How do you get three mathematicians and a physicist into a Mini? Two in the front, two in the back - along with four sets of luggage and a crate of wine. It was a tight squeeze, even without the booze. At the time - July 1985 - I had no idea that this memorable journey down the coast of California through the redwood forests would divert my research into uncharted territory, although it did take a book review and a visit to the rodeo to complete the transition. The result? Three eureka moments, out of the blue, over a period of about 15 years.
The first happened in the Mini, on the way home from a mathematics conference in Arcata, a small town about 200 miles north of San Francisco. The mathematicians had intended to fly, but the physicist, Lenny Pismen, had rented a car, so we decided to use that instead. I'm not sure why he opted for a Mini: probably because he had planned to travel on his own.
Years later, I rented another one to convey my wife, our two sons and a close friend - very close - around Yellowstone National Park for a week. We chose the car because it was cheap, and also the only one on offer.
But back to California. The four of us reckoned it would be more fun to travel together, and decided that we'd just about fit. Edgar Knobloch, now professor of physics at the University of California, Berkeley, joined Lenny in the front. Marty Golubitsky (today distinguished professor of mathematics and physical sciences at Ohio State University) and I were squashed in the back, along with most of the luggage.
Marty and I had started working together a couple of years earlier, researching the symmetry in dynamical systems - pattern formation in anything that changes over time, according to fixed rules. To pass the time when we weren't hopping out of the car to look at giant trees, we started thinking about the patterns that form when you hook a number of identical units together into a ring.
We had already sorted out a general method for approaching that sort of question, so we sat and tried to work out how the story must go. There wasn't room to write anything down, so we did the whole thing in our heads.
Suddenly, the entire story came together: the problem went from impossible to obvious in a split second. By the time we stopped at Marty's favourite winery in the Napa Valley to pick up a crate of his favourite wine, the problem was solved. We went out for Vietnamese crab at Fisherman's Wharf, spent the night on Edgar's floor, and nearly missed our flight to Houston because it was the day of the San Francisco Marathon and half the streets were closed. Later, we redid the calculations properly, filling in the details, and wrote them up as our contribution to the published proceedings of the conference.
We discovered that the typical patterns for these systems were "travelling waves", where successive units around the ring would do exactly the same thing, but with a time delay.
The simplest example is to hold your arms out sideways and let them dangle from the elbows. Now let the dangling arms swing to and fro. Typically, they either synchronise, with both arms moving in the same way, or they anti-synchronise, with the left arm doing the exact opposite of the right. There are other more esoteric patterns, too.
At the time, we thought of these results as pure mathematics, a nice example of our general techniques; part of a bigger picture that we were slowly piecing together. It didn't occur to us to look for applications outside the discipline, except for a few rather artificial ones, such as electronic oscillator rings, which really just turned the maths into hardware. And it might never have occurred to us were it not for my second eureka moment.
Quite by coincidence, a few years later, New Scientist magazine sent me a book to review. I still have it: it's called Natural Computation: Selected Readings (MIT Press, 1988), edited by Whitman A. Richards, today a professor at the Massachusetts Institute of Technology's Artificial Intelligence Laboratory. It's a collection of technical articles in which engineers explain how to take inspiration from nature - using the biology of the eye to design computer vision, or create control systems modelled on the human hand.
The article that particularly caught my eye had the catchy title, "Some properties of regularly realisable gait matrices", and it was about how animals move. Engineers were starting to become interested in how to make robots with legs, which in principle should be much better than wheels for many tasks: finding unexploded ordnance on army firing ranges; decommissioning nuclear power stations; exploring Mars. But the mechanics of legged locomotion lagged far behind that of the wheel, so the engineers were taking notes from animal movement.
In its introduction, the article told me some basic stuff that I hadn't thought about before. Animals move in a variety of patterns, called "gaits", repeating the same sequence of movements over and over again in a series of "gait cycles".
A horse, for example, can walk, trot, canter or gallop - four distinct gaits, each with its own characteristic pattern. In the walk, the legs move in turn, and each hits the ground at successive quarters of the gait cycle. So the sequence goes left-back, left-front, right-back, right-front, all equally spaced, over and over again. The trot is similar, but one diagonal pair of legs hits the ground first, and the other pair does so half a gait cycle later. So in both gaits, all four legs do essentially the same thing, but with specific differences in timing.
Now, where had I seen that kind of thing before? In those rings of identical units that Marty and I had sorted out while bumping our way down Route 101 in Lenny's rented Mini. This, it turned out, was the second eureka moment ... but I didn't realise it at the time.
