So far in this series we’ve been looking at what happens when we roll circles on circles:
• In Part 1 we rolled a circle on a circle that’s the same size.
• In Part 2 we rolled a circle on a circle that’s twice as big.
• In Part 3 we rolled a circle inside a circle that was 2, 3, or 4 times as big.
In every case, we got lots of exciting math and pretty pictures. But all this pales in comparison to the marvels that occur when we roll a ball on another ball!
You’d never guess it, but the really amazing stuff happens when you roll a ball on another ball that’s exactly 3 times as big. In that case, the geometry of what’s going on turns out to be related to special relativity in a weird universe with 3 time dimensions and 4 space dimensions! Even more amazingly, it’s related to a strange number system called the split octonions.
The ordinary octonions are already strange enough. They’re an 8-dimensional number system where you can add, subtract, multiply and divide. They were invented in 1843 after the famous mathematician Hamilton invented a rather similar 4-dimensional number system called the quaternions. He told his college pal John Graves about it, since Graves was the one who got Hamilton interested in this stuff in the first place… though Graves had gone on to become a lawyer, not a mathematician. The day after Christmas that year, Graves sent Hamilton a letter saying he’d found an 8-dimensional number system with almost all the same properties! The one big missing property was the associative law for multiplication, namely:
The quaternions obey this, but the octonions don’t. For this and other reasons, they languished in obscurity for many years. But they eventually turned out to be the key to understanding some otherwise inexplicable symmetry groups called ‘exceptional groups’. Later still, they turned out to be important in string theory!
I’ve been fascinated by this stuff for a long time, in part because it starts out seeming crazy and impossible to understand… but eventually it makes sense. So, it’s a great example of how you can dramatically change your perspective by thinking for a long time. Also, it suggests that there could be patterns built into the structure of math, highly nonobvious patterns, which turn out to explain a lot about the universe.
About a decade ago I wrote a paper summarizing everything I’d learned so far:
But I knew there was much more to understand. I wanted to work on this subject with a student. But I never dared until I met John Huerta, who, rather oddly, wanted to get a Ph.D. in math but work on physics. That’s generally not a good idea. But it’s exactly what I had wanted to do as a grad student, so I felt a certain sympathy for him.
And he seemed good at thinking about how algebra and particle physics fit together. So, I decided we should start by writing a paper on ‘grand unified theories’—theories of all the forces except gravity:
The arbitrary-looking collection of elementary particles we observe in nature turns out to contain secret patterns—patterns that jump into sharp focus using some modern algebra! Why do quarks have weird fractional charges like 2/3 and -1/3? Why does each generation of particles contain two quarks and two leptons? I can’t say we really know the answer to such questions, but the math of grand unified theories make these strange facts seem natural and inevitable.
The math turns out to involve rotations in 10 dimensions, and ‘spinors’: things that only come around back to the way they started after you turn them around twice. This turned out to be a great preparation for our later work.
As we wrote this article, I realized that John Huerta had a gift for mathematical prose. In fact, we recently won a prize for this paper! In two weeks we’ll meet at the big annual American Mathematical Society conference and pick it up.
John Huerta wound up becoming an expert on the octonions, and writing his thesis about how they make superstring theory possible in 10-dimensional spacetime:
The wonderful fact is that string theory works well in 10 dimensions because the octonions are 8-dimensional! Suppose that at each moment in time, a string is like a closed loop. Then as time passes, it traces out a 2-dimensional sheet in spacetime, called a worldsheet:
In this picture, 'up' means 'forwards in time'. Unfortunately this picture is just 3-dimensional: the real story happens in 10 dimensions! Don't bother trying to visualize 10 dimensions, just count: in 10-dimensional spacetime there are 10 – 2 = 8 extra dimensions besides those of the string’s worldsheet. These are the directions in which the string can vibrate. Since the octonions are 8-dimensional, we can describe the string’s vibrations using octonions! The algebraic magic of this number system then lets us cook up a beautiful equation describing these vibrations: an equation that has ‘supersymmetry’.
For a full explanation, read John Huerta’s thesis. But for an easy overview, read this paper we published in Scientific American:
This got included in a collection called The Best Writing on Mathematics 2012, further confirming my opinion that collaborating with John Huerta was a good idea.
