The program EasyFractal makes it possible to calculate different Mandelbrot and Julia sets very rapidly. The program was programmed in Omikron Basic in its entirety. First, the data is stored in a word field and then displayed in a user window using the EasyGem command "Display_Data". This results in rather beautiful and quite often 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 millions of colors), because the detailed colorings are then much more shown to advantage.
After the program has been started, one window is already open. This window shows the well-known Mandelbrots Fractal. This picture is NOT stored somewhere but is really computed in real-time. It is therefore also possible to scroll around on the virtual screen any which way by clicking on the control elements of the window or by using the cursor keys.
The increments may be further adjusted with the number keys 0 to 9. The increments are calculated according to the formula "Increments = 2^Key". This means that if you were to press the number 3, the increment would be 2^3=8.
The information bar indicates the current mouse position; however, not as usually in pixels but as a real and imaginary part (REAL and IMAG) 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 Mandelbrots fractal), keep the Alt-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 Alt-key.
New features of the Macintosh operating system, which were introduced with system 8, are also used by Easy Fractal, if the corresponding system version is available.
- Proportional Window Sliders: The size of the window sliders ist proportional to the ratio of the size of the shown document part and the size of the whole document.
- Smart Scrolling: The content of the window follows immediately the movement of the window slider.
- File Navigation Services: This is a new file select box indroduced with system 8.5. It supplies several new features as e.g. to open more than one file at a time.
-Some minor errors have been removed.
New in Version 1.20
- There are three new iteration functions, which also may be modified as desired using two freely selectable parameters.
- The dialogs New ... and Settings ... were combined and equipped with additional setting options.
- Now it is also possible to save or print virtual images as 'PICT.' If you have the appropriate hardware, you can now also print out pictures with EasyFractal, which are the size of a poster.
- Blocks may also be changed subsequently.
- Areas marked with a block can now be calculated specifically.
- Undo function for zoom with up to 64 stages.
- Superfluous new calculations and a few minor redraw errors have been eliminated.
Note: Starting immediately, please use the Alt key for zooming instead of the Shift key, because the Shift key has been assigned the function of changing blocks.
This short tutorial will serve to familiarize you with the most important functions of EasyFractal. More in-depths information and details can be found in the manual following the tutorial.
We now wish you a lot of fun and success
exploring the depths of the fractal universe!
The 10 minute tutorial contains the following five instructional steps:
A. Zooming into a Fractal
B. Scrolling through the Infinite Fractal Depths
C. Fantastic Kaleidoscope
D. Calculating New Mandelbrot and Corresponding Julia Sets
After the start of the program, the best-known fractal, the Mandelbrots Fractal, is automatically calculated. The Mandelbrots Fractal is a symmetrical figure, which seems to be positioned on its side.
1 - Now please click on a spot in the green border area of the black figure. It is best to click on the distinct indentation where the "bottom side" of the symbol seems to be. While clicking in this area, hold down the Alt-key and zoom into the fractal.
2 - Repeat this procedure a few times. Make sure that you do not zoom into any of the black areas, because the spot you use to click on with your mouse will become the center of your picture next time you zoom.
3 - When zooming into Mandelbrots Fractal try "hitting" a pretty rosette. You will notice that the deeper you zoom into the fractal, the more interesting the structures become, but it also takes longer to calculate.
4 - After finding a rosette, you will notice that it contains a large black area in its center. This black hole we now want to fill with content. Choose the menu option Settings ... in the Fractals menu. Place the cursor into the input field at Iterations and replace the 256 with 1024. Confirm with OK. After a short calculation period, new structures will appear in your picture where the black area had been previously. You may now zoom into the new structures as described in step 1.
5 - Now stop. Use the mouse to click into the center of the picture while simultaneously pressing the Ctrl-key. Keep both keys depressed and watch the reverse zoom. However, make sure to release both keys before Mandelbrot's Fractal disappears into the infinite depths of the fractal universe...
6 - Now use the mouse to draw a small frame over one of the small, thickened areas on the antenna of Mandelbrot's Fractal. Then select the menu item Calculate from the Edit menu. The marked area will now be enlarged in such a way as to fit just so into the existing window. Repeating this process allows you to examine specific areas that are of interest to you. Incidentally, you can also change a block afterwards using the mouse by keeping the Shift key depressed.
1 - Choose the menu option Open ... in the File menu and load the picture "Mandala" from the folder "Fractals". We would like to take this opportunity to point out once more that all files in the folder "Fractals" are not true pictures but rather only the information about a fractal, which EasyFractal needs when loading the file to calculate the respective picture.
