I had a slinky as a child and loved it. Initially because if you stretched it out a little bit and peered inside, it looked similar to this (alas no Tom Baker would appear at the other end). And then because it walked down stairs. All by itself. How did it do that? In that respect, a slinky is a fantastic tool for understanding physics. What is going on in a slinky that makes it able to walk down stairs or in the above instance a treadmill? Or perhaps a simpler question: what happens if you hold a slinky up so that its coils stretch down to just above the ground and then drop it?

Understanding that can lead onto quite sophisticated concepts and theory. My favourite part of the treadmill video is this bit. The slinky is walking towards the edge. It’s going to fall off. No, wait, it turns around and starts walking the other way! And then it seems to do the same trick the other side. And then the other side, again and again. Not only is it walking, but it’s controlling itself in order to remain on the treadmill. OK, I shouldn’t get carried away. Such teleological language isn’t appropriate.  But there is this feeling of agency on the part of the slinky. It’s ‘trying’ to stay on the treadmill.

The other day I posted about complexity. Complex systems almost by definition exhibit emergent behaviour. Emergence, much like complexity is a slippery concept. If you’re not careful, emergence can simply mean ‘something happens that you didn’t expect’. While there is this surprising element of an emergent system (who would have thought thousands of starlings would produce such complex, interacting shapes?), a deeper notion of emergence involves an appreciation that our understanding of a system on one level may not give us much information about what is going on in other levels.

Is the self-regulation of the slinky on the treadmill an emergent property? It is in the sense that the slinky wasn’t designed to do that. No one thought up and then created the slinky in order to change direction on a treadmill. But more importantly the self-regulation can only be understood by looking at not just the slinky, but how it interacts with its environment. You could understand almost all there is to understand about how a slinky walks down a stair, but still fail to predict its self-regulating walk along a treadmill.

So how does it do it? Perhaps you realised it as soon as you saw it (It took me a few rewinds of the video), but the slinky’s trick is to take advantage of the boundary between the continually moving treadmill track and the stationary edge. There is something of a raised edge between either side of the track that is keeping it within bounds. When a ‘foot’ of the slinky lands on the edge, it tends to fall back onto the treadmill track. But there is another quite subtle element. Go back to the 9 second mark. As the slinky walks towards the nearside edge it eventually walks off of the track so that one foot is half-on/half-off. This foot is not travelling as fast as the other foot that is left behind. The foot that is still on the track travelled further to the right ‘downstream’. This means the next step will be more inline with the stationary edge (the angle subtended by the two feet and the stationary track will be more actue). The more the slinky lands off the moving track, the more it will move back to a path that is directly downwards. But at one point it makes quite a sharp right turn and heads off to the other side. What happened there?

Again, it was taking advantage of the difference between the moving track and stationary edge. One foot was half-on/half-off. The other foot stepped over and landed with more of itself on the moving track. So it moved faster than the foot that was behind it. This then increased the angle made by the feet and the stationary edge and so the slinky moved back towards the centre of the track. Such a simple system (a coiled spring and a moving walkway), such sophisticated behaviour. I’ve given an explanation, but I can’t say I really understand what’s going on. Maybe you have a better account?

The study of dynamical systems often involves understanding how these slight differences can produce profound changes. Take a Lorenz Attractor. It emerges from a very simple mathematical model – the Lorenz system only consists of three equations. But these three equations are coupled – they interact with each other. Sometimes changing the initial conditions and running the model forward doesn’t seem to have much of an effect. Sometimes a tiny change can lead to radically different behaviour.

It’s not so much that the danger is in the detail but that sometimes it’s the seemingly unimportant or irrelevant features or interactions in a system that are critical to its behaviour. How do we know what the important interactions are? Quite often we don’t so we explore the behaviour of the system with simulation and analysis. Little insights that are hard won. I must admit that sometimes I do wonder if the effort is worth it. But every now and them I’m rewarded with a better understanding of something. That’s a good feeling. Makes me want to strut my stuff. Move over slinky.

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