I'm part of some organized activity. There's this dude that definitely is our groups identity element. He's nice though. Zero initiative, but there's not a bad bone in his body.
We need more John Conway's. Why make something so dense and complex even more difficult with letter-number combinations when you can just name stuff BABY MONSTER
It feels like with "the monster" humanity has learned the answer to something without being able to even ask the question. Almost as if we've stepped into territory that isn't meant for us yet. Of course the given name adds to the lovecraftian feeling of it and I love stuff like this.
21:15 "The universe doesn't really care if its final answers look clean, they are what they are by logical necessity with no concern over how easily we'll be able to understand them" - I love this sentence! This video was amazing. It's just so cool that structure arises when studying symmetries themselves. Your videos always inspire in me a sense of awe and wonder, but this video was something else. Thank you for all the time and effort you put into this.
It looks possibly to have taken inspiration from Niel Degrasse Tyson's quotation, something to the effect of, "the universe is under no obligation to make sense to you."
@Dr Deuteron no, it is just you who don't want to think about the monster so you make it sound scary, even though the monster has always been part of reality & doesn't hurt us
Just got new earbuds and while trying them on, suddenly my google assistant starts reading my notifications. Of course, it had to read this never-ending title for me lmao
Hahaha that's awesome. Did it work out the name of the number (with variations on duodecicillion etc... maybe somebody can give us the actual name), or just read out the digits?
Honestly, there's something beautiful about the way 3b1b explains things. At around 4:28, he explains that the permutations of 101 different objects would amount to 9x10^159. However, instead of simply saying 'this is roughly the same as the number of atoms in the universe squared', he says 'if every atom in the universe had a mini universe inside of it, that would be how many sub-atoms there would be.' Take the time to appreciate the time he took to make these numbers just a bit more interesting!
It's a nice metaphor since humans suck at understanding multiplication intuitively. A big number multiplied by a big number just equates to a bigger number for us, we're bad at telling the difference.
It was we who he helped us be more interested. I do love the care he takes with words. I DID take the time to appreciate the video he made. Now I have decided not to take the time to correct your last sentance, maybe you might like to correct it yourself, then I would know that you have been taking the time to appreciate that it takes time to help show that many things can be fascinating once we have taken the time that it takes to appreciate them. Grok is a good word. Lets call the monster Grok.
the way you explain the concept of group as the concept of number "3" is really mind opening and important. A lot of people having trouble with math because it's seem so conceptual and they always try to link it to something more grounded, but to be good at math they need to approach math as how conceptual it is. Eventually of course math is used to help real life problem but it's not always straight forward, so you need to think about it in the world of math itself. It's like when I started coding and at first my mind will work only with what the final UI or graphic display on the screen, but slowly my mind would think purely of what happen data wise and not really the final representation.
it does help to start with a concrete problem and then shift it into the abstract though. you don't learn multiplication starting with anything but the natural numbers
A quote from a Pratchett novel comes to mind, when a wizard tries to explain how a mysterious cabinet works. 'Yes. The box exists in ten or possibly eleven dimensions. Practically anything may be possible.' 'Why only eleven dimensions?' 'We don't know,' said Ponder. 'It might be simply that more would be silly.'
@@jetison333 "Making Money" by Terry Pratchett. Although most of the book is not related to that quote. Still a fantastic book. It's the sequel to "Going Postal" so that may be a good place to start.
"It's the size... of the monster" is such a scary way to put it. What the hell, math?! And then you get to the end of the video and there's a monster, a baby monster, and a happy family.
i honestly love it; it's a big indication of how the universe doesn't really care if it makes sense to us. if anything, to me it seems to actively resist being understood, like it doesn't like humans poking and prodding at it so it conjures up stuff like quantum physics and the monster group as if to say "ha! explain THIS". it's amusing and oddly charming, even if it is just me personifying reality itself.
Math always felt so incredibly clean, structured and sterile to me. The idea that something so bizarre could exist in it, with nobody understanding why, kinda terrifies me too. It feels anomalous, like something that shouldn't even exist. Gives me SCP vibes to be honest. Remember Theta Prime?
This is IMO the best video 3b1b has produced by far. Amazing explanations, visualizations, stories, everything. At first everything went over my head, but after learning a bit about group theory, this video is so cool.
"The universe doesn't really care if its final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them." Elegantly stated. As a grad student in mathematical physics, this definitely lines up with my experience!
Wow, seems incredible, because math independently can feel so elegant and beautiful , but the most amazing thing is when it’s applied on the natural ways of reality, you can actually see it shift, and basically “become a huge mess”. Really tells you something about the way math serves us , and the way it explains reality simultaneously Wish you best of luck!
@@amitbenjam Thats the thing. The field in question here has nothi g to do with physical reality. This is a result of first order logic. Its purely mathematical.
I as a student in an entirely different field, hate solving equations in mathematics....but something always draws me into the theorems that it provides.
you will find that you obtain gains even by taking that exact same class you took. I've learned through exp that you could take the same course and even use the exact same text and do the exact same chapters and still make gains. I can't even recall a time in my undergrad where i felt like I mastered any text book so well it was rendered useless...
@@dasmartretard Really on point. I've authored textbooks and learned much more about the subject by simply focusing so deeply on the foundations. Even new ideas come together more clearly for me on subsequent editions. I'm convinced that just a solid introductory book can make someone much more competent in applied work than the typical PhD student gets after a dozen courses.
As a graduate student who studies the briefly-mentioned “theory of modular forms” and knows about the monstrous moonshine conjecture (my advisor proved a related ‘moonshine conjecture’) this video was truly wonderful and the best conceptual introduction to group theory I’ve seen. Well done!
I'm interested in understanding the statement of the 'Modularity theorem' and also getting some idea of what the Langlands program is about. Can you recommend some books or lectures? (I don't have a Mathematics degree)
I just graduated with a Master's degree in Mathematics, my thesis was on modular forms stuff. Who was your advisor (if you don't mind sharing)? I've attended the Automorphic Forms Workshop a couple of times, I've probably at least heard their name. For everyone else, I think Kilford's book on Modular forms was somewhat readable with a decent background in group theory (maybe some number theory) and knowing kind of what Fourier expansions are. For anyone outside of math, you need to learn what proofs are first, and then a few more things. Diamond and Shurman's book on Modular Forms is a beast. I kind of managed to get through the first chapter and maybe a few pieces of the second chapter, although I tried to do that before Kilford, so I think it would be easier now. Also, I think some experience with complex analysis might be necessary that I wasn't fully solid on. Modular forms was not an easy topic for me to just jump into. Progressing from no math beyond Calculus to modular forms, here are some books I might recommend: Learn proofs: A Transition to Advanced Mathematics, Doud and Nielsen (free and well written, highly recommend) Learn group theory: A First Course in Abstract Algebra, Fraleigh Representations and Characters of Groups, James and Liebeck (all solutions at the back of the book, it's amazing; also learn a bit about generalizations of this 196,883 thing) Galois Theory, Cox (not strictly necessary to get to modular forms, but you do get to see what's going on with the roots of polynomials business) Abstract Algebra, Dummit and Foote (if you really want to go hard; also, just the first fourteen chapters - after that it goes into Algebraic Geometry and other topics; also you might want to at least take a glance since Kilford touches on exact sequences and Dummit and Foote has a section on them) Learn complex analysis: Understanding Analysis, Stephen Abbott (probably best to start with Real Analysis) Fundamentals of Complex Analysis with Applications to Engineering and Science, Saff and Snider Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable, Ahlfors (this is probably the standard text, although I used the one by Stein and Shakarchi, which was alright) Some basic ideas of Number Theory might not be bad, maybe something like Fundamentals of Number Theory by LeVeque, and maybe Introduction to Analytic Number Theory by Apostol, seeing as modular forms in Kilford are mostly treated in the sense of Analytic Number Theory. Then Kilford. Going through all of those books is probably not fully necessary, but since when is stopping to smell the roses a bad thing in math?
I like your final quote "Fundamental objects are not necessarily simple. The universe doesn't really care if it's final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."
Well, how does one "show" it, exactly? Constructing it, or even defining it, is not really straight forward. For a long time, the only result or description of it was a proof of its existence.
I might be wrong but I think photons are in 5D space, as far as special/general relativity is concerned anyway. The basic idea which when you go into detail is totally different (going off memory, may be wrong) is photons reveal what is in 3d space, travel through time (4d) and they are in the 5d. Surely, much much more could be unknown about em and time is controversial to begin with but im sure thats the consensus in modern physics so far; although anyone could come along and massively correct me plus, time is apparently, according to some, just a result of the universe expanding or whatever. So, photons apparently dont actually move, they stay static in one part of space time, but space time moves really fast and its as if the light travels as fast as is possible in space, but its just travelling at the fastest (known) possible speed in the universe; the rate at which it 'expands'. I think some people claim that in one dimension, there is a pulling force, like a gravity, to the universe and then in another dimension theres like an opposite pull - not so much a push but pulling in the other, or another, direction - and basically, the waves that photons make are a result of one dimension pulling, causing a peak in the wave (the light wave travelling 'up'), while the other dimension pulls the other way, causing a trough in the wave (the light wave travelling 'down'). Idk if one dimension, would be space, or time, or electromagnetism. Now, it's a bit like when you grab two edges of a piece of paper and pull them apart. The paper rips into two pieces in around 0.27 of a second. On the macro level (watching in real time) it's as if it's a clean break, it was pulled near enough equally in both directions and torn apart. Yet, if you were to slow it down to an extreme level, like 20000 times slower or something and magnify by about the same, you would see the left hand pulls the paper diagonally to the left, then the right hand pulls it to the right, very quickly. These pulls to the left and to the right happen very frequently, very quickly and there are very much of them, as the tear travels down the paper; from whence the tear started and whence the tear must end (the other end of the paper). and when you observe time at a very slow instance, almost freezing time in effect, you can see the up, down, up, down, up, down, up, down of the photon. Really, if you consider the photon is staying still, and this is all happening very fast, perhaps at the speed of which the universe is 'expanding', then it's like you can see the different pulling dimensions acting as the 'left hand' and the 'right hand'. But, it's not 2d like the paper, it's 3d or 4d or some higher amount of dimensions. The light wave, travelling in one direction, is kinda like the tear in the paper. You can see how the universe is tearing, being pulled left to right in the patterns of the light wave. Yet, just like how the paper was near enough a clean break, torn apart in 0.27s, this pulling and tearing of the universe is near enough a clean break (we'll assume; scientists have found that the universe is likely expanding at a faster rate in one direction than another direction(saw Zach Star mention an article about it in a hypersphere universe video) but that's a story for another day haha) so whether or not we see the pattern, the pulling left to right on the microscopic, micro-timed level, the result is still near enough the same. We see the light, pretty much in an instant, before we can even notice we see it. If there is a plentiful amount of photons, which they're most usually is, we see a constant stream of photons and we visualize the object that emits the photons.
