Today, many members of the YouTube math community
are getting together to make videos about their favorite numbers over 1,000,000, and we're encouraging you - the viewers to do the same. Take a look at the description for details. My own choice is considerably larger than a
1,000,000, roughly 8×10^53. For a sense of scale, that’s around the
number of atoms in the planet Jupiter, so it might seem completely arbitrary, but what I love is that
if you were to talk with an alien civilization or a super-intelligent AI that invented math
for itself without any connection to our particular culture or experiences, I think both would
agree this number is something very peculiar and that it reflects something fundamental. What is it, exactly? Well, it’s the size of the monster, but
to explain what that means, we're gonna need to back up and talk about group theory. This field is all about codifying the idea
of symmetry. For example, when we say a face is symmetric,
what we mean, is that you can reflect it about a line, and it’s left looking completely the same; it’s
a statement about an action that you can take. Something like a snowflake is also symmetric, but in
more ways. You can rotate it 60 degrees, or 120 degrees, you can flip it along various different
axes, and all these actions leave it looking the same. A collection of all of the actions like this taken
together is called a “group”. kind of... at least... groups are typically defined more abstractly
than this, but we’ll get to that later. Take note, the fact that mathematicians have co-opted
such an otherwise generic word for this seemingly specific kind of collection should give you some
sense how fundamental they find it. Also take note; we always consider the action of doing nothing
to be part of a group, so if we include that do-nothing action, the group of symmetries of a snowflake
includes 12 distinct actions. It even has a fancy name, “D6”. The simple group of symmetries that only has two elements acting on a face also has a fancy name, “C2”. In general there’s a whole zoo of groups,
with no shortage of jargon to their names, categorizing the many different ways that something
can be symmetric. When we describe these sorts of actions, there’s
always an implicit structure being preserved. For example, there are 24 rotations that I can apply to a cube that leave it looking the same, and those 24 actions taken together do indeed constitute a group. But if we allow for reflections, which is kind
of a way of saying the orientation of the cube is not part of its structure we intend to preserve, you get a bigger group, with 48 actions in total. If you loosen things further and consider the faces to be a little less rigidly attached, maybe free to rotate and get shuffled around, you would get a much larger set of actions. And yes, you could consider these symmetries in the sense that they leave it looking the same, and all these shuffling rotating actions
do constitute a group, but it's a much bigger and more complicated group. The large size in this group reflects the
much looser sense of structure which each action preserves. The loosest sense of structure is if we have
a collection of points and consider any way that you could shuffle them, any permutation, to be a
symmetry of those points. Unconstrained by any underlying property that needs to be preserved, these permutation groups can get quite large. Here, it's kind of fun to flash through every possible permutation of 6 objects and see how many there are. In total, it amounts to 6!, or 720. By contrast, if we gave these points some structure, maybe making them the corners of a hexagon and only considering the permutaions that preserve how far apart each one is from the other, we get only the 12 snowflake symmetries we saw earlier. Bump the number of points up to 12, and the
number of permutations grows to about 479 million. The monster that we’ll get to is rather
large, but it’s important to understand that largeness in and of itself isn’t all
that interesting with groups; the permutation groups already make that easy to see. If we were shuffling 101 objects,
for example, with the 101 factorial different actions that can do this, we have a group with a size of around 9Ă—10^159. If every atom in the observable universe had a copy of that universe inside itself, this is roughly how many sub-atoms there would be. These permutation groups go by the name S sub n, and they play an important role in group theory. In a certain sense, they encompass all other
groups. And so far you might be thinking: “ok, this intellectually playful enough, but is any of this actually useful?” One of the earliest applications of group
theory came when mathematicians realized that the structure of these permutation groups
tells us something about solutions to polynomial equations. You know how in order to find the two roots
of a quadratic equation, everyone learns a certain formula in school? Slightly lesser known is the fact that there’s also
a cubic formula, one that involves nesting cube roots with square roots in a larger expression. There’s even a quartic formula for degree
4 polynomial, which is an absolute mess. It's almost impossible to write without factoring things out. And for the longest time mathematicians struggled
to find a formula to solve degree 5 polynomials. I mean, maybe there's one, but it's just super complicated. It turns out though, if you think about the group
which permutes the roots of such a polynomial, there’s something about the nature of this
group that reveals that no quintic formula can exist. For example, the 5 roots of the polynomial
you see on screen now, they have definite values, you can write out decimal approximations, but what you can never do, is write those exact values
by starting with the coefficients of the polynomial and using only the four basic operations of arithmetic together with radicals, no matter how many times you nest them. And that impossibility has everything to do
with the inner structure of the permutation group S5. A theme in math through the last two centuries
has been that the nature of symmetry in and of itself can show us all sorts of non-obvious
