3Blue1Brown - 2020-08-19
An introduction to group theory (Minor error corrections below) Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: https://3b1b.co/monster-thanks Timestamps: 0:00 - The size of the monster 0:50 - What is a group? 7:06 - What is an abstract group? 13:27 - Classifying groups 18:31 - About the monster Errors: *Typo on the "hard problem" at 14:11, it should be a/(b+c) + b/(a+c) + c/(a+b) = 4 *Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do comprise tens of thousands of pages, but no one paper was quite that crazy. Thanks to Richard Borcherds for his helpful comments while putting this video together. He has a wonderful hidden gem of a channel: https://youtu.be/a9k_QmZbwX8 You may also enjoy this brief article giving an overview of this monster: http://www.ams.org/notices/200209/what-is.pdf If you want to learn more about group theory, check out the expository papers here: https://kconrad.math.uconn.edu/blurbs/ Videos with John Conway talking about the Monster: https://youtu.be/jsSeoGpiWsw https://youtu.be/lbN8EMcOH5o More on Noether's Theorem: https://youtu.be/CxlHLqJ9I0A https://youtu.be/04ERSb06dOg The symmetry ambigram was designed by Punya Mishra: https://punyamishra.com/2013/05/31/symmetry-new-ambigram/ The Monster image comes from the Noun Project, via Nicky Knicky This video is part of the #MegaFavNumbers project: https://www.youtube.com/playlist?list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo To join the gang, upload your own video on your own favorite number over 1,000,000 with the hashtag #MegaFavNumbers, and the word MegaFavNumbers in the title by September 2nd, 2020, and it'll be added to the playlist above. ------------------ These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
"We always consider the action of doing nothing to be part of the group" - 1:41
My favorite quote
Me
So in other words, Luigi Wins By Doing Absolutely Nothing was fated by the laws of mathematics itself to be a thing you could do.
@Mnnvint hahaaha
@Alexander Schäfer same idea for comment
This is why I have friends.
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.'
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
Sounds like he is sucking up to Witten there.
Pratchett was an onteresting writer
"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!
@I need no channel youtube! 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.
Base 10 in the first place is completely unnecessary, if you think about it, there is nothing special about the number 10, we only use base 10 since we have 10 fingers
When the teacher cares more about answer cleanliness than the actual universe itself
@Gabus 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.
This is a gorgeous explanation of the monster, presented in a way I can almost understand. My father would have loved this.
@jack wiśniewski Terrible joke I must say.
Sorry for your loss... 🙏
@Stan Downer I see.... 😒
@Stan Downer Yes he does! From a biography:
Finally we note that Conway has been married to wife Diana since 2001 and has a son Gareth born 2001. Their home is in Princeton, New Jersey, USA. With previous wives he has sons Oliver born 1988 and Alex born 1983; daughters Susan born 1962, Rose born 1963, Elena born 1965 and Ann-Louise born 1968. He has three grandchildren: John, Ellen and Joseph Wayman. He also has two great-grandchildren.
@Achuthan Karnnan He's not lying, Conway did have a son named Gareth.
He did the math, he did the monster math.
He did the math, It was an abstract smash!
ad
The monster math, it was a textbook smash
This makes me wish YouTube had a Karma system like Reddif
😁😆😆🤌💯👍
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.
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!
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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...
@Derek Mizumoto Well said
@Derek Mizumoto 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.
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.
@chaser Hi. Want some science-recommandatios?
@chaser pretty sure thats not true he just wanted to make a block out of smaller blocks.
me: "ayo alien, is the nummber 8x10^53 interesting?"
alien: "yea for real i love the nummber"
@Pablo escobar You are a disgrace to the Pablo community
me: "what about number 69?"
alien: "lmao"
@Pablo escobar why r u geh
@Pablo Pereyra he is
@Pablo Pereyra you tell him
This makes me feel like maths and physics is getting closer and closer to the source code of the universe
@Dániel Martínez in binary language of computation u have that 1 + 1 cant be equal to 2, as 2 dont even exist in the set of numbers considered, so logic is very mutable and not that fundamental. What the video described is not permutation of logic, but permutation of pure information, its much more fundamental things.
@Matthijs van Duin I'm honestly curious if it's possible to have a different logic base. Like to us, it's unfathomable, but perhaps there is a way, just impossible in our reality. It's impossible to prove otherwise.
@Graham Ward
> Since logical conclusions seem to be independent from the universe we inhabit
Impossible to prove. We have no other universes to compare our experience against.
Thing is, math is not the language of reality. Math is a precise language for describing the thoughts that people can have. It just happens that we can have thoughts that are really useful in forming mental models that describe the reality we currently experience predictively. That shouldn't be too surprising, since our existence is predicated upon and integrally tied in with this reality, and making predictions within this reality is the purpose of our nervous system.
