Associahedra: The Shapes of Multiplication

Let's say you have a list of three numbers and you want to multiply them.

There's only one way to do it.

You just multiply them.

But I'd like to tell you about an example where there are infinitely many ways to multiply three things.

[MUSIC PLAYING] In our last episode, we talked about different properties of multiplication.

Associativity and commutativity are the most familiar.

But they're just two of many.

We also saw it's possible to multiply things that aren't numbers.

And in that case, we may not have some properties, like associativity.

But that's not a bad thing.

In fact, it's a beautiful thing.

We looked at a type of multiplication from topology called loop concatenation.

And we discovered a way to multiply loops in a topological space.

A loop is a continuous function from the interval of length 1 into a topological space that send 0 and 1 to the same point.

Intuitively, we can think of a car driving along some path where at the end of one second, the car goes back to where it started.

And if we have two loops a and b that both start and stop at the same point in space, we can multiply them.

What are the multiplication instructions?

Simply go around a in the first half second, then go around b in the last half second.

So each car travels at twice the original speed.

Now, we didn't have to define it this way.

Instead of dividing the interval in half.

We could've sliced it in infinitely many other ways and defined that to be the multiplication instruction.

Go around a during the red interval and go around b during the blue interval.

But let's just choose to split the interval in half.

So these instructions let us view the wedge of two loops as one single loop, the product of a and b.

But this multiplication is not associative.

To see this, suppose we have three loops, a, b, and c, and let's compare the two ways we can multiply them.

This is just the two ways we can put parentheses around two of the letters.

These two loops are not the same, because the time in which the red, blue, and yellow cards do their traveling is different.

And here's where it gets interesting.

There is a way to get from one loop to the other.

We simply have to ask the red car to slow down a little and the yellow car to speed up.

Think about it.

To get from the first picture to the second, we just have to spend more time traveling along a and less time on c. So just lengthen the red interval and shrink the yellow interval.

So even though the two loops aren't equal, we can get from one to the other by making a continuous adjustment over time.

And this adjustment has a name.

It's called a homotopy.

Intuitively, it's given by this animation.

Mathematically we can express it as a continuous function of two variables, say s and t. In fact, we can think of a homotopy as giving us a path from one loop to the other.

The left point corresponds to the loop where the red car travels in the first quarter second.

The right point corresponds to the loop where the red car travels in the first half second.

And every point in between corresponds to an intermediate option.

So let's recap.

Loop concatenation is not associative.

Therefore, ab times c and a times bc are not the same.

However, they are the same up to homotopy.

In other words, there is a continuum of ways to multiply three loops and this continuum provides a homotopy between the two extremes, which we can represent as a line segment that joins two vertices.

The upshot is that a line segment encodes all the ways we can multiply three loops together.

But what happens if we multiply four loops, a, b, c, and d?

Well, there are five different ways to put parentheses around four letters.

So there are five different ways to concatenate four loops.

And none of them were equal, because multiplications not associative.

But there are homotopies, and hence, path between them.

For instance, this point corresponds to the loop ab times c times d, where we travel around a and b in the first two eighths of a second, then around c in the next quarter of a second, then around d in the last half second.

Moreover, there is a path from here to here.

It's given to us by homotopy that's very similar to the one we saw earlier.

Just go around a a little slower and c a little faster.

A similar homotopy gets us from here to here.

And another one gets us from here to here.

So there are five total vertices and four of them can be connected by this orange path.

But here's what's cool, we could've taken a different route.

Notice, there's a homotopy from this point to that point.

Simply spend more time on a and b and less time on d. Then we, can get from here to here by spending even more time on a and less time on c and d. So we have two paths, two homotopies connecting these two points.

These paths aren't the same.

But the last time we came across two things that weren't the same, we discovered that there was a path between them.

So what do you think?

Is there a path between these two paths.

What would that even look like?

Think about it.

If you have two literal paths, how do you get from one to the other?

Well, you can walk here or here or here or here.

You have infinitely many choices.

There is a continuum of individual paths connecting the two original paths.

And we can think of that continuum as forming a path between paths.

So back to the pentagon.

A point on the purple path represents a loop in which the four cars each travel at a certain speed and the same goes for a point on the orange path.

And you can get from one point to the other, simply by adjusting the speeds.

For example, there is a path from here to here.

Just go around a little slower and c a little faster.

Since we can do this between any two points, there is a continuum of paths between these two paths.

And this continuum fills in the face of the pentagon.

In other words, we have a homotopy between two homotopies.

All right, let's recap.

Each vertex of this pentagon represents a way of multiplying for loops.

Each edge represents a homotopy, or a path between these ways.

And the face of the pentagon represents a homotopy between the final two paths.

The upshot now is that a pentagon encodes all of the ways that we can multiply four loops together.

But why stop at four?

What happens if you multiply five loops?

Well, there are 14 ways to put parentheses around five letters.

So you end up with a polygon that has 14 vertices, 21 edges, 9 faces, some of which are pentagons and some are rectangles, and one solid interior.

The edges represent homotopies between the loops.

The faces represent homotopies between those homotopies.

And the solid interior represents a homotopy between the homotopies between the homotopies.

And you know the story doesn't stop at five.

It goes on for forever, literally.

The set of all loops that start and stop at the same point in a topological space, like a donut, form what's called an A infinity space.

The A stands for associativity and the infinity reminds us of the eventually infinite string of homotopies.

And The sequence of shapes that we get, a line segment, a pentagon, a polyhedron with nine faces and so on are called the associahedra.

For every integer n bigger than or equal to 2, each different way of putting parentheses around n letters gives us a different vertex of the n minus 2 dimensional associahedron, usually called Kn.

And when n equals 2, there's only one way to put parentheses around two letters.

So you get a point called K2.

And there are two ways to parenthesize three letters, five ways to parenthesize four letters, 14 ways to print size 5 letters, and 42 ways to print say six letters, and so on.

These numbers 2, 5, 14, 42 are called the Catalan Numbers.

And they show up a lot in mathematics.

In fact, combinatorialist Richard Stanley wrote a book, appropriately called "Catalan Numbers," which contains over 200 examples of where this special sequence shows up.

So needless to say, associahedra play a big role in combinatorics.

But in this episode, we're dealing mainly with typology.

And it turns out that the only topological spaces with an A infinity structure are, in some sense, loop spaces.

This is a key result in a branch of math called Homotopy Theory, which uses loops and spheres and higher dimensional spheres to study topological spaces.

It was proved by mathematician Jim Stasheff in the early 1960s, where he introduced the associahedra as a way to organize all of the homotopies that come with loop concatenation.

More recently, A infinity spaces and their algebraic cousins, called A infinity algebras, have found application not just in homotopy theory, but also in algebra, geometry, and mathematical physics too.

In fact, Stasheff's work on associahedra paved the way for a more general tool called an operad, which is a gadget that some mathematicians use to keep track of all of the different types of multiplication.