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TIMESTAMP 08/11/2009. The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful 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.).
It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).
What's the formula of this thing?
There are a few subtle variations, which mostly end up producing the same kind of incredible detail. Listed below is one version. 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 can be 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.