At the moment, we roll a die to select which anchor point we move towards. We may end up rolling the same anchor point over and over again.

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So let's add the constraint:

  • If we roll the same anchor point as we had on the last iteration, we roll again until we select a different one.
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I've labelled this the 'no repeat' constraint, as no anchor point is repeated.

When playing the chaos game with this constraint and four anchor points, we get a remarkable fractal structure.

Chaos game played over 10,000,000 iterations with four anchor points, going halfway towards a random anchor at every iteration but with constraint that the same anchor point is different at every iteration.

Five anchor points also produces an especially beautiful image.

Chaos game played over 10,000,000 iterations with five anchor points, going halfway towards a random anchor at every iteration but with constraint that the same anchor point is different at every iteration.

There are other constraints we could explore too.

Expanding on the 'no repeat' constraint, we can introduce another rule

  • If we roll the same anchor point as we had on the last iteration or any nearest neighbours , we roll again until we select a different one.
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This constraint only works if there are five or more anchor points. If there are three anchor points, then every anchor point is of equal distance from eachother. If there are four, then the game will just ping pong between two corners.

With five anchor points, we get another remarkably different fractal structure

Chaos game played over 10,000,000 iterations with five anchor points, going halfway towards a random anchor at every iteration but with constraint that we cannot repeat the same anchor point or either nearest neighbour

We can also explore a constraint that is in the inverse of this.

  • If we roll the same anchor point as we had on the last iteration or any furthest away anchors , we roll again until we select a different one.
Chaos game played over 10,000,000 iterations with five anchor points, going halfway towards a random anchor at every iteration but with constraint that we cannot repeat the same anchor point or any anchor point that is the furthest away.
Below is a table of all the polygons I've run with all these different constraints.
Triangle Square Pentagon Hexagon Heptagon Octagon Nonagon Decagon
Normal
No Repeat
No Repeat Nearest
No Repeat Furthest