@Grant Jacobson to answer your question: even if light is in a lower dimension, a 1d point could travel through 3d space and hit near enough exactly the vertex(corner) of a cube. This 1D point would then be reflected along another path in 3D space(4D spacetime if you wanna get technical). In a similar way, whether photons are 3D, 5D or More Dimensional, if a photon was to travel directly into a vertex, a corner of the monster, it would be deflected and reflected along a different path in it's own dimensions; or perhaps the photon would be reflected into different dimensions to the ones that it was originally in, then it's path would continue there. I can only assume The Monster, being some form of physical object, in space, would have oodles and oodles, trillions of corners. I can also only assume, that these corners must reach the edge of many other dimensions because... they're corners. With it being such a highly dimensional being, these corners must crossover into many and lots of different dimensions, so it goes to suggest that they must probably crossover every, or most, dimensions, including the ones that light travels in. Now, if enough light hits enough corners, travelling from (starting from) enough different dimensions (assuming it can even be reflected/deflected into hitting the corners of the monster; covering all grounds)(and assuming light could even act the same within it's presumed finite dimensional range in any set of dimensions, that's a pondering for another day;[*]), then, some of the monster could be illuminated and visualized. Maybe the light could travel along edges, not just corners, or perhaps even it would be hard for us to distinguish between corners and edges at that high level of dimensions; this way we could possibly see more of it. Through seeing the corners we could get some kind of glimpse of what the shape of it could be, from some presumably very limited angle; yet, we could estimate a full picture of the monster by calculating symmetry patterns in what we do see. * -[perhaps the higher the dimension the light travels in, the higher range of dimensions it occupies. Wherein, the range of dimensions it occupies as it occupies higher dimensions could expand at a proportional/formulaic rate as the range of dimensions it occupies gets higher] TL;DR: To the one person who may have actually been interested in this: Thank you. You're welcome!
Sometimes I lull myself into believing that Grant is a normal human being, and then I see a video like this, and I remember that we are speaking with higher-dimensional beings.
Yeah, I learned a "lot" of math during my CS/info-engineering studies, and I loved group theory and Galois fields, but maaan - the Monstrous Moonshine Conjecture
I'm not THAT much of a math person, but when you showed how the square permutations are the same as the dot permutations, my mind was blown in the best possible way
Blowing your mind in the best possible way is one of the most beautiful things about learning pure mathematics imo. It comes from a deeper and through understanding, and it's very satisfying.
It is kinda funny that you said square instead of cube because the group of the symmetries of a square is not isomorphic to the group of permutations of 4 elements i.e. there no 1 to 1 mapping between those 2 groups.
I remember reading "Symmetry and the monster" about 15 years ago, and fell in love with the monster group -- and due to one of the later chapters in it, the next video queued up after this one is now "Hamming codes and error correction ". Classic work.
You gave essential intuition for one of the most interesting and complicated fields of modern math since it can pop up almost anywhere. Wow. Your video can actually serve as a first step into studying this branch.
Grahams number be like in case you dont get it, grahams number was originally created as a number of dimensions involved in a math problem. Mathematicians had to define that massive number of dimensions to solve the problem.
That 11 dimensions relate to 1+1+1+8, and the 196,560 of the Leech lattice relates to 24, where bosonic M-theory is in 27=1+1+1+24 dimensions. There are physical theories with 196,884 degrees of freedom that contain both the 196,883 and the 196,560 of the Leech.
This has got to be one of my favorite 3Blue1Brown videos. I love the way you present just how fundamental groups are. One line in particular I just love: "This is asking something more fundamental than 'what are all the symmetric things?' It's a way of asking, "what are all the ways that something can be symmetric?'".
Things like this studied by maths are actually more fundamental than the universe itself. If another universe were to exist with completely different physics yet housing some form of entities of sufficient intelligence, they would eventually come to these same results, since they are consequences of logical necessity rather than properties of the universe we inhabit.
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@@MatthijsvanDuin what you just said blows my mind, could you, please, explain more about it? Cant logic work diferent in other universes? Truly interesting coment!
@ In a sense, logic works independent from the universe. To reach a logical conclusion, you start with some set of assumptions, and then combine those axioms to reach some sort of conclusion. Because we exist in the universe, our starting assumptions are typically tied to the universe, but the conclusions we reach are only tied to those assumptions. We could start with an assumption that is untrue like "glass breaks when you hit it with a soft object" and reach the conclusion "if I hit a window with a feather, it will break". This conclusion is not true in our universe, but the logic is correct given the starting assumption. Since logical conclusions seem to be independent from the universe we inhabit, it seems likely that if there were another universe with different physical laws, logic would still work independently from those laws. An intelligent species in that universe would likely still recognize Pythagoras' theorem as true, even if they didn't have any use for planar geometry, since they could still recognize that if they started with certain assumptions, Pythagoras' theorem would be a necessary result. Similarly, if these alternate universe mathematicians started studying symmetry, they would also eventually discover the Monster, since the Monster is tied only to the idea of symmetry, and not to any physical reality of symmetry.
"Not 4 dimensions, not 5, but we'll have to go to..." Me: Hmm the next number is six so then it should be si..." *sees the dimension counter started to increase more than 100000 My brain: ight imma head out
Indeed. Before this part, I was following along rather nicely, understanding most of the things being said. The same can't be said of the parts after this, and I am just as confused as I ever was watching most of 3blue1brown's videos on more advanced topics.
15:04 that’s interesting, in the Galois theory course I did this year, we didn’t do composition series. Instead, we showed that insolubility of a polynomial by radicals is implied by the Galois group being insoluble. Using the fact that Sn is insoluble for n>=5, you’re basically done.
Just pure admiration to the way you explain such abstract concepts in a such elegant, clear and concise way. Thank you for your time and effort. Much love, appreciation and respect for you.
I wish mathematicians were more creative with naming and I didn't have to relearn the definition of "normal" and "regular" in about fifty different contexts.
Can I just say that the abstraction analogy was so genius, it gave me the chills. It also raised a particular question for me: if our ability to understand the abstraction of numbers early on is a result of us being drilled with sheets and sheets of basic addition/multiplication homework, would it be possible for someone to grind through sheets of basic group theory problems and end up having a clear understanding of the abstraction of groups?
@@totolamenace I agree. It just seems crazy. I feel like I could never wrap my head around group theory well enough to work with them so easily, but clearly, that's what those mathematicians also thought when they were in high school (maybe).
To think that everyone at some point in their life feel the kind of confusion towards something like "3" that is similar to the kind of confusion I now feel towards something like "S5", just blows my mind.
@@FareSkwareGamesFSG The important thing is to get used to abstraction but also making things understandable and fitting things in a context, like the group actions stuff that was talked about here. It turns out all groups come from symmetries of objects but how complicated those objects are depends on the group. There is a certain truth to von Neumann's quote (which I think was a sort of joke) "In mathematics you don't understand things. You just get used to them."
Man, I am the worst human being in math. I don't even remember the multiplication tables. That said, the fact that I could get a grasp of what you were saying in this video tells a lot about how good of a teacher you are! I wish my high school teachers had been a fraction as fascinating as you were in this. Loved the video! thanks!
Wait.. So you are telling me that throwing you into a class room with 30 others and drilled constantly on outdated theories/tactics didnt interest you?!
You can pass calculus 1 without memorizing most of the multiplication table I have memorized a few multiplication problems and got fairly fast at solving the rest which was good enough for me.
dont worry im sure even the stereotypical "math wiz" would be confused by this, most of this seems pretty esoteric and you probably have to be not only smart but also pretty invested in the subject
The part about cycling three elements around (and ending up where you started if you keep doing it) jumped out at me, because that shows up a lot in Rubik's cube solves.
Yes, moves you can make on a Rubik’s cube, as well as sequences of moves, where two sequences which change the configuration in the same way are treated as equivalent, are elements of a group.
I love your videos. I’ve always been a little intimidated for the level of abstraction of some mathematical concepts, but you can explain many of them more intuitively, with elegance and also generating more interest. Thank you and please keep doing this great work. :)
its a great story idea. Though, i guess its already been done, considering all the "we are in a simulation" conspiracy theories. or the quintara marathon, lol. Just an experiment by a committee of extra-dimensional scientists to see if they can make a universe capable of developing life. oh snap, isnt that a rick and morty bit too hahaha
How does one even begin to ask questions like that? You'd first have to believe that there is an answer that is a finite number (instead of infinite possibilities). How do you you figure THAT out?
Mathematician here. Let me assure you that this "number line of hardness" is not a walk in the park. Probably logarithmic scale. The third point, where it says "a/(b+c)+b/(c+a)+c/(b+a)=4" (btw, he made a little typo in the last denominator) as in "find (all) positive integers a,b,c that fulfill this equation", is already mind-bogglingly hard. And the fourth point is Fermat's Last Theorem. Formulated in 17. century, it took until 1994 to finally prove it.
Lone Starr yeah, I picked up on that as well. Would you say 3b1bs assessment of the difficulty in finding all the simple groups is accurate? As in is it significantly harder than Fermat’s last theorem?
This is the first of your videos where I felt more clueless than ever, usually I feel like getting closer to God when I watch your videos but not in this one... the monster beat me xD (probably because I never read a single line of text about group theory hahahahaha while otherwise I am familiar with other branches of mathematics in varying degrees)
I'm honored to have a good friend who helped prove the umbral moonshine conjecture, but it's far, far beyond my understanding. I definitely remember group theory from linear algebra and vector spaces, and they are absolutely beautiful, but my impression from my own reading was that finite group theory gets very complicated very quickly - just as with chemistry analogy! Lovely video as always. Thank you!
This is why I love Maths and Physics. "They are what they are by logical necessity." That's mindblowingly fascinating but at the same time like super trivial. Cause of course things can't be what they can't be.
So I looked up the monster group on wiki, and then further to the classification of finite simple groups, and my god that is very impressive. People call this the greatest intellectual achievement of humanity and it's not an overstatement.
Might be over my head but just in case you didn't understand, he most likely meant "high level" as in "simplified, asbtracted" and not as in "advanced" :)
This is really fascinating. The more I learn about math, the more I realize the ways we can transform reality into abstract symbols that we can use to find patterns. This a pretty broad generalization, but it has huge implications. We can express just about *any* abstract concept in *any* particular facet of the entire lived experience of a human, or to *any* small detail of the entire universe in a defined, processable manner. How freaking insane is that? The more I learn, the more I can see the patterns between things I’d never thought to connect to each other, and I still don’t even know what I don’t know yet. I’m not sure I have any other words for that than “awesome” in the most literal sense.
This is why I donated the man 1000 UCO :) like I said boss do not spend until 2024 haha What a teacher.. you remind me of someone mate keep it up this is blowing my mind how good you are I didnt sub years ago, my bad ;) but I left math and dodged programming out of lazymind. Now i be a relic :) peace all 54 digits must be one hell of a big result I cant wait to see the ending
*The beauty of this channel is that it makes these advanced math concepts feel approachable to those with no experience. We all appreciate these videos, keep doing what you do!*
Lanthanoids and actinoids? Yeah, there are 14 of each if you count either La/Ac or Lu/Lr to be part of the main group 3, and 15 of each if you treat group 3 as only having the two elements Sc and Y. So there's either 28 or 30 rare earths, and 26 sporadic groups. _Almost_ ...
Lantanoids and Actinoids are only written down separately to make the periodic table look more clean. In reality, they fit there perfectly. They just make the table wider.
I'm at the end of my CS Ba. with maths as secondary subject and just this last semester I was told off-handedly that we know what *all* of the fields are. This blew my mind. But nothing in all my university education has come close to giving me an intuitive understanding of groups like this video. hats off to you, Mr 1Brown, my subscription is long overdue!
