As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows.

The Julia set is one of the best known examples of a fractal. It is a structure with an infinite amount of fine detail: you can zoom in on the edge of the fractal forever, and it will continue to reveal ever-smaller details.

The Julia sets, defined by the equation (\ref{julia}), can take all kinds of shapes, and a small change in $c$ can change the Julia set very greatly. In 1979, with the help of computer, B. B. Mandelbrot studied the Julia sets and tried to classify all the possible shapes and came up with a new shape: the Mandelbrot Set.

Apr 12, 2025 · The "filled-in" Julia set is the set of points which do not approach infinity after is repeatedly applied (corresponding to a strange attractor). The true Julia set is the boundary of the filled-in set (the set of "exceptional points").

May 28, 2023 · The set we obtain with this equation is known as the Julia set. In fact, there is a different Julia set for almost every \(c\). Similarly as we did for the Mandelbrot set, we obtain a sequence of complex numbers \(z_n\) with \(n=0,1,2,\ldots\).

A Julia (named after G. Julia) set is the boundary of the sets of unbounded and bounded iterates of the family of functions \[f_a(x) = x^2 + a \tag{1} \label{eq1}\] where $a$ is fixed and $x_0$ varies about the complex plane $x + yi$.

Julia sets are certain fractal sets in the complex plane that arise from the dynamics of complex polynomials. These notes give a brief introduction to Julia sets and explore some of their basic properties. 1. The Filled Julia Set. Consider a polynomial map f : C ! C, such as f(z) = z2 1. What are the dynamics of such a map?

The boundary of the set of points that do NOT escape to infinity is called the Julia set of the function f. Julia set may be a continuous curve, or totally disconnected set, depending on the value of c.

The Julia sets, defined by the equation (\ref{julia}), can take all kinds of shapes, and a small change in $c$ can change the Julia set very greatly. In 1979, with the help of computer, B. B. Mandelbrot studied the Julia sets and tried to classify all the possible shapes and came up with a new shape: the Mandelbrot Set.

Here are six Julia sets and their corresponding locations in the Mandelbrot set: Rendering Fractals. Of course we can zoom around and explore the fractal details of both Julia and Mandelbrot sets. There are also many ways to render and colorize these fractals to give them more aesthetically interesting looks.