Pictures of Julia and Mandelbrot Sets/Foreword
About This Book
This Wikibook deals with the production of pictures of Julia and Mandelbrot sets. Julia sets and Mandelbrot sets are very well-defined concepts. The most natural way of colouring is by using the potential function, though it is not actually the method most usually used. The book explains how to make pictures that are completely faultless in this regard. All the necessary theory is explained, all the formulas are stated and some words are said about how to put the things into a computer program.
The subject is primary pictures of Julia and Mandelbrot sets in their "pure" form, that is, without artificial intervention in the formula or in the colouring. Exceptions to this rule are techniques such as field lines, landscapes and critical systems for non-complex functions, that appeal to artistic utilization.
The book should not contain theory that has nothing to do with the pictures. Nor should it contain mathematical proofs. For our purposes faultless pictures are enough evidence of correct formulas, because the slightest error in a formula generally leads to serious errors in the picture.
If you find that something in this book ought to be explained in more details, you can either develop it further yourselves or advertise for that on the discussion page.
If you add a new picture or replace an illustration by a new one, it should be of best possible quality and have a size of about 800 pixels. Draw it twice or four times as large and diminish it.
Julia and Mandelbrot sets
Julia sets depend on a Rational Function
If a complex rational function is entered in the computer program and submitted to a certain iterative procedure, you get a colouring of the plane called a Julia set (although it is the domain outside the Julia set that is coloured). However, in order to get a picture that has aesthetic value the function must have a certain nature. It must either be constructed in a specific way to ensure that the picture is interesting, or it must contain a parameter, a complex number, that can vary. Being able to vary a parameter increases our chances of finding an interesting Julia sets for some value of the parameter.
What do we mean by 'interesting'? Essentially that the iterative procedure behaves in a somewhat chaotic way. If the iterative procedure's behaviour at each point is easily predicted for a particular function by behaviour of nearby points, then the Julia set for that function is not very crinkly, and rather uninteresting.
A Mandelbrot set is an Atlas to the Related Julia sets
If we vary the parameter in our rational function we can produce a kind of 'map' of values that lead to interesting Julia sets. Values of the complex parameter correspond to points in the plane. The set of points that lead to interesting Julia sets gives us some information about the structure of the Julia sets for the parameter value at each point. Such a set is called a Mandelbrot set. The Mandelbrot set can be regarded as an atlas of the Julia sets.
The difference between the Mandelbrot set for the family and "its" Julia sets, is that the structure of the Mandelbrot set varies from locality to locality, while a Julia set is self-similar: the different localities are transformations of each other.
Sometimes you will prefer the more complex picture of the varying structure of the Mandelbrot set. Sometimes you will prefer the pure structure of the Julia set. Sometimes you will draw the Julia set because the drawing of the Mandelbrot set is slow for certain functions.