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#mathart

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#FractalFriday (It's already Saturday)

Elephant valley of the Mobius Mandelbrot set.

Formula: \(z_{n+1}=f(z_n^2+c,1)\)

where \(f(z, a)\) is defined as (C-like pseudocode):

Complex f(Complex z, Real a) {
Real dist = abs(re(z)) - a;
if (re(z) > a) {
re(z) = -a + dist;
im(z) = -im(z);
} else if (re(z) < -a) {
re(z) = a - dist;
im(z) = -im(z);
}
return z;
}

Coloring is done by using three very similar colormaps, mapped to triangle inequality average values.
The final color is produced by interpolating between these three colors using the so called "atom domains".

Generated using my GLSL Shadertoy fractal generator: shadertoy.com/view/33sSRf

Continued thread

And here is the same visualization style applied to two other map projections.

Equirectangular keeps the scale constant on the equator and meridians. But the squares close to the poles are stretched.

Gall-Peters preserves the relative areas. The poles are even more stretched.

While the Mercator projection fails to preserve the relative sizes, it does map squares to (almost) squares, which made the Mercator visualization look very nice.