The beautiful Mandelbrot set has become one of the glories of the computer age. But its incredible complexity does not so much reflect the capacity of Your microprocessor as the immense richness of the number system. To appreciate this You will have to study the aritmetics behind the set.

The Mandelbrot set is generated by calculating two simple formulas over and over again. Expressed in terms of ordinary cartesian coordinates it runs like this:

Choose a point in the plane, with coordinates P,Q say.

Calculate the coupled formulas:

Xn+1 = Xn*Xn-Yn*Yn+P

Yn+1 = 2*Xn*Yn+Q

starting with Xo = Yo = 0

(If You are familiar with the complex number system this looks much simpler: Zn+1 = Zn*Zn+C. That's why P and Q are called Real and Imaginary in the program box)

That is, You calculate the formulas and put the results back into them. The results Xn,Yn will form a series.
The Mandelbrot set is defined as all the points P,Q for wich the series Xn,Yn is finite; that is |Xn+Yn| < when n goes to .

That's all. The program Intelligent Mandelbrot will calculate the formulas for You. Whith its help You can study the interesting behaviour of the series Xn,Yn.

Inside the set the series will either converge to some point, or oscillate between several points.
Outside the set it will, per definition, explode into infinity, but it may take lots of iterations before that happens. You will see the series jump around in a seemingly random manner, and then, suddenly, disappear into cosmos.
At the border of the set things will become totally messy. The series will, in fact, be chaotic.

In the graphical representations of the Mandelbrot set the computer will give different colours to different points, according to how their corresponding series behave. More specifically, the number of iterations needed before the series starts running away to infinity will decide the colour of the pixel of the corresponding point.

Whith Intelligent Mandelbrot You can watch the numbers in the series move around. You will not see the beautiful colours. Instead You will be able to understand the mechanism behind the beauty!

The easiest way to use it is to follow some line parallell to one of the axes. That is, You keep either X or Y fixed, and move till You find the border of the set. You may then change direction and see what happens. It's really fascinating!

You should note that Intelligent Mandelbrot allows You to choose other starting conditions than Xo = Yo = 0. In fact, for any value of P;Q You may generate the set of all Xo,Yo for which the corresponding series are finite. These are the Julia sets, of which there are infinitely many. So now You have got Intelligent Julia for the same price...


Written by Lars Rosenberg