Chapter 7: Mechanisms in Programs and Nature

Section 4: Chaos Theory and Randomness from Initial Conditions

Spinning and tossing [as sources of randomness]

Starting with speed v, the speed of the ball at time t is simply v - a t, where a is the deceleration produced by friction. The ball thus stops at time v/a. The distance gone by the ball at a given time is x = v t - a t^2/2, and its orientation is Mod[x, 2 Pi r]. For dice and coins there are some additional detailed effects associated with the shapes of these objects and the way they bounce. (Polyhedral dice have become more common since Dungeons & Dragons became popular in the late 1970s.) Note that in practice a coin tossed in the air will typically turn over between ten and twenty times while a die rolled on a table will turn over a few tens of times. A coin spun on a table can rotate several hundred times before falling over and coming to rest.

From Stephen Wolfram: A New Kind of Science [citation]