Monster Group (John Conway) - Numberphile
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- čas přidán 1. 06. 2024
- The Monster Group explained. Conway playlist: bit.ly/ConwayNumberphile
More links & stuff in full description below ↓↓↓
Featuring John Conway (Princeton University) and Tim Burness (University of Bristol).
Brown papers and Numberphile artwork: bit.ly/brownpapers
More Conway on this topic: • Life, Death and the Mo...
And Conway on Game of Life, etc: bit.ly/JohnConway
A little extra bit from Tim: • Monster Group (a littl...
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Videos by Brady Haran
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"One thing I'd really like to know before I die is why the monster group exists" RIP John Conway
This was the first thing that came to my mind as soon as I knew about his faith. RIP
@@matteovergani3474 What faith? Wikipedia says he was an atheist.
@@glmathgrant he means fate I guess
Grant Fikes he believed in the usefulness of mathematical research. All people have to believe in something to give meaning to their lives. Whether ones mission in life is determined by a higher power is beyond debate. Either one chooses to believe that or not, and whether or not that makes any difference..... one might have to ask Augustin Cauchy, a firm Catholic, or Ramanujan who actually said his mathematics was a gift from a goddess
Did he die?
Conway is an interesting guy. He comes across as an old and wise mathematician who has seen terrifying things that he can't quite explain to regular people
what terrifying things? i like terrifying things in math
@@TheMultiRaphael studying math is terrifying by definition (source: i have three people with PhD in mathematicics and like four engineers in my family and i might go for a minor in math myself, majoring in political science)
It is super frustrating to find something in math that blows your mind, or just feels super profound, and you want to explain it to someone... and then you start thinking about all the stuff you'd need to explain to even begin to talk about the profound thing, and then you consider the average attention span of your friends and . . . And you start to feel alone.
Ketamine helps with the abyss
Boy this is a fun reply chain
12:12 I love the transition from "okay, so it's very very complicated" to Conway just "It's like Christmas tree ornaments"
"It's very difficult to explain."
"I like to think of them as Christmas tree ornaments."
Then you realize he is referencing a 26 dimension tree ornament.
and a 196,883 dimensional ornament :)
or a grahams number dimensional ornament :0
That's nothing compared to my boi infinite dimensions : D
most people like jewelery
Very cool and very well explained. "In mathematics you never understand anything, you just get used to things."
@Electro_blob even in a low level, like addition and multiplication, if you think about it.
- John von Neumann
@@l.3ok if you think about it, multiplication is the first time you experience fast-growing numbers
Unfortunately Too Late To Help Me Pass Calculus 2 At GeorgiaTech 20 Years Ago...
🤣🥊❣️
@@glyph241 lol I just saw a georgia tech student's speaking presentation rn
I once went to Princeton and was thinking about moving there... hoping for a friendly sign that I should be there...
There was John standing on the corner scratching his head staring at pigeons...
To this day it is one of my most vivid memories.
I take it that convinced you to move there?
@@PC_Simo It convinced him to dedicate his life to staring at pigeons, in order to complete Conway's work in that field. Until he has completed this work, he will be unable to reply to your comment.
@@omp199 Apparently so 😅.
from a 2-dimensional equilateral triangle to 196883-dimensional monster. Boy, that escalated quickly.
Yea they didn't exactly expand slowly enough to follow from start to conclusion!
I would like to at least have seen the first of the 26 if it were any simpler
@@kailomonkey Welp you replied to a 5 year old comment so I'll go ahead and reply to a 7 month old reply.
13:33
A group of that size is still massive and wouldn't simplify the explanation at all.
@@user-rv9vk8by5i It's never too late to comment :)
@@kailomonkey The M11 group is the smallest and only one anyone has attempted to do a visual representation of to my knowledge but still looks like a big freaky mess.
I wish I had this man's talent for drawing straight lines.
