The complex Julia set is typically calculated in 2D with a constant complex seed value. However, if you consider the set of all possible Julia sets (all possible seed values), you get the complete 4D Julibrot set. The Mandelbrot set is contained as a 2D slice of the 4D Julibrot set. You can also take 3D slices of the 4D Julibrot set as shown here. The left picture shows what the Mandelbrot set looks like if you vary the real part of the starting value along the third dimension and the right picture shows what it looks like when you vary the imaginary part. Technically, this is not a hypercomplex fractal, but it’s still worth mentioning here.

Here is some Mathematica code:

`(* runtime: 18 seconds, increase n for higher resolution *)`

n = 100; Julibrot[z0_, zc_] := Module[{z = z0, i = 0}, While[i <12 && Abs[z] < 2, z = z^2 + zc; i++]; i];

image = Table[z = 1.5; While[z >= 0 && Julibrot[z, x + I y] < 12, z -= 3/n]; z, {y, -1.5, 1.5, 3/n}, {x, -2, 1, 3/n}];

ListDensityPlot[image, Mesh -> False, Frame -> False,PlotRange -> {0, 1.5}]

### Links

- Julia Set in Four Dimensions – good explanation, by Eric Baird
- Juliabrot Discussion – on Fractal Forums
- cross-section showing Mandelbrot set – by Ramiro Perez
- Alien Interface – animated Julibrot with orbit trapping, by David Makin

I think Fractint once said CLifford A. Pickover discovered an identical fractal to the Julibrot, which he called a “Chaotic Distance Repeller Tower,” making him the unknown Julibrot discoverer. Pickover should share the hono(u)r with Mark Petersen.