The original Mandelbrot is an amazing object that has captured the public’s imagination for 30 years with its cascading patterns and hypnotically colorful detail. It’s known as a ‘fractal’ - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you ‘zoom’ into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).

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. Use n=8 to find the exact object in this article.