
Mandelbrot Set
The Mandelbrot Set, named after Benoit B. Mandelbrot who discovered it, is the set of points on the complex plane that remains bounded to the set when an iterative function is applied to them. The iterative function in question is a nonlinear relation where,
z_{n+1} = z_{n}^{2} + c
The set is built by iterating the function for each point on the complex plane and setting z_{0} = 0 and c to the complex point being considered. Since c is a complex point it will be in the form x + yi.
If, through the iterative process, z_{n} tends to infinity then we say that c is not part of the Mandelbrot set, otherwise we say that c is part of the Mandelbrot set. To help us decide how to classify the points we use a well known result that if z_{n} of z_{n} (i.e. the distance of z_{n} from the origin) is greater than 2 then the iterative process will tend to infinity.
Of particular interest to us is the graphical representation of the Mandelbrot set, where we colour each point depending on where its attractor lies. Points which remain bounded within the Mandelbrot set are coloured white, while points which break away and tend to infinity are coloured depending on the number of iterations required to break away from the Mandelbrot set (i.e. z_{n} > 2). The graphical representation of the Mandelbrot set thus segments the complex plane into two regions  the Mandelbrot set and the rest of the complex plane. The two regions are separated by an infinitely crumpled boundary (i.e. zooming in on the boundary of the regions still shows a crumpled boundary) which shows us how "chaos and order cooperate to create intricately beautiful images with selfreplicating patterns". Mandelbrot referred to this crumpled boundary as having a fractal structure similar to that seen in natural boundaries such as coastlines.
An interesting replicating pattern that occurs on the boundary between the two regions is that of the image of Mandelbrot set itself. In other words, zooming in at particular locations of the Mandelbrot set will show miniature copies of the Mandelbrot set, and rezooming on the copy to the analogous location will reveal another copy of the set.

Julia Curves
During the First World War, two French mathematicians, Gaston Julia and Pierre Fatou, had already studied such boundaries with a fractal structure. Although Julia and Fatou managed to sketch some of their output graphically, they could not achieve the detail that Mandelbrot (and other people after him) obtained using computers. Known as Julia Curves, Julia and Fatou described the behaviour of an iterative function on the complex plane. The iterative function turns out to be the same as the one used in the generation of the Mandelbrot set, however c and z_{0} are initialised at different values. In generating the Mandelbrot set c is varied depending on the point that is being iterated, while in generating a Julia Curve v is kept constant and z_{0} is varied depending on the point being iterated. Thus the Julia Curve is parameterised by c, and for every c on the complex plane we get the Julia Curve K_{c}. Collectively, all the Julia Curves K_{c} are known as the Julia Set.
The graphical representation of the Julia Curve is obtained in a similar way to which the Mandelbrot Set is obtained. If z_{n} of z_{n} (i.e. the distance of z_{n} from the origin) is greater than 2 then the iterative process will tend to infinity. Points which are bounded within the 2 region are coloured white, while points which break away from this region are coloured depending on the number of iterations required to break away.

Relationships between the Mandelbrot Set and the Julia Curves
Since the Julia Sets are parametrised by a complex point c, we can say that with one image Mandelbrot managed to create a catalogue of all the Julia sets which indexes and characterises all the Julia sets. Of particular interest to us is the existence of characteristics that can be portrayed graphically to the user. One of these characteristics is defined in a theorem by Fatou and Julia which states that for a point c within the Mandelbrot Set (i.e. c does not escape to infinity under the iterative function) the corresponding Julia set K_{c} is said to be connected. The converse applies, so that if a point c lies outside of the Mandelbrot Set then the corresponding Julia Curve K_{c} is disconnected and we obtain what is known as a Cantor Set.
A second theorem by Tan Lei discusses the similarity that exists between the Mandelbrot Set M and the Julia Curve K_{c} both zoomed in at point c. As explained in The Beauty of Fractals: Images of Complex Dynamical Systems,
If you look at M with a microscope focused at c, what you see resembles very much what you see if you look at the Julia set K_{c} with the same microscope still focused at c, and the resemblance tends to become perfect (except for a change of scale) when you increase the magnifying power.

Further Reading
The Mandelbrot Set and Julia Sets
The Mandelbrot and Julia sets Anatomy


