The monster (early view)

Such a great piece!

Amazing video. I know exactly what you mean by frustrated by Group Theory (was an undergrad grabbing a GT textbook and quickly gave up, lol). Wish I saw your video back then. Probably I would be doing math right now!

Love this video Grant! I considered being a double major with math in college (along with astronomy). Math 5 was the first required class at Caltech for the math major. The first week was super easy, and I don't think I took it quite seriously enough. The second week I didn't understand anything. That was the end of my aspirations to double in math. Anyway, this video was great, and really helped me develop some appreciation for what it was I missed out on all those years ago. :)

Mike Jarvis

Basically, because the prime factorization has 5^9: 2^46 · 3^20 · 5^9 · 7^6 · 11^2 · 13^3 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 Pretty arbitrary-seeming, but no more so than anything else about the prime factorization.

Mike Jarvis

Thanks for this beautiful video. You had me hooked with the notion that an alien civilization/AI would find this number interesting and fundamental. I had trouble understanding what was going on at 13:18, with the 2^x+y = 2^x * 2^y. It seems like you're visually articulating the idea that addition of exponents is equivalent to multiplying the bases, but I can't quite figure out the way the two number lines interact.

spacediver

Why does the monster end with 9 zeros? That is strange to me.

that's one bit of reading I had delayed for a decade you did for me. thank you so much, can we have some follow up on the string theory relation please.

Entertaining, enlightening... enterlightening

Beautifully video! The 20 minutes went by like seconds! It was that intriguing

Do we have any idea what this 193,883 dimensional object is? A polytope? A Rubik's cube? Something else entirely?

Max Goldstein

At 17:10 there's a typo in "preserve"

Dan Davison

That's a good question, and actually incredibly complicated to give a straight answer to. You may enjoy this brief article: http://www.ams.org/notices/200209/what-is.pdf Note how it ends: "Unfortunately none of these definitions is completely satisfactory. At the moment all constructions of the algebraic structures above seem artificial; they are constructed as sums of two or more apparently unrelated spaces, and it takes a lot of effort to define the algebraic structure on the sum of these spaces and to check that the monster acts on the resulting structure. It is still an open problem to find a really simple and natural construction of the monster vertex algebra."

3blue1brown

Interesting. I wonder if he declined because it felt too complicated to describe with more depth than "the largest simple sporadic group" in half a lecture, or for other reasons.

3blue1brown

In some sense, even the experts see their understanding as too shallow. You may enjoy this brief article: http://www.ams.org/notices/200209/what-is.pdf

3blue1brown

Fair point. In my view, the video is "about" introductory group theory, but the monster makes for a nice hook. With the aim of being as approachable as possible, I am not aware of a definition of the monster beyond "the largest simple sporadic group" which wouldn't be buried in jargon. If you know of one that doesn't involve laying out vertex algebras or some such thing, do let me know :)

3blue1brown

Thanks for the note, I think I fixed it.

3blue1brown

I was debating whether to take an aside to define it, but as it was the mention of those particular families was an aside away from the point about sporadic groups, so it felt like too much. One way to view it is to number all the elements. Start off doing so in order, so that whenever i < j, element i is to the left of j. Now after the shuffle, we want to count how many pairs are out of order, how often is element i to the right of element j when i < j? When that count is even, we say the permutation has even parity, likewise for odd. But it takes a little thinking to convince yourself this is a group (why can't two even permutations combine to make an odd?).

3blue1brown

The second point would involve a lot more detail and quite a bit of learning for me, to be honest. As a slightly related note, you may be curious to learn about the close connection this corner of math has with sphere packing in higher dimensions, which at times also has nice connections to error correction.

3blue1brown

Ah, interesting. I'm loosely familiar with Hossenfelder but have yet to read any of her books. Do you have any recommendations?

3blue1brown

Certainly worthy, but ultimately the cuts have to be made somewhere. Honestly, a full video on this corner of Galois theory should be made....someday.

3blue1brown

Very neat! Did you end up studying something related to moonshine?

3blue1brown

That's great to hear, it sounds like you're exactly the person I had in mind while making this video. There's a lot that can be said about Conway. Have you seen any of the Numberphile videos with him? The podcast they did about him after he died was also great.

3blue1brown

Thanks for the note, I did think about this. After listening back, I decided to stand by the phrasing that the groups Sn "encompass" all the others; I think with the visuals and context it should be clear that's not saying all groups are Sn. As you mention, this is, of course, a reference to Cayley's theorem, but maybe I'll save a deeper look into that theorem and why it matters for another video.