Please make an “Essence of Group Theory”. We, the undergraduate math students, need your help Edit: "We, the undergrads of America, need your help" was purely phrased that way as a reference to old Uncle Sam Posters with similar phrasing. I absolutely was not trying to exclude undergrads outside America. I literally just thought it made for cool phrasing, but I changed it.
not gonna lie i can live with not seeing the essence of calculus/linear algebra as I was studying them, but not having a "the essence of group theory" as an undergrad would make me extremely envious of future undergrads that have that series
Search up for Visual Group theory on youtube, by Professor Macauley. Once you get a grasp of what Group Theory is, you can watch the playlist of Richard Borcherds, who was mentioned here. I think the course taught by Richard Borcherds is a tad bit more complicated and I think he mentioned himself it is for very ambitious undergrad students or first-year graduate.
To make sure I understand this correctly: the chart at 18:39 reflects that the set of finite simple groups is *countable* and that it consists of 18 countable infinite subsets (with possible overlap) as well as 26 countable finite subsets? I ask because I would have assumed that the answer to the question "how many simple groups are there" would be just "it's infinite"... Which turns out to be true, it's just that there are some groups that can be extended indefinitely by a simple algorithm and some that by their nature can only exist in spaces of certain specific dimensions-one of which is the "monster group", so named because it's the biggest of the non-infinite families. Really really interesting. In some ways it's a baffling thing to come to, but when I think about it I realize it's befuddling in a familiar way. It makes me think of prime numbers. Prime numbers emerge so simply from basic arithmetic, and yet they exhibit behaviour that can't be fully described with arithmetic alone. The sequence of primes is in a sense "arbitrary": there's no rigid pattern to them, only some fuzzy tendencies. But in another way, the sequence of primes is anything but arbitrary: there is no accident, it is exactly what it must be. I think that, to someone who is building mathematics from the ground up, prime numbers would be the first sign that things can't be kept in control, that there is no way to prevent simple rules from leading to surprising results. It would be the first indication that there are things like the "monster group" waiting in the mists ahead.
The fact that the set of finite simple groups is countable is pretty easy to prove. The first infinite family mentioned is the set of cyclic groups of prime order, and we know there are infinitely many primes. It's also at most countable because there can only be finitely many groups of a given size (just enumerate all the possible times-tables). That there are 18 infinite families and 26 exceptions is much harder to prove, of course.
I think each of the 26 exceptions is not a finite set, but rather a single group. Out of all finite simple groups, there are 18 countably infinite families, plus 26 groups that don't fit into any family.
@@sarahbell180 Why not? It looks like Gödel coding might well interest the OP, given his observation of unexpected complex behaviour from simple arithmetic rules.
Dont worry, no one really dies like most people think we do. The essence from him will be born again! No one can get away from this existence. At least not so fast and easily as many of us wish we could be.
This is the clearest intro to group theory I've ever encountered that also explains why group theory isn't trivial but wonderful and mysterious. Great job!
Eight hundred and eight sedecillion, seventeen quindecillion, four hundred and twenty four quattordecillion, seven hundred and ninety four tredecillion, five hundred and twelve duodecillion, eight hundred and seventy five undecillion, eight hundred and eighty six decillion, four hundred and fifty nine nonillion, nine hundred and four octillion, nine hundred and sixty one septillion, seven hundred and ten sextillion, seven hundred and fifty seven quintillion, five quadrillion, seven hundred and fifty four trillion and three hundred and sixty eight billion.
Fun fact: For the equation at 14:07 (a/(b+c) + b/(c+a) + c/(a+c) = 4), the smallest coordinate of the smallest integer solution is 80 digits long. You can find it using an elliptic curve.
I've already concluded with the Euler constant that no matter how abstract we get we will never get rid of arbitrary constants. If anything, I hope that one day group theory will be able to derive geometrical and algebraic constants from fundamental logic.
I think you're mixing math and physics - all constants in math are very well defined, and we know how they arise. In physics, constants and principles are usually made up to correct the formulas into observations and nobody knows why exactly those numbers, nor do they apply everywhere or just here (anthropic principle)
@@HorukAI This whole video id about a mathematical constant and nobody knows why it's exactly that. I mean, we can calculate it, but that doesn't explain much.
@@tacticalassaultanteater9678 sorry this was not a constant, but a cardinality of a particular finite simple group. My point is that constants in mathematic indeed do arise from deduction process (logic) while in physics it's often an equation scalar based on observation, or mathematical necessity added so that it fits to hypothesis (think inflation)
@@HorukAI Numbers like the gravitational constants aren't even proper constants, they are derived from the quantities we pick. When I say constant I mean precisely values like π, Euler's number or the cardinality of the monster group. Artifacts of logic, independent of our circumstances, universal, yet hard to explain why their values are exactly what they are.
You actually usually don't get to name your own conjectures or theorems. Suppose your paper contains a new result labeled Theorem 1. After you paper becomes very important and discussed a lot, people will not want to keep calling it, Farrell's 2028 Theorem 1. They'll name it something catchy and says something about the result. For example, whoever proved the fundamental theorem of calculus didn't call it the fundamental theorem. It was the people who began using it who named it that. And clearly, it was named like this because it's a "fundamental" theorem.
This is possibly the most incredible video I've seen in a while. I've just finished a course in Galois theory, and it almost feels like 3blue1brown is following my own mathematical journey. I'm truly privileged to live in this time. Thanks Grant for producing a work of art!
This has to be one of the best explanations of group theory I've ever seen. This might be somewhat surprising, but I've never really thought about the connection between symmetry and group theory, although I've worked with group theory a lot... (as a cryptographer, though, not as a mathematician). It was never explained to me like this. Really cool to see.
So informative! The fact that these videos are so synoptic makes them all the easier to understand. So often lecturers can be rather reductionistic in their approach, and students suffer as a result.
Yes, there are only two steps needed to do chemistry: 1. Find the periodic table 2. Do all the chemistry Jokes aside, I was lost during the first few minutes but I followed through till the end. What kind of charm is this, Grant??
@@aliince9372 Jokes aside refers to the sarcastic statements that this comment made before saying, "jokes aside." It is a relatively commonly used phrase in english.
I like how he explained everything super clearly, but also made it clear that we're missing most of the picture. That's really hard to do, but he just made it look effortless.
I had to take Modern Algebra I and II for my Bachelor's, and I actually had to drop Modern I the first time I took it because the professor tried to use teach about symmetry instead of from the definitions. I spent almost half a semester stumbling and flailing, unable to visualize or wrap my head around it, and unable to memorize the Cayley tables for different groups. When I tried again with a different professor, who taught based around the formal definitions, everything immediately clicked and it became one of my favorite classes I'd ever taken. One of the things I love about math is that there's usually more than one way to think about or approach a problem. So while I'm glad that other people learned from the symmetry group lens of it, it took me a minute to get over how weird it felt to hear someone say the definitions weren't how they learned it.
I have to agree, the algebraic definition of a group (and in general of algebraic structure) is way nicer to me than an example using symmetries. I think that well known sets of numbers will always be the best example for algebraic structures, after which you are ready to learn the formal definition and only after this you can go an translate this to real stuff, because now you don't need to figure out how two symmetries combine, you can simpy check they actaully do it as you expected.
Yes, that's the thing. No one who actually does algebra, or at least have taken a graduate sequence, think first and foremost about symmetries when they are solving a problem. It's mostly when you have to convince non-math people that why math is could be nice.
There are very efficient (linear time) algorithms for generating all permutations of a given set of points. I think Wikipedia even includes the python code for that algorithm. So you just have to add the animation. It's great that he put in that work. If you know about this/ have done similar things before, this is doable within 30min. If you have to do research first, I'd estimate about 1-2h of work.
@@danielyuan9862 It’s so very fun for me, so i tell you: I have the hobby to spread science by asking people for watch-suggests and also offer the same.
Grant, there is a suddenly stop in 10:33, and then goes to another subject. I never commented in your videos, but man, thank you for making the best Math channel in UA-cam, and always explaining such a complicated matter in such a simple way. Greetings from Brazil.
Unexpected existential crisis. I'm- *awful* with math and stuff, but towards the very end. . . It felt like we'd stumbled across a tesseract. Or a cross section of a cube. Seeing "coincidences" in unrelated areas in math, and in string theory, and this massive number that relies on a huge dimension to make sense- it feels like we've got a very small glimpse of only part of it. I dunno if I'm making any sense. It's four in the morning and I'm half asleep. Still. A lot to think about. Thanks for the quality vid.
As a math student and math educator, I am so incredibly glad this exquisite video can produce such an intense reaction on someone that admittedly is “awfull” (big quotes there, don’t truly believe that) and could therefore lose its interest quite easily. It proves to us all that this spark we see can be shared to others that haven’t tasted its sweetness during the period one usually expects to.
Hi! You are making perfect sense, this kind of thing has happened before in history. Maybe not to this extreme, but "math monsters" that at first are just there and "don't make sense" eventually are made sense of. For example Imaginary/complex numbers arose from math and were kinda "nonsensical" to mathematicians at the time, but they are fairly well understood by scores of college students of our time. (that pattern holds for pretty much every extension of the natural numbers actually) In any case, I wanted to say that I agree with Bernardo Picão... if you got that feeling of marvel from this video, that's a really cool thing this video is achieving, and maybe you can now see why some people love math, it's sometimes beautiful in a way you can't explain. I've been a fan of 3b1b from the beginning for this very same reason, he is really good at communicating the core beautiful ideas behind math, even if you don't understand anything at the very least you can glimpse it. He started with college math that I understood quite well, but I just had so much fun watching his videos and showing them to some people. In the case of the monster apparently even our most advanced mathematicians can barely glimpse it... On another note, I've always wondered if there's a limit to how complex of a concept can we as humanity understand? I mean, as we are now, there certainly should be... you learn the condensed product of generations of great gifted people who built upon each other... lifetimes of past great people pondering upon a subject and after many years you can maybe start working on new math to maybe (big maybe) advance it a little. Will there be a time when a lifetime is not enough? When new math is only for the most gifted of humans? Maybe eventually to keep advancing we'll have to really look into teaching efficiently so we have enough time... I figure someone probably thought along this lines before, but never heard of it (I'd like to hear about it).
@@azlastor A lovely response. Let's wait and see if we can specialize in what we want, or if mathematics will become so big we need to coordinate the fields to specialize ourselves in. Orchestrating all that would be a sight.
Something about Grant's calm reasonable voice is enchanting. Conveys his awe of the subject without hype. Gently encourages futher inquiry. Congratulations, buddy! You are a great friend to humanity.
I love your videos, fortunately for me my father is a mathematician so whenever I don’t understand something I can discuss it with him till I understand. Thank you for the great moments!
This is amazing. I remember learning about group theory, at the beginning it was super confusing but became way clearer at the end, I ended up loving this field of maths. This is completely different approach of presenting it, amazing work.
As a student familiar with contest math, I was pleasantly surprised by how detailed the difficulty scale at 14:07 is. The first item is just 1 + 1, the second item is a problem from this year’s AMC test, the third item is probably an olympiad inequality problem, and the fourth is Fermat’s Last Theorem. Adding these difficulty markers really emphasizes the point “this is a hard problem” better than words could (if you’re familiar with the problems, that is).
That's the thing about abstract algebra. A set can consist of literally anything. An operation on that set can be almost anything. That's all you need for an algebraic structure, so, in that sense, a lot of things have the same structure. Groups in particular are pretty common, so a lot of things are a lot more similar than they seem at first blush!
