If you do not know of the Mandelbrot set, I highly recommend you read my post on the subject first or any other article, before tackling this topic. That said, the Julia sets have a lot in common with the Mandelbrot set. For example, they both use the same iterative equation, *f(z) = z^2 + c*. For the Mandelbrot set, *z *is initially zero and *c *is the complex position of every point iterated. For a Julia set, *z *is initially the complex position of the point and *c *is the same randomly selected complex value for every point. This value determines which Julia set someone will get when iterating the equation. This is why they are called the Julia *sets*, because there is an infinite number of complex points that can be chosen for the value of *c*, each giving a unique Julia fractal. Here are some examples of Julia sets:

You may notice that no individual Julia fractal is as interesting, varying, or complex as the Mandelbrot set. This is because the Mandelbrot set can be thought of as a “map” of all of the Julia sets. The Julia fractal determined by a certain point *c* mimics the visual patterns and behavior of the Mandelbrot set at that point *c*. My analogy is that a Julia set is comparable to the “derivative” of the Mandelbrot set at the point where it was chosen. Just as a tangent line diverges from a curve, but is nonetheless equal to it at the point chosen, a Julia set diverges from the Mandelbrot set as it gets further away from *c*, but is equal to it at *c*. These images are excellent representations of this relationship:

The image above is split into a grid of Julia sets, whose point *c* is equal to its position in the image. This produces the familiar image of the Mandelbrot set since each Julia set reflects the characteristics of it at the point *c* it was chosen. This video gives a great explanation:

Further explanation of Julia sets can be found here and in the next post, I will be showing how one can make a Julia and Mandelbrot set explorer.