3blue1brown

Added a brief note to this effect at the end.

3blue1brown

Always love seeing your content, and this video is great as always - but I was still left with the question "What *is* the Monster?" I feel it would be useful for the conclusion to more directly state or reiterate the definition. I *just* watched the video, and I know that the Monster is something to do with groups and/or symmetry. If I were taking a test, I'd say it's a specific finite simple group. If I really dig, I think it's the largest sporadic finite simple group. But... what's that? I mean, what defines it? It's the number of permutations possible in.. some set while retaining some property of the set. Is it possible for the conclusion to be less abstract? (This isn't snarky - I'm literally asking if it is possible).

This will be the video I point to for anybody who will ask me "what group theory is about?" or "what do you think is most beautiful in math?" The analogy between simple groups and primes/elements is very well presented. Thanks, Grant. This is good stuff for all the curious minds.

I studied group theory under Bob Griess as an undergrad, and towards the end of the semester a student asked if he could devote part of a lecture to the monster group. He declined! It's been a source of fascination since then, and this video is a good addition to explaining the mystery, which I've resolved that I will never grok.

Thanks for the nice introduction to the monster group. I was wondering this group since John Conway talked about it in one of the Numberphile videos (Life, Death and the Monster). He wondered why it exists and it's beautiful, it should be no coincident, but he didn't describe it in the interview (because the interview focused on his life, not a specific problem). I was interested in the monster group since, but I didn't have a clue. This video sheds the light on it for me. Of course, I see my understanding is too shallow, but I have one step near to the monster group. Thank you.

Hitoshi Yamauchi

I like the group theory overview. I think the Monster isn't really well enough defined in the video to consider this video to be "about" the Monster. But I still like it!

Gabe

Amazing content, as always! I understand you are going to re-record (parts of?) the video anyways, so maybe this is irrelevant, but there is a minor hick-up in the audio at 5:19 that sounds like to takes of audio were stitched together at not quite the right point. Something you might want to iron out if you're going for absolute perfection ;).

I find myself really wanting a definition of parity to get the animation of A5 @ 17:15. Perhaps that is the intent as you hint strongly at it with your addition of color. I haven't looked it up yet as I find your explanations a more satisfying jumping off point so I will see if you add anything about that first.

Benjamin Bailey

The things you learn! <3 My pet peeve is the use of the Dutch apostrophe (100's) when a standard s does the job

Paolo Torelli

A few minor points to an otherwise outstanding video: - around 3:15 perhaps it'd be good to add colours to the permutted elements, for clarity and to be consistent to later representation of permutation in the video? - Not sure if that'd require too much background, but I'd enjoy a little more explanation around how these groups were found, escpecially the sporadic ones!

I like your conclusion at the end, you've probably heard of Sabine Hossenfelder but she makes a similar point. That in physics many people are too focused on how natural laws "should" look because the math is nice or simple, but that they do not even entertain the possibility that maybe the universe is just "messy" sometimes and there is no simpler way to phrase things ;)

17:10 "parity" → “parity” (proper quotation marks)

Brought back good memories of the year-long Abstract Algebra class I did at the University of Padova on my year abroad from UC Berkeley. Well done, Grant! Time to dust off that Galois Theory book I bought but never made time yet to dig into... Thanks for the inspiration.

Steve Muench

~11:52 rotation axis of the middle cube looks like a diagonal (but in fact it's the y axis) — so slightly different projection would look better

IMHO it's worth mentioning that not only structure of S_5 allows to prove unsolvability of degree 5 equations, but structure of S_4 and S_3 correspond to formulas for roots of degree 4 and degree 3 equations (Galois theory is not only for impossibility proofs…).

~16:05 «(prove you have them all.)» → «(Prove…» (capitalization)

Oh man this is just out of this world. This level of Pure Math simplified and told in the form of a story and one that interests one to leave everything and jump right in !! This is mind boggling and very satisfying at the same time. Dear BlueBrown, you made my day.

Glad that you took this one over a million. The group order of the Monster actually has a relation to the number of atoms in the universe (also called Dirac's large number hypothesis). I pointed this out in my book "The mathematical reality" https://www.amazon.com/dp/B0849ZXQB1 (I'm happy to send it to anyone interested). The relation remains speculative, though.

