This manual pertains to version 2.03 of the program.

Fractal images may be created by the general iteration: **x**_{n+1} ← **F(x**_{n}**,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 i^{2} = -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: **x**_{n+1} ← **x**_{n}^{2}** + c**, starting with **x**_{0} = 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 **x**_{n+1} ← **x**_{n}^{2}** + 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

5) push the "Spread" button.

With the "Rotate" button you can shift a series of boxes one place. To do so, execute the previous sequence but now push "Rotate" instead of "Spread".

Push the "OK" button when you're done editing.

To create a sequence of bitmaps for an animation, select the first image for the series followed by pushing the "Set start" button or choose *Set end* from the *Animation* menu. Select the second image that defines the end of the sequence and push the "Set end" button or choose *Set end* from the *Animation* menu. Now the beginning and the end of the sequence are defined. Then launch *Create* from the *Animation* menu. A dialogue box with several options comes up:

The number of bitmaps to create from start to end can be set, the number of iterations and the power at the beginning and the end of the sequence can also be chosen. Mandel will interpolate values for the intermediate images. The xmin, xmax, ymin and ymax values have been set by the *Set start* and *Set end* commands, but may be adapted here. If "Keep in RAM" is checked, all images will remain visible in the program. The images will be removed from the program interface as soon as the bitmap file is saved to disk if this option remains unchecked (recommended). If "Colour rotation" is checked, the entire colour palet will be rotated one place for each new bitmap as in the video below.

The *Mandel>Iteration path* menu item relies on the presence of one of the programs of the Serf Software Suite. When you select this option from the menu or right-mouse click on a fractal image, a dialogue box appears:

If you now push the "Iteration graph" button, the program creates two columns of data for the real and imiginary parts of the variable **x**_{n}. The columns will contain the evolution of **x**_{n} for each iteration starting with **c** as defined in the upper two editboxes in the dialogue box. If **x** becomes larger than 4 or if itermax is reached it stops. The program then calls (using JRTalk.dll as an intermediate) your Serf Software program to display the graph. Note that the coordinates in the dialogue box were set to the default value of 0 (zero) in this example. They would show other coordinates if you had right-mouse clicked on an image. The push button "New image" uses also the second pair of coordinates to create a new fractal image and is therefore very similar to *New* from the *File* menu..