2 - After the picture has been computed, please click on one of the arrows from the scrollbar at the side or bottom and inspect the area around the fractal rosette. All new picture details are recalculated in real-time when scrolling around in an EasyFractal picture. In theory you may scroll infinitely to the right or left, to the top or bottom; however, this is restricted by the limited size of the virtual screen. It is possible, nevertheless, to set a new center of the virtual screen with each zoom step and with that explore almost the entire fractal universe.
Tip: Zoom out of the fractal "Mandala" (mouse click + Ctrl-key). After about 70 zooms, you can see from which part of Mandelbrots Fractal the "Mandala" picture originates. (More information about setting zoom depths, etc. can be found in the manual below.)
1 - Return to the starting point of the mandala rosette (or load the picture once more if you cannot hit the center).
Choose the menu option Edit Palette ... in the Edit menu, if the palette editor is not open yet.
2 - Enter the following values into the palette editor:
Start color = 0, Dest color = 255. Now go to Rotation and click on one of the two arrows on the right side next to the corresponding input field. Keep the mouse button depressed. Rotating the color palette has a colorful kaleidoscopic effect on the opened picture window.
3 - Load a new color palette. For this, you need to click on the menu option Load Palette ... in the Edit menu; then double-click on the file "Ease.CTAB" in the folder "ColorTables". The new color palette will have an immediate effect on the top picture, which suddenly will appear to look quite different. The color bar of the palette editor features the new color combinations in detail. Return to Rotation, cycle the new colors a bit forward and back, and watch the constantly changing images and structures of the picture, which are caused by the changing color schemes.
1 - You already did calculate a Mandelbrot set, Mandelbrots Fractal to be exact, which is loaded right at the start of the program. In order to try out a different formula, choose the menu option New ... in the File menu.
A dialog box opens. Enter a name for your new fractal and look at the popup menu below the name. Z^2 is the default setting and results in the well-known Mandelbrots Fractal. Now choose the third function X^2-Y^2,X*Y*2+Y and confirm with OK.
2 - A somewhat unshapely solid body appears with quite different structures in its green border area than the baroque ornaments known from the Mandelbrots Fractal.
3 - Now let the program calculate a Julia set to a point of this Mandelbrot set. Use the option Make Julia in the Fractals menu. When clicking on this option, the cursor will turn into a pair of crosshairs as soon as a window with a Mandelbrot set is at the top and the mouse cursor has been moved across the picture. Choose the center of the picture with the mouse. It is best to begin in the green border area where the colors seem to appear in a more uniform manner (left side).
4 - When clicking with the mouse the dialog box "New Fractal" appears again, which you only need to confirm at this point by selecting OK. The new window now depicts a Julia set that corresponds to the just chosen point of the Mandelbrot set. You may also zoom in and out of this picture, change color palettes, as well as rotate colors. Try out additional points in the Mandelbrot set and watch how the attached Julia sets change as well.
Tip: To find pretty and esthetically pleasing spots in this Mandelbrot set, turn your attention to smooth, less turbulent areas when zooming in.
(If you cannot find anything suitable, load the fractal "Wave" in the folder "Fractals". It depicts such a spot. Zoom into the eye of the spiral and try to let the program calculate a deeper depth for the black hole in the center as described in step A 4 (Iterations).
1 - Open the picture "Galaxy" from the folder "Fractals".
2 - Go to the palette editor and activate the Start color. Use your mouse to click on an especially bright color in the color bar of the palette editor. The chosen color number (also: color index) and the corresponding hue are now indicated at the top.
3 - Activate the Dest color and again choose a color from the color bar. Choose a dark color this time, one that is about 4 cm removed from the first chosen color.
4 - Click on Color gradient.
5 - Now manually set the Start color to 0 and the Dest color to 255. Go to Rotation and keep the mouse button depressed while clicking on one of the two arrow keys to test the new color environment.
6 - Now once again click on Start Color and choose a new color from the color bar. Now select the button Pick Color and then use the pipette cursor to choose any color from the entire desktop. After clicking with the mouse, this color is now indicated in the color bar and depicted as a thin vertical line.
7 - Repeat the same process with the Dest color, with the exception that this time you do not go to Pick Color but to Change Color instead. The Mac OS color picker opens and you may choose any color you desire.
8 - Click on Color gradient and check the result as described in step 5.
9 - Repeat these steps as often as it takes to attain a color environment that is to your liking. Then save your new color palette with the menu option Save Palette ... from the Edit menu. This palette may now also be used for other fractals.
Tip: In order to obtain a smooth and continuous color gradient, you should set your computer to "millions of colors".
