Monday, May 02, 2011

Randomness and rules

It seems to me that “randomness” is characterised by three things:
1. A “mixed-upness” of results (for example, the digits 1, 2, 3, 4, 5, and 6 are all equally likely on a die).
2. Laws which constrain this “mixed-upness” (for example, that a die has no “memory,” that numbers are not “due to come up,” and that the sum of many dice is very close to normally distributed).
3. The fact that, apart from those laws, the results are unknown to us.
This random walk displays all three characteristics, as it wanders unpredictably across a 2-dimensional grid:

A random walk in 2 dimensions (image by “Zweistein”). Given enough time, a return to the start will almost certainly occur. This is not true in 3 dimensions.

Chaos theory is famous for displaying randomness constrained by rules:

Two random trajectories on the shape called the Lorenz attractor (image by “Hellisp”). Both trajectories trace out the same general shape, but the details of the two paths differ unpredictably.