The idea that the fundamental rules of logic might be different in another universe shouldn't be too surprising, because kinda by definition we don't know what we don't know. The idea that our nervous systems have limits and some thoughts are (practically or theoretically) impossible for us to have, and some concepts are (practically or theoretically) impossible for us to understand is emotionally uncomfortable but something we readily accept when applied to every other living thing. We are smarter than every other animal, sure. But just because a kangaroo can jump highest of all the animals doesn't mean there's no height limit.
@Dániel Martínez If you can imagine different universes where fundamental constants are shaped differently, that means you have a multiversal understanding of physics which means that your understanding is beyond a single universe. This is just a daily occurance in mathematics.
@teenspirit1 Wow. Thanks for your answer. So could you put an example of "fundamental constants shaped differently"? Because you said this is commo in maths, how do you think in a different logic? Or logic stays the same while physics change? Whats the point of having a understanding beyond a single universe and what does that means?
Some names or links would be appreciated too :)
Excuse my ignorance heheh.
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!
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.
@zildyanVH 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.
@Lőrinc Bethlenfalvy 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)
@zildyanVH A cardinality is a number and a number not dependent on any variable is a constant.
@zildyanVH 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.
@Lőrinc Bethlenfalvy well the natural numbers themselves are defined as cardinality of sets on the most fundamental level. So every number is a constant :D
But I get your point, mine was just that there is nothing strange in pi - and that you cannot go deeper through the rabbit hole to find out why exactly that number describes relation between radius and circumference of a circle - and not some else. I hope I'm wrong, but I guess we people always want deeper meaning, I just think sometimes there isn't one.
What is strange is on how many places and formulas it shows up, but in the end you understand that there are always some circular translations included like in Euler's formula, or in Schrödinger's equation.
"What's the most important thing in math?"
"Coming up with funny names."
Like the WIMPs and MACHOs In astrophysics
@Ian Visser The biggest issue with modern science is that we had to name everything in English, a language that lacks the ability to make compound words which means that every time a new concept comes up we have to use some existing noun for it. If just any other Germanic language had become the Lingua Franca of the modern world we wouldn't have this issue as we could easily just come up with names that describe the thing we're talking about and if they got too long and complex we could just contract them.
@hedgehog3180 Actually everything in science is named in Latin, then translated to everything else. Nothing has english names to start with.
Hmmm...
That seems accurate
@Darryl Johnson Oh you're a mathematician?
Name all normal things 😎
6:15 one thing that I thought was interesting is that you don’t actually need a quintic function to find the roots of 5th degree polynomials because differentials of polynomials have x intercepts corresponding with the peaks and valleys of its parent function.
"The answer is absurd" Made me laugh way harder than it should have after being that invested.
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Yes welcome to math, would you like to buy a bridge?
@willisverynice help me im crying 😂
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!
Personally, I adored the abstract nature of algebra and the way we got introduced to it, much more than I’d care about this but this is a great explanation for non-mathematicians! I’m about to send this to all my friends and family ahahahah
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 :)
This was like watching anime without subtitles. I didn’t understand a thing but it was gorgeous.
I don't even understand spoken English correctly but I liked it
It was honestly enlightening and it piqued my curiosity for group theory - there were parts that I didn't understand, but that just enhanced my desire to learn more.
@Jared Jones yh eventhough I only understood it vaguely, it's still pretty cool, idk sorts ominous in a way how there's this number that just pops up in unrelated fields. Definitely peaks my interest
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.
Hey I took a course in group theory. Now I understand a few Japanese words enough to not always having to read subtitles. :)
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.
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. :)
I love group theory! Did my thesis on Sylow Theorems, but I want to learn more about groups.
Welcome to the group of figuring it out
Ah, I was taught groups as a generalization of regular numbers. "A group is anything that behaves like addition or multiplication on their own". This geometry approach might help me appreciate the discussions about modern more complex finite groups used in cryptology.
That definition I beleive comes from Boolein logic, and primitive Boolein operators?
I can certainly see such an interpretation being useful in computer-cryptography.
Wow. I watched this video before beginning this academic year and found it very interesting (without much understanding of it though). Now, after almost a semester of studying group theory, I watched it again. So many things started to click on my mind. Thank you SO much for this.
I love group theory! The chemical applications of group theory course I had in grad school was where all the questions I had throughout my entire undergrad degree seemed to have been answered. It’s got so many applications and I’m definitely no expert (as this video has proven beyond any shadow of a doubt) but I love it!
My favorite number above 1,000,000:
10^10^10^10^10^1.1
It's so big that if you cut it in half, it would be written the same. If you were to measure time with it, it would not matter if you used seconds or years because it would make a completely negligible amount of difference compared to how long a time it is either way
(Curtesy to GameTheory for my discovering this number)
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?'".
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.
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. <3
Thanks for the book mention, I think I'll check it out.
@Juel Herbranson 🙂
Wow, I’m 4 weeks into my modern algebra course and this is making so much sense now. Literally the ‘click’ moment happened when you were explaining isomorphisms 🤯🤯
Now I need to know, since these sporadic groups are among the "exceptions" to normal patterns of symmetry, what can this very highly dimensioned abstraction do that anything lower can't?