@@SaberToothPortilla That´s why we learn so much about this topic in Materials Engineering graduation, the group theory have such a power to help us dealing with the interaction between atoms that is hard to conceive other efficient way to deal with these questions...
15:35 for a moment there I thought to myself “atomic structure? Surely it would have been better to say something like ‘fundamental’ or ‘prime’ structure to be more mathematical?” However then I remembered that ‘atomic’ comes from ‘indivisible’ and is thus a perfect way to describe it! Well done sir Edit: 16:54 ‘it’s like the universe was designed by committee’ hahaha
The other reason I like the atom -> molecule analogy slightly more than the prime -> integer one is that just knowing the simple groups does not tell you all the ways they can combine. Just as molecular structure involves a lot more information than a list of atoms within it, knowing all the ways simple groups can combine is an unfathomably hard question.
@3Blue1Brown Group theory is also massively important to chemistry, since simple chemical structures can be defined as groups you can lean on group theory the derive fascinatingly specific properties of their atomic composition. Things like bond strength, molecular orbital energy levels, & so on can be calculated, and used to estimate applied properties like IR spectra without performing instrumental analysis. Molecular symmetry & group theory by Carter is where I learned all this stuff back in College, & I would recommend paging though it if your interested :)
This is absolutely brilliant! I have been looking for an introduction to Group theory that would help me understand some of the foundations of Galois representations to try to grasp even a very general understanding of the maths underlying the proof of Fermat’s last theorem. But most video content on Galois groups assume so much knowledge already that I couldn’t make any headway, until I found this one, so thanks! I loved your channel anyway, as an amateur maths enthusiast :)
"Hey, Raphael, we need some number for this constant. What did God give us?" "I don't know, someone lost the sheet. Just smash the keyboard a few times, it should work."
I loved my group theory course. I find abstraction easy and calculation hard. For me, group theory was the easiest math class I ever took, given what I already knew going in compared to what I had to learn in the class. Linear Algebra, on the other hand, was the hardest math class I ever took, at least for me. Linear Algebra was my first ever college math class (actually, my very first class on the first day of college) and Group Theory was the last one I ever took as a senior.
I wish this was how I was taught group theory in undergraduate before I had to go through all the dirty deeper details in grad school. I am a teacher now and your videos have become inspiration on how I teach.
Next week I'm going to teach basics of group theory, group operations and the character table for undergraduate chemistry students. I remember the first time I had contact with the subject, I got as excited as I got confused and graspless about it. I never could make sense of WTF was a character table, a representation, an irred, and stuff (at the time). Now, as a teacher, I felt compelled to give my students the opportunity to at the least understand these things, to then show them its massive applications on chemistry. This video helped me enormously on the task! Much appreciated for such a wonderful, plain, straight to the point presentation on a subject no undergrad or grad teacher ever wondered to offer on the subject.
*me, who passed grade 11 university math so barely that my teacher literally stopped me in the hallway to make sure I didn’t take math in grade 12/higher education* : ah yes, the perfect video that I’ll definitely understand
![Lp. Сердце Вселенной #60 РОЖДЕНИЕ ЛОЛОЛОШКИ [Финал] • Майнкрафт](http://i.ytimg.com/vi/YoR0pAV9FVQ/mqdefault.jpg)
"We always consider the action of doing nothing to be part of the group" - 1:41
My favorite quote
Story of every group project! Now you have the excuse: "I am the group's identity element, I act on nothing and I am necessary" :)
@@alexanderschafer8979 "Next time, how about we have a semigroup project!"
That one truly hits home ;)
I'm part of some organized activity. There's this dude that definitely is our groups identity element. He's nice though. Zero initiative, but there's not a bad bone in his body.
Part of the group project more like
Other Mathematicians: "Polynomials, Permutations, Quaternions"
John Conway: "Monster, Baby Monster, Happy Family"
You are short of pariahs!
We need more John Conway's. Why make something so dense and complex even more difficult with letter-number combinations when you can just name stuff BABY MONSTER
Monstrous moonshine!
CoronaVirus took away one of the greatest man of our time
dog Need more help naming polytopes, there are weird shortenings like “griddip” or “groh”
I'm a simple man. I see an outrageously large number in the video title, I click.
how's the KFC austin
ayy it's Austin love your vids
Ayyy Austin how's that birdwatching going
Heyyyyy! I heard your sister in law got married.
Eyyy
It feels like with "the monster" humanity has learned the answer to something without being able to even ask the question. Almost as if we've stepped into territory that isn't meant for us yet. Of course the given name adds to the lovecraftian feeling of it and I love stuff like this.
Your comment genuinely made me teary-eyed, ngl
You share the same number as the monster, Jupiter. Seems sus to me.
"humanity has learned the answer to something without being able to even ask the question" literally the plot of hitchhiker's guide
"Forty-two," said Deep Thought, with infinite majesty and calm.
Given that Horrible Problems Lovecraft himself "didn't have the constitution for math", do we have the "constitution" for this?
21:15 "The universe doesn't really care if its final answers look clean, they are what they are by logical necessity with no concern over how easily we'll be able to understand them" - I love this sentence!
This video was amazing. It's just so cool that structure arises when studying symmetries themselves. Your videos always inspire in me a sense of awe and wonder, but this video was something else. Thank you for all the time and effort you put into this.
Indeed, it WAS something else. One of his very best; and the bar was sky high to begin with
Let's not discuss platonism vs constructivism.
It looks possibly to have taken inspiration from Niel Degrasse Tyson's quotation, something to the effect of, "the universe is under no obligation to make sense to you."
@Dr Deuteron He looks like a gentle giant to me, so maybe he cares about our universe. Let's just be careful not to go near baby monster lol
@Dr Deuteron no, it is just you who don't want to think about the monster so you make it sound scary, even though the monster has always been part of reality & doesn't hurt us
15:54 reminds me of the "how to draw an owl" joke:
step 1: draw a circle
step 2: draw the rest of the owl
This is the essence of LISP
Its TRIVIAL and BASIC
@Benjamin Schocket-greene my proof: 23 pages long
Homestuck..
@@tsanguine Damn crazy how the comment had nothing to do with that. Gonna have to send you to the shadow realm for that one
Just got new earbuds and while trying them on, suddenly my google assistant starts reading my notifications. Of course, it had to read this never-ending title for me lmao
Weird flex but ok...
@@kavinbharathirm9478 You're meme-flexing, and it's misplaced. Earbuds can be had for a few bucks.
Which earbuds btw?
wf 1000xm3, so not exactly cheap though
Hahaha that's awesome. Did it work out the name of the number (with variations on duodecicillion etc... maybe somebody can give us the actual name), or just read out the digits?
Honestly, there's something beautiful about the way 3b1b explains things. At around 4:28, he explains that the permutations of 101 different objects would amount to 9x10^159. However, instead of simply saying 'this is roughly the same as the number of atoms in the universe squared', he says 'if every atom in the universe had a mini universe inside of it, that would be how many sub-atoms there would be.' Take the time to appreciate the time he took to make these numbers just a bit more interesting!
It's a nice metaphor since humans suck at understanding multiplication intuitively. A big number multiplied by a big number just equates to a bigger number for us, we're bad at telling the difference.
I'm glad you were here too, though, so I understood he meant atoms squared.
1:47 identity
It was we who he helped us be more interested. I do love the care he takes with words. I DID take the time to appreciate the video he made. Now I have decided not to take the time to correct your last sentance, maybe you might like to correct it yourself, then I would know that you have been taking the time to appreciate that it takes time to help show that many things can be fascinating once we have taken the time that it takes to appreciate them. Grok is a good word. Lets call the monster Grok.
the difference between fiction and reality is that fiction has to make sense.
Would that I could like a comment twice.
That is a beautiful quote
Stories have a beginning, a middle, and an end. Reality doesn’t.
Lawrence D’Oliveiro technically, reality does have those, but only 1 of each. (hint: scope = universe)
@@maxlife459 It’s not clear that the universe has a beginning or an end.
Any chance of an "Essence of Group Theory" series? I would love that!
underrated comment
As a physics student I really need that
I would really love that!
plhease
Sounds like fun
Thanks to the math community for having good and reasonable events often, it's always a pleasure to see math collabs.
Lmao you say math collabs like it’s a song! I can’t wait for “Monstrous Moonshine Conjecture” by Mc-Kay feat. John Conway
@@carlsagan1377 Lol
@@carlsagan1377 You'll be waiting the rest of your life, he died in April.
@@maxwellsequation4887 Oftenly isn't a word, you can just use often.
Xeridanus I know, and to COVID, too. It’s a damn shame. RIP
the way you explain the concept of group as the concept of number "3" is really mind opening and important. A lot of people having trouble with math because it's seem so conceptual and they always try to link it to something more grounded, but to be good at math they need to approach math as how conceptual it is. Eventually of course math is used to help real life problem but it's not always straight forward, so you need to think about it in the world of math itself. It's like when I started coding and at first my mind will work only with what the final UI or graphic display on the screen, but slowly my mind would think purely of what happen data wise and not really the final representation.
I feel like UA-cam is robbing me of your comment cuz it ends on "on the screen, but slowly my mind ..."
it does help to start with a concrete problem and then shift it into the abstract though.
you don't learn multiplication starting with anything but the natural numbers
Someone should give him a medal for making such an abstract theory so beautiful and entertaining, and yet extremely educational! ❤️
Т
He is the one who more than deserves it !
He did the math, he did the monster math.
I have no idea who you are or where you are, but I want you to know I love you, just for this comment. Stay beautiful, my friend.
@@pcdm43145 Awww. Thanks! You too!
yes monster math should be a thing
Nice.
It was a graveyard graph!
Conclusion: "The universe is under no obligation to make sense to you!"
Sigma approves this message
hmm it would appear that the universe is singing to me
What is that melody?
Evidence suggests there's Math ahead
A Unexpected But Not unwelcomed Outcome
A quote from a Pratchett novel comes to mind, when a wizard tries to explain how a mysterious cabinet works.
'Yes. The box exists in ten or possibly eleven dimensions. Practically anything may be possible.'
'Why only eleven dimensions?'
'We don't know,' said Ponder. 'It might be simply that more would be silly.'
do you know what book thats from? it seems interesting.
@@jetison333 "Making Money" by Terry Pratchett. Although most of the book is not related to that quote. Still a fantastic book. It's the sequel to "Going Postal" so that may be a good place to start.
Pratchett is my absolute favorite.
Me: how did you get so strong?
Mathematician: every time I find a new dimension, do 1 push-up
Me: Jesus Christ
Yes but
"It's the size... of the monster" is such a scary way to put it. What the hell, math?!
And then you get to the end of the video and there's a monster, a baby monster, and a happy family.
"The Monster At The End Of This Video"
The scary part, as far as I can tell, is not how BIG the Monster is, but the fact that it is NOT infinite... At some point, it just... stops.
And that's where the grouping stops. It can't go any further than The Monster boss, because calculation past that is impossible. Wow man, just epic.
This was like watching anime without subtitles. I didn’t understand a thing but it was gorgeous.
Right it was frustrating I watched the whole thing and was like wait that’s it
yeah the video is so interesting and it's so annoying that i still don't really understand it also my dyslexia doesn't help either...
can relate
"""""""anime""""""
But i mean... there are subtitles
"Nothing has given me the feeling that I understand why the monster is there."
