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~Rd, the slope of the double-logarithmic plot gives d. Note that d converges to about 1.7 as the picture gets finished.

A Java applet appears here.

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