wavy iteration bands that just touch each other, for hybrid quadratic Julia set where the c value varies during iteration (e.g. looping several values).

calculation inner loop snippet:

```
vec2 C1 = C;
C = c[iteration % count];
vec2 origin = C1 + vec2(100.0, 0.0);
float o = length(origin);
float e = (1.0 + 2.0 * o) / 2.0;
float r = sqrt(e * e - o * o + o) + e;
if (length(z - origin) > r) break; // escaped
z = cSqr(z) + C;
```

#JuliaSet #hybrid #EscapeTime #fractals #MathArt #CreativeCoding #GLSL #ComplexDynamics #maths

don't ask me how I derived r, I forgot to save my workings from a couple of years ago

tune 100+0i to taste (may break when small)

inspired by tavis' deep field #nebulabrot #DeepZoom images on #fractal #fractals forums, I did a little shader that for each c in the complement of the #MandelbrotSet M, colours according to how often z <- z^2 + c hits a given small target disc , weighted by derivative (as a proxy for point density).

it looks as though the hit sources are distributed everywhere near the boundary of M, which i think i can prove for target discs outside a sufficiently large esape circle, but i'm not sure how for discs nearer M. intuitively, by the time any cell pair in binary decomposition of exterior escapes, it covers an annulus with radii R, R^2, so any disc outside R will be hit by some region in every cell pair.

#math #maths #proof #ComplexDynamics

I’ve been fascinated by the #MandelbrotSet and #Fractals since my college days 😍:

“The Quest To Decode The Mandelbrot Set, Math’s Famed Fractal”, Quanta (https://www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-famed-fractal-20240126/).

On HN: https://news.ycombinator.com/item?id=39145513

#JuliaSet #ComplexDynamics #Iterators #Mathematics #Maths

Figured out how to plot wakes implicitly.

Given a wake with parameter ray angles $s_-, s_+$, for each pixel $c$ in the image trace the dynamic rays at those angles towards the Julia set: if and only if they land together, then $c$ is in the wake.

An application of Theorem 2.5 from https://arxiv.org/abs/1709.09869 "A survey on MLC, Rigidity and related topics" by Anna Miriam Benini.

Previously I had been tracing the two parameter rays into a polygonal boundary and filling that using rasterization, to do: benchmark and compare the two methods in various scenarios.

#MandelbrotSet #JuliaSet #fractal #maths #ComplexDynamics #ExternalRays