s_{0}=1.0, s_{i+1}=-0.5*s_{i}

Center-1.416536+-1.07538i; Zoom =8384

The first of two zooms.

The zooms are not much different from zooms for the quadratic case. As s_{i+1} approaches 0, the iteration step get close to the usual iteration.

s_{0}=1.0, s_{i+1}=-0.5*s_{i}

Center-1.416536+-1.07538i; Zoom =8384

The first of two zooms.

The zooms are not much different from zooms for the quadratic case. As s_{i+1} approaches 0, the iteration step get close to the usual iteration.

s_{0}=1.0, s_{i+1}=0.5*s_{i}

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.

s_{0}=0.0, s_{i+1}=s_{i}+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.

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 25 steps.

Center:-1.2101-0.41833i; Zoom = 32

Found a mini in a crack.

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 25 steps.

Center:0.0+0.0i; Zoom =0.5

At 25 steps between resets there is a small capture area in the nose.

One zoom coming up for this one, a peek at a brown spot in the wings.

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 20 steps.

Center:-0.98714+0.296305i; Zoom = 2048

Another zoom. The color entropy dial is turned down a little.

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 20 steps.

Center:-1.92826+0.00031i; Zoom = 32768

Near the stable areas there is interesting chaos. Here is the first of two zooms

s_{0}=0.0, s_{i+1}=s_{i}+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 z^{2}+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.

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 15 steps.

Center:-1.854112+0.001739i; Zoom = 13421

Third zoom of Sequence Fractals Part III #11. Close to yesterday’s image Sequence Fractals Part III #13, a little above the real axis

s_{0}=0.0, s_{i+1}=s_{i}+0.1. Reset after 15 steps.

Center:-1.8625+0.0i; Zoom = 52

Second of three zooms into Sequence Fractals Part III #11

This one is near the tip of the Pinocchio nose.