We can add more complexity to the rules of what we do when we roll the die.
To recap, the rules are currently:
If we roll red, we move halfway towards the red anchor point
If we roll green, we move halfway towards the green anchor point
If we roll blue, we move halfway towards the blue anchor point
We'll keep the first two rules the same, but we'll make a small addition to the last rule:
- If we roll blue, we move halfway towards the blue anchor point and we rotate the point about an angle θ around the blue anchor point.
This means that in 2/3 of the cases, we'll produce the same structure.
Doing this with three anchor points, the Sierpiński triangle is transformed into this structure:
Chaos game played over 10,000,000 iterations with three anchor points going half-way towards the anchor point, and rotating about an angle of 90 degrees if the blue point is selected.
Chaos game played over 10,000,000 iterations with three anchor points going half-way towards the anchor point, and rotating about an angle of 180 degrees if the blue point is selected.
Why is this happening? This fractal structure doesn't seem to resemble the Sierpiński triangle at all.
The behaviour is more obvious if we vary the angle smoothly from 0 to 360 degrees.