The trick is to imagine the line first, put the pen on the starting point and then follow the imagined line (fixating the endpoint) _without hesitation_; i.e. not stopping the pace or lifting the pen.
It works for me, but then again im left-handed, so it might not be the same.
Hi Present Perfect!
Would not waste this to my 1 wish......
Furry
@@GeneralKenobi69420 duh
"This is quite a difficult thing to explain..." - "I think of them as christmas tree ornaments."
:D
What Conway said was interesting here, "It's absolutely amazing. Incredible! ..It's the fact that the theorem is true - apparently, and we don't know why it's true."
In science, scientists often have models or theories that they can't totally understand. But that's the fault of the theory not the scientists. But in math, the theorem IS the explanation, it is the perfect description. So for him to say that we don't understand even after we've gotten the theorem. That really is peculiar
No. The proof / theorem can conclude via proof by contradiction. Most of the time such a proof doesn't give you a why, just that it has to be true.
@@tpat90 Ah I hadn't considered that.
Math and science have different epistemologies.
@@tpat90 I always found proofs by contradiction, to be particularly spooky. They tell you something about reality that can't be true, but that's it. It gives you almost nothing else, so you're still left not knowing why it can't be true or why it's true. You just know that it has to be.
@@zualapips1638 At the basic and fundamental level, saying something is not as it appears to be in a particular world is just as informative as saying the opposite. I don't know, but if you actually spend time breaking down the contradicting statement you've acquired from your proof down to its axiomatic state, then you know just as much were you to work backward from the opposite of your final destination as you traced from your contradicting statement.
I. Freaking. Love. This channel... Imagine the world, where Divinci or Newton could sit and have a conversation about their intellectual interests and the world could listen in. Numberphile came late but it found the party for sure
Rest in Peace, John. Thank you for sharing your beautiful ideas with us. "We do care" :)
RIP John. I hope people will follow in your footsteps of genius and continue your interesting work!
...what?
@@yvesnyfelerph.d.8297 He died a couple weeks ago to COVID-19
I found amazing, how two different conversations were merged together, and it kinda completed each other.
Numberphile hasn't done so many of these lately, but it's a fascinating style.
Dr. Conway is a really interesting man
Too bad that he made it quite clear that this interest is not mutual ;)
Dan Hunt was* :(
As if 4-11-2020, past tense on that remark
RIP Dr. Conway.
he was RIP Conway
This is the greatest introduction to group theory I've ever heard! Well done!
You know...there's a youtube channel called 3Blue1Brown. Go check it out, you'll see what I mean.
But really, this IS an AMAZING introduction. All I'm saying is that calling it THE best introduction might not be correct.
@@livintolearn7053 He said it's the best he's heard. You can't exactly tell him he's wrong there, even if you think there's something better out there. Even if he had seen 3b1b's group theory vid, that wouldn't invalidate his opinion. So he is well within his right to say that. That being said, I do agree personally that 3b1b's explanation is absolutely brilliant.
Mr. Conway is on a different level. You can tell he's not even all there, like his mind isn't even to be bothered by such trivial conversation, just brilliant
His mind is slowly making its way into the 196,883rd dimension
Now when you said it, Conway reminds me of one scientist in Star Trek The Next Generation who had seen other dimensions. They even look very similar
Watch out you are cutting air with such edge
as soon as I heard of his death, I thought of this video. I hope he managed to undserstand the monster group.
We will never know.... :-(
I like to believe he just found a way to the 200.000th dimension to hang out with those monsters and he just went on to solve more problems with them :') RIP
@@Joghurt2499 What a magnificent thing you said!
"The one thing I'd really like to know about is why the Monster Group exists."
"I'd like to understand what the Hell is going on."
RIP John Conway. I don't understand what you did for Mathematics but I love that you are so comfortable with your limitations.
I go into this video thinking, "maybe I can solve this one day'.
Finish the video and I`m like 'skrew dis I`m not dealing with 200,000 dimensions'.
