Julia sets

Julia sets are created from the iteration of a 2D map, in this case the analytic map z^2 + c. For each pixel of the image there is a corresponding z value which is that pixel's location in the complex plane, or quaternion space. For each image, a c value is chosen and the map is iterated as follows: for each pixel, set z to the corresponding value for that pixel, and calculate z^2 + c. This new value replaces z for the next iteration. The calculation is done again and again, each time yielding a new z value, but always with the same c value. Eventually, the z value will either converge onto a periodic orbit or diverge to infinity. How long it takes to do this is measured by the number of iterations it takes to escape a circle of radius 2, and the number of iterations is used to color the pixel.


Like complex numbers but with an extra pair of imaginary axes, the quaternions are a further generalization of the complex field that gives up commutativity; formally it's a division ring. A complex number can be represented by a + b*i, a quaternion by a + b*i + c*j + d*k. Quaternions can be added, subtracted, multiplied, etc., and in particular, the formula z^2 + c can be iterated, but now z and c are 4-dimensional. As an example, the fractal called 368a.gif was created as a slice parallel to the r i plane (which is the same as the complex plane) but .4 in the j direction and 0 in the k direction. The constant c is (r, i, j, k) = (0.368, 0.1, 0.03, 0.01). The zoom frame is as follows: left -0.122, right -.05, bottom 0.37, top 0.53. For further info on quaternions, go to the library and read about them. :-)
Back to the Fractal page.