facts about the other objects we that we study. To give just a hint of one of the many many ways
this applies to physics, there’s a beautiful fact known as Noether’s theorem saying that
every conservation law corresponds to some kind of symmetry, a certain group. So all these fundamental laws like conservation
of momentum and conservation of energy each correspond to a group. More specifically, the actions we should be able to apply to a setup, such that the laws of physics don't change. All of this is to say that groups really are
fundamental, and the one thing I want you to recognize right now is that they’re one
of the most natural things that you could study. What could be more universal than symmetry? So you might think the patterns among groups
themselves would somehow be very beautiful and symmetric. The monster, however, tells a different story. Before we get to the monster, though, at this
point some mathematicians might complain that what I’ve described so far are not groups,
exactly, but group actions; and the groups are something slightly more abstract. By way of analogy, if I mention the number
“3”, you probably don’t think about a specific triplet of things, you probably
think about 3 as an object in and of itself, an abstraction, maybe represented with a common symbol. In much the same way, when mathematicians
discuss the elements of a group, they don’t necessarily think about specific actions on
a specific object, they might think of these elements as a kind of thing in and of itself, maybe
represented with symbols. For something like the number three, the abstract
symbol does us very little good unless we define its relation with other numbers, for example
the way it adds and multiplies with them. For each of these, you could think of a literal
triplet of something, but again most of us are comfortable, probably even more comfortable,
using the symbols alone. Similarly, what makes a group a group are
all the ways that its elements combine with each other. And in the context of actions, this has a very vivid meaning; what we mean by combining is to apply one action after the other, read from right to left; if you flip a snowflake
about the x-axis, then rotate it 60 degrees counterclockwise, the overall action is the
same as if you had flipped it about this diagonal line. All possible ways that you could combine two elements of a group like this defines a kind of multiplication. That is what really give a group its structure. Here, I’m drawing out the full 8x8 table of the symmetries of a square. If you apply an action on the top row and follow it
by an action from the left column, it’ll be the same as the action in the corresponding grid
square. But if we replace each of these symmetric actions with something purely symbolic, well, the multiplication table still captures the inner structure of the
group, but now it’s abstracted away from any specific object it might act on, like
a square, or roots of a polynomial. This is entirely analogous to how the usual
multiplication table is written symbolically, which abstracts away from the idea of literal
counts. Literal counts, arguably, would make it clearer what’s
going on, but since grade school we all grow comfortable with the symbols. After all, they’re less cumbersome, they free
us up to think about more complicated numbers, and they also free us to think about numbers in new and very different ways. All of this is true of groups as well, which are best understood, as abstractions. above the idea of symmetry actions. I’m emphasizing this for two reasons, one
is that understanding what groups really are gives a better appreciation for the monster. And the other is that many students learning about
groups for the first time can find them frustratingly opaque, I know I did. A typical course starts with this very formal and abstract definition, which is that a group is a set – any collection of things – with a binary operation
– a notion of multiplication between those things – such that this multiplication
satisfies four special rules, or axioms. And all this can feel, well, kind of random,
especially when it isn’t made clear that all these axioms arise from the things that must be obviously true when you’re thinking about actions and composing them. To any students among you with such a course in the future, I would say if you
appreciate that the relationship groups have with symmetric actions is analogous to the
relationship numbers have with counts, it can help to keep a lot of the course a lot more grounded. An example might help to see why this kind
of abstraction is desirable. Consider the symmetries of a cube and the
permutation group of 4 objects. At first, these groups feel very different;
you might think of the one on the left as acting on the 8 corners in a way that preserves
the distance and orientation structure among them, but on the right we have a completely
unconstrained set of actions on a much smaller set of points. As it happens, though, these two groups are
really the same, in the sense that their multiplication tables will look identical. Anything that you can say about one group will
be true of the other. For example, there are 8 distinct permutations
where applying it three times in a row gets you back to where you started (not counting
the identity). These are the ones that cycle three elements together. There are also 8 rotations of a cube that
have this property, the various 120 degree rotations about each diagonal. This is no coincidence. The way to phrase this precisely is to say
there’s a one-to-one mapping between rotations of a cube and permutation of four elements
which preserves composition. For example, rotating 180 degrees about
the y-axis, followed by 180 about the x-axis, gives the same overall effect as rotating 180 degree around the z-axis; remember, that’s what we mean by a product of two actions. And if you look at the corresponding permutations, under a certain one-to-one association,
this product will still be true, applying the two actions on the left gives the same overall effect as