Bricks Of Awesome You know you're screwed when you're rounding off the number of dimensions.
True, I'm barely coping with three here...
I see your through your thin façade to your Odobenable interior, you secret walrus!
If it was easy someone would have done it already, lol ;)
You got this man, I believe in you!
"First of all, it has the, do nothing element"
*cue triangle doing nothing*
Jackson Kehoe the do nothing is called the identity. It’s like multiplying by 1. The identity acting on its self gives the identity.
Rest in peace John Conway :(
John passed away but his work will continue to inspire many 💙
Man, group theory is the coolest field of mathematics. I wish there were more uses for it in my everyday life; it was by far my favorite course in uni. So far, I've only really seen it used in database theory, but I'd love to see it elsewhere
It’s fundamental in physics
Pete's very cool Monster Group painting is full of little gems and the original is available at: bit.ly/brownpapers
"Group Theory Legend". Very apt.
Rest in peace my guy a truly interesting and inspiring mathematician
Just want to say I love the longer, more detailed videos you guys have been doing lately. The Riemann Hypothesis, -1/12, and now this. For a layman with an interest in mathematics, these videos are deep enough to draw you in and get thinking about the concepts involved, but not so technical or esoteric as to completely scare away the non-professional. Great stuff!
rest in peace :(
I'm enjoying this a lot!
very good!
Rest in peace John Conway.
A modern genius 🌹
John reminds me of the famous Nietzsche quote: "Whoever fights with monsters should beware that he does not become one. If you gaze long enough into the abyss, the abyss will gaze back into you."
We will miss you, John
I hope, that in the end, he sat down with the creator and he finally explained it to him.
Rest in symmetrical peace, mister.
Gems, gems are truly outrageous; they are truly, truly outrageous.
Yet another intriguing video! Please make more videos with Dr. Conway, these are great!
He finds these videos boring. :P
Oskar Mamrzynski I don't think he even watches the videos… he just finds my questions boring! :)
But he was still kind enough to answer them all.
Numberphile thats all you can ask from a interviewee isnt?
@@numberphile, Will there be future videos with Conway in it? :)
@@numberphile This dude is that smart huh...
RIP John Conway
Welcome if looked this up after watching Grants video about his mega favourite number
Thanks for the welcome, happy to be here :)
Real chads come here from Conway’s death wish
Now we're never going to find out what's going on with the Monster Group. RIP John Conway.
KakarotSC RIP indeed, only 299 some other mathematicians mentioned in this video still working on what he was working on.
...nobody cares. Absolutely nobody.
@@yvesnyfelerph.d.8297 Sounds like someone did their PhD against their will lol. Maybe some people care.
Nah, that's not true. In fact, when the Atlas was published, Conway stopped with his work in Group theory while many others continued their work on it.
Hey Brady, great job as usual. I really enjoy the videos you've done interviewing amazing mathematicians such as Mr. Conway. Is there a possibility that we could see more of these types of videos?
The last line John Conway said in the video is, he really want to know why monster group exists.
He died in 2020 due to COVID induced pneumonia. Rest In Peace, professor.
Now I get what 3blue1brown was talking about in his last video
This one caught my by surprise. Poor guy was remembered almost entirely for The Game of Life. I hope his other contributions to maths lives on too.
His introduction title was "Group Theory Legend". He is one of the monsters of group theory.
It won't be for a while until Numberphile will have another perfect cube number of videos... Cherish this moment.
HOW DID YOU GUYS GET JOHN FREAKING CONWAY AND NOT SHOUT IT FROM THE HEAVENS?!
Very clear explanation, seems like a great teacher.
I started watching this thinking I could use this information for my paintings.
I am now a little terrified of the next part of my life.
How did it work out?
John Conway is the man!
Really great video, and well explained.
Came back to this video afterany years, RIP John Conway.
The monster group seems frustrating to the non-initiated that I am because based on how it's explained, it doesn't make (intuitive) sense that it would stop abruptly. I got the same feeling about Heegner numbers and 163. There's the list, and there's nothing else, and it's not a matter of searching harder.
