Mandelbulbs are three-dimensional Mandelbrot fractals. They are created using mathematical renderings in a three-axis, artificially colored raytrace. The more you zoom in, the more lush detail is brought out. With the Mandelbulbs, you get an eerie, organic image that looks like a photograph of an alien world.
From Skytopia.com:
What's the formula of this thing?
Similar to the original 2D Mandelbrot , the 3D formula is defined by:
z -> z^n + c
...but where 'z' and 'c' are hypercomplex ('triplex') numbers, representing Cartesian x, y, and z coordinates. The exponentiation term is defined by:
{x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }
...where:
r = sqrt(x^2 + y^2 + z^2)
theta = atan2( sqrt(x^2+y^2), z )
phi = atan2(y,x)
And the addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by:
{x,y,z}+{a,b,c} = {x+a, y+b, z+c}
The rest of the algorithm is similar to the 2D Mandelbrot!
Here is some pseudo code of the above:
r = sqrt(x*x + y*y + z*z )
theta = atan2(sqrt(x*x + y*y) , z)
phi = atan2(y,x)
newx = r^n * sin(theta*n) * cos(phi*n)
newy = r^n * sin(theta*n) * sin(phi*n)
newz = r^n * cos(theta*n)
...where n is the order of the 3D Mandelbulb.
Skytopia has some absolutely amazing images from Mandelbulbs. Because the modeling software did not exist fully, he had to modify some open-source code to handle the Z coordinates. Fascinating.