MANDELBROT PLOT

Probably the most famous fractal of all is that generated at the edge of the Mandelbrot set, a unique, almost organic series of shapes that plummets, unresolved, to the realms of the infinitely small.

It is precisely this uncertainty at the boundary that provides the fascination of the Mandelbrot set. It somehow defies common sense to believe that no matter how much you magnify the edge (by examining ever smaller areas) it will never resolve into a definite boundary, it will always be possible to magnify the resulting tiny areas still further and produce new, equally complex, patterns.

In fact the limits are imposed not by nature but by the minimum size of number with which our computers can work, and the speed with which they can perform the necessary calculations.

The basic formula used to produce these numbers is extremely simple. The reason it takes so long is that it has to be iterated over and over again to find out whether or not the number belongs to the Mandelbrot set or lies outside it.

For some numbers this takes only a few iterations, for others it can take hundreds of millions of iterations. Since it's necessary to limit the number of iterations performed on any particular point to some reasonable limit (in order to get a result while you're still alive to see it), you can only ever achieve an approximation of the true shapes; however, this is still extremely pretty.

The formula used by Mandelbrot was simple; it's z = z*z + c which is iterated a certain number of times. If z becomes infinitely large then it isn't a member of the set, whereas if it stays finite then it is. In practise it is found that if z becomes greater than two then it will probably become infinite, so normally z <= 2 (less than or equal to two) is the limit tested for.

It is near the edge of the set, where an enormous number of iterations are required to decide whether or not a particular point falls within the set that provide the uncertainty, and the patterns. In fact only by performing an infinite number of iterations can it be decided if certain points belong in the set, and clearly there isn't time for an infinite number of calculations so we can never finally define the edge.

Now, you've begun to suspect that there must be something strange about this little equation (z = z*z + c), and there is: both z and c are complex numbers. An example of a complex number is the square root of -1. Since any number multiplied by itself produces a positive number it's hard to write it any other way, and furthermore if you try to resolve the sum 3+(square root of -1) it's difficult to see how else to write it, and in fact you can't; a complex number thus has to be written in two parts.

In line with this idea the complex number z is treated as a + jb, where j is the strange part (called the imaginary component).

By taking a as the x co-ordinate and b as the y co-ordinate we can plot a point on the screen, and to decide what we plot it's necessary to work out the value of z. If this value stays below 2 then we leave the location blank, otherwise we plot to it.

So how do we actually calculate this value? The answer is in geometry. The value we want is the length of a hypotenuse of a right angled triangle with sides real(z) and imaginary(z) so we can use Pythagoras' theorem to solve it: value (z) = square root(real(z*z) imaginary(z*z)). As you can see, this calculation is performed in line 260.

This program makes use of Lawrence Simons' plot and screensave routines. First you need to type in the Mandelbrot listing (call it something like mandle.bas) and then the other programs have to be merged with it.

The plot routine is in issue 23 (August 88) and as explained at the time you only need the lines from 50000 onward - don't include the rest or they will overwrite the Mandelbrot listing. The other routine required is also by Lawrence Simons (thanks again Lawrence - we love you) and appeared in issue 28 (January 89) with four missing lines appearing in issue 29 (February 89). You must only use the lines from 54000 onwards and merge it with the rest of the program.

If you don't have these issues, shame on you, contact the Old Barn at Somerton (see the contents page for the address).

Once you have the three parts of the program assembled - make sure you save a complete version before running it - simply type run and follow the prompts.

First of all you will be asked it you want to Save or Load. Type in S or L. Obviously you can't load any pictures until you've generated them so type S.

What you are asked for next are two numbers on the x axis and two on the y axis (remember, x is across and y isn't). The range of numbers you can use are shown on the picture of the complete set and the area thus defined is what will be displayed on the screen.

Once you've provided the numbers you'll be asked for the number of iterations; the bigger this number is the better the detail will be but the longer it will take to draw. The pictures you see here were generated overnight using 40 iterations.

The final piece of information is the name of the file the picture will be saved to. This allows you to build up a library of images of the Mandelbrot set which you can use to brighten up programs and amaze your friends.