Here’s a cute connection between topological entropy, braids, and the golden ratio. I learned about it in this paper:

• Jean-Luc Thiffeault and Matthew D. Finn, Topology, braids, and mixing in fluids.

### Topological entropy

I’ve talked a lot about entropy on this blog, but not much about topological entropy. This is a way to define the entropy of a continuous map from a compact topological space to itself. The idea is that a map that mixes things up a lot should have a lot of entropy. In particular, any map defining a ‘chaotic’ dynamical systems should have positive entropy, while non-chaotic maps maps should have zero entropy.

How can we make this precise? First, cover with finitely many open sets Then take any point in apply the map to it over and over, say times, and report which open set the point lands in each time. You can record this information in a string of symbols. How much information does this string have? The easiest way to define this is to simply count the total number of strings that can be produced this way by choosing different points initially. Then, take the logarithm of this number.

Of course the answer depends on typically growing bigger as increases. So, divide it by and try to take the limit as Or, to be careful, take the lim sup: this could be infinite, but it’s always well-defined. This will tell us how much new information we get, on average, each time we apply the map and report which set our point lands in.

Of course the answer also depends on our choice of open cover So, take the supremum over all finite open covers. This is called the **topological entropy** of

Believe it or not, this is often finite! Even though the log of the number of symbol strings we get will be larger when we use a cover with lots of small sets, when we divide by and take the limit as this dependence often washes out.

### Braids

Any braid gives a bunch of maps from the disc to itself. So, we define the **entropy of a braid** to be the minimum—or more precisely, the infimum—of the topological entropies of these maps.

How does a braid give a bunch of maps from the disc to itself? Imagine the disk as made of very flexible rubber. Grab it at some finite set of points and then move these points around in the pattern traced out by the braid. When you’re done you get a map from the disk to itself. The map you get is not unique, since the rubber is wiggly and you could have moved the points around in slightly different ways. So, you get a bunch of maps.

I’m being sort of lazy in giving precise details here, since the idea seems so intuitively obvious. But that could be because I’ve spent a lot of time thinking about braids, the braid group, and their relation to maps from the disc to itself!

This picture by Thiffeault and Finn may help explain the idea:

As we keep move points around each other, we keep building up more complicated braids with 4 strands, and keep getting more complicated maps from the disc to itself. In fact, these maps are often chaotic! More precisely: they often have positive entropy.

In this other picture the vertical axis represents time, and we more clearly see the braid traced out as our 4 points move around:

Each horizontal slice depicts a map from the disk (or square: this is topology!) to itself, but we only see their effect on a little rectangle drawn in black.

### The golden ratio

Okay, now for the punchline!

**Puzzle.** Which braid with 3 strands has the highest entropy per generator? What is its entropy per generator?

I should explain: any braid with 3 strands can be written as a product of generators Here switches strands 1 and 2 moving the counterclockwise around each other, does the same for strands 2 and 3, and and do the same but moving the strands clockwise.

For any braid we can write it as a product of generators with as small as possible, and then we can evaluate its entropy divided by This is the right way to compare the entropy of braids, because if a braid gives a chaotic map we expect powers of that braid to have entropy growing linearly with

Now for the answer to the puzzle!

**Answer.** A 3-strand braid maximizing the entropy per generator is And entropy of this braid, per generator, is the logarithm of the golden ratio:

In other words, the entropy of this braid is

This fact was proved here:

• D. D’Alessandro, M. Dahleh and I Mezíc, Control of mixing in fluid flow:

A maximum entropy approach, *IEEE Transactions on Automatic Control* **44** (1999), 1852–1863.

So, people call this braid the **golden braid**.

What does it mean? I don’t know. The 3-strand braid group is called , and its center is $\mathbb{Z},$ and its quotient by its center is the famous group . I wrote a long story about this:

• John Baez, This week’s finds in mathematical physics (week 233).

There’s also a strong connection between braid groups, certain quasiparticles in the plane called Fibonacci anyons, and the golden ratio. But I don’t see the relation between these things and topological entropy! So, there is a mystery here—at least for me.