You may well be aware of the diagrams of the Mandelbrot Set as they appear in many forms on many Fractal diagram computer programs and on a variety of websites. It looks like this:
You may also have met the iterative formula that produces the diagrams.
It is zr+1 = z r2 + c where z is a complex variable and c is a complex constant.
In fact I could find no answers to the question anywhere, so I had to produce my own software to find the answers myself.
I started with n = 3, 4, 5, etc and got the following results:
 |
 |
 |
| n = 3 | n = 4 | n = 5 |
|
 |
 |
| Fig (i) | Fig (ii) | Fig (iii) |
 |
 |
 |
| n = 2.1 | n = 2.2 | n = 2.5 |
Somehow the Mandelbrot Set with n = 2 seems to be 'evolving' into the diagram with n = 3 !
For these reasons I now refer to these sets as "Mandelbrot-Plus" Sets
I often think people visualise the Mandelbrot diagrams as a contour map of a country surrounding a lake, which is the set itself. The contours (colours) seem to represent interesting thing like mountains, but the surface of the lake is flat and uninteresting.
That was how I visualised it until I became interested in how the iterative process leads to a bounded sequence within the main Mandelbrot-Plus Sets. In other words is there a pattern to the points in the complex plane (orbits) as the sequence progresses?
This led me to look at the orbits corresponding to different points in the Mandelbrot Set and found that, for instance within the large 'circular sea' the sequence tends to a single point (a different point for each value of c), while in the left hand smaller 'bay' the sequences are attracted to 2 points, oscillating between them eventually, while in the top and bottom 'inlets' the attractor consists of 3 points...... and so on for the smaller and smaller inlets.
This can be seen in the following diagrams, where this time the colours inside the Mandelbrot-Plus Sets represent the number of points that form the attractor:
 |
 |
 |
| n = 2.0 | n = 3.0 | n = 2.0 (enlarged) |
In the righthand figure, which is an enlargement of part of the figure on the left, three points should be noted
To see this we need to look at orbits for points close to the boundary between two regions. for instance points on either side of the boundary between the red and yellow regions in the right-hand diagram above:
 |
 |
 |
| 2 positions | in RED region | in YELLOW region |
In this part of the red region the orbit consists of points cycling between positions on each of the three incoming paths, gradually working in towards their point of intersection, which is the single attractor.
As you get nearer to the boundary the process of convergence gets slower and slower until eventually it stops short of the point of intersection, and instead the process is attracted to three distinct points, one on each of the branches. This is what happens to points in the yellow region.
On the boundary the process would tend to a single point, but it would take an infinite number of iterations to reach it.
I am interested in what the actual boundaries between regions look like.
The yellow region above appears to 'overflow' into what one would expect to be red. This is because by the given number of iterations I used to stop the process the successive positions of the points in the orbit were moving inwards so slowly that they appeared to have settled to 3 distinct points. By increasing the number of iterations defining the process the boundary' recedes' towards the narrows joining the two areas of the Mandelbrot Set.
This leaves me with a fundamental and (for me) an unanswered question
I demonstrated most of this software at the ICME-7 congress in Quebec in 1992 and at ICME-8 in Seville in July 1996, as part of a group of teachers and educators from the UK in a workshop on "Chaos in Mathematics Education" and many people showed considerable interest in the subject. What I hope will now happen is that:
This allows you to test out for yourself the reasoning behind the production of the diagrams and encourage you to explore the concepts of Attractors etc:, by looking at the "contours" inside the Mandelbrot-Plus sets. It is this exploring for yourself which is the essence of learning!