My heart leapt in my chest when i saw that you did a video on the monster! Monstrous moonshine was a central part of my grad school application (as a bewildering and beautiful bridge between seemingly unrelated disciplines), so it holds a very special place in my heart, and you always put so much care and thought into your videos that i knew you'd do it justice. The introduction to group theory was fantastic as well — i've had difficulty in the past explaining group theory to my friends and family, so this will be an invaluable resource for future efforts toward the same 😊

Dan Kinch

Thanks for the catch, fixed.

3blue1brown

Fascinating video. For someone who never ventured into the more abstract levels of math, this was surprisingly approachable. I especially loved how you used geometry, symbols, and numbers to explain groups and group actions. Also unrelated, I’ve always been intrigued by John Conway, mostly for game of life but also for his less well-known but perhaps more significant contributions to math. RIP :(. Perhaps cover game of life in a future video?

this was real refreshing to watch. got set theory coming up so algebra is on the back burner, but this one was on my mind for a long time. thanks! edit/ps: around 15:58 when it says 100s of mathematicians, there is an odd fade in

This timing is serendipitous too... I’ve been obsessed with the Rubik’s cube lately and that led me to brush up on group theory and open back up my copy of Dummit and Foote.

Eric Severson

This might be your best video yet. So nice to see nice visualization of abstract algebra. I have a real love-hate relationship with the subject; it was my first upper division math class and it was taught terribly so it was all lost on me and that steered me further away from pure math. I wish your channel existed back then.

Eric Severson

Awesome! Perhaps my favourite video of yours! Make sure to define the term "identity" as the "do nothing action" around 1:45. You use the term later at 11:15.

Using the atom analogy, all the simple groups are atoms. The monster is the biggest simple group, that is, it's the biggest atom! You can't add more atoms to an atom. It's freakishly large for something that's supposed to be an "atom", something simple enough that it can't be broken down into simpler parts. The giant number is the size of the group - the number of "actions" in the group. There's a related number, 193883, which is in some sense the minimum number of dimensions required for the monster to exist.

Phlosioneer

This video is a good reference when trying to explain what "Abstract Algebra" (or at least group theory) is about to someone utterly confused by a standard definition referring to "algebraic structures" and jargon like groups, rings, fields, modules, homomorphisms, etc. The analogy with arithmetic abstraction and is great. We can talk about how stuff works without needing to invoke specific incarnations of objects by abstracting away the particulars and using a common language/symbology that applies broadly.

Alexis Olson

I feel like Cayley's theorem might deserve a quick mention. Although not stated, I got the impression that you were saying that every group is isomorphic to a symmetric group proper rather than the actual theorem that any group can be characterized as a *sub*group of a symmetric group (which explains why there isn't just one infinite family of groups).

Alexis Olson

So after all of that, I'm not sure I know what that big number is. It's the number of atoms in the monster? It's not the sum of the size of those atoms, somehow? Why can't I just add more atoms to get something bigger?

Max Goldstein

I've been up for 30 hours straight working on my own problems and was getting ready to crash, and so I almost skipped watching this tonight, but after seeing the graphic of the cute monster, I couldn't help pressing play. This video was so great that I watched it twice. Wonderful, as always! I can't wait to watch it again tomorrow with my 15-year-old son who also loves your channel. Thank you!

Should have left this one till morning coffee...

Allen Lorenz

I would also add that there's a connection between the monster moonshine result and string theory, which just adds to the "why are you doing this nature, why does this group exist, and how can it pop up randomly in physics?" factor

Phlosioneer

;)

3blue1brown

Ah, good to know, I'll lengthen in.

3blue1brown

I did have fun with that. It makes me want to use the same effect in future videos. Thank god for John Conway's naming instincts!

3blue1brown

Ooh, good catch, thanks!

3blue1brown

Thanks! I'll fix both of those.

3blue1brown

The second-to-last screen (right before the credits, mentioning the larger project) is too fast. I wasn't able to read even half of it before it moved on to the credits. Otherwise, fantastic as always.

Phlosioneer

FYI "12,000 page paper" spoken, but graphic says "1200 pages" @ 16:05

Beautiful video, I love your explanation of group abstraction by tying it into number abstraction. Two editing remarks: at 13:54, there’s a noticeable dip in the audio. And at 20:08, you mention what sounds like “macase peers” (assuming its a name) while showing john conway. Not sure if I heard/understood that incorrectly though.

Tyler Wolfe

Fantastic video, as always, but I REALLY love what you did with the pacing. Keeping on returning to The Monster, reminding the viewers what they're waiting for, but without ruining the suspense with a partial explanation a the beginning. Genuinely fantastic

Kai Salmon

Nice to see hexagons getting some representation here.