Sequence Fractals Part IV #15

$ z^2+c*s_i$
s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
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 '*s_i' formula, almost all of these look like a circle. That makes some sense. The sequence s_i 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. |*s_i| = 1.0. If r > 0, |c| = r, then the adder at each step, +c*s_i has |c*s_i| = r. The c*s_i 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.