2.8 One more roll of the dice

Returning to the start of this week’s topic: the ‘perfect’ die – is its roll chaotic? This question was explored in 2012 by a team of mathematicians in Poland, who used a high-speed camera to capture the trajectory of a die roll at a rate of 1500 frames a second. They found that, in addition to knowing the initial position and velocity of the die, the most important factors are the friction and bounciness of the landing surface. Air resistance on the other hand can be disregarded.

Energy is dissipated each time the die bounces. The number of bounces before the die comes to rest depends on the die’s interaction with the surface. On a high-friction rigid surface, the amount of energy dissipated on impact with the surface is low, and the die will tend to bounce around more. On a smooth, low-friction, or soft, surface, the amount of energy dissipated on impact with the surface is higher, resulting in fewer bounces. When a die is rolled on a surface like this, if the initial conditions can be established with sufficient accuracy, then it’s theoretically possible for the outcome of the throw to be predicted. But with more bounces, the outcome becomes more chaotic, and harder to predict.