Wandering in Space
In a random walk in three dimensions, a walker has six directions in which to move, chosen randomly, with each step. Instead of flipping a coin or rolling a tetrahedral die (see "Walking a Line" and "Walking on a Grid"), you can use a regular cubic die to decide your direction step by step.
Cubic dice have six sides.
The resulting paths, however, differ in important ways from those your see in one and two dimensions.
Unlike the case for one- and two-dimensional random walks, even after taking infinitely many steps, the chances of returning to the starting point is only about one in three. There's so much space available in three dimensions that a walker has more chance of wandering far afield than in one or two dimensions.
In fact, the mathematics of a three-dimensional random walk affords an important lesson for anyone who is lost in space (see "Galactic Gridlock"). Unless you happen to make it home again within your first few steps, you're likely to end up lost forever. There are simply too many ways to wander off.
Example of a three-dimensional random walk.
NEXT: Random Quivers
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