### A Math Puzzle Coming From Chemistry

#### Posted by John Baez

I posed this puzzle a while back over on Azimuth, and nobody solved it. Maybe it was too mathematical for most people there—it seems to be a problem in Klein geometry, actually. But also it’s a bit tricky: now that I think about it more, I’m not sure how to solve it either!

So, maybe you can help.

Suppose we have an ethyl cation. We’ll pretend it looks like this:

As I explained before, it actually doesn’t—not in real life. But never mind! Realism should never stand in the way of a good puzzle.

Continuing on in this unrealistic vein, we’ll pretend that the two black carbon atoms are distinguishable, and so are the five white hydrogen atoms. As you can see, 2 of the hydrogens are bonded to one carbon, and 3 to the other. We don’t care how the hydrogens are arranged, apart from which carbon each hydrogen is attached to. Given this, there are

$2 \times \displaystyle{ \binom{5}{2} = 20 }$

ways to arrange the hydrogens. Let’s call these arrangements **states**.

Now draw a dot for each of these 20 states. Draw an edge connecting two dots whenever you can get from one state to another by having a hydrogen hop from the carbon with 2 hydrogens to the carbon with 3. You’ll get this picture, called the Desargues graph:

The red dots are states where the *first* carbon has 2 hydrogens attached to it; the blue ones are states where the *second* carbon has 2 hydrogens attached to it. So, each edge goes between a red and a blue dot. And there are 3 edges coming out of each dot, since there are 3 hydrogens that can make the jump!

Now, the puzzle is to show that you can also get the Desargues graph from a different kind of molecule. Any molecule shaped like this will do:

The 2 balls on top and bottom are called **axial**, while the 3 around the middle are called **equatorial**.

There are various molecules like this. For example, phosphorus pentachloride. Let’s use that.

Like the ethyl cation, phosphorus pentachloride also has 20 states… but only if count them a certain way! We have to treat all 5 chlorines as distinguishable, but think of two arrangements of them as the same if we can rotate one to get the other. Again, I’m not claiming this is physically realistic: it’s just for the sake of the puzzle.

Phosphorus pentachloride has 6 rotational symmetries, since you can turn it around its axis 3 ways, but also flip it over. So, it has

$\displaystyle{ \frac{5!}{6} = 20}$

states.

That’s good: exactly the number of dots in the Desargues graph! But how about the edges? We get these from certain transitions between states. These transitions are called **pseudorotations**, and they look like this:

Phosphorus pentachloride really does this! First the 2 axial guys move towards each other to become equatorial. Beware: now the equatorial ones are no longer in the horizontal plane: they’re in the plane facing us. Then 2 of the 3 equatorial guys swing out to become axial.

To get from one state to another this way, we have to pick 2 of the 3 equatorial guys to swing out and become axial. There are 3 choices here. So, we again get a graph with 20 vertices and 3 edges coming out of each vertex.

**Puzzle.** Is this graph the Desargues graph? If so, show it is.

I read in some chemistry papers that it is. But is it really? And if so, why? David Corfield suggested a promising strategy. He pointed out that we just need to get a 1-1 correspondence between

• *states* of the ethyl cation and *states* of phosphorus pentachloride,

together with a compatible 1-1 correspondence between:

• *transitions* of the ethyl cation and *transitions* of phosphorus pentachloride.

And he suggested that to do this, we should think of the split of hydrogens into a bunch of 2 and a bunch of 3 as analogous to the split of chlorines into a bunch of 2 (the ‘axial’ ones) and a bunch of 3 (the ‘equatorial’ ones).

It’s a promising idea. There’s a problem, though! In the ethyl cation, a *single* hydrogen hops from the bunch of 3 to the bunch of 2. But in a pseudorotation, *two* chlorines go from the bunch of 2 to the bunch of 3… and meanwhile, two go back from the bunch of 3 to bunch of 2.

And if you think about it, there’s another problem too. In the ethyl cation, there are 2 distinguishable carbons. One of them has 3 hydrogens attached, and one doesn’t. But in phosphorus pentachloride it’s not like that. The 3 equatorial chlorines are just that: equatorial. They don’t have 2 choices about how to be that way. Or do they?

Well, there’s more to say, but this should already make it clear that getting ‘natural’ one-to-one correspondences is a bit tricky… if it’s even possible at all!

We could try solving the problem using the ideas behind Felix Klein’s ‘Erlangen program’. The group of permutations of 5 things, say $S_5,$ acts as symmetries of either molecule. For the ethyl cation the set of states will be $X = S_5/G$ for some subgroup $G.$ You can think of $X$ as a set of structures of some sort on a 5-element set. The group $S_5$ acts on $X$, and the transitions will give an invariant binary relation on $X$. For phosphorus pentachloride we’ll have some set of states $X\prime = S_5/G\prime$ for some other subgroup $G\prime$, and the transitions will give an invariant relation on $X\prime$.

We could start by trying to see if $G$ is the same as $G\prime$—or more precisely, conjugate. If they are, that’s a good sign. And if not, it’s bad: it probably means there’s no ‘natural’ way to show the graph for phosphorus pentachloride is the Desargues graph… so maybe it’s not true.

I could say more, but I’ll stop here.

## Re: A Math Puzzle Coming From Chemistry

You could and did – there was an extra sentence or two that you edited out from the Azimuth version!

You probably had your reasons… :-)