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 item New ..., a dialog box to enter the name of the fractal will open. This dialog box may also be used to determine the real as well as virtual size of the new window and a variety of other settings. For the time being, it is best to leave the default maximal values (32766). Lower values have the result that the picture is not quite as sensitive to the movements of the window sliders. Some functions can be modified using the parameters P0 and P1. This can be recognized by the fact that these parameters are explicitly listed in the iteration formula of the pop-up menu. In this case, the parameters have to set to values that make sense; otherwise, you will receive
nothing but black areas.
The speed, at which the picture is being calculated, depends very heavily on the chosen function. Z^2 works quite fast, while Z*TAN(Z) is comparatively slower due to the complicated tangents calculations.
The mouse arrow will be changed to the Berkhan cursor during the calculation process. This tells you that the program is busy with the calculation. Therefore, do not think that your computer might be frozen or has crashed, if the Berkhan cursor is sometimes visible for longer periods of time. If the estimated calculation time exceeds 2 seconds, a progress bar will also be displayed. The remaining parameters in this dialog box are described later when the menu item Settings ... is being discussed.
Saving Fractals:
When you have found a nice picture than it is also possible to save it, of course. To save the picture, choose the menu item Save as .... A dialog box will open, giving you the choice between two file types.
Save as FRAC saves only the parameters generated by the chosen picture as well as the color palette assigned to the window. Depending on the length of the color palette, such a picture then requires only a few kilobytes of file space on the hard disk. This means that you can easily store tens of thousands of pictures using this file type, without your hard drive running out of space. However, pictures saved in this way can only be loaded again by EasyFractal.
Save as PICT saves the entire virtual screen of the uppermost window in the PICT format. The thus saved picture can then be loaded into other programs and edited further. Pictures in the PICT format cannot be loaded again by EasyFractal.
Note: Since the virtual picture is usually rather large and therefore cannot be saved as a PICT, you first have to reduce the virtual picture size to more realistic dimensions. This can be achieved either by adjusting the size directly in this dialog box or you can change the size already at a previous time, using the menu item Settings ... to determine a smaller virtual screen.
Of course, you may also print out your pictures. The usual menu items Page Setup ... and Print ... are at your disposal to print your picture. Again, the virtual picture has to be first dimensioned to a suitable size, either previously or in the opening dialog box.
The only existing standard functions present in this menu are the commands Undo Palette / Undo Zoom or Copy, respectively. The latter copies a picture to the clipboard to import it into other programs. Just draw the familiar frame with your mouse button. The marked area can extend across the entire virtual screen. The marked area is transferred to the clipboard using the appropriate menu item.
Once you have defined a block with your mouse, you can still change this block at this time. For this, move your mouse close to a corner and press the Shift key. The block corner closest to the mouse now jumps to the mouse position. As long as the Shift key and the mouse button are depressed, this corner will follow the mouse movement.
The menu item Calculate can be used to enlarge the area marked with the block in such a way as to fit it just so into the existing window. This function also makes it possible to examine some specifically chosen areas. The menu option Undo Zoom allows you to revert the magnification at any time in up to 64 stages.
The lower portion of the menu features the function Load palette ..., Save palette as ..., Edit Palette ..., and Change Palette Size .... 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.
Edit Palette ...
This comprehensive palette editor offers many practical and useful functions to color fractals to suit individual tastes. Its biggest advantage is that all colored areas of a picture are assigned to one corresponding color number. The color of this color number can easily be changed using the palette editor. Only then exists the possibility to obtain the fractal structures of a picture.
Start color
Dest color
Here you indicate the color numbers with which to determine a specific interval in your color palette and to which you may later apply different functions (e.g., color gradient, color rotation, etc.). All colors of a color palette belonging to a picture are assigned specific numbers (indices), which are depicted in order in the color bar of the palette editor.
Pick index
Pick index may be used to determine a color number. The cursor changes into a pipette as soon as the mouse is dragged across the color bar. If the mouse button is clicked, then the chosen color number is automatically entered into the currently active number field (Start Color or Dest Color). You may also index a color number from a fractal picture in the same way. For this, just click into the desired picture and choose a color with the pipette. The color you have chosen using Pick index will be depicted in the palette editor on the right side next to the color number field and the small arrow keys. The arrow keys also offer you the option to set the color number. Alternatively, you can enter the color number into the color number field directly. Please note that you cannot just indicate any high number, because the number depends on the type of loaded palette and your loaded fractal picture, respectively. In order to determine the number of possible colors just click on the picture and go to the menu option Settings ... in the Fractals menu. The number of iterations indicated there corresponds with the maximum number of colors that may be used for this picture. If your color palette attached to the picture initially only contains 256 colors, then this indicates that the color palette has been activated four times on top of one another if an iteration number of, e.g., 1024 exists. This makes for an often rather fuzzy and tangled up appearance in the case of more detailed structures. If you would like to view or change the number of colors in your color palette, then choose the menu option Change Palette Size ... in the Edit menu. The default setting initially depicts the size of the current color palette. If you were now to enter, e.g., 1024 and confirm this with OK, then this would change not only the colors of your picture but also, of course, change the color bar in the palette editor, which now has been expanded by a black area, something you can see if you scroll to the right.
