java - How do you create and render a 3D Fractal? -

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so i've been assigned recursive art project ap cs class , have bunch of spare time, i've decided try little bit more ambitious.

my plan create 3d fractal, either rendered , shaded in visualization gl, or represented via spatially mapping respective equations' outputs points on cube , drawing those. if explanation seems unclear, please check out links @ bottom images. now, don't need fractal able modified in-program. need render single bufferedimage, i'll putting directly on jframe.

my experience in java, far project goes, bit limited. i've drawn mandelbrot , julia set fractals before, i have little no experience drawing/rendering objects in 3d in java. stuff can , figure out myself though, no worries here.

thus, question: how 1 map fractal should in 2nd dimension (e.g. log(no. of subdivided entities)*log(side length of subdivision) = 2) to 3rd dimension (e.g. log(no. of subdivided entities)*log(side length of subdivision) = 3)? i'm lost trying mathematically work out, , believe there more organized approach go circumventing lot of math exists.

also, if aware of a structured approach render 2d fractal, drawn formula, , render in 3d, provided respective formula provided (power raised), please let me know. i've heard of ray tracers, no idea are, brief summary cool.

here links pictures of result want obtain:

http://2008.sub.blue/assets/0000/4575/power8_large.jpg

https://www.youtube.com/watch?v=rk8jhcvlctu

it looks image example of mandelbulb. similar iteration formula mandlebrot set using 3d points , novel idea of raising 3d point power means.


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