Anyway: string theory sounds fancy, but for many years I’d been tantalized by the relationship between the octonions and a much more prosaic physics problem: a ball rolling on another ball. I had a lot of clues saying these should be a nice relationship… though only if we work with a mutant version of the octonions called the ‘split’ octonions.
You probably know how we get the complex numbers by taking the ordinary real numbers and throwing in a square root of -1. But there’s also another number system, far less popular but still interesting, called the split complex numbers. Here we throw in a square root of 1 instead. Of course 1 already has two square roots, namely 1 and -1. But that doesn’t stop us from throwing in another!
This ‘split’ game, which is a lot more profound than it sounds at first, also works for the quaternions and octonions. We get the octonions by starting with the real numbers and throwing in seven square roots of -1, for a total of 8 dimensions. For the split octonions, we start with the real numbers and throw in three square roots of -1 and four square roots of 1. The split octonions are surprisingly similar to the octonions. There are tricks to go back and forth between the two, so you should think of them as two forms of the same underlying thing.
Anyway: I really liked the idea of finding the split octonions lurking in a concrete physics problem like a ball rolling on another ball. I hoped maybe this could shed some new light on what the octonions are really all about.
James Dolan and I tried hard to get it to work. We made a lot of progress, but then we got stuck, because we didn’t realize it only works when one ball is 3 times as big as the other! That was just too crazy for us to guess.
In fact, some mathematicians had known about this for a long time. Things would have gone a lot faster if I’d read more papers early on. By the time we caught up with the experts, I’d left for Singapore, and John Huerta, still back in Riverside, was the one talking with James Dolan about this stuff. They figured out a lot more.
Then Huerta got his Ph.D. and took a job in Australia, which is as close to Singapore as it is to almost anything. I got a grant from the Foundational Questions Institute to bring John to Singapore and figure out more stuff about the octonions and physics… and we wound up writing a paper about the rolling ball problem:
Whoops! I haven’t introduced G2 yet. It’s one of those ‘exceptional groups’ I mentioned: the smallest one, in fact. Like the octonions themselves, this group comes in a few different but closely related ‘forms’. The most famous form is the symmetry group of the octonions. But in our paper, we’re more interested in the ‘split’ form, which is the symmetry group of the split octonions. The reason is that this group is also the symmetry group of a ball rolling without slipping or twisting on another ball that’s exactly 3 times as big!
The fact that the same group shows up as the symmetries of these two different things is a huge clue that they’re deeply related. The challenge is to understand the relationship.
There are two parts to this challenge. One is to describe the rolling ball problem in terms of split octonions. The other is to reverse the story, and somehow get the split octonions to emerge naturally from the study of a rolling ball!
In our paper we tackled both parts. Describing the rolling ball problem using split octonions had already been done by other mathematicians, for example here:
• Robert Bryant and Lucas Hsu, Rigidity of integral curves of rank 2 distributions.
• Gil Bor and Richard Montgomery, G2 and the “rolling distribution”.
• Andrei Agrachev, Rolling balls and octonions.
• Aroldo Kaplan, Quaternions and octonions in mechanics.
We do however give a simpler explanation of why this description only works when one ball is 3 times as big as the other.
The other part, getting the split octonions to show up starting from the rolling ball problem, seems to be new to us. We show that in a certain sense, quantizing the rolling ball gives the split octonions! Very roughly, split octonions can been as quantum states of the rolling ball.
At this point I’ve gone almost as far as I can without laying on some heavy math. In theory I could show you pretty animations of a little ball rolling on a big one, and use these to illustrate the special thing that happens when the big one is 3 times as big. In theory I might be able to explain the whole story without many equations or much math jargon. That would be lots of fun…
… for you. But it would be a huge amount of work for me. So at this point, to make my job easier, I want to turn up the math level a notch or two. And this is a good point for both of us take a little break.
In the next and final post in this series, I’ll sketch how the problem of a little ball rolling on a big stationary ball can be described using split octonions… and why the symmetries of this problem give a group that’s the split form of G2… if the big ball has a radius that’s 3 times the radius of the little one!
I will not quantize the rolling ball problem—for that, you’ll need to read our paper.