# How to use Mandel.exe?

This manual pertains to version 2.03 of the program.

Fractal images may be created by the general iteration: xn+1F(xn,c)
where F is a non-linear function of the complex numbers x and c. 'n' stands for the iteration number.
Complex numbers are two dimensional and are written for instance as x = a + ib, where a is the real part and b the imiginary part. i has the particular propery in that its square is -1. Hence i2 = -1. Many non-linear functions can produce fractals (see for instance "The beauty of fractals" by Peitgen and Richter, Springer Verlag, ISBN 3-540-15851-0). The most well-known fractal set is due to B. Mandelbrot. Mandelbrot fractals are obtained by iterating: xn+1xn2 + c, starting with x0 = 0 and varying the constant c systematically. To create the image shown in the top left corner of this page, the real part of c was varied between -2 and +1 and the imaginary part between -1.2 and 1.2. Note that the image is symmetrical relative to the real, horizontal, axis. For some starting values, c, the iteration converges to a single stable point (i.e. as for c = -0.6 - 0.16i in the figure below). This point is called an "attractor". For other values of c the iteration indefinitely oscillates between two or more values (as for c = -0.73 - 0.165i in the middle graphs). The values for c for which the latter two situations occur are shown in black in the top left image.
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It can be shown that x tends to infinity with increasing iterations once its absolute value becomes larger then 4. The iteration numbers, n, for which abs(x) gets larger than 4 are shown as different colours in the top left image. It takes only a single iteration if c = -2 + i (dark red) to reach abs(x)>4.It takes several more for other values of c.

When launching Mandel.exe a window containing the complete Mandelbrot set is shown. To zoom into the set, draw a rectangle on this image and push the "new" button (shown in a red circle in the figure below) or launch the File>New command from the menu.

One may successively zoom into the images until the limit of the floating point resolution (80 bit or 24 decimals) is reached.
Several options for the creation of images are available. Select Parameters from the Mandel menu to acces these. A dialogue box will then come up:

The maximum number of iterations (corresponding to black), and the image dimensions can be set. Set the "Colour bar" option if you wish to include the iteration colour coding in the image. The "Default" button resets all parameters to the default setting shown here. As said, the Mandelbrot set is created using the xn+1xn2 + c iteration. In this iteration x is squared (power = 2). It is possible to set the power to a value between 1 and 10 (not including 1, because the equation would not be non-linear anymore). Setting the power to a value other than 2 will slow down the calculations quite a bit as they become more intricate. Three examples with a power of 1.01, 4 and 10 are shown underneath:

To change the colour palette for subsequent images, choose Change Palette from the Mandel menu. The next dialogue box pops up:

To save or load a 256-colour palette, push the "save" or "load" buttons. A second window comes up requesting a "*.mpl" file name. Do not change the "*.mpl" extension. To revert to the program-defined default palette, push "default". To change the RGB (red, green, blue) values of one of the palette entries, click on a little coloured box in the palette. The selected box is highlighted by a white square around it and the box marked "old" now displays the selected colour along with its index number. Next move the Red, Green and Blue slide bars to change the colour. The new colour appears in the "new" box. To enter the new colour in the palette, push the "OK" button. To copy the colour from one index to another, carry out the following sequence:
1) Click the little box to be copied,
2) push the "from" button,
3) click the little box to copy to,
4) push the "to" button and finally
5) push the "Copy" button.
To create a gradient of colours carry out the following sequence:
1) Click the little box that contains the starting colour,
2) push the "from" button,
3) click the little box where you wish to end the gradient,
4) push the "to" button and finally