That quote is honestly viscerally terrifying.
That's some Cthulu shit
I was worried since I saw the Numberphile video with John Conway, and I'm still worried after watching this.
@@staglomagnifico5711 the way he said "oh no, it's not arbitrary" has sort of haunted me since. RIP Sir John Conway
i honestly love it; it's a big indication of how the universe doesn't really care if it makes sense to us. if anything, to me it seems to actively resist being understood, like it doesn't like humans poking and prodding at it so it conjures up stuff like quantum physics and the monster group as if to say "ha! explain THIS". it's amusing and oddly charming, even if it is just me personifying reality itself.
Math always felt so incredibly clean, structured and sterile to me. The idea that something so bizarre could exist in it, with nobody understanding why, kinda terrifies me too. It feels anomalous, like something that shouldn't even exist. Gives me SCP vibes to be honest. Remember Theta Prime?
This is IMO the best video 3b1b has produced by far. Amazing explanations, visualizations, stories, everything. At first everything went over my head, but after learning a bit about group theory, this video is so cool.
Imo the best vid was the Alice and Bob one.
"The universe doesn't really care if its final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."
Elegantly stated. As a grad student in mathematical physics, this definitely lines up with my experience!
For some reason this sounds like a happy 70 year old grandpa who studied maths
Wow, seems incredible, because math independently can feel so elegant and beautiful , but the most amazing thing is when it’s applied on the natural ways of reality, you can actually see it shift, and basically “become a huge mess”.
Really tells you something about the way math serves us , and the way it explains reality simultaneously
Wish you best of luck!
@@amitbenjam Thats the thing. The field in question here has nothi g to do with physical reality.
This is a result of first order logic.
Its purely mathematical.
@@ineednochannelyoutube5384 incredible
I as a student in an entirely different field, hate solving equations in mathematics....but something always draws me into the theorems that it provides.
I just did an entire semester on group theory, and yet every second of this video had something for me to take away. Brilliant stuff!
you will find that you obtain gains even by taking that exact same class you took. I've learned through exp that you could take the same course and even use the exact same text and do the exact same chapters and still make gains. I can't even recall a time in my undergrad where i felt like I mastered any text book so well it was rendered useless...
@@dasmartretard Well said
@@dasmartretard Really on point. I've authored textbooks and learned much more about the subject by simply focusing so deeply on the foundations. Even new ideas come together more clearly for me on subsequent editions. I'm convinced that just a solid introductory book can make someone much more competent in applied work than the typical PhD student gets after a dozen courses.
As a graduate student who studies the briefly-mentioned “theory of modular forms” and knows about the monstrous moonshine conjecture (my advisor proved a related ‘moonshine conjecture’) this video was truly wonderful and the best conceptual introduction to group theory I’ve seen. Well done!
I'm interested in understanding the statement of the 'Modularity theorem' and also getting some idea of what the Langlands program is about. Can you recommend some books or lectures? (I don't have a Mathematics degree)
@@MrAlRats
As one getting a maths degree I'm also interested in references.
commenting incase someone mentions references for me to add to my ever growing reading list.
@@MrAlRats Knapp "Elliptic curves", it's covered in the last chapter or near the end of the book.
I just graduated with a Master's degree in Mathematics, my thesis was on modular forms stuff. Who was your advisor (if you don't mind sharing)? I've attended the Automorphic Forms Workshop a couple of times, I've probably at least heard their name.
For everyone else, I think Kilford's book on Modular forms was somewhat readable with a decent background in group theory (maybe some number theory) and knowing kind of what Fourier expansions are. For anyone outside of math, you need to learn what proofs are first, and then a few more things.
Diamond and Shurman's book on Modular Forms is a beast. I kind of managed to get through the first chapter and maybe a few pieces of the second chapter, although I tried to do that before Kilford, so I think it would be easier now. Also, I think some experience with complex analysis might be necessary that I wasn't fully solid on. Modular forms was not an easy topic for me to just jump into.
Progressing from no math beyond Calculus to modular forms, here are some books I might recommend:
Learn proofs:
A Transition to Advanced Mathematics, Doud and Nielsen (free and well written, highly recommend)
Learn group theory:
A First Course in Abstract Algebra, Fraleigh
Representations and Characters of Groups, James and Liebeck (all solutions at the back of the book, it's amazing; also learn a bit about generalizations of this 196,883 thing)
Galois Theory, Cox (not strictly necessary to get to modular forms, but you do get to see what's going on with the roots of polynomials business)
Abstract Algebra, Dummit and Foote (if you really want to go hard; also, just the first fourteen chapters - after that it goes into Algebraic Geometry and other topics; also you might want to at least take a glance since Kilford touches on exact sequences and Dummit and Foote has a section on them)
Learn complex analysis:
Understanding Analysis, Stephen Abbott (probably best to start with Real Analysis)
Fundamentals of Complex Analysis with Applications to Engineering and Science, Saff and Snider
Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable, Ahlfors (this is probably the standard text, although I used the one by Stein and Shakarchi, which was alright)
Some basic ideas of Number Theory might not be bad, maybe something like Fundamentals of Number Theory by LeVeque, and maybe Introduction to Analytic Number Theory by Apostol, seeing as modular forms in Kilford are mostly treated in the sense of Analytic Number Theory.
Then Kilford.
Going through all of those books is probably not fully necessary, but since when is stopping to smell the roses a bad thing in math?
I like your final quote "Fundamental objects are not necessarily simple. The universe doesn't really care if it's final answers look clean; they are what they are by logical necessity, with no concern over how easily we'll be able to understand them."
Typical horror film technique: don’t show the monster to preserve suspense.
Well, how does one "show" it, exactly? Constructing it, or even defining it, is not really straight forward. For a long time, the only result or description of it was a proof of its existence.
no plz cuz Grant really likes it, let him appear the monster plz
I might be wrong but I think photons are in 5D space, as far as special/general relativity is concerned anyway. The basic idea which when you go into detail is totally different (going off memory, may be wrong) is photons reveal what is in 3d space, travel through time (4d) and they are in the 5d. Surely, much much more could be unknown about em and time is controversial to begin with but im sure thats the consensus in modern physics so far; although anyone could come along and massively correct me
plus, time is apparently, according to some, just a result of the universe expanding or whatever.
So, photons apparently dont actually move, they stay static in one part of space time, but space time moves really fast and its as if the light travels as fast as is possible in space, but its just travelling at the fastest (known) possible speed in the universe; the rate at which it 'expands'.
I think some people claim that in one dimension, there is a pulling force, like a gravity, to the universe and then in another dimension theres like an opposite pull - not so much a push but pulling in the other, or another, direction - and basically, the waves that photons make are a result of one dimension pulling, causing a peak in the wave (the light wave travelling 'up'), while the other dimension pulls the other way, causing a trough in the wave (the light wave travelling 'down'). Idk if one dimension, would be space, or time, or electromagnetism.
Now, it's a bit like when you grab two edges of a piece of paper and pull them apart. The paper rips into two pieces in around 0.27 of a second. On the macro level (watching in real time) it's as if it's a clean break, it was pulled near enough equally in both directions and torn apart.
Yet, if you were to slow it down to an extreme level, like 20000 times slower or something and magnify by about the same, you would see the left hand pulls the paper diagonally to the left, then the right hand pulls it to the right, very quickly.
These pulls to the left and to the right happen very frequently, very quickly and there are very much of them, as the tear travels down the paper; from whence the tear started and whence the tear must end (the other end of the paper).
and when you observe time at a very slow instance, almost freezing time in effect, you can see the up, down, up, down, up, down, up, down of the photon. Really, if you consider the photon is staying still, and this is all happening very fast, perhaps at the speed of which the universe is 'expanding', then it's like you can see the different pulling dimensions acting as the 'left hand' and the 'right hand'. But, it's not 2d like the paper, it's 3d or 4d or some higher amount of dimensions.
The light wave, travelling in one direction, is kinda like the tear in the paper. You can see how the universe is tearing, being pulled left to right in the patterns of the light wave.
Yet, just like how the paper was near enough a clean break, torn apart in 0.27s, this pulling and tearing of the universe is near enough a clean break (we'll assume; scientists have found that the universe is likely expanding at a faster rate in one direction than another direction(saw Zach Star mention an article about it in a hypersphere universe video) but that's a story for another day haha) so whether or not we see the pattern, the pulling left to right on the microscopic, micro-timed level, the result is still near enough the same. We see the light, pretty much in an instant, before we can even notice we see it.
If there is a plentiful amount of photons, which they're most usually is, we see a constant stream of photons and we visualize the object that emits the photons.
@Grant Jacobson to answer your question: even if light is in a lower dimension, a 1d point could travel through 3d space and hit near enough exactly the vertex(corner) of a cube. This 1D point would then be reflected along another path in 3D space(4D spacetime if you wanna get technical).
In a similar way, whether photons are 3D, 5D or More Dimensional, if a photon was to travel directly into a vertex, a corner of the monster, it would be deflected and reflected along a different path in it's own dimensions; or perhaps the photon would be reflected into different dimensions to the ones that it was originally in, then it's path would continue there.
I can only assume The Monster, being some form of physical object, in space, would have oodles and oodles, trillions of corners.
I can also only assume, that these corners must reach the edge of many other dimensions because... they're corners. With it being such a highly dimensional being, these corners must crossover into many and lots of different dimensions, so it goes to suggest that they must probably crossover every, or most, dimensions, including the ones that light travels in.
Now, if enough light hits enough corners, travelling from (starting from) enough different dimensions (assuming it can even be reflected/deflected into hitting the corners of the monster; covering all grounds)(and assuming light could even act the same within it's presumed finite dimensional range in any set of dimensions, that's a pondering for another day;[*]), then, some of the monster could be illuminated and visualized. Maybe the light could travel along edges, not just corners, or perhaps even it would be hard for us to distinguish between corners and edges at that high level of dimensions; this way we could possibly see more of it.
Through seeing the corners we could get some kind of glimpse of what the shape of it could be, from some presumably very limited angle; yet, we could estimate a full picture of the monster by calculating symmetry patterns in what we do see.
* -[perhaps the higher the dimension the light travels in, the higher range of dimensions it occupies. Wherein, the range of dimensions it occupies as it occupies higher dimensions could expand at a proportional/formulaic rate as the range of dimensions it occupies gets higher]
TL;DR: To the one person who may have actually been interested in this: Thank you. You're welcome!
@Pybro by the way, I like this use of the definition here, very refreshing. thanks
Sometimes I lull myself into believing that Grant is a normal human being, and then I see a video like this, and I remember that we are speaking with higher-dimensional beings.
@Juho Grohn Actually the representation was 196,883^2. So he is much bigger than just 196,883.
Yeah, I learned a "lot" of math during my CS/info-engineering studies, and I loved group theory and Galois fields, but maaan - the Monstrous Moonshine Conjecture
It is comments like this that put a smile on my face every time I open UA-cam. Thanks.
LOL!!! :D Same here! A lingo that lower-dimensional beings like us are not destined to understand!
The higher-dimensional beings only speak to you, you can never speak to them ;)
I'm not THAT much of a math person, but when you showed how the square permutations are the same as the dot permutations, my mind was blown in the best possible way
same dude same
THAT is what non math people should be seeing im math classes, not boring formulas without context
cube
Blowing your mind in the best possible way is one of the most beautiful things about learning pure mathematics imo.
