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Fractal patterns can be found beyond the second dimension, and when exploring the third dimension, you can see some really amazing patterns. The fundamental recursive patterns are the same, but some interesting work arounds have to be done in order to render efficiently and extend the complex plane.
Hypercomplex numbers
Hypercomplex numbers
Rather than using complex numbers(a + bi), hypercomplex, or sometimes called triplex numbers, are used instead. The actual math behind hypercomplex numbers can be really confusing, and deserves its own article just to explain them. If you are interested in how they work, check out the wikipedia page for them (https://en.wikipedia.org/wiki/Hypercomplex_number). For the sake of keeping this concise, all you need to know is that they are a way of representing Cartesian (x,y,z) coordinates as one term.
Spherical Coordinates
Spherical Coordinates
Another concept used when creating an algorithm to generate three dimensional fractals is spherical coordinates. While they aren't required, and sometimes can be slower, they make the algorithm much more concise and simple to understand. Spherical coordinates are the three dimensional way of describing a point in space adjacent to polar coordinates. If you ever took a trigonometry, you might be familiar with polar coordinates, but if you aren't, it's just another way to represent a 2d point, but instead of x,y, it uses radius and theta, or the angle. With spherical coordinates, another variable, phi, is added to represent the other rotation.
Paul Nylander and Daniel Whites Formula
Paul Nylander and Daniel Whites Formula
In 2007, a thread was started on fractal forums(link) in search of a 3D equivalent to the mandelbrot set. Before then, no true equivalent had been found that had the self-similar properties that are found in the Mandelbrot set. Many different iterations were tested before one formula was revealed that seemed different than the others. It used a geometric based aproach using the spherical coordinates described just before. From my research, it doesn't seem to have a particular proof, or direct relationship to the mandelbrot set. What does remain the same is that both formulas use a defined power(2 in the case of the mandelbrot set, and 8 in the mandelbulb set) on a point in space and adding the first term. The formula for the mandelbulb is below(as illustrated by Daniel White here: link)
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)
The results produced by this formula are truly breathtaking:
Wait... how did we go from a formula to an image??
Well, all of the images shown above use the formula described in conjunction with a rendering algorithm called raymarching. It uses some really clever techniques to create a purely math based approach to rendering, which is incredibly efficient.
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