Fractal Domains Fractal Domains is a program designed to draw fractals produced by iterating on the complex plane. There are many other programs used by sophisticated fractal designers but Fractal Domains is probably one of the easiest to use that still produces images which are print worthy. Fractal Domains is shareware. That means you can use it for free, but you can’t save your settings unless you pay a small fee and register the program. If you download and use the program at home you don't need to pay for it, but if you want to work on a fractal for a while and come back to the same image and work on it later, you'll need to save the settings and that requires you to pay for it.. Information about registering is available at the Fractal Domains website. The fee is about $20 US. The fact that you can't save your settings means that each time you open up the program you will need to start from scratch. Nevertheless, you can always take screen shots which you can save as files. The program only works on a Mac and the creator has no plans in the near or distant future to write a PC version. Saving your image Before saving any image you must first render the image. Under the Fractal menu choose Render Image. A pop up menu comes up. First, you must click on Anti-Alias: this will smooth out the edges of your color regions. This is so important I shall rewrite that phrase
ALWAYS REMEMBER TO ANTI-ALIAS.
If you don't anti-alias, then this will be immediately obvious to anyone seeing your image. Colored regions will have scraggly, choppy edges and there will be random bits of "noise" scattered throughout your picture that detract from the beauty of your fractal.
Once you have rendered your image and remembered to click the Anti-alias button, you’re ready to save. You can give it a jpg suffix so that the computer saves the image as a picture file. If you create on disk, the image will save to a destination of your choice. If you Create Image in Memory, you will need to save it before closing the program.
The Quadratic Maps
Adjusting your parameters:
1. Click on FILE/New Fractal/Mandelbrot. The Mandelbrot set will pop up. You may want a Julia set instead. To get one, choose the J-arrow icon on the left-hand side of the image window. Now, when you click on a point in or around the M-set, the associated Julia set pops up in a separate window. It also inherits the same parameter settings as your M-set, which is very convenient. If you don’t like the Julia set, close it without saving and click somewhere else around the M-set to produce another Julia set window.
2. Click on Fractal/Parameters. The pop-up menu allows you to change various drawing elements. First, in the Area menu, change the width and height to 600 to get a bigger window. Hit Apply. Always hit apply once you’ve made a few changes.
3. Go to the Dwells menu; the dwell limit is the maximum number of iterations the computer will try for each initial condition. You will need to increase this if you zoom in many times. The Dwell Method is the various ways the computer decides if a point has escaped. Try different ones to get different effects. The Exterior region split chops up the escaping set by the speed of escape.
4. In the Orbit menu, click the Orbit Trap button. There are many options to choose from. Simply vary these options and see if you like the result. The Trap Extent choice determines the width of the trap. If you change this to a number like 0.1 you get very thick stalks. Always hit Apply once you’ve made a change because you never know how it will affect your image.
5. Close the Parameters menu by hitting OK.
6. If you like a certain part of your image you can zoom in to investigate. On the left-hand side of the image window are several controls. Click on the magnifying glass. Your cursor turns into a magnifying glass and now you can click on any part of your image and instantly zoom in. To zoom out, click on the crossed arrows on the left-hand side of the image window then click somewhere on your image. You can use the cross on the left-hand side to click and drag on a portion of your image and thereby enlarge it. The oval icons on the left-hand side allow you to shift the image sideways in a particular direction.
Fiddling with the color options
1. If you click on the Palette menu, a pull-down menu of choices comes up. These are pre-set color maps. Try some of them out.
2. You can try your luck by letting the computer randomly choose a selection of colors. Click on the Windows menu and choose the Randomizer option. A pop-up menu emerges with a click-bar at the top. When you click on this bar, the computer randomly varies your color choices. BEWARE: there is no way to reverse this process! Any changes you make in this menu are permanent. Try varying the options in Distribution and Regions. You may find that fewer choices are better in which case you should limit the number of random colors assigned at the top of the menu. When you close this pop-up menu, the changes are saved.
3. You can individually adjust any color choice within your image. Choose the pencil option on the left-hand part of your image window then click on a point in your image. The Color Map Editor pop-up menu comes up. The colors in this menu are the speed colors assigned to the escaping points. Slower points are on the right and fast ones are on the left. Find the color you want to change and use the arrow to click on the triangle indicator below the color. Below the color band, the color squares pop up with the escape time numbers above them. Click on the color, a color wheel comes up which will allow you to choose a new color and brightness. You can even click and drag the color you like to the squares at the bottom of the menu and thereby save that color for future use. Once you click on OK, the color is transferred to the region you chose on your color band. This isn’t reversible. Notice at the top of this menu you can chose to adjust color in the escaping set or in the orbit traps. To add a new color where there was none, you click on the arrow + icon at the top left then click below a portion of your color band. It creates a new triangle where you can place a color of your choice.
Rational Maps
Rational Maps allow you to get away from the
quadratic maps we have studied to investigate the parameter set and Julia sets of any rational function you can imagine. A rational function is a function like
Notice that Fractal Domains uses the
variable instead of . This is standard usage for complex functions. Don't be put off by this; just pretend it's an .
A rational function is basically a polynomial over a polynomial but you can use complex coefficients. These will give you your most original and striking images. What you create has probably never been seen before since the number of options is literally infinite.
Go to the File menu, chose New Fractal, then Rational. A new window pops up with the parameter set for the cubic map . You can do everything to this image that you can do with the regular quadratic Mandelbrot set including creating orbit traps, adjusting the size of the image, anti-aliasing, and creating associated Julia sets. See above for hints about using the color menu and rendering.
However, there is one extra option: making up your own function. To do this, follow these steps:
1. Click on Fractal/Parameters. You will see the usual menu choices. The extra menu option is Formula. When you click on this there are options to change the numerator and denominator of your formula. To change the numerator, click on Edit next to the numerator bar. To add powers of z, increase the Power number with the up/down switch. Then choose Add Term. You can give your z a coefficient by choosing a Real and Imaginary part to the coefficient. When you are done making your choices choose OK. Then you can do the same with the denominator. There are various other options for you to choose from. Try some of these out.
2. Although not essential, it's usually best for the degree on the numerator to be several degrees different from the denominator.
3. Now investigate, zoom-in, play with colors, and create something no one has seen before.
4. Remember you've got both a parameter set and Julia sets to play with.