You should make more videos on Group Theory or Abstract Algebra in general......you can't just excite us about something and never address it again
Group Theory is a bit too deep for the average Numberphile viewer.
@@user-kh5tv9rb6y true
Absolute Legend. Massive inspiration for generations to come.
A request: It would be very interesting to see 3D cross-sections of the object whose symmetries are the monster group
Amazing stuff, thanks for sharing!
I didn't understood a thing about "Montser Group"… I hope it is simply because I was concentrated at eating a grapefruit while listening to this video…
I think I'll re-listen to this video another day… When I'll be more concentrated.
To that grapefruit, you are a Monster!
this is probably my favourite from numberphile and i've seen many
john conway is so incredible
Cool, now I know what I can to decorate my next high-dimensional Christmas party with!
You're all invited by the way. It's just outside Paris: you just follow the Allée des Bouleaux until you get to the Parc de Bagatelle, then you turn left, follow that street for roughly 100m and then you go straight $@#(* until you see the large tesseract. Turn $@#(* again and at the hyperroundabout just look around and you can already see my high-dimensional fractal mansion. It's easy to find, really, since you know, in higher dimension most street corners are orthogonal.
I followed your instructions and am now in Flatland. I may be a few years late.
Is that the same thing I once heard referred to as the "Tarski Monster"? Is it the 196,883-D object?
Fred
Yea,
Very nicely explained.
not
love this channel
Thank you for the deep dive to the realm of the Monster 👍👍👍
And i thought it would be great for the continuation to this video if you could make one about monstrous supergravity theories, monstrous m-theory and exceptional Yang-mills theories and the magic star algebra❤
Hopefully I'm going to Bristol for Uni next year :)
This is a great motivation for why groups are important.
More Conway please! What a dude!
Very helpful 🙏, thank you
I love how "Group Theory Legend" is John Conway's formal job title. LOL!
So... simple group is some kind of "prime" in terms of group theory?
Yes
It means a group having no non-trivial quotient group (identifying certain elements into equivalence classes, respecting group multiplication), or equivalently having no non-trivial *normal* subgroup. Groups G naturally split into short exact sequences 0→N→G→Q→0 with Q a quotient group and N a normal subgroup, unless G is simple, then either one of N or Q must be G, and the other one 0 (trivial group). N is the kernel of the map to Q, and Q is the cokernel of the map from N. All normal subgroups are invariant under conjugation with any element from the larger group, while other subgroups are not. For groups, all quotients are normal but not all subs. It is different for Hopf Algebras (which have non-trivial co-multiplication and co-identity instead of merely plain copying and forgetting, but are otherwise similar to groups), and still different for monoids, where not all quotients look like quotients from a set-theoretic framework (/Z+ being a quotient of |N+ e.g.), but things become clearer from the category theoretic pov, using monomorphisms and epimorphisms, plus their normal variants.
When factoring groups into quotient groups and normal subgroups, simple groups are the prime objects. They may not be when factoring into two disjoint and spanning subgroups (factoring the order of the group) say through Zappa-Szep product.
I really like the use of Conway's sink for composition.
At 3:52, you've written the product with a first, then R120. Note that the composition of functions or in this case, multiplication of transformations does not commute. The version written on the paper is the correct version unless you define this not as composition, but left-to-right application of the transformations.
More Group Theory!!!
Thanks for the video, Brady.
"The number of dimensions we're talking about here is 196,883, so it is a very difficult thing to picture on your mind"
Yes it is.
Wow, the non-monster sporadic groups aren't exactly small either, are they? I was hoping that there was some toy example of a sporadic group that can be visualized but given that the smallest sporadic group already needs 10 dimensions to fully show its symmetries ... not gonna happen :/
Amazing.
Brady, can you do a video on E8, and its relation to the theory of everything?
