# Random Walk

A seed particle is placed in the middle of the red circle. Then, a new
particle is started at a random position on the circle. For every time
step, it does a random move one step in any of four (von
Neumann neighbourhood) or eight (Moore neighbourhood) possible directions.
If at the new position there is a neighbouring cell (4/8 neighbours
considered), the particle stays there. Otherwise, it does the next step.
Particles which leave the circle are eliminated.

The right hand side of the graphic shows the fractal dimension **d**
calculated from the picture by taking into account circles of increasing
radius **R** around the center and counting the number of particles
**N** within each circle. Because **N~R**^{d}, the slope
of the double-logarithmic plot gives **d**. Note that **d** converges
to about 1.7 as the picture gets finished.

Peter Uelkes
Last modified: Sun Jun 23 10:43:25 CEST 2002