Group theory, abstraction, and the 196,883-dimensional monster

Today, many members of the YouTube math community are getting

together to make videos about their favorite numbers over 1 million,

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 million, roughly 8x10 to the 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 that 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 going to 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 the actions like this taken together is called a group.

Kind of, at least.

Groups are typically defined a little 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 of how fundamental they find it.

Also take note, we always consider the action of doing nothing to be part of the 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 is 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 a kind of way of saying

that the orientation of the cube is not part of the 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 of 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 we consider

any way 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 six 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 permutations that

preserve how far apart each one is from the other,

well then we only get 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 we'll get to is rather large, but it's important to understand

that largeness in and of itself is not that interesting when it comes to 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 9x10 to the 159.