back left foot) hits the ground half a cycle later than the front left foot. This is a half-period phase shift.
We knew that synchrony and phase shifts are common in symmetric networks. In fact, we had worked out the only plausible symmetric network that could explain all of the standard gaits of four-legged animals. And we’d sort of assumed, because we couldn’t think of any other reason, that symmetry was also necessary for synchrony and phase shifts to occur.
Then Marty’s postdoc Marcus Pivato invented a very curious network that had synchrony and phase shifts but no symmetry. It had sixteen nodes, which synchronized in clusters of four, and each cluster was separated from one of the others by a phase shift of one quarter of a period. The network was almost symmetric at first sight, but when you looked closely you could see that the apparent symmetry was imperfect.
To us, Marcus’s example made absolutely no sense. But there was no question that his calculations were correct. We could check them, and we did, and they worked. But we were left with a nagging feeling that we didn’t really understand why they worked. They involved a kind of coincidence, which definitely happened, but “shouldn’t have.”
While Marty and Marcus worked on other topics, I worried about Marcus’s example. I went to Poland for a conference and to give some lectures, and for the wholeof that week I doodled networks on notepads. I doodled all the way from Warsaw to Krakow on the train, and two days later I doodled all the way back. I felt I was close to some kind of breakthrough, but I found it impossible to write down what it might be.
Tired and fed up, I abandoned the topic, shoved the doodles into a filing cabinet, and occupied my time elsewhere. Then one morning I woke up with a strange feeling that I should dig out the file and take another look at the doodles. Within minutes I had noticed that all the doodles that did what I wanted had a common feature, one that I’d totally missed when I was doodling them. Not only that; all of the doodles that didn’t do what I wanted lacked that feature. At that moment I “knew”
what the answer to the puzzle was, and I could even write it down symbolically. It was neat, tidy, and very simple.
The trouble with that kind of knowledge, as my biologist friend Jack Cohen often says, is that it feels just as certain when you’re wrong. There is no substitute for proof. But now, because I knew what to prove and had a fair idea of why it was true, that final stage didn’t take very long. It was blindingly obvious how to prove that the feature that I had observed in my doodles was sufficient to make happen everything I thought should happen. Proving that it was also necessary was trickier, but not greatly so. There were several relatively obvious lines of attack, and the second or third worked.
Problem solved.
This description fits Poincaré’s scenario so perfectly that I worry that I have embroidered the tale and rearranged it to make it fit. But I’m pretty sure that it really did happen the way I’ve just told you
What was the key insight? I’ve just looked through my notes from the Warsaw–Krakow train, and they are full of networks whose nodes have been colored. Red, blue, green . . . At some stage I had decided to color the nodes so that synchronous nodes got the same color. Using the colors, I could spot hidden regularities in the networks, and those regularities were what made Marcus’s example work. The regularities weren’t symmetries, not in the technical sense used by mathematicians, but they had a similar effect.
Why had I been coloring the networks? Because the colors made it easy to pick out the synchronous clusters. I had colored dozens of networks and never noticed what the colors were trying to tell me. The answer had been staring me in the face. But only when I stopped working on the problem did my subconscious have the freedom to sort it out.
It took a week or
We knew that synchrony and phase shifts are common in symmetric networks. In fact, we had worked out the only plausible symmetric network that could explain all of the standard gaits of four-legged animals. And we’d sort of assumed, because we couldn’t think of any other reason, that symmetry was also necessary for synchrony and phase shifts to occur.
Then Marty’s postdoc Marcus Pivato invented a very curious network that had synchrony and phase shifts but no symmetry. It had sixteen nodes, which synchronized in clusters of four, and each cluster was separated from one of the others by a phase shift of one quarter of a period. The network was almost symmetric at first sight, but when you looked closely you could see that the apparent symmetry was imperfect.
To us, Marcus’s example made absolutely no sense. But there was no question that his calculations were correct. We could check them, and we did, and they worked. But we were left with a nagging feeling that we didn’t really understand why they worked. They involved a kind of coincidence, which definitely happened, but “shouldn’t have.”
While Marty and Marcus worked on other topics, I worried about Marcus’s example. I went to Poland for a conference and to give some lectures, and for the wholeof that week I doodled networks on notepads. I doodled all the way from Warsaw to Krakow on the train, and two days later I doodled all the way back. I felt I was close to some kind of breakthrough, but I found it impossible to write down what it might be.
Tired and fed up, I abandoned the topic, shoved the doodles into a filing cabinet, and occupied my time elsewhere. Then one morning I woke up with a strange feeling that I should dig out the file and take another look at the doodles. Within minutes I had noticed that all the doodles that did what I wanted had a common feature, one that I’d totally missed when I was doodling them. Not only that; all of the doodles that didn’t do what I wanted lacked that feature. At that moment I “knew”
what the answer to the puzzle was, and I could even write it down symbolically. It was neat, tidy, and very simple.
The trouble with that kind of knowledge, as my biologist friend Jack Cohen often says, is that it feels just as certain when you’re wrong. There is no substitute for proof. But now, because I knew what to prove and had a fair idea of why it was true, that final stage didn’t take very long. It was blindingly obvious how to prove that the feature that I had observed in my doodles was sufficient to make happen everything I thought should happen. Proving that it was also necessary was trickier, but not greatly so. There were several relatively obvious lines of attack, and the second or third worked.
Problem solved.
This description fits Poincaré’s scenario so perfectly that I worry that I have embroidered the tale and rearranged it to make it fit. But I’m pretty sure that it really did happen the way I’ve just told you
What was the key insight? I’ve just looked through my notes from the Warsaw–Krakow train, and they are full of networks whose nodes have been colored. Red, blue, green . . . At some stage I had decided to color the nodes so that synchronous nodes got the same color. Using the colors, I could spot hidden regularities in the networks, and those regularities were what made Marcus’s example work. The regularities weren’t symmetries, not in the technical sense used by mathematicians, but they had a similar effect.
Why had I been coloring the networks? Because the colors made it easy to pick out the synchronous clusters. I had colored dozens of networks and never noticed what the colors were trying to tell me. The answer had been staring me in the face. But only when I stopped working on the problem did my subconscious have the freedom to sort it out.
It took a week or
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