It comes from a deeper and through understanding, and it's very satisfying.
It is kinda funny that you said square instead of cube because the group of the symmetries of a square is not isomorphic to the group of permutations of 4 elements i.e. there no 1 to 1 mapping between those 2 groups.
I remember reading "Symmetry and the monster" about 15 years ago, and fell in love with the monster group -- and due to one of the later chapters in it, the next video queued up after this one is now "Hamming codes and error correction
". Classic work.
Thanks for the book mention, I think I'll check it out.
@@jherbranson 🙂
You gave essential intuition for one of the most interesting and complicated fields of modern math since it can pop up almost anywhere. Wow. Your video can actually serve as a first step into studying this branch.
I found this to be a little info-and-reference dense to be a single first step into an extremely complex field, kinda a weird flex
Physicists: 11 dimensions... That's a lot...
Mathematicians: Haha dimensions go brrrrrrrrr
69 dimensions is pretty fun
Grahams number be like
in case you dont get it, grahams number was originally created as a number of dimensions involved in a math problem. Mathematicians had to define that massive number of dimensions to solve the problem.
@@lilapela haha
@@rhealastname266 nice
That 11 dimensions relate to 1+1+1+8, and the 196,560 of the Leech lattice relates to 24, where bosonic M-theory is in 27=1+1+1+24 dimensions. There are physical theories with 196,884 degrees of freedom that contain both the 196,883 and the 196,560 of the Leech.
Grant: "To maintain some hope of sanity..."
Me: thank you so much
Annddd... it's gone :(
This has got to be one of my favorite 3Blue1Brown videos. I love the way you present just how fundamental groups are. One line in particular I just love: "This is asking something more fundamental than 'what are all the symmetric things?' It's a way of asking, "what are all the ways that something can be symmetric?'".
This makes me feel like maths and physics is getting closer and closer to the source code of the universe
we are coded in source 8, not so buggy
Things like this studied by maths are actually more fundamental than the universe itself. If another universe were to exist with completely different physics yet housing some form of entities of sufficient intelligence, they would eventually come to these same results, since they are consequences of logical necessity rather than properties of the universe we inhabit.
@@MatthijsvanDuin what you just said blows my mind, could you, please, explain more about it? Cant logic work diferent in other universes? Truly interesting coment!
@ I guess it's plausible that we could live in a universe in which there were no dark energy and still have logic work the same.
@ In a sense, logic works independent from the universe. To reach a logical conclusion, you start with some set of assumptions, and then combine those axioms to reach some sort of conclusion. Because we exist in the universe, our starting assumptions are typically tied to the universe, but the conclusions we reach are only tied to those assumptions. We could start with an assumption that is untrue like "glass breaks when you hit it with a soft object" and reach the conclusion "if I hit a window with a feather, it will break". This conclusion is not true in our universe, but the logic is correct given the starting assumption.
Since logical conclusions seem to be independent from the universe we inhabit, it seems likely that if there were another universe with different physical laws, logic would still work independently from those laws. An intelligent species in that universe would likely still recognize Pythagoras' theorem as true, even if they didn't have any use for planar geometry, since they could still recognize that if they started with certain assumptions, Pythagoras' theorem would be a necessary result. Similarly, if these alternate universe mathematicians started studying symmetry, they would also eventually discover the Monster, since the Monster is tied only to the idea of symmetry, and not to any physical reality of symmetry.
"Monstrous Moonshine Theory" would be a hell of a band name
I will be using that thank you very much
Probably a post rock band name
You have to do Math Rock with this name. :)
@@kjl3080 Keep us posted
Justin Golden ok
"Not 4 dimensions, not 5, but we'll have to go to..."
Me: Hmm the next number is six so then it should be si..."
*sees the dimension counter started to increase more than 100000
My brain: ight imma head out
I had the exact same reaction lol
Indeed. Before this part, I was following along rather nicely, understanding most of the things being said. The same can't be said of the parts after this, and I am just as confused as I ever was watching most of 3blue1brown's videos on more advanced topics.
Oh i wish i could see your reaction when you learn about hyperbolic temporal dimensions.
@@racheline_nya Well I'm looking forward to it
"head out" haHAAA
15:04 that’s interesting, in the Galois theory course I did this year, we didn’t do composition series. Instead, we showed that insolubility of a polynomial by radicals is implied by the Galois group being insoluble. Using the fact that Sn is insoluble for n>=5, you’re basically done.
Just pure admiration to the way you explain such abstract concepts in a such elegant, clear and concise way. Thank you for your time and effort. Much love, appreciation and respect for you.
"What's the most important thing in math?"
"Coming up with funny names."
Google is a # and googleplex is that # squared
I wish mathematicians were more creative with naming and I didn't have to relearn the definition of "normal" and "regular" in about fifty different contexts.
a sUbseT oF a mETrIC SPaCe M Is callED ClOPen WHEN its BOtH oPen AnD cLosed
@@chadpatrick6795 that is wrong
@@Elyzeon. google it
Can I just say that the abstraction analogy was so genius, it gave me the chills. It also raised a particular question for me: if our ability to understand the abstraction of numbers early on is a result of us being drilled with sheets and sheets of basic addition/multiplication homework, would it be possible for someone to grind through sheets of basic group theory problems and end up having a clear understanding of the abstraction of groups?
I'd say it's the case for those mathematicians. They manipulate groups with ease, similar to what we would do with numbers.
@@totolamenace I agree. It just seems crazy. I feel like I could never wrap my head around group theory well enough to work with them so easily, but clearly, that's what those mathematicians also thought when they were in high school (maybe).
To think that everyone at some point in their life feel the kind of confusion towards something like "3" that is similar to the kind of confusion I now feel towards something like "S5", just blows my mind.
I agree: "Genius" is an apt term for that analogy. I wonder how many eyes were opened by the care taken in explaining that the way he did
@@FareSkwareGamesFSG The important thing is to get used to abstraction but also making things understandable and fitting things in a context, like the group actions stuff that was talked about here. It turns out all groups come from symmetries of objects but how complicated those objects are depends on the group. There is a certain truth to von Neumann's quote (which I think was a sort of joke) "In mathematics you don't understand things. You just get used to them."
This is truly one of the best videos describing overview of group theory and its recent developments in the past few decades. Thank you!
Man, I am the worst human being in math. I don't even remember the multiplication tables. That said, the fact that I could get a grasp of what you were saying in this video tells a lot about how good of a teacher you are! I wish my high school teachers had been a fraction as fascinating as you were in this.
Loved the video! thanks!
Wait..
So you are telling me that throwing you into a class room with 30 others and drilled constantly on outdated theories/tactics didnt interest you?!
@@jakechamberlain2206 aka school
You can pass calculus 1 without memorizing most of the multiplication table I have memorized a few multiplication problems and got fairly fast at solving the rest which was good enough for me.
Really? I had to atleast get to an introduction to proofs class before I could understand this video
dont worry im sure even the stereotypical "math wiz" would be confused by this, most of this seems pretty esoteric and you probably have to be not only smart but also pretty invested in the subject
This is a gorgeous explanation of the monster, presented in a way I can almost understand. My father would have loved this.
Could it be that your dad's name was John?
Your father was an amazing man
Sorry for your lost man…
Your father was most likely a cool nerd
@@Mipetz38 Did you check the last name?
The part about cycling three elements around (and ending up where you started if you keep doing it) jumped out at me, because that shows up a lot in Rubik's cube solves.
Maybe this is an urban legend, but I read that Rubik invented his cube while trying to explain group theory to his students.
@M J He was in architecture.
Yes, moves you can make on a Rubik’s cube, as well as sequences of moves, where two sequences which change the configuration in the same way are treated as equivalent, are elements of a group.
@@rchaser Hi. Want some science-recommandatios?
@@rchaser pretty sure thats not true he just wanted to make a block out of smaller blocks.
I love your videos. I’ve always been a little intimidated for the level of abstraction of some mathematical concepts, but you can explain many of them more intuitively, with elegance and also generating more interest. Thank you and please keep doing this great work. :)
17:02 "It's like the universe was designed by committee" brilliant
I thought he said "comedy"
I think this committee is really fucking up the design of the universe. God: "Hold my beer".
It's really external to this universe. This applies to all possible universes.
its a great story idea. Though, i guess its already been done, considering all the "we are in a simulation" conspiracy theories. or the quintara marathon, lol. Just an experiment by a committee of extra-dimensional scientists to see if they can make a universe capable of developing life. oh snap, isnt that a rick and morty bit too hahaha
14:08 “This question turns out to be hard - exceedingly hard.”
Yep. I wholeheartedly and immediately believe you.
How does one even begin to ask questions like that? You'd first have to believe that there is an answer that is a finite number (instead of infinite possibilities). How do you you figure THAT out?
@@rayniac211 There are infinite possibilities. But we can still structure them.
4 is the answer I can betcha
Mathematician here. Let me assure you that this "number line of hardness" is not a walk in the park. Probably logarithmic scale. The third point, where it says "a/(b+c)+b/(c+a)+c/(b+a)=4" (btw, he made a little typo in the last denominator) as in "find (all) positive integers a,b,c that fulfill this equation", is already mind-bogglingly hard. And the fourth point is Fermat's Last Theorem. Formulated in 17. century, it took until 1994 to finally prove it.
Lone Starr yeah, I picked up on that as well. Would you say 3b1bs assessment of the difficulty in finding all the simple groups is accurate? As in is it significantly harder than Fermat’s last theorem?
*Next video: But what is the meaning of life? A visual introduction*
I know the meaning of life but I won't tell it to you :) Give me permanent citizenship
I just wanted to like your comment, but I figured the topic "the meaning of life" makes 42 a suitable number of likes.
@@skraemerLP rip it exceeded 42 :(
3blue1brown is a higher being who can understand things he can't even explain to us.
This is the first of your videos where I felt more clueless than ever, usually I feel like getting closer to God when I watch your videos but not in this one... the monster beat me xD (probably because I never read a single line of text about group theory hahahahaha while otherwise I am familiar with other branches of mathematics in varying degrees)
I'm honored to have a good friend who helped prove the umbral moonshine conjecture, but it's far, far beyond my understanding. I definitely remember group theory from linear algebra and vector spaces, and they are absolutely beautiful, but my impression from my own reading was that finite group theory gets very complicated very quickly - just as with chemistry analogy! Lovely video as always. Thank you!
This is why I love Maths and Physics. "They are what they are by logical necessity." That's mindblowingly fascinating but at the same time like super trivial. Cause of course things can't be what they can't be.
the beautiful thing is why are things that are necessitated simultaneously surprising and fascinating?
If things were different, they wouldn’t be the same
You might be slightly missing the point.
It is what it is, or else it would be different.
So I looked up the monster group on wiki, and then further to the classification of finite simple groups, and my god that is very impressive. People call this the greatest intellectual achievement of humanity and it's not an overstatement.
15:27 'This is a super high level description of course, with about a semester worth of details missing' It made my day.
Care for Bitcoin investment tips for good returns?
That is the comment, I was looking for.
