Cyclic Driven IFS

3-Cycle in a different order

What happens if we repeat the pattern

T₃ followed by T₁ followed by T₂

instead of

T₁ followed by T₂ followed by T₃?

The first few points are shown on the left. Click the picture for an animation. The right shows the limiting pattern. Does it look familiar? It should; it is the pattern generated by repeating T₁ followed by T₂ followed by T₃.

Click to animate.

To understand why this is so, write the first several terms of both sequences

123123123123123123123123123123...

312312312312312312312312312312...

The second sequence is just the first shifted two terms to the left. The first sequence is the same as the second, but starting from T₂T₁(0.5, 0.5) instead of from (0.5, 0.5). As with the fractals generated by regular IFS, here the final pattern does not depend on the starting point. We prefer to start with (0.5, 0.5) because this is the most neutral choice.

Can you find an order of cycling through 1,2, and 3 that converges to a different triple of points? Here is the answer.

Return to Cyclic IFS.