I mentioned the connection between gaits and maths in my review, as a throwaway line, and added: "Does anyone want to fund an electronic cat?" Within a few days, I received a response. Jim Collins, a young biomechanicist - someone who applies mechanics to biology, especially medicine - was visiting the University of Oxford for a year, and he telephoned me.
"I can't fund an electronic cat, but I know people who can. Can I come to see you?" Thus began a collaboration that lasted ten years.
Jim told me that gaits are thought to be produced by relatively simple circuits in animals' nervous systems, known as central pattern generators, or CPGs. They are located not in the brain, but in the spine. And they set the basic rhythms, the patterns, for the movement of muscles that cause animals to walk, trot, canter, gallop or whatever.
No one had actually seen a CPG at this time. Their existence was inferred, and in some quarters they were a bit controversial. However, the evidence was quite strong. What no one knew was the exact structure of the CPG circuitry, namely what sort of network the nerve cells involved made. So we did the best we could, and worked out what the maths that Marty and I developed implied for the patterns generated by CPGs, assuming various plausible structures for them.
We started with quadrupeds ("A horse is a quadruped, and quadruped's Latin for beast, as everybody that's gone through the grammar knows," as Wackford Squeers informs Nicholas Nickleby in Charles Dickens' novel), and then went on to show that similar ideas applied to hexapods.
There had already been a lot of work done on the mechanics of legged locomotion. Our approach was more abstract, trying to infer the structure of a hypothetical CPG from the patterns observed in the legs. But it had an interesting consequence: the same network of nerve cells, operating under different conditions, was capable of generating all of the most symmetric quadruped gaits - and, very likely, less symmetric ones, such as the canter and the gallop.
There were some puzzles. None of the networks that we proposed was completely satisfactory, for various technical reasons that I couldn't formulate very clearly but which bugged me nevertheless. This came to a head when I gave a seminar about the work during a visit to Houston, and Marty and one of his students, Luciano Buono, managed to put their finger on the precise difficulty.
After a brainstorming session, we convinced ourselves that any workable CPG for quadrupeds must have at least eight units, whereas Jim and I had been convinced that one unit per leg - four in total - was the natural choice. Buono became so interested in the problem that he took it up for his PhD. The upshot was that the most natural CPG for quadrupeds should have eight units, two for each leg: one to work the muscles that flex the leg, the other to work those that extend it.
This network had a curious implication. Along with all the standard gait patterns, it predicted one that we had never encountered before. In this gait, which we named the "jump", the two rear legs hit the ground together, and then the two front legs hit the ground together one quarter of the way through the gait cycle. If it had been halfway through the gait cycle, this gait would have been a very standard one - the bound, used, for example, by a dog when running fast. But one quarter of the way through the gait cycle was a real puzzle, especially since no legs hit the ground halfway or three quarters of the way through. It was as though the animal was suspended in mid-air.
We came to this conclusion late in the afternoon. The Houston Livestock Show and Rodeo was in town, and, Texas being Texas, we had seats booked for that evening. So we went to the Astrodome and watched cattle being roped and buggies being raced. Then came the bucking broncos.
The horses were trying to throw their riders off their backs, and the riders were trying to stay on for as long as they could, which was often just a few seconds. Suddenly Marty and I looked at each other and started counting. The horse that we were watching was jumping into the air, both back feet giving it a push, then both front, then it hung in the air ... it looked very much as if the difference in timing was one quarter of the full gait cycle.
This was the third eureka moment. It took Marty six weeks to get hold of a copy of the video footage of that precise horse. He counted the number of video frames between the back legs and the front ones hitting the ground, and it was very close to one quarter of the number of frames on a complete cycle. We had found our missing gait.
Later, we discovered that two other animals, the Norway rat and the Asia Minor gerbil, also employ this unusual gait. We found several other features of real gaits that were predicted by our eight-unit CPG, including one involving centipedes. Of course, none of this proved that our theory was correct, but it did mean that it passed several tests that might have proved it wrong.
This sequence of eureka moments had an impact on more than just my research. It gave me a wonderful topic for public lectures, and I've used it at everything from science-fiction conventions to major conferences. It even featured in my 1997 Royal Institution Christmas Lectures on BBC Two, where we enacted several gait patterns using volunteers from the audience.
The lesson I learnt from all this was not just that strange coincidences sometimes have a bearing on research. It was that if you talk to lots of other scientists and mathematicians, you greatly increase the chances of such coincidences occurring. It's much like the phrase usually attributed to Gary Player, the professional golfer. When told that most things in golf are a matter of luck, he replied: "Yes. And the more I practise, the luckier I get."