The church of St Mary Redcliffe is a stone’s throw from my new home in Bristol, and with its soaring gothic architecture is well worth including on any visit to the city. But for a mathematical tourist there’s an extra treat: the ‘chaotic pendulum’ in the north transept.
The pendulum could be considered the symbol of predictability and regularity, serving as the essential component of timekeeping devices for hundreds of years. Yet simple modifications yield devices that instead exhibit chaotic behaviour, the typical example being the double pendulum, where one pendulum is attached to the end of another. The chaotic pendulum at St Mary’s is believed to be unique in design: water is continually pumped into the crossbeam, causing it to tip one way or the other to let it flow out again; but it’s not generally possible to predict which way it’ll go next!
It’s important to note that this isn’t because the motion is random: there are rules governing its behaviour, and if you could return the chaotic pendulum to exactly the same state as it was at the start of the video, then it’d do exactly the same again. However, in chaotic systems tiny changes can be amplified into disproportionate effects, whilst large changes might have no impact. This leads to what is known as sensitive dependence on initial conditions. Think of catching a train, where if you’re a second early then you arrive at your destination on time, but if you’re a second late you’ll have to wait for the next one, so your two second delay grows into half an hour. On the other hand, whether you arrive at the station ten minutes or ten seconds early makes no difference to when you arrive at the other end. Thus your arrival time at the destination depends on your arrival time at the station in a highly non-linear way.
So if your attempt to reset the chaotic pendulum were even slightly flawed, then the video would be useless for making predictions. Similarly, even if you could build a completely accurate mathematical model, any readings taken from the real pendulum would contain errors, and so after a while the predictions of the model would diverge from the real behaviour. Fortunately for a pure mathematician like myself it’s more interesting to build that model, capturing the essence of all chaotic pendulums, than to be able to predict the activities of a single example!
The Clifton scientific trust has this to say about the lessons that can be drawn from the chaotic pendulum:
Some people look to science for certainties on which to base their lives. Increasingly we realise our knowledge can never provide certainty, even for this simple machine. The world is a more wonderful and a more surprising place than we could have imagined.
I’d agree with their conclusion, but not for their reasons… A truly random world, for example, would be constantly surprising. What’s wonderful about ours is that seemingly complicated phenomena can be described by simple rules – complexity can emerge from the interaction of otherwise easy-to-understand parts, rather than requiring some irreducibly complex explanation (such as a creator). Whilst the existence of chaos might restrict our ability to predict, it shouldn’t discourage us from trying to explain.