Physicists: 11 dimensions... That's a lot...
Mathematicians: Haha dimensions go brrrrrrrrr
@Rhea Last Name nice
@Nick Maslov The original comment was way way better than this. Nice try though.
@Conner Canales That other reason to give your live for your country...
Jim Frazier Yes , that brought the same thought to my mind , only similar to an Einstein being able to write a correct formulae becomes next challenge !! Truly the impossible only comes back to that we can’t explain. Greater quantities like our oceans have ( X — formulae ) to us endless, repeat cycles or ones we can’t yet get head around. How long or wavy is a piece of string ?etc
11!?
I just learned about Permutations, as well as other relevant math items, last semester and it feels nice knowing whilst also understanding what the video is talking about
17:04 no this is exactly what chemistry did, “these all are alike, those? those are extra”
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.
The parts of periodic table that do not fit this nice pattern are not a part of chemistry even in uni. There is a table of isotopes that has neutron and positron counts on its axes, and then there are nuclear isomers that differ from usual atoms by the way neutrons and protons swirl inside of a nuclei (metastable isomers) or weird shape (fission isomers), even though the number of nucleons is the same. They are randomly all over the table, with no apparent pattern.
@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
@The Examiner We are simply not prepared for the ultra wide chad periodic table.
It would be absolutely amazing if you could make a video elaborating on Noether's Theorem 😍 With your ability to deliver different concepts and ideas I have a feeling it will be outstanding
Groups are so beautiful. I did my masters thesis on 'The Solubility of a product of Subgroups of Relatively Prime Order'.
I had the best time researching groups!
I agree! I'm a physics grad student studying group theory, and it's spectacular! I hope you're doing well in life. :)
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.
@Huy Truong That's racist!
Your comment may read as a compliment and a joke but all I smell is zero confidence.
@The Major Objects that the Monster group acts on are 196883 dimensional. But to represent an element of the monster group you need to represent it as a 196883x196883 matrix (or a 196882x196882 matrix over a field of 2 elements), which I figure is where that 196883^2 comes from
@nucular you mean that the representation vector space dimension is 196883?
@Monojit Chatterjee right, yeah
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
@Daniel Yuan funny guy
Damn it, when my highschool math teacher asked us what our favorite mathematical theory is, I should have put "monsterousmoonshine"
Yeah I half wish I was still in school just to see my math teachers face as I talk about the Monstrous Moonshine Conjecture
Humans: Interesting number
Alien civilization: Very interesting
Super AI: Overflow!
I'm so grateful for your work in explaining these concepts so clearly. I don't have the time to study these things in detail.
Any chance of an "Essence of Group Theory" series? I would love that!
I need that.
Please, somebody tell him, that would be something so lovely, so beautiful, that... It just needs to pop out into the existance, I really love his videos
yes it will be nice!
Socratica has an intro to abstract algebra series
Grant would absolutely kill it covering the concepts of abstract algebra, factorization domains, group theorems, rings, etc. Just imagining it makes me excited. If he simplified calculus he can simplify abstract algebra to a wide audience. If only it was a more popular subject.
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.
"The universe doesn't really care if it's final answers look clean" – Sabine Hossenfelder.
Amazing visuals. Truly makes you appreciate the beauty of mathematics. Keep up the great work Grant.
I did some group theory back in the day and never felt like I grasped it conceptually as well as I have now that I've seen this vid, awesome content as always 3b1b
Other Mathematicians: "Polynomials, Permutations, Quaternions"
John Conway: "Monster, Baby Monster, Happy Family"
He had to create new words out of nothing.
@Chen X idk his books seem to be much more pictorial than the average math textbook. Super rigorous and boring is probably what you would think of normal math textbooks.
oh boy you haven't even scratched the surface
read "Winning Ways for your Mathematical Plays" by Berlekamp, Conway, and Guy which is peak whimsical math naming
@Paul Williams a Marius Sophus Lie!
John Conway's Mountrous Moonshine Conjucture, coming to theathers
This channel always lures me in with a fun cool concept and destroys me with a freshman course in theoretical mathematics
God, as a high school student, this one of the most challenging videos I’ve ever watched. My brain hurts.
That’s making space for more Info 😳
You should watch the videos on 'Tree 3' from Numberphile's channel. I've never felt such a sense of doom. Not as challenging, just bizarre.
Omnitroph - 2020-08-22
the difference between fiction and reality is that fiction has to make sense.
Lapid Palid - 2021-12-29
now that's a great quote
willisverynice - 2021-12-31
Then it is just reality.
Math Account - 2022-01-01
r/im14andthisisdeep
A Convenient Myth - 2022-01-25
"Truth, of course, must of necessity be stranger than fiction, for we have made fiction to suit ourselves."
- G.K. Chesterton, "Heretics"
Protoka - 2022-02-10
Reality makes perfect sense. It's just not obliged to make sense to us.