of the code for the colliding blocks. github.com/3b1b/manim/tree/master/old_projects/clacks
Understood the first 10 minutes ... after that I just zoomed out and thought how beautiful this pattern is
One of your best videos.
I remember how Harry Potter books made some kids into reading books.
Well this kind of videos can make kids into liking maths.
Bravo!
I took number theory the previous semester in uni and now i can see the point of it. Brilliant job! The visualization of such theoretical problems is so helpful
Asking questions is the key to math and science. One well placed "Why?" can open any number of doors.
Why do
Why?
@@RailwayPenguin why why?
[Ed Boy Intensifies]
Indeed. Curiosity may have killed the cat, but it also made us consider the possibility that the cat may in fact also be alive too until one opens the box and the possibilities collapse.
I have no clue what he’s saying but this is Therapeutic
You can understand if you want to though, I think he speaks and explains concepts as slowly as possible.
Its pretty trivial tho, Half of the video and still haven't learn anything new
Differential Calculus soft whispering ASMR
should have taken pre calc
rawnak i found it primed me for sleep 😴
I don’t understand why people would not like this
Same kind of people that watch Beavis and Butt-head.
I was wondering the exact same thing
Just some person its not interesting
@@shandy_2001 but then they should pick another video. the easy to understand visual preparation is teaching in its highest art. those videos pose a reference for how to spread the ideas of Mathematics to the general public/audience. there's nothing to dislike about that, imho.
You missinterpret the thumb down, those people are nice Austrialians who want to put a thumb up
(sorry it's a bad yoke)
What you said toward the end about accidentally rediscovering things people learned in the past bringing an intrinsic value to them that simply being taught lacks was...completely true.
It reminds me of this time once in which I tried to use multidimensional arrays to represent the possible results of a series of coinflips and accidentally discovered that the number of heads has pascal's triangle embedded into it.
i know this is gonna sound cheesy but, your explanations and the amazing visuals on this video are so outstandingly clear!! i've never found a channel on youtube so far as great as conveying knowledge and information as you are, the way you approach the content and explain it is truly marvelous. so thanks for making these awesome high quality videos :)
Never before in my life have I been so captivated by something I understand so little.
Lucky you! Most of us have experiences of quite sudden heart captivation by some newcomer to us , and we understand ourselves and others so little!
Don't ask me why, but in French we use to pronounce the "ch" in "Dirichlet" like a "k". So it is pronounced "diriklé" and not "dirichlé".
In Russian, we use "Дирихле".
Put it into the Google Translator and listen to the voice
Additionally, I've never heard anyone stress the second syllable as this narrator is doing.
Why tho
@@hbushnell Dirichlet (despite the Pierre and Lejeune in P.G.L. Dirichlet) was a German mathematician. So the usual tendency is to pronounce the 'ch' closer to a 'k' sound than an 'sh' sound. I myself as a native English speaker say it more like Dearie Clay (emphasis on first and third syllables). I consider the narrator's pronunciation somewhat unusual.
@@toddtrimble2555 Aaah, correct. I was convinced he was French, because that name really sounds French. That explains this pronounciation. Thanks!
This video has had me thinking about this for a long, long time. I love how it is so easy for the mind to "see" the rays. But the "spirals" are just discontinuous dots that happen to be near each other, and the human brain is compelled to connect the dots. I wonder how often we get fooled by things like sub atomic particle data or astrophysics data that is really discontinuous, and assign some kind of relation that might not really be there, but "kind of" explains the data so we stop looking for other explanations. Fine job on the whole idea, and the presentation. Many thanks.
Im starting to appreciate TH-cam recommendations
Ok now i want a 20 minutes video of the prime numbers plotted in the graph zooming out ...
Wild Abra used teleport.
…4 4 used TM55!
Poor computer
A wild Abra appeared
Go, 1
Wild Abra used 1 is not a prime number
It's super effective
1 commited suicide
Go, 0
Wild Abra used Teleport
Wild Abra fled
yes, i do learn more effectively when it's something i voluntarily click on to learn. I like learning when it's interesting, not when i'm stuck doing an hour of homework to "review" things i can already do perfectly well.
Yep. At 29 I learn more on TH-cam every day than I ever learned in school 😅
2:25 "But when you zoom out" + the visualization + the music
Goosebumps!