  
                               EasyFractal Manual
            (Demo program for Omikron Basic and the EasyGem Library)
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The program EasyFractal makes it possible to calculate different Mandelbrot and Julia sets very rapidly. First, the data is stored in a byte field and then - using the EasyGem command "Display_Data" - displayed in a user window. The results are quite often rather beautiful and esthetically pleasing pictures, also called fractals. Even though the program will also run without problems in a 256 color mode, it is recommended to choose a higher pixel depth (thousands or million of colors), because the detailed colorings are then much more shown to advantage. 
After the program has been started, one window is already open, which shows the well-known standard Mandelbrot Fractal. This image is NOT stored somewhere on the computer but is being calculated in real time. Therefore, it is also possible to scroll around on the virtual screen any which way, by clicking on the control elements of the window or pressing the cursor keys.
The step rate can be defined with the number keys 0-9. Pressing one of the number keys, the step rate is calculated via the formula "Step_Rate = 2^Key". If you, for instance, press the "3", then the step rate is 2^3=8. 

The information bar indicates the current mouse position; however, not as usually in pixels but as a real and imaginary part of a complex number (Z=Zr+i*Zi), since the calculation of Mandelbrot and Julia Sets does after all take place within the complex number plane. If you should find one area especially interesting (e.g., at the edge of the Mandelbrot Fractal), keep the shift-key depressed and then use the mouse to click on the point you would  like to examine further. This point will then move to the center of the screen. Zooming into the picture also takes place at that particular location. This process will continue as long as the mouse button is depressed. Of course, you may also zoom out again. For zooming out, select the ctrl-key instead of the shift-key.

EasyFractal offers a menu bar wit three menu titles, which we will now discuss in detail:

File Menu
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This menu offers the conventional functions to create, open, save, and print documents as well as the function to end and exit the program.

If you select the menu option "New ...," a dialog box opens in which you can name the fractal, choose between several iteration functions, and set the real as well as the virtual dimension of your new window. Initially, we recommend accepting the virtual maximum value (32766). Smaller values will have the effect that the finished picture will not react quite as sensitive to the movements of the window sliders. 

The speed used to calculate the picture is very much dependent on the chosen function. "Z^2" is very fast, while "Z*TAN(Z)" is comparatively slower due to the complex tangent calculation. During the calculation time, the mouse arrow will switch to the Berkhan Cursor, which indicates that the program is  working. Consequently, you should not think that your computer has crashed if  this Berkhan Cursor is visible for a longer period of time.

Two functions are available for data saving:  
1. "Save as 'FRAC' ..." will store only the parameters, which generate the chosen picture as well as the color palette assigned to the window. Depending on the size of the color palette, such a picture may need only a few kilobytes of space on your hard disk drive. Therefore, you may use this option to save easily thousands of pictures without having to worry about free space on your hard disk drive. However, pictures stored with this option can only be loaded again with EasyFractal.
2. "Save as 'PICT' ..." will save the actual visible part of the frontmost  window in PICT format. The picture may then be loaded and edited in other  programs. Pictures using the PICT format cannot be loaded again with  EasyFractal.

Of course, you may also print out your pictures. The usual menu options "Page Setup ..." and "Print ..." are at your disposal. However, only the area of the uppermost window visible on the screen will be printed.  
 
Edit Menu
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The command "Copy" is the only standard function contained in this menu. This command will enable you to copy parts of the picture to the clipboard, which then can be inserted into other programs. As usual, keep the mouse button depressed and create a frame. The marked area may encompass the entire virtual window; however, the parts located outside of the visible area will not be copied. Later, when pasting the copied area, these areas will then be transparent. Selecting the corresponding menu option pastes the marked area to the clipboard. 

The lower portion of the menu features the function "Load palette ..." and "Save palette . . .." These may be used to assign an individual color palette to every window, which will then display one and the same picture in a variety of different color ranges. The folder "ColorTables" contains a small selection of color palettes. 

Fractals Menu
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If a fractal window is in the topmost position, you may use the menu option "Settings ..." to modify the parameters necessary for the calculation of this fractal.
It opens a dialog box, which features the name of the fractal at the top. The interpolation range may be set secondly. In order to achieve the highest calculation speed possible, intermediate points are determined using the calculus of interpolation. A larger interpolation range shortens the calculation time; however, the picture depicts fewer details. Thus, only values between 1 and 8 serve any purpose. The optimal value depends also slightly on the fractal itself. 
Thirdly, you can set the number of maximum iterations. This number indicates the maximum number of times a calculation loop for a picture point should be executed until the process is interrupted and the color is set to zero (black in the default palette). While seeking interesting pictures, it is recommended to use an initial value of 255 or less, since this will speed up the calculation process. If a pretty spot still containing black areas has been discovered, it is possible to improve the detail accuracy by applying a higher value (e.g., 1023 or 4095).

Both of the two following fields determine the virtual size of the picture.  
The next field contains the value for the total zoom in percent. The scale was chosen in such as way that the traditional Mandelbrot Fractal will fit just so at a resolution of 320 x 200 pixels. 
NOTE: Starting with an overall zoom of 10 billion percent, the picture begins to become slightly blurred because computers can only calculate with a finite precision (approx. 15 to 16 decimal places). In the case of a zoom setting that is too high, those points that are next to each other can no longer be distinguished as individual points.
 
When zooming into a picture, the enlargement occurs in predetermined increments. The next field will now indicate the factor of each increment. The higher the number, the bigger the zoom effect. A number smaller than 1 reverses the process. 

The next four fields are mainly for advanced users who would are experienced in the theory of the Mandelbrot and Julia Sets. Remember, the calculation takes place in the complex number plane. 
The next two values indicate which complex numbers corresponds to the center of the virtual screen. 
The fields "Real part of C" and "Imaginary part of C" are relevant only for Julia Sets. As we know, each Mandelbrot Set has a correspondingly defined  Julia Set and both of these fields contain exactly the real and imaginary part  of this complex constant C. 
If you have chosen values for these constants from other sources (books, magazines, etc.), you may enter them here and use them to calculate the corresponding fractal. 

The last item gives you the opportunity to influence the appearance of the generated pictures. From the point of view of pure mathematics, this place should always be occupied by an infinite or at least a rather large number, however, if one's interest rests less with pure mathematics than with pretty pictures, it is also possible to experiment with different values. 
For example: The value 1.4 with a Julia set of Z^2 yields seemingly very spatial structures. The rule of thumb is that larger numbers yield rather more baroquely appearing pictures, while smaller values result in rather non-ornate areas with smooth edges.

The picture will immediately be recalculated using the new values if you exit the dialog box after clicking on "OK." 

The second menu option in the "Fractal Menu" serves to generate Julia Sets. This option may be chosen only if the uppermost window is a window with a Mandelbrot Set. If you now select this menu option, the menu entry is marked with a check mark and the mouse cursor is switched to crosshairs as soon as it is located over a window containing a Mandelbrot Set. Now you may use this to click into the Mandelbrot Set, which opens the "New ..." dialog box. Clicking on "OK" triggers the opening of a new window displaying the Julia Set that corresponds to this point. Although every point in a Mandelbrot Set has a corresponding Julia Set, really interesting pictures are obtained only when clicking into the turbulent areas at the edge of the Mandelbrot Fractal.

We wish you many joyful hours with this little program, which is also available as a source code, meaning that you may modify it to your heart's desire and, for example, add new iteration functions.

  
Your Omikron Basic Team
 
    