How to change the colors assigned to the color numbers is described in the next section.
Pick Color
The pipette cursor appears when choosing this button. It may be used over the entire desktop surface to pick up a color from any area, whether it be the menu bar, a fractal picture, or a desktop background color.
One mouse click suffices to add the chosen color immediately to the color bar of the palette editor where it is depicted as a thin vertical line. At the same time, it will also be shown in the small color field of the Start color and Dest color, respectively, depending which one is currently active.
Note: The function can be canceled by pressing the right mouse button or alternatively by pressing the command key in combination with the left mouse button.
Change Color
This button may also be used to assign a new color to a color number. The color picker dialog box of the operating system appears featuring the known functions.
The following functions concern only an interval of your color palette, which has been defined with the color number of the Start color and the Dest color.
Color gradient
This function is used to determine a color gradient, which has been defined with the interval set with the Start color and Dest color color of your color palette.
Step Size
This is used to set the size or rhythm of the increment steps of a function.
(If the function Color gradient is set to a certain Step size, the result is, for example, a silvery looking color effect, that is if the Step size is 2 and all even numbers run from a light to a dark color and all uneven numbers run in the opposite direction from black to white.)
Example for Color gradient with Step size 2:
1. Color gradient:
Step size = 2
Start Color No. 2 = light green
Dest Color No. 200 = dark green
2. Color gradient:
Step size = 2
Start Color No. 3 = black
Dest Color No. 201= white
Rotation
Rotation maybe used to determine the ideal setting for a fractal's color palette. By slowly cycling through the colors, the picture constantly changes its appearance. Just click on one of the two little arrows when using Rotation and keep the mouse button depressed. You will see how the colors of the current palette are cycled through one by one, which leads to some very interesting optical effects with some pictures. If you want to incorporate your entire color palette, you have to set the Start color to 0 and the Dest color to the highest number of your palette. The rotation process may also include smaller intervals of the palette without the other colors being moved. To effect this, you only have to indicate the appropriate interval for the Start color and Dest color. Another option for rotation manipulation arises when the Step size is set to 2, for example. In this case the indices are rotated by 2 numbers in each step. Just try out and see for yourself, which setting is the most interesting one for your picture.
Inverting
The chosen interval range of the color palette is replaced by the corresponding complementary colors.
Monochrome
The chosen interval range of the color palette is replaced by the corresponding gray tones.
Note: Choosing the menu item Undo Palette will cancel the last palette operation.
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.
A dialog box will open, showing the fractal name at its top. The position of the radio buttons will tell you whether this is a Mandelbrot or Julia set.
The pop-up menu below that shows the applied iteration function and the values chosen for parameters P0 and P1.
Further down, you can adjust the Interpolation range. In order to enable the fastest calculation possible, intermediary points are determined using interpolation. A wider interpolation range shortens the calculation time; however, the picture has fewer details. Therefore, only values between 1 and 8 serve any purpose. The optimal value also depends somewhat on the fractal itself.
You can set the number of maximal Iterations behind the interpolation range. This number indicates the maximum numbers of calculation loops for a picture point, until the process is interrupted and the color set to zero (black in the standard palette) is inserted. For the search for interesting pictures, it is recommended to use an initial value of 256 or less, since the calculations will then be faster. Once a nice spot still containing black areas has been discovered, it is possible to improve the detail accuracy by using a higher value (e.g., 1024 or 4096).
The following four fields determine the Real size and the Virtual size of the picture.
Zooming into a picture is done in specific increments. The next field now indicates by which factor the picture is to be enlarged from one increment to the next. The higher the number, the larger the increments. A number smaller than 1 reverses the process.
For the Total magnification in percent, the scale was chosen in such a way as
to fit the classic Maqndelbrot's Fractal just so at a resolution of 320 x 200 pixels.
Note: Starting with an Total magnification 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 Fractals 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, and if you are interested in programming, you should visit the Omikron Basic homepage on the internet. There you can also find the source code for EasyFractal 1.00, meaning that you may modify it to your heart's desire and, for example, add new iteration functions.