the one on the right. When you have a correspondence where this remains true for all products, it’s called an “isomorphism”, which is maybe the most
important idea in group theory. This particular isomorphism between cube rotations
and permutations of four objects is a bit subtle, but for the curious among you, you
may enjoy taking a moment to think hard about how the rotations of a cube permute its four
diagonals. In your mathematical life, you'll see more examples of a given group arising from seemingly unrelated situations, and as you do, you’ll get a better sense for what group theory is all about. Think about how a number like 3 is not really about a particular triplet of things, it’s about all possible triplets of things. In the same way a group is not really about
symmetries of a particular object, it’s an abstract way that things even can be symmetric. There are even plenty of situations where
groups come up in a way that doesn’t feel like a set of symmetric actions at all, just
as numbers can do a lot more than count. In fact, seeing the same group come out of
different situations is a great way to reveal unexpected connections between distinct objects;
that’s a very common theme in modern math. And once you understand this about groups,
it leads you to a natural question, which will eventually lead us to the monster: What
are all the groups? But now you’re in a position to ask that question in a more sophisticated way: What are all the groups up to isomorphism, which
is to say we consider two groups to be the same if there’s an isomorphism between them. This is asking something more fundamental
than what are all the symmetric things, it’s a way of asking what are all of the ways that
something can be symmetric? Is there some formula or procedure for producing
them all? Some meta-pattern lying at the heart of symmetry
itself? This question turns out to be hard. Exceedingly hard. For one thing, there's the division between infinite groups, for example the ones describing the symmetries of a line or a circle, and finite groups,
like all the ones we’ve looked at up to this point. To maintain some hope of sanity, let’s limit
our view to finite groups. In much the same way that numbers can be broken
down in their prime factorizations, or molecules can be described based on the atoms within
them, there's a certain way that finite groups can be broken down into a kind of composition of smaller groups. The ones which can’t be broken down any
further, analogous to prime numbers or atoms, are known as the “simple groups”. To give a hint for why this is useful, remember
how we said that group theory can be used to prove that there’s no formula for a degree-5 polynomial
the way there is for quadratic equations? Well, if you're wondering what that proof actually looks like, it involves showing that if there were some kind of mythical quintic formula, something which uses only radicals
and the basic arithmetic operations, it would imply that the permutation group on 5 elements decomposes into a special kind of simple group, known fancifully as cyclic groups of prime
order. But the actual way that it breaks down involves a different kind of simple group, a different kind of atom, one which polynomial solutions built up from radicals
would never allow. That is a super high level description, of
course, with about a semester’s worth of details missing, but the point is that you have this
really non-obvious fact about a different part of math whose solution comes down to
finding the atomic structure of a certain group. This is one of many different examples where understanding the nature of these simple groups, these atoms, actually matters outside of group theory. The task of categorizing all finite groups breaking down into two steps: One - find all the simple groups, and two -
find all the ways to combine them. The first question is like finding the periodic
table, and the second is a bit like doing all of chemistry thereafter. The good news is that mathematicians have
found all the finite simple groups.. Well, more pertinent is that they proved that the ones that they found are, in fact, all the ones out there. It took many decades, tens of thousands of
dense pages of advanced math, hundreds of some of the smartest minds out there, and
significant help from computers, but by 2004 with a culminating 12,000 pages to tie
up the loose ends, there was a definitive answer. Many experts agree: this is one of
the most monumental achievements in the history of math. (sigh) The bad news, though, is that this answer is absurd. There are 18 distinct infinite families of
simple groups, which makes it really tempting to lean into the periodic table analogy; but
groups are stranger than chemistry because there are also 26 simple groups that are just left
over that don’t fit the other patterns. These 26 are known as the sporadic groups. That a field of study rooted in symmetry itself
has such a patched together fundamental structure is... I mean it’s just bizarre! It’s like the universe was designed by committee. If you’re wondering what we mean by an infinite
family, examples might help: one such family of simple groups includes all the cyclic groups with prime order; these
are essentially the symmetries of a regular polygon with a prime number of sides, but where you’re not allowed to flip the polygon over. Another of these infinite families is very
similar to the permutation groups that we saw earlier, but there’s the tiniest constraint on how
they're allowed to shuffle n items. If they act on 5 or more elements, these groups
are simple, which incidentally is heavily related to why polynomials with degree 5 or
more have solutions that can’t be written down using radicals. The others 16 families are notably more complicated, and I’m told that there’s at least a little ambiguity in how to organize them into cleanly distinct
families without overlap. But everyone agrees is that the 26 sporadic groups
stand out as something very different. The largest of these sporadic groups is known, thanks to John Conway, as the monster group, and its size is the number I mentioned at the start. The second largest, and I promise this isn’t
a joke, is known as the baby monster group.
