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left side, binary decomposition of every 3rd iteration before z->z^2+c (starting from 0) escapes a tiny radius (~0.21) (I think this radius is related to the derivative w.r.t. z of the periodic attractor somehow)

right side, binary decomposition of every 1th iteration before z escapes a larger radius (2) (this radius is the minimal escape radius for the quadratic Mandelbrot set)

the image is of the period 3 island, rotated 90 degrees from the usual view

wondering if this might possibly be useful for computing external angles: going to the cusp on the left is .(0) and on the right is .(011) (where 0 is light and 1 is dark); the other angles of the minibrot are tuned by .(011) and .(100) relative to the top level set... dunno if it's possible to get better than O(n^2) cost though...

· · Web · 1 · 0 · 0

works for distance estimates too, here are some nested units:

```
460 core

"TwoD.frag"

vec3 color(vec2 p, vec2 dx, vec2 dy)
{
Dual1cf c = dual1cf(complexf(p), 0);
Dual1cf z = dual1cf(complexf(0));
c.d[0] = mul(c.d[0], length(vec4(dx, dy)));
float Rr = 1000.0;
float R = 0;
float rr = 0.2117;
float r;
float sr = 0.08;
float s = 0;
for (int i = 0; i < 1000; ++i)
{
if (Rr <= length(z.x) && R == 0.0)
{
R = tanh(clamp(2.0 * length(z.x) * log(length(z.x)) / length(z.d[0]), 0.0, 4.0));
break;
}
else
if (i % 3 == 0 && rr <= length(z.x) && r == 0.0)
{
r = tanh(clamp(2.0 *length(z.x)/rr * log(length(z.x)/rr) / length(z.d[0]), 0.0, 4.0));
}
else
if (i % 6 == 0 && sr <= length(z.x) && s == 0.0)
{
s = tanh(clamp(2.0 * length(z.x)/sr * log(length(z.x)/sr) / length(z.d[0]), 0.0, 4.0));
}
z = add(sqr(z), c);
}
if (length(z.x) <= Rr) return vec3(0.0);
if (0.0 < s && s < 0.5 && r < 0.5) return vec3(1.0, 1.0, 0.0);
if (0.0 < r && r < 0.5 && R < 0.5) return vec3(1.0, 0.0, 0.0);
if (0.0 < R && R < 0.5) return vec3(0.0, 0.0, 1.0);
return vec3(1.0);
}
```

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