Classic Iterated Function Systems

Sierpinski Gasket Chaos Games with Rotations

The usual Sierpinski gasket chaos game is based on moving half way towards one of three fixed points. We can modify the rules to incorporate rotations of 180° around one or more of the fixed points. In each game a point moves halfway toward a randomly chosen fixed point. If that fixed point is a rotation point, however, then the point also rotates 180° around that fixed point. That is, you flip around the fixed point to the exact opposite side.

Choose how many rotation fixed points to use and play the chaos game here!

How to Play. The small gray dot is the initial seed. Click the "Next 1" button to randomly select a fixed point. A new point will be plotted in the same color as the corresponding fixed point, with a line connecting the old point with the new point. Keep clicking "Next 1" to plot the orbit, one point at a time. To see the points generated more quickly (without the line!), click either the "Next 10" or "Next 1000" buttons to plot the next 10 or 1000 points, respectively. Click "Clear" to start over with a new initial seed.

You might notice a few spurious points that are clearly not part of the limiting fractal, especially if the seed point starts outside the bounding polygon. These are points from the beginning of the orbit. But once a point is inside the bounding polygon, all subsequent points will stay inside the polygon. It is the reason that many chaos game programs will compute the first 10 or 20 points in the orbit before starting to plot the subsequent orbit points. By that time the orbit will consist of points very close to the final fractal.