Might be over my head but just in case you didn't understand, he most likely meant "high level" as in "simplified, asbtracted" and not as in "advanced" :)
This is really fascinating. The more I learn about math, the more I realize the ways we can transform reality into abstract symbols that we can use to find patterns. This a pretty broad generalization, but it has huge implications. We can express just about *any* abstract concept in *any* particular facet of the entire lived experience of a human, or to *any* small detail of the entire universe in a defined, processable manner. How freaking insane is that? The more I learn, the more I can see the patterns between things I’d never thought to connect to each other, and I still don’t even know what I don’t know yet. I’m not sure I have any other words for that than “awesome” in the most literal sense.
I've been going through this process. It's a bit of a rabbit hole...
Looking at literally everything: It's all symmetry groups?
3B1B: Always has been.
lmao
This is why I donated the man 1000 UCO :) like I said boss do not spend until 2024 haha
What a teacher.. you remind me of someone mate keep it up this is blowing my mind how good you are I didnt sub years ago, my bad ;) but I left math and dodged programming out of lazymind. Now i be a relic :) peace all
54 digits must be one hell of a big result I cant wait to see the ending
*The beauty of this channel is that it makes these advanced math concepts feel approachable to those with no experience. We all appreciate these videos, keep doing what you do!*
17:04 no this is exactly what chemistry did, “these all are alike, those? those are extra”
Lanthanoids and actinoids? Yeah, there are 14 of each if you count either La/Ac or Lu/Lr to be part of the main group 3, and 15 of each if you treat group 3 as only having the two elements Sc and Y.
So there's either 28 or 30 rare earths, and 26 sporadic groups. _Almost_ ...
Lantanoids and Actinoids are only written down separately to make the periodic table look more clean. In reality, they fit there perfectly. They just make the table wider.
@1 2 Unstable elements can be literally anything doe, as they need not be stable.
Some chemists may be concerned with the symmetry of chemistry... while we are concerned with the chemistry of symmetry
@@theexaminer4906 We are simply not prepared for the ultra wide chad periodic table.
I'm at the end of my CS Ba. with maths as secondary subject and just this last semester I was told off-handedly that we know what *all* of the fields are. This blew my mind. But nothing in all my university education has come close to giving me an intuitive understanding of groups like this video. hats off to you, Mr 1Brown, my subscription is long overdue!
Please make an “Essence of Group Theory”. We, the undergraduate math students, need your help
Edit: "We, the undergrads of America, need your help" was purely phrased that way as a reference to old Uncle Sam Posters with similar phrasing. I absolutely was not trying to exclude undergrads outside America. I literally just thought it made for cool phrasing, but I changed it.
yeeeees, pleaseeee. I really need it and I think lots of other math students too
We, the undergrads of the fuckin planet Earth, need your help
@@AlejandroFernandez-mq3jl yes!!!
not gonna lie i can live with not seeing the essence of calculus/linear algebra as I was studying them, but not having a "the essence of group theory" as an undergrad would make me extremely envious of future undergrads that have that series
Search up for Visual Group theory on youtube, by Professor Macauley. Once you get a grasp of what Group Theory is, you can watch the playlist of
Richard Borcherds, who was mentioned here. I think the course taught by Richard Borcherds is a tad bit more complicated and I think he mentioned himself it is for very ambitious undergrad students or first-year graduate.
To make sure I understand this correctly: the chart at 18:39 reflects that the set of finite simple groups is *countable* and that it consists of 18 countable infinite subsets (with possible overlap) as well as 26 countable finite subsets?
I ask because I would have assumed that the answer to the question "how many simple groups are there" would be just "it's infinite"... Which turns out to be true, it's just that there are some groups that can be extended indefinitely by a simple algorithm and some that by their nature can only exist in spaces of certain specific dimensions-one of which is the "monster group", so named because it's the biggest of the non-infinite families.
Really really interesting. In some ways it's a baffling thing to come to, but when I think about it I realize it's befuddling in a familiar way. It makes me think of prime numbers. Prime numbers emerge so simply from basic arithmetic, and yet they exhibit behaviour that can't be fully described with arithmetic alone. The sequence of primes is in a sense "arbitrary": there's no rigid pattern to them, only some fuzzy tendencies. But in another way, the sequence of primes is anything but arbitrary: there is no accident, it is exactly what it must be.
I think that, to someone who is building mathematics from the ground up, prime numbers would be the first sign that things can't be kept in control, that there is no way to prevent simple rules from leading to surprising results. It would be the first indication that there are things like the "monster group" waiting in the mists ahead.
The fact that the set of finite simple groups is countable is pretty easy to prove. The first infinite family mentioned is the set of cyclic groups of prime order, and we know there are infinitely many primes. It's also at most countable because there can only be finitely many groups of a given size (just enumerate all the possible times-tables).
That there are 18 infinite families and 26 exceptions is much harder to prove, of course.
I think each of the 26 exceptions is not a finite set, but rather a single group. Out of all finite simple groups, there are 18 countably infinite families, plus 26 groups that don't fit into any family.
You should check Godel's Incompleteness Theorem, I think you will love it
@@mertaliyigit3288
This really doesn't apply here.
@@sarahbell180 Why not? It looks like Gödel coding might well interest the OP, given his observation of unexpected complex behaviour from simple arithmetic rules.
me: "ayo alien, is the nummber 8x10^53 interesting?"
alien: "yea for real i love the nummber"
underrated comment
don't speak to the alien that way. you will get vaporized so fast for being that gay.
@@DrakonlordEreshkihal-vf5o You are a disgrace to the Pablo community
me: "what about number 69?"
alien: "lmao"
@@DrakonlordEreshkihal-vf5o why r u geh
I lack the words to express how blown my mind is, and I'm amazed at how well you explained it conceptually... it's just so much to understand...
RIP John Conway. He said he wanted to understand the Monster before he died.
Dont worry, no one really dies like most people think we do. The essence from him will be born again!
No one can get away from this existence. At least not so fast and easily as many of us wish we could be.
C: RUBBISH!
Says the voice from beyond.
oh! I guess that proves your point !
Maybe we'll just regroup again
God's teaching it to him right now...
@@bitterlemonboy How do you know?
Grant is great in all of these: Understanding the math, coming up with a clever storyline, and being able to visualize it all. Truly unique.
This is the clearest intro to group theory I've ever encountered that also explains why group theory isn't trivial but wonderful and mysterious. Great job!
18:01 RIP John Conway, sadly a victim of COVID-19.
:(((((
Did he really die _of_ covid or with? Well doesn't matter he is dead anyway. *sadface*
musashi939 jeez don’t
Damn it. :(
musashi939 yes he did
C'mon guys. It's his favourite number and none of you even cared to read all the digits.
TOTALLY UNDERRATED COMMENT
Thanks bro. 😂❤️🙈
Eight hundred and eight sedecillion, seventeen quindecillion, four hundred and twenty four quattordecillion, seven hundred and ninety four tredecillion, five hundred and twelve duodecillion, eight hundred and seventy five undecillion, eight hundred and eighty six decillion, four hundred and fifty nine nonillion, nine hundred and four octillion, nine hundred and sixty one septillion, seven hundred and ten sextillion, seven hundred and fifty seven quintillion, five quadrillion, seven hundred and fifty four trillion and three hundred and sixty eight billion.
@@ML-xp1kp ahaha you took your time, you really took your time
I honestly don't have the energy to do so.
man I need some coffee.
Fun fact: For the equation at 14:07 (a/(b+c) + b/(c+a) + c/(a+c) = 4), the smallest coordinate of the smallest integer solution is 80 digits long. You can find it using an elliptic curve.
There's a typo in it.
I've already concluded with the Euler constant that no matter how abstract we get we will never get rid of arbitrary constants. If anything, I hope that one day group theory will be able to derive geometrical and algebraic constants from fundamental logic.
I think you're mixing math and physics - all constants in math are very well defined, and we know how they arise. In physics, constants and principles are usually made up to correct the formulas into observations and nobody knows why exactly those numbers, nor do they apply everywhere or just here (anthropic principle)
@@HorukAI This whole video id about a mathematical constant and nobody knows why it's exactly that. I mean, we can calculate it, but that doesn't explain much.
@@tacticalassaultanteater9678 sorry this was not a constant, but a cardinality of a particular finite simple group. My point is that constants in mathematic indeed do arise from deduction process (logic) while in physics it's often an equation scalar based on observation, or mathematical necessity added so that it fits to hypothesis (think inflation)
@@HorukAI A cardinality is a number and a number not dependent on any variable is a constant.
@@HorukAI Numbers like the gravitational constants aren't even proper constants, they are derived from the quantities we pick. When I say constant I mean precisely values like π, Euler's number or the cardinality of the monster group. Artifacts of logic, independent of our circumstances, universal, yet hard to explain why their values are exactly what they are.
When I grow up I wanna be a mathematician for the sole reason to be able to make names like _the monstrous moonshine conjecture._
All the best! You made being a mathematician sound even more fun!
I just wanna come up with a conjecture so I can call it Conjecturey McConjectureFace.
You actually usually don't get to name your own conjectures or theorems.
Suppose your paper contains a new result labeled Theorem 1. After you paper becomes very important and discussed a lot, people will not want to keep calling it, Farrell's 2028 Theorem 1. They'll name it something catchy and says something about the result.
For example, whoever proved the fundamental theorem of calculus didn't call it the fundamental theorem. It was the people who began using it who named it that. And clearly, it was named like this because it's a "fundamental" theorem.
Or Dimensional Ejaculation
rather than calling it « ejaculation » you can give it your name
This is possibly the most incredible video I've seen in a while. I've just finished a course in Galois theory, and it almost feels like 3blue1brown is following my own mathematical journey. I'm truly privileged to live in this time. Thanks Grant for producing a work of art!
This has to be one of the best explanations of group theory I've ever seen. This might be somewhat surprising, but I've never really thought about the connection between symmetry and group theory, although I've worked with group theory a lot... (as a cryptographer, though, not as a mathematician). It was never explained to me like this. Really cool to see.
@@sandrapadilla5 Ah, I see, you are just a spam account that has copied the name of the channel here.
@@MrFair >cryptographer spots bot
My fucking sides
So informative! The fact that these videos are so synoptic makes them all the easier to understand. So often lecturers can be rather reductionistic in their approach, and students suffer as a result.
Yes, there are only two steps needed to do chemistry:
1. Find the periodic table
2. Do all the chemistry
Jokes aside, I was lost during the first few minutes but I followed through till the end. What kind of charm is this, Grant??
...jokes? What jokes?
@@aliince9372 Jokes aside refers to the sarcastic statements that this comment made before saying, "jokes aside." It is a relatively commonly used phrase in english.
The way Grant makes his videos:
1. Find a topic
2. Make an incredible video about it
I like how he explained everything super clearly, but also made it clear that we're missing most of the picture. That's really hard to do, but he just made it look effortless.
This video was 22 minutes long?
That felt like at most 7.
Grant is a legendary explainer, managing to present very brain-hurting topics engaging
The world needs more people like Grant teaching
So, it felt like one pi-th as long...? Interesting apparent compression rate. :)
@@QmcometdudeShardMaster was gonna comment the same
Yeah I watched the 20 minuters and I was too focused in trying to understand something to notice the time pass
Same. But partly because 1.75X speed haha
Your use of teasing and suspense is masterful. These videos have some of the best pacing and production value I've ever seen. Thank you
I had to take Modern Algebra I and II for my Bachelor's, and I actually had to drop Modern I the first time I took it because the professor tried to use teach about symmetry instead of from the definitions. I spent almost half a semester stumbling and flailing, unable to visualize or wrap my head around it, and unable to memorize the Cayley tables for different groups. When I tried again with a different professor, who taught based around the formal definitions, everything immediately clicked and it became one of my favorite classes I'd ever taken.
