3D Mandelbrot Set

3D Mandelbrot Set, based on Daniel White’s formula for squaring a 3D hypercomplex number

Volumetric, imax=24

This is my favorite hypercomplex fractal, based on Daniel White’s creative formula for squaring a 3D hypercomplex number by applying two consecutive rotations. I’m not sure how mathematically meaningful this is, but it is stunningly beautiful. Here is some Mathematica code demonstrating a simple way to render this 3D fractal as a depth map by slowly marching tiny cubes (voxels) toward the boundary:

(* runtime: 1 minute, increase n for higher resolution *)
n = 100; Norm[x_] := x.x; Square[{x_, y_, z_}] := If[x == y == 0, {-z^2, 0, 0}, Module[{a = 1 - z^2/(x^2 + y^2)}, {(x^2 - y^2)a, 2 x y a, -2 z Sqrt[x^2 + y^2]}]];
Mandelbrot3D[pc_] := Module[{p = {0,0, 0}, i = 0}, While[i < 24 && Norm[p] < 4, p = Square[p] + pc; i++]; i];
image = Table[z = 1.5; While[z >= -0.1 && Mandelbrot3D[{x, y, z}] < 24, z -= 3/n]; z, {y, -1.5, 1.5, 3/n}, {x, -2, 1, 3/n}];
ListDensityPlot[image, Mesh -> False, Frame -> False, PlotRange -> {-0.1, 1.5}]


Global Illumination, imax=24

I had to write my own isosurface ray-tracer in order to render these fractals. The above image was made to look like a cloud of smoke using a volumetric technique described by Krzysztof Marczak. The image on the left was made using James Kajiya’s path tracing method for Global Illumination (GI).

Global Illumination and Participating Media Links

Higher power variations of this fractal can be rendered based on the following formula:
{x,y,z}n = rn{cos(nθ)cos(nφ),sin(nθ)cos(nφ),-sin(nφ)}
r=sqrt(x2+y2+z2), θ=atan(y/x), φ=atan(z/sqrt(x2+y2))







Minibrot, imax=20

Mandelbrot zoom, imax=24


2 Responses to “3D Mandelbrot Set”

  1. 1 Shears
    August 29, 2009 at 3:47 pm

    I get error in Mathematica: SetDelayed::write: Tag Norm in Norm[x_] is Protected.

    • 2 nylander
      August 29, 2009 at 4:16 pm

      You are using a newer version of Mathematica that already has Norm defined. Just remove that line and it should work (unless there are other changes to the newer version of Mathematica that I don’t know about).

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