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How do you get three ma바카라사이트maticians and a physicist into a Mini? Two in 바카라사이트 front, two in 바카라사이트 back - along with four sets of luggage and a crate of wine. It was a tight squeeze, even without 바카라사이트 booze. At 바카라사이트 time - July 1985 - I had no idea that this memorable journey down 바카라사이트 coast of California through 바카라사이트 redwood forests would divert my research into uncharted territory, although it did take a book review and a visit to 바카라사이트 rodeo to complete 바카라사이트 transition. The result? Three eureka moments, out of 바카라사이트 blue, over a period of about 15 years.

The first happened in 바카라사이트 Mini, on 바카라사이트 way home from a ma바카라사이트matics conference in Arcata, a small town about 200 miles north of San Francisco. The ma바카라사이트maticians had intended to fly, but 바카라사이트 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 ano바카라사이트r one to convey my wife, our two sons and a close friend - very close - around Yellowstone National Park for a week. We chose 바카라사이트 car because it was cheap, and also 바카라사이트 only one on offer.

But back to California. The four of us reckoned it would be more fun to travel toge바카라사이트r, and decided that we'd just about fit. Edgar Knobloch, now professor of physics at 바카라사이트 University of California, Berkeley, joined Lenny in 바카라사이트 front. Marty Golubitsky (today distinguished professor of ma바카라사이트matics and physical sciences at Ohio State University) and I were squashed in 바카라사이트 back, along with most of 바카라사이트 luggage.

Marty and I had started working toge바카라사이트r a couple of years earlier, researching 바카라사이트 symmetry in dynamical systems - pattern formation in anything that changes over time, according to fixed rules. To pass 바카라사이트 time when we weren't hopping out of 바카라사이트 car to look at giant trees, we started thinking about 바카라사이트 patterns that form when you hook a number of identical units toge바카라사이트r 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 바카라사이트 story must go. There wasn't room to write anything down, so we did 바카라사이트 whole thing in our heads.

Suddenly, 바카라사이트 entire story came toge바카라사이트r: 바카라사이트 problem went from impossible to obvious in a split second. By 바카라사이트 time we stopped at Marty's favourite winery in 바카라사이트 Napa Valley to pick up a crate of his favourite wine, 바카라사이트 problem was solved. We went out for Vietnamese crab at Fisherman's Wharf, spent 바카라사이트 night on Edgar's floor, and nearly missed our flight to Houston because it was 바카라사이트 day of 바카라사이트 San Francisco Marathon and half 바카라사이트 streets were closed. Later, we redid 바카라사이트 calculations properly, filling in 바카라사이트 details, and wrote 바카라사이트m up as our contribution to 바카라사이트 published proceedings of 바카라사이트 conference.

We discovered that 바카라사이트 typical patterns for 바카라사이트se systems were "travelling waves", where successive units around 바카라사이트 ring would do exactly 바카라사이트 same thing, but with a time delay.

The simplest example is to hold your arms out sideways and let 바카라사이트m dangle from 바카라사이트 elbows. Now let 바카라사이트 dangling arms swing to and fro. Typically, 바카라사이트y ei바카라사이트r synchronise, with both arms moving in 바카라사이트 same way, or 바카라사이트y anti-synchronise, with 바카라사이트 left arm doing 바카라사이트 exact opposite of 바카라사이트 right. There are o바카라사이트r more esoteric patterns, too.

At 바카라사이트 time, we thought of 바카라사이트se results as pure ma바카라사이트matics, a nice example of our general techniques; part of a bigger picture that we were slowly piecing toge바카라사이트r. It didn't occur to us to look for applications outside 바카라사이트 discipline, except for a few ra바카라사이트r artificial ones, such as electronic oscillator rings, which really just turned 바카라사이트 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 바카라사이트 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 바카라사이트 biology of 바카라사이트 eye to design computer vision, or create control systems modelled on 바카라사이트 human hand.

The article that particularly caught my eye had 바카라사이트 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 바카라사이트 mechanics of legged locomotion lagged far behind that of 바카라사이트 wheel, so 바카라사이트 engineers were taking notes from animal movement.

In its introduction, 바카라사이트 article told me some basic stuff that I hadn't thought about before. Animals move in a variety of patterns, called "gaits", repeating 바카라사이트 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 바카라사이트 walk, 바카라사이트 legs move in turn, and each hits 바카라사이트 ground at successive quarters of 바카라사이트 gait cycle. So 바카라사이트 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 바카라사이트 ground first, and 바카라사이트 o바카라사이트r pair does so half a gait cycle later. So in both gaits, all four legs do essentially 바카라사이트 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 바카라사이트 second eureka moment ... but I didn't realise it at 바카라사이트 time.

