Sequence Fractals Part III #21


s0=1.0, si+1=0.5*si
Center:0.0+0.0i; Zoom =0.5

Yesterday’s sequence reset after 30 steps. Without a reset, arithmetic sequences head off to infinity. Fractal pictures with them just have some ghostly shadows, no chaos. The first few orbit steps for small numbers bounce around, but soon everything heads off to infinity. Only light shadows from the early iteration remain.

The same holds for geometric sequences that go to infinity. But not all geometric sequences blow up. Here is one that converges to 0. Usually the fractals built on these sequences, such as this one, have a large capture set, and no need to reset after a few steps.

Sequence Fractals Part III #20


s0=0.0, si+1=si+0.1. Reset after 30 steps.
Center:0.0+0.0i; Zoom =0.5

At 30 steps between resets the white capture set is gone.

I found some cracks that host minis, similar to yesterday’s post. Sequence Fractals Part III #19 Much smaller than yesterday’s, but after adjusting the scale they were so similar they do not deserve a separate post.

Sequence Fractals Part III #15


s0=0.0, si+1=si+0.1. Reset after 20 steps.
Center:0.0+0.0i; Zoom = 0.5

Change the sequence to reset to after ever 20 steps.

The capture set is very small. That is not surprising. Each step we are taking the usual z2+c iteration and kicking it a little to the right. On every 20th step the orbit gets pushed 1.9 units to the right. It may be more surprising that anything remains in the capture set.