What We Cannot Know: Explorations at the Edge of Knowledge. Marcus Sautoy du

region filled with one colour. There is complexity at all scales.

      A one-dimensional example of such a picture can be cooked up as follows. Draw a line of unit length and begin by colouring one half black and the other white. Then take half the line from the point 0.25 to 0.75 and flip it over. Now take the half in between that and flip it over again. If we keep doing this to infinity then the predicted behaviour around the point at 0.5 is extremely sensitive to small changes. There is no region containing the point 0.5 which has a single colour.

      There is a more elaborate version of this picture. Start again with a line of unit length. Now rub out the middle third of the line. You are left with two black lines with a white space in between. Now rub out the middle third of each of the two black lines. Now we have a black line of length ¹⁄₉, a white line of length ¹⁄₉, a black line of length ¹⁄₉, then the white line of length ¹⁄₃ that was rubbed out on the first round, and then a repeat of black–white–black.

      You may have guessed what I am going to do next. Each time rub out the middle third of all the black lines that you see. Do this to infinity. The resulting picture is called the Cantor set, after the German mathematician Georg Cantor, whom we will encounter in the last Edge, when I explore what we can know about infinity. Suppose this Cantor set was actually controlling the outcome of the pendulum in my desktop toy. If I move the pendulum along this line, I find that this picture predicts some very complicated behaviour in some regions.

      A rather strange calculation shows that the total length of the line that has been rubbed out is 1. But there are still black points left inside: ¹⁄₄ is a point that is never rubbed out, as is ³⁄₁₀. These black points, however, are not isolated. Take any region round a black point and you will always have infinitely many black and white points inside the region.

      What do the dynamics of my dice look like? Are they fractal and hence beyond my knowledge? My initial guess was that the dice would be chaotic. However, recent research has turned up a surprise.

       KNOWING MY DICE

      A Polish research team has recently analysed the throw of a dice mathematically, and by combining this with the use of high-speed cameras they have revealed that my dice may not be as chaotic and unpredictable as I first feared. The research group consists of father-and-son team Tomasz and Marcin Kapitaniak together with Jaroslaw Strzalko and Juliusz Grabski, and they are based in Lódź. In their paper in the journal Chaos, published in 2012, the team draw similar pictures to those for the magnetic pendulum, but the starting positions are more involved than just two coordinates because they have to give a description of the angle at which the cube is launched and also the speed. The dice will be predictable if for most points in this picture when I alter the starting conditions a little the dice ends up falling on the same side. I can think of the picture being coloured by six colours corresponding to the six sides of the dice. The picture is fractal if however much I zoom in on the shape I still see regions containing at least two colours. The dice is predictable if I don’t see this fractal quality.

      The model the Polish team considered assumes the dice is perfectly balanced like the dice I brought back from Vegas. Air resistance, it turns out, can be ignored as it has very little influence on the dice as it tumbles through the air. When the dice hits the table a certain proportion of the dice’s energy is dissipated, so that after sufficiently many bounces the dice has lost all kinetic energy and comes to rest.

      Friction on the table is also key, as the dice is likely to slide in the first few bounces but won’t slide in later bounces. However, the model explored by the Polish team assumed a frictionless surface as the dynamics get too complicated to handle when friction is present. So imagine throwing the dice onto an ice rink.

      I’d already written down equations based on Newton’s laws of motion for the dynamics of the dice as it flies through the air. In the hands of the Polish team they turn out not to be too complicated. It is the equations for the change in dynamics after the impact with the table that are pretty frightening, taking up ten lines of the paper they wrote.

      They discovered that if the amount of energy dissipated on impact with the table is quite high, the picture of the outcome of the dice does not have a fractal quality. This means that if one can settle the initial conditions with appropriate accuracy, the outcome of the throw of my dice is predictable and repeatable. This predictability implies that, more often than not, the dice will land on the face that was lowest when the dice is launched. A dice that is fair when static may actually be biased when one adds in its dynamics.

      But as the table becomes more rigid, resulting in less energy being dissipated and hence the dice bouncing more, I start to see a fractal quality emerging.

      Moving from (a) to (d), the table dissipates less energy, resulting in a more fractal quality for the outcome of the dice.

      This picture looks at varying two parameters: the height from which the dice is launched and variations in the angular velocity around one of the axes. The less energy that is dissipated on impact with the table, the more chaotic its resulting behaviour and the more it seems that the outcome of my dice recedes back into the hands of the gods.

       DOES GOD PLAY DICE?

      What of the challenge to define God as the things we cannot know? Chaos theory asserts that I cannot know the future of certain systems of equations because they are too sensitive to small inaccuracies. In the past gods weren’t supernatural intelligences living outside the system but were the rivers, the wind, the fire, the lava – things that could not be predicted or controlled. Things where chaos lay. Twentieth-century mathematics has revealed that these ancient gods are still with us. There are natural phenomena that will never be tamed and known. Chaos theory implies that our futures are often beyond knowledge because of their dependence on the fine-tuning of how things are set up in the present. Because we can never have complete knowledge of the present, chaos theory denies us access to the future. At least until that future becomes the present.

      That’s not to say that all futures are unknowable. Very often we are in regions which aren’t chaotic and small fluctuations have little effect on outcomes. This is why mathematics has been so powerful in helping us to predict and plan. Here we have knowledge of the future. But at other times we cannot have such control, and yet this unknown future will certainly impact on our lives at some point.

      Some religious commentators who know their science and who try to articulate a scientific explanation for how a supernatural intelligence could act in the world have intriguingly tried to use the gap that chaos provides as a space for this intelligence to affect the future.

      One of these religious scientists is the quantum physicist John Polkinghorne. Based at the University of Cambridge, Polkinghorne is a rare mind who combines both the rigours of a scientific education with years of training to be a Christian priest. I will be meeting Polkinghorne in person in the Third Edge when I explore the unknowability inherent in his own scientific field of quantum physics. But he has also been interested in the gap in knowledge that the mathematics of chaos theory provides as an opportunity for his God to influence the future course of humanity.

      Polkinghorne has proposed that it is via the indeterminacies implicit in chaos theory that a supernatural intelligence can still act without violating the laws of physics. Chaos theory says that we can never know the set-up precisely enough to be able to run deterministic equations, and hence there is room in Polkinghorne’s view for divine intervention, to tweak things to remain consistent with our partial knowledge but still influence outcomes.

      Polkinghorne is careful to stress that to use infinitesimal data to effect change requires a complete holistic top-down intervention. This is not a God in the detail but by necessity