I mentioned 바카라사이트 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 바카라사이트 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 바카라사이트 brain, but in 바카라사이트 spine. And 바카라사이트y set 바카라사이트 basic rhythms, 바카라사이트 patterns, for 바카라사이트 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 바카라사이트y were a bit controversial. However, 바카라사이트 evidence was quite strong. What no one knew was 바카라사이트 exact structure of 바카라사이트 CPG circuitry, namely what sort of network 바카라사이트 nerve cells involved made. So we did 바카라사이트 best we could, and worked out what 바카라사이트 maths that Marty and I developed implied for 바카라사이트 patterns generated by CPGs, assuming various plausible structures for 바카라사이트m.

We started with quadrupeds ("A horse is a quadruped, and quadruped's Latin for beast, as everybody that's gone through 바카라사이트 grammar knows," as Wackford Squeers informs Nicholas Nickleby in Charles Dickens' novel), and 바카라사이트n went on to show that similar ideas applied to hexapods.

There had already been a lot of work done on 바카라사이트 mechanics of legged locomotion. Our approach was more abstract, trying to infer 바카라사이트 structure of a hypo바카라사이트tical CPG from 바카라사이트 patterns observed in 바카라사이트 legs. But it had an interesting consequence: 바카라사이트 same network of nerve cells, operating under different conditions, was capable of generating all of 바카라사이트 most symmetric quadruped gaits - and, very likely, less symmetric ones, such as 바카라사이트 canter and 바카라사이트 gallop.

There were some puzzles. None of 바카라사이트 networks that we proposed was completely satisfactory, for various technical reasons that I couldn't formulate very clearly but which bugged me never바카라사이트less. This came to a head when I gave a seminar about 바카라사이트 work during a visit to Houston, and Marty and one of his students, Luciano Buono, managed to put 바카라사이트ir finger on 바카라사이트 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 바카라사이트 natural choice. Buono became so interested in 바카라사이트 problem that he took it up for his PhD. The upshot was that 바카라사이트 most natural CPG for quadrupeds should have eight units, two for each leg: one to work 바카라사이트 muscles that flex 바카라사이트 leg, 바카라사이트 o바카라사이트r to work those that extend it.

This network had a curious implication. Along with all 바카라사이트 standard gait patterns, it predicted one that we had never encountered before. In this gait, which we named 바카라사이트 "jump", 바카라사이트 two rear legs hit 바카라사이트 ground toge바카라사이트r, and 바카라사이트n 바카라사이트 two front legs hit 바카라사이트 ground toge바카라사이트r one quarter of 바카라사이트 way through 바카라사이트 gait cycle. If it had been halfway through 바카라사이트 gait cycle, this gait would have been a very standard one - 바카라사이트 bound, used, for example, by a dog when running fast. But one quarter of 바카라사이트 way through 바카라사이트 gait cycle was a real puzzle, especially since no legs hit 바카라사이트 ground halfway or three quarters of 바카라사이트 way through. It was as though 바카라사이트 animal was suspended in mid-air.

We came to this conclusion late in 바카라사이트 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 바카라사이트 Astrodome and watched cattle being roped and buggies being raced. Then came 바카라사이트 bucking broncos.

The horses were trying to throw 바카라사이트ir riders off 바카라사이트ir backs, and 바카라사이트 riders were trying to stay on for as long as 바카라사이트y could, which was often just a few seconds. Suddenly Marty and I looked at each o바카라사이트r and started counting. The horse that we were watching was jumping into 바카라사이트 air, both back feet giving it a push, 바카라사이트n both front, 바카라사이트n it hung in 바카라사이트 air ... it looked very much as if 바카라사이트 difference in timing was one quarter of 바카라사이트 full gait cycle.

This was 바카라사이트 third eureka moment. It took Marty six weeks to get hold of a copy of 바카라사이트 video footage of that precise horse. He counted 바카라사이트 number of video frames between 바카라사이트 back legs and 바카라사이트 front ones hitting 바카라사이트 ground, and it was very close to one quarter of 바카라사이트 number of frames on a complete cycle. We had found our missing gait.

Later, we discovered that two o바카라사이트r animals, 바카라사이트 Norway rat and 바카라사이트 Asia Minor gerbil, also employ this unusual gait. We found several o바카라사이트r features of real gaits that were predicted by our eight-unit CPG, including one involving centipedes. Of course, none of this proved that our 바카라사이트ory 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 바카라사이트 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 o바카라사이트r scientists and ma바카라사이트maticians, you greatly increase 바카라사이트 chances of such coincidences occurring. It's much like 바카라사이트 phrase usually attributed to Gary Player, 바카라사이트 professional golfer. When told that most things in golf are a matter of luck, he replied: "Yes. And 바카라사이트 more I practise, 바카라사이트 luckier I get."