A basic explanation of fractals

      A plane consists of points P which will be either represented by their real Cartesian co-ordinates ( x , y ) or more usually by the complex number Z = x + iy. You do not need to understand complex numbers to enjoy fractals.
    The points in the plane are subjected to a transformation by means of a formula or procedure which converts any point P₀ into its image point P₁. This process is then repeated to convert P₁ into P₂ and so on, thus creating the sequence of points P₀ P₁ P₂ P₃ . . .   which is called the orbit of the initial point P₀ . Each conversion is called an iteration, so n iterations will get you as far as the point P _n .

              Julia sets
      The orbits themselves are sometimes the interest, but more often they are divided into two classes based on the long term behaviour of the orbit.
    Sometimes the orbit will eventually go further from the origin than any set distance and never return. Such orbits 'diverge to infinity' and the set of initial points for which this happens is the 'exterior'. The boundary of this exterior set is usually very complicated - in fact fractal.
    The points of this boundary make up the Julia set. The whole plane excluding the Julia set is known as the Fatou set. These are named after the mathematicians who first studied them.
    Sometimes the Julia set is just a disconnected cloud of points, known as 'Fatou dust'. It is this situation which defines the outside of the Mandelbrot set, see below. At other times the Julia set is one or more continuous fractal lines, usually enclosing an 'interior' set.

        Julia Fractals
      Sometimes just the Julia set is drawn, but usually the exterior is coloured in steps which show how many iterations n are needed before P _n is further from the origin than a specified maximum distance. This is the 'escape to infinity' fractal. The interior is then just left black or some other plain colour.
 

Note that the diagrams on this page do not down-load.

            Finite Attractors
      However, the interior set is also of interest. When the orbit remains finite, several possible behaviours are possible.
    Sometimes the orbit approaches a fixed point A, eventually always being within any specified close distance from A  - no matter how small. In this case A is an attractor, and the set of all points whose orbit is attracted to A is called the basin of A. In the third diagram above, the point A is highlighted and the basin of A fills the whole of the interior set. The colour is stepped according to the number of iterations required to come within a specified small distance from A.
    There can be more than one attractor, in which case each basin is coloured distinctively and stepped in shades of that colour by Genfract.
   Sometimes, instead of approaching a single point, the orbit eventually approaches an attractive cycle A₁,A₂,A₃, ... A_k. From any of these points, the orbit simply runs round the other points in the cycle before returning to the start to do it all again. This is called a periodic attractor of period k.   The interior set can also be coloured to show the number of iterations required to come close to the cycle.
 

This Julia set is just a cloud of disconnected points.

This one is connected and encloses an attractor of period 3, highlighted.

This is called a dendrite. It is connected but encloses nothing.

            Mandelbrot sets
      The transformation of the plane is usually effected by a formula applied to the complex number Z = (x,y). This formula often contains a complex constant C = (p,q), thus new Z = f ( Z , C ). You don't need to understand that, but the point is that different values of the constant will produce different Julias. These are divided into two types - those which are connected, and those which are not. If you plot the value of C as a point (p,q), you can mark the ones which give a connected Julia one way and the others another way. The first of these is a Mandelbrot set, and the others ( which give a Julia which is just a cloud of disconnected points ) is its exterior.
      For any particular value of the constant, it would not be possible to try all starting values Z₀ to see whether the orbit diverges for all of them except for a cloud of disconnected points. Luckily, it is only necessary to try one particular value for Z₀ - called the critical value. This is quite often zero, but is actually a value of Z for which the derivative of F ( Z ) is zero.
    So, the method is :-  For each value of  C = (p,q), calculate the orbit of the critical value for Z₀. If this diverges mark it as the exterior of the Mandelbrot set, otherwise mark as belonging to the Mandelbrot set.
    The exterior is usually coloured to show how many iterations are needed for the orbit to get beyond some set distance from the origin. Just outside the Mandelbrot boundary, the shapes are extraordinarily convoluted - and this is where you get the best enlarged pictures.
 

    Home page

    Example outputs

    Critical values and compound Mandelbrots

    An explanation of Volterra- Lotka fractals

    Description and examples of Volterra-Lotka fractals

    A short description of what Genfract can do

    Down-load Genfract  (version 3.1, zip 291 kb)

    e-mail :-brian.edwards@felicite-parmentier.freeserve.co.uk