That is brilliant video intro to group theory. Can you do a video on Schurr’s Lemmas please please. Thank you
Bradys editing skills are extrordinary
RIP John.
2:34 "so how many triangles have you drawn in your life?"
"all of them"
Nice to hear Conway speak.
Conway has serious Gandalf energy
Aww, no more videos in this chain? I hoped for this to keep going and then eventually end up at the beginning in a very logical, symmetric way.
Nice mini-series though :)
Hard to make a video concluding something which doesn’t have a conclusion sadly
I like to think that when Conway died he went somewhere he could see his 196883 dimensional Christmas tree. I hope he gets to hold it in his hand and play around with it, watch as it’s symmetric properties become clear.
The Monster Structure is the true shape of Azathoth.
Jonathan Morgan This name is familiar, what it is?
*****
Lovecraft
Jonathan Morgan Oh yeah thanks
honestly, this number seems like it should appear in more technobabble fiction.
@@alveolate, so what's the problem then? :)
Thank you, thank you, thank you!
Does the Group Monster have any relevance to String Theory?
And wouldn't a circle have an infinite amount of symmetries? That seems bigger to me than the Monster.
An nice example of Plato's Forms ....and an example of Kant's synthetic a priori knowledge all rolled into one monster! ...
Are we sure there are only 26 sporadic groups or is it still possible someone will find another one, the Godzilla group even bigger than the Monster?
There is a proof (of 10000+ pages) that the simple groups are exactly the non sporadic group plus this list of 26 groups, so there should be no more.
This proof is in thousands of different math papers written by hundred of mathematicians. Such a long proof might have a mistake (very likely), and such mistake might "hide" a sporadic group (a little unlikely, but not impossible)
Pretty sure there's a proof that there are no others.
ML is the way forward
@@Ziplock9000 Literally never true for problems like this.
@@treefittycents LITERALLY it already has. Google for some breakthroughs in ML relating to this
I enjoy your videos very much guys, I thought a simple like wasn't enough to say.
I love the artist's rendering of the monster group! Bottle of moonshine next to the drummer, cheeky!
Insane. And fascinating. I want to know more.
The question to the answer is: How many roads must a man walk down.
Great vid!
excellent!!!
I'm curious about that periodic table. Hopefully we can see a video that describes some of those groups.
I'd love to know more examples of the symmetries inside the monster group, because so far, it still feels very abstract to me. I know its size can be described as 2 3 5 7 11 13 17 19 23 29 31 41 47 59 71 so which subgroup of symmetries does each of those numbers relate to?
Hey Numberphile! I really love your videos, and they have inspired me to try to pursue a career in advanced math. I'm in 9th grade, I program in C#, and I'm learning Trig and Linear Algebra, and already know the basics of calculus. How should go about pursuing this career? What would be a good job?
I would wait it out a bit and see what field of mathematics you're most interested in first. A suitable career can vary wildly depending on that.
How far are you ^^
Would have been worthwhile to talk about how the mathematician found the monster and the orher monster.
Specific point of interest for the earlier part of the video where he is explaining that R and S have to stay in their own group when you use combinations from one of them.
But you can make a rotation type of movement using only mirrored symmetry though right? In the triangle, if we follow it counterclockwise starting from 1 it's 1-2-3. Now take the original triangle he drew with 1 at the top, 2 at the left, and 3 at the right. Swap 1 and 3, now the order counterclockwise starting at 1 is 1-3-2, not a rotation yet, but if we then swap any of the other two sides (so anything but reversing the first action) we get something guaranteed to read 1-2-3 counterclockwise, meaning we just made a rotational symmetry using only the mirroring subgroup.
I know this looks like a lot of text, but if you actually write it out you'll see it works.
Yes, any two reflections result in a rotation.
This is even true of continuous reflections and rotations in 3-D space, so long as the two "mirrors" are not parallel.
If they are, you get a translation.
Fred
I will never understand abstract algebra, but thanks for showing up, John!