Sequence Fractals Part IV #15


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.

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