Month: August 2021

Sequence Fractals Part IV #16

Posted on by earl
359.jpg

z^2+c*s_i
s0=1.0, si+1=(-0.96+0.28i)*si
Center:0.121+0.419i; Zoom = 32

Here is a zoom into the top edge of the circle in the previous picture, Sequence Fractals Part IV #15.

I call these jigsaw puzzle fractals. They does not have the normal behavior where just a few points around the edge of the capture set escape each step, resulting in smooth colors and swirls. Instead large chunks escape on distinct iteration steps. The result looks like fitting together distinctly colored puzzle pieces.

Here are some earlier examples of “jigsaw puzzle” fractals: Sequence Fractals Part III #27, Sequence Fractals Part III #38

Sequence Fractals Part IV #15

Posted on by earl
358.jpg

z^2+c*s_i
s0=1.0, si+1=(-0.96+0.28i)*si
Center: 0+0i; Zoom = 0.5

Next up: Non-periodic sequences. This one is based on the 7-24-25 Pythagorean triple. The Pythagorean triples provide an easy way to find rational coordinates that generate an irrational angle. The sequence moves around the circle with radius 1.0 by a rotation of an irrational angle.

I used a similar sequence with z^2+c+s_i earlier, Sequence Fractals Part III #27 for example.

With the ‘*si‘ formula, almost all of these look like a circle. That makes some sense. The sequence si is dense in the unit circle. Informally that means it is everywhere. Formally, pick any point on the circle and your favorite small \epsilon and there is a point in the sequence within \epsilon of your point.

All c values (pixels) with a common radius will behave similarly. |*si| = 1.0. If r > 0, |c| = r, then the adder at each step, +c*si has |c*si| = r. The c*si are dense in the circle with radius r. For the most part, only |c| matters; all c’s on a given circle have the same eventual fate, capture or escape. It is only near this circular capture/escape border the angle of c makes a difference.

If the colors were adjusted they would look like the sun with solar flares.

Sequence Fractals Part IV #9

Posted on by earl
352.jpg

z^2+s_i*c
si = 1, 1.2650895i, (repeating)
Center:-0.035238+0.0.816426i , Zoom = 1600

Almost the same setup as yesterday’s picture Sequence Fractals Part IV #8, with a very small value change to the second value in the sequence.

My first attempt after changing the sequence value resulted in an almost totally white image. I had to increase the escape iteration count 10 fold to start seeing the details.

With the high resolution and large iteration count, these take over a day to render. I had to turn off anti-alias to have something ready to publish today.

Sequence Fractals Part IV #8

Posted on by earl
351.jpg

z^2+s_i*c
si = 1, 1.26509i, (repeating)
Center:-0.035238+0.0.816426i , Zoom = 1600

Zoom into the bridge near the center of yesterday’s picture, Sequence Fractals Part IV #7.

Note: Yesterday’s picture is a bit misleading. There appears to be a white bridge (points in the capture set) between the two bodies. Upon increasing the iteration count, I discovered that the two bodies were not actually connected. Most of the points in this area escape, but they take a very long time before deciding to leave.