3D Mandelbrot

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The mandelbulb

The 2D Mandelbrot set has been an icon since 1980. Extending it to a genuine third dimension turned out to be harder than it looked. The complex numbers don't extend cleanly to 3D — the next sensible algebra up is the quaternions, which live in 4D — so for thirty years there was no agreed answer to what a 3D Mandelbrot even was.

Daniel White and Paul Nylander cracked it in 2009 by sidestepping the algebra and using spherical coordinates instead. Square the angles, raise the radius to the eighth power, add the original point, iterate. The result is the mandelbulb — recognisably a Mandelbrot, recognisably three-dimensional, and absolutely covered in detail you'd never see from the outside.

The first renders took hours per frame and lived as still images on a fractal forum. There was no flying through one. This is the same fractal, the same formula, raymarched per pixel in your browser tab. Every pixel runs a signed-distance estimator, marches a ray through 3D space until it hits the surface, then shades the surface with two lights and a soft shadow. Sixty times a second.