One of the things I love about math is that there's usually more than one way to think about or approach a problem. So while I'm glad that other people learned from the symmetry group lens of it, it took me a minute to get over how weird it felt to hear someone say the definitions weren't how they learned it.
I have to agree, the algebraic definition of a group (and in general of algebraic structure) is way nicer to me than an example using symmetries.
I think that well known sets of numbers will always be the best example for algebraic structures, after which you are ready to learn the formal definition and only after this you can go an translate this to real stuff, because now you don't need to figure out how two symmetries combine, you can simpy check they actaully do it as you expected.
Yes, that's the thing. No one who actually does algebra, or at least have taken a graduate sequence, think first and foremost about symmetries when they are solving a problem. It's mostly when you have to convince non-math people that why math is could be nice.
RIP John Conway. You will be missed
;(((
you knew him ?
@@taopaille-paille4992 I didn't know him, but he was one of the most important mathematicians in modern times.
He had no Abel prize or Fields medal though. Outside of top 50 then.
Oh no! I met him during a book tour about 3 years ago. He looked healthy then...
I want to take a second to appreciate that the creator of this video made each of these frames and put them in one by one 3:24
Pretty sure he actually makes a program to do it (don't remember which video I think I heard this from)
Grant uses python to create his animations which he has open-sourced!
he uses a python library (that he also created) called manim, aka math animation. it’s really cool, you should check it out on github!
There are very efficient (linear time) algorithms for generating all permutations of a given set of points.
I think Wikipedia even includes the python code for that algorithm.
So you just have to add the animation.
It's great that he put in that work.
If you know about this/ have done similar things before, this is doable within 30min. If you have to do research first, I'd estimate about 1-2h of work.
@@sebastianjost What's the name of the algorithm?
Thanks!
"the cyclic groups of prime order" sounds like the name of a secret society
watch out for C5
@@danielyuan9862 Too bad there is no C4
@@lilyyy411 C4 exists, but it is not a simple group, as C4 has C2 as a nontrivial subgroup
@@gabriellasso8808 No, we ran out of C4 so now we're resorting to just shooting everyone
@@danielyuan9862 It’s so very fun for me,
so i tell you: I have the hobby to spread science by asking people for watch-suggests and also offer the same.
Mathmaticians: *exist*
Computers: *chuckle* I'm in danger.
*smells the air* "what are you cooking?"
Mathematician:"oh nothing it's my computer going through a couple of equations I wrote this morning"
Where's the warm air conditioning coming from?
Oh, that's my computer working through a couple equations I wrote this morning
@Aditya Sharma it was born, just to suffer
I'm pretty sure they get off on it
Try getting a mathematician to do the mental processing a computer does. At least the computer can crash and reset. The human mathematician just dies.
Grant, there is a suddenly stop in 10:33, and then goes to another subject.
I never commented in your videos, but man, thank you for making the best Math channel in UA-cam, and always explaining such a complicated matter in such a simple way.
Greetings from Brazil.
this is one of my all-time favorite videos on youtube, i keep coming back to rewatch it
Unexpected existential crisis. I'm- *awful* with math and stuff, but towards the very end. . . It felt like we'd stumbled across a tesseract. Or a cross section of a cube. Seeing "coincidences" in unrelated areas in math, and in string theory, and this massive number that relies on a huge dimension to make sense- it feels like we've got a very small glimpse of only part of it. I dunno if I'm making any sense. It's four in the morning and I'm half asleep. Still. A lot to think about. Thanks for the quality vid.
As a math student and math educator, I am so incredibly glad this exquisite video can produce such an intense reaction on someone that admittedly is “awfull” (big quotes there, don’t truly believe that) and could therefore lose its interest quite easily. It proves to us all that this spark we see can be shared to others that haven’t tasted its sweetness during the period one usually expects to.
I don't know if you watched that video, but colliding a small block and a big block enough times produces pi.
...like stumbling across an inkling of a Grand Unified Theory. I dig it.
Hi! You are making perfect sense, this kind of thing has happened before in history.
Maybe not to this extreme, but "math monsters" that at first are just there and "don't make sense" eventually are made sense of. For example Imaginary/complex numbers arose from math and were kinda "nonsensical" to mathematicians at the time, but they are fairly well understood by scores of college students of our time. (that pattern holds for pretty much every extension of the natural numbers actually)
In any case, I wanted to say that I agree with Bernardo Picão... if you got that feeling of marvel from this video, that's a really cool thing this video is achieving, and maybe you can now see why some people love math, it's sometimes beautiful in a way you can't explain.
I've been a fan of 3b1b from the beginning for this very same reason, he is really good at communicating the core beautiful ideas behind math, even if you don't understand anything at the very least you can glimpse it. He started with college math that I understood quite well, but I just had so much fun watching his videos and showing them to some people. In the case of the monster apparently even our most advanced mathematicians can barely glimpse it...
On another note, I've always wondered if there's a limit to how complex of a concept can we as humanity understand? I mean, as we are now, there certainly should be... you learn the condensed product of generations of great gifted people who built upon each other... lifetimes of past great people pondering upon a subject and after many years you can maybe start working on new math to maybe (big maybe) advance it a little.
Will there be a time when a lifetime is not enough? When new math is only for the most gifted of humans? Maybe eventually to keep advancing we'll have to really look into teaching efficiently so we have enough time... I figure someone probably thought along this lines before, but never heard of it (I'd like to hear about it).
@@azlastor A lovely response. Let's wait and see if we can specialize in what we want, or if mathematics will become so big we need to coordinate the fields to specialize ourselves in. Orchestrating all that would be a sight.
i am not going to lie, this is the most BEAUTIFUL video that i have ever seen on youtube. Looking forward to new videos!!
Something about Grant's calm reasonable voice is enchanting. Conveys his awe of the subject without hype. Gently encourages futher inquiry. Congratulations, buddy! You are a great friend to humanity.
This is exactly the content I want while I’m drinking
Same
I don't need to drink while watching this video (I'm under 21 anyways lol)
Drunk maths are best
@@danielyuan9862 funny guy
I love your videos, fortunately for me my father is a mathematician so whenever I don’t understand something I can discuss it with him till I understand. Thank you for the great moments!
The feeling when you love the animation but you don't understand anything:
Yes that's me
exactly. it feels so right but that's it. like I have such a loose grasp on something that's so solid.
This is amazing. I remember learning about group theory, at the beginning it was super confusing but became way clearer at the end, I ended up loving this field of maths. This is completely different approach of presenting it, amazing work.
As a student familiar with contest math, I was pleasantly surprised by how detailed the difficulty scale at 14:07 is. The first item is just 1 + 1, the second item is a problem from this year’s AMC test, the third item is probably an olympiad inequality problem, and the fourth is Fermat’s Last Theorem. Adding these difficulty markers really emphasizes the point “this is a hard problem” better than words could (if you’re familiar with the problems, that is).
Didn't it take Whitehead and Russell two volumes of Principia Mathematica to prove 1+1=2?
the fact that the only problem on this scale I can wrap my head around is 1+1 makes me sad and optimistic at the same time - so much to learn
Thanks
Best introduction to the subject I've ever seen. Made me fall in love with group theory once again. Thank you Mr. Grant! ❤️
Excuse me, did he just talk about Group theory, prime numbers, chemistry, the periodic table, Noether's theorem *AND* string theory in a single video
No, he was just showing the simetries between fields of knowledge. It was also part of group theory *-
Yes, he did talk about Group theory.
That's the thing about abstract algebra.
A set can consist of literally anything.
An operation on that set can be almost anything.
That's all you need for an algebraic structure, so, in that sense, a lot of things have the same structure. Groups in particular are pretty common, so a lot of things are a lot more similar than they seem at first blush!
@@SaberToothPortilla That´s why we learn so much about this topic in Materials Engineering graduation, the group theory have such a power to help us dealing with the interaction between atoms that is hard to conceive other efficient way to deal with these questions...
Hes playing math nerd bingo
15:35 for a moment there I thought to myself “atomic structure? Surely it would have been better to say something like ‘fundamental’ or ‘prime’ structure to be more mathematical?” However then I remembered that ‘atomic’ comes from ‘indivisible’ and is thus a perfect way to describe it! Well done sir
Edit: 16:54 ‘it’s like the universe was designed by committee’ hahaha
The other reason I like the atom -> molecule analogy slightly more than the prime -> integer one is that just knowing the simple groups does not tell you all the ways they can combine. Just as molecular structure involves a lot more information than a list of atoms within it, knowing all the ways simple groups can combine is an unfathomably hard question.
3Blue1Brown good point! Keep up the great work, your videos are simply amazing
He's just being flowery with description, I would think that the quotations and context would make that obivous
@3Blue1Brown Group theory is also massively important to chemistry, since simple chemical structures can be defined as groups you can lean on group theory the derive fascinatingly specific properties of their atomic composition. Things like bond strength, molecular orbital energy levels, & so on can be calculated, and used to estimate applied properties like IR spectra without performing instrumental analysis. Molecular symmetry & group theory by Carter is where I learned all this stuff back in College, & I would recommend paging though it if your interested :)
This is absolutely brilliant! I have been looking for an introduction to Group theory that would help me understand some of the foundations of Galois representations to try to grasp even a very general understanding of the maths underlying the proof of Fermat’s last theorem. But most video content on Galois groups assume so much knowledge already that I couldn’t make any headway, until I found this one, so thanks! I loved your channel anyway, as an amateur maths enthusiast :)
So, the monster is the spaghetti code of the universe
Flying spaghetti monster confirmed.
// Magic number below. Don't delete.
@@allan710 // don't try to understand how it works, just take it for granted
lol
@@darthdagnanslowerlegs5754 // we don't have the budget to invest in constants
"Hey, Raphael, we need some number for this constant. What did God give us?"
"I don't know, someone lost the sheet. Just smash the keyboard a few times, it should work."
I loved my group theory course. I find abstraction easy and calculation hard. For me, group theory was the easiest math class I ever took, given what I already knew going in compared to what I had to learn in the class.
Linear Algebra, on the other hand, was the hardest math class I ever took, at least for me. Linear Algebra was my first ever college math class (actually, my very first class on the first day of college) and Group Theory was the last one I ever took as a senior.
I wish this was how I was taught group theory in undergraduate before I had to go through all the dirty deeper details in grad school. I am a teacher now and your videos have become inspiration on how I teach.
Next week I'm going to teach basics of group theory, group operations and the character table for undergraduate chemistry students. I remember the first time I had contact with the subject, I got as excited as I got confused and graspless about it. I never could make sense of WTF was a character table, a representation, an irred, and stuff (at the time).
Now, as a teacher, I felt compelled to give my students the opportunity to at the least understand these things, to then show them its massive applications on chemistry. This video helped me enormously on the task! Much appreciated for such a wonderful, plain, straight to the point presentation on a subject no undergrad or grad teacher ever wondered to offer on the subject.
*me, who passed grade 11 university math so barely that my teacher literally stopped me in the hallway to make sure I didn’t take math in grade 12/higher education* : ah yes, the perfect video that I’ll definitely understand
If you like it, don't give up.