[monster: don't talk to me or my son ever again] Together with the baby monster 19 of the sporadic groups are in a certain sense children of the monster, and Robert Griess called these 20 the “happy
family”. He called the other 6, which don’t even
fit that pattern, the pariahs. As if to compensate for how complicated the
underlying math here is, the experts really let loose on their whimsy while naming things. Let me emphasize, having a group which
is big is not that big of a deal. But the idea that the fundamental building
blocks for one of the most fundamental ideas in math come in this collection that just
abruptly stops around 8×10^53? That’s weird. Now, at this point, that I introduced groups as
symmetries, a collection of actions, you might wonder what it is that the monster acts on. What object does it describe the symmetries of? Well, there is an answer, but it doesn’t fit into
2 or 3 dimensions to draw. Nor does it fit into 4 or 5, instead to see
what the monster acts on we’d have to jump up to... wait for it... 196,883 dimensions. Just describing one of the elements of this
group takes around 4 gigabytes of data, even though plenty of groups that are way bigger have a
much smaller computational descriptions. The permutation group on 101 elements was,
if you’ll recall, dramatically bigger, but we could describe each element with very little
data, for example a list of 100 numbers. No one really understands why the sporadic
groups, and the monster in particular, are there. Maybe in a few decades, there will be a clearer
answer, maybe one of you will come up with it, but despite knowing that they’re deeply
fundamental to math, and arguably to physics as well, a lot about them remains
mysterious. In the 1970s, mathematician John McKay was
making a switch to studying group theory to an adjacent field, and he noticed that a number very similar to this 196,883 showed up in a completely unrelated context, or at least, almost. A number one bigger than this was in the series expansion of a fundamental function in a totally different part of math, relevant to these
things called modular forms and elliptic functions. Assuming that this was more than a coincidence
seemed crazy, enough that it was playfully deemed “moonshine” by John Conway, but
after more numerical coincidences like this were noticed, it gave rise to what became
known as the monstrous moonshine conjecture. Whimsical names just don’t stop. This was proved by Richard Borcherds in 1992,
solidifying a connection between very different parts of math that at first glance seemed
crazy. 6 years later, by the way, he won the Fields
Medal, in part for the significance of this proof. And related to this moonshine is the connections between the monster and string theory. Maybe it shouldn’t come as a surprise that something
that arises from symmetry itself is relevant to physics, but in light of just how
random the monster seems at first glance, the connection still elicits a double-take. To me, the monster and its absurd size is
a nice reminder that fundamental objects are not necessarily simple. 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.
This is unsubstantiated hearsay, but I remember a story about how someone in conversation with Conway remarked that it was surprising to them that the Monster group was so large, to which Conway responded "Really? You might as well be surprised that it's so small."
It's easy to forget that there are a heck of a lot of numbers out there - maybe we're being greedy by expecting interesting ones to be small and nice.
edit: a word
3b1b is one of the few people who just constantly deliver work of such high quality that it's hard to understand how he has enough time in the day without having sub-universes in each of his atoms.
I genuinely think he will be part of those people who are named as references by future mathematicians and scientists, as the love he's got for his subject is contagious. I did shudder reading "groups of Lie type", but I'm barely even an applied mathematics person so no surprise there..
I'm also always glad to see Noether's theorem because it's one of those simple and profound bits of knowledge that bridge the gap between the beauty of mathematics and the beauty of physics, and a bit sad to see it taking just a tiny fraction of this video (understandably though).
Brilliant!
One mistake I found: I don't think many people consider the j invariant part of Galois theory, rather than number theory or the theory of modular forms). (And calling it that understates how strange McKay's connection is, because Galois theory is very closely related to finite group theory.)
I wished he had said a little bit more about what "simple" means, specifically the definition using group actions: A group is simple if whenever it acts on something, if more than one element of the group acts trivially, they all act trivially. One could give an example like D_6 acting on both sides of the snowflake, with the rotations preserving the slides, and the reflections swapping the two sides, so D_6 is not simple.
What does it say about me that, while I didn't recognize the number, I guessed what it was anyway?
Math YouTube is on fire today... Four minutes after 3B1B posted this one, Numberphile posted one titled 569936821221962380720
I recommend reading Symmetry and the Monster if you want to get some more info.
So glad I took intro to abstract algebra, all throughout the video I was like " Hey, I did that!" , also anyone else got the goosebumps?
This is part of a maths community thing MegaFavNumbers, started by James Grime (I think). The whole playlist is here: https://www.youtube.com/playlist?list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo
One of my favorite math exchanges on YouTube is from one of the Numberphile videos with Conway. Brady asks about the Monster Group and why it is so huge. Conway immediately responded by asking why it is so small. Why do the sporadic groups just . . . stop? The biggest one is small enough that you can write it down in ordinary decimal notation.