On our bulletin board, we are making a lovely homemade coarse picture of a Julia set. Today you will take advantage of the computer's penchant for tedium and make finer pictures. The computer will do exactly what you did to make the picture. It will try a bunch of different seeds, see how long it takes them to escape, and color the complex plane accordingly. Its pictures look better only because it uses smaller pixels.
To start out, we will use the Complex Numbers program from before. Log on to your account and you should see it in your applications window. From the main menu, pick Recursive Arithmetic. For your parameter value, pick .36, .1. Then for your seed, use 0,0. You will see the complex plane, with a dot at the origin-the location of your seed. Hit a key and the recursion will start. It will take your seed and square it, and then add the constant c. Hit another key and it will take that value, square it, and add c. Continue hitting keys and watch where the dots end up after many iterations. Does it converge to a fixed point, fall into a cycle, or bounce around chaotically? Try this for a few different seeds and see what impact that has on the behavior of the sequence. Some of the seeds may escape-that means they aren't members of the Julia Set.
Hit the "c" key and change the value of your parameter. Then do the same experiments as you did for .36, .1. See if you can find some different cycles. When you pick a particular value for c, you will see it light up in yellow on the picture of the Mandelbrot Set at the bottom of your screen. Different places in the Mandelbrot Set lead to different types of behavior.
Quit the Recursive Arithmetic screen and go back to the main menu. This time, select Phase Space. Here you will see the same sequence of dots that you saw before, but they will be drawn automatically and you will see them against a background of the Julia Set. Try c = -.5, .6 and see what kind of cycle you get.
If you pick your c value from the main region of the Mandelbrot Set, you will see the sequences converge to an equilibrium value. The way that they converge can vary widely, however. They tend to spiral in, but depending on your value of c, that spiral will have very different shapes. Sometimes you can see little arms
Now we will move on and use the Fractint program to make even finer Julia Sets. It does the same thing that my program does, but much faster because they have a special algorithm for speeding up the arithmetic. Close out Complex Numbers and select Fractint from your applications window. If you are doing this lab from home, you can go to the Fractint Home Page and download Fractint onto your own computer. Here is the link: http://spanky.triumf.ca/www/fractint/fractint.html.
As the credits scroll, hit enter to get the fractal program's main menu. Go to Select Fractal Type and pick Julia. After you hit enter, it will ask you to fix the value of parameter c. Use the same c value you had last time: -.11,.87 for B Period and -1,.26 for D Period. The computer will give you two separate prompts for each part of your complex number. It will then draw you a picture of the Julia set that corresponds to that value of c. The dark blue on the inside is the Julia set itself (trapped seeds), while the other colors represent different rates of escape. Each pixel on the screen represents a different seed.
While the picture is on the screen, hit the Delete key. This sends you to Select Video Mode, where you should pick F4 and hit enter. Leave the fractal type and video mode set that way for the whole lab.
You can magnify parts of the picture and look for more and more detail. The true picture is infinitely detailed-never smooths out. The computer, of course, has limitations on the precision of its arithmetic. Hit Page Up repeatedly to get the magnification window the size you want it. Then use arrow keys to locate it properly.
When you are ready to see a different Julia set, you can change the parameter settings. First you need to zoom back out. Do that by hitting the backslash (\) key, as many times as it takes to get back to your original image. Then hit the z key to reset the value of parameter c. Try real = -.8 and imaginary = .1. It has some neat shapes upon shapes upon shapes--smaller and smaller replicas. This is the phenomenon known as "self-similarity."
With the Julia Set picture on the screen, hit the spacebar. This takes you out to the Mandelbrot Set, your catalog of possible Julia Sets. As we discussed in class, the Mandelbrot Set displays all of the various choices you might make for your parameter c. Changing the value of c gives you a different Julia Set. You can see that in action by hitting the spacebar once again. A little sketch of the corresponding Julia Set will appear in the lower right corner of the screen. Use your arrow keys or your mouse to move around in the Mandelbrot Set and the Julia Set sketch will change accordingly. When you get to a sketch that you want to see in more detail, hit the spacebar again and you will return to a full screen of Julia, drawn with your new choice of parameter c.
Investigate a variety of values for parameter c, and see how many different types of Julia sets you can discover. Some of them are pretty boring, just squished circles, while others are very intricate. For what values of c do you get a connected Julia set? When does it break apart into separate pools? When is it nothing more than scattered dust?
You might also keep your eye on some particular seed value, say 0,0. When is the origin included in the Julia set and when is it not? This is what builds the Mandelbrot set. If you are interested in knowing more about how that is constructed, call me over.
As you find interesting things, note down the value of c and give some description of the Julia set you got. ( You will find a table for this purpose below.)
| Value of parameter c | Description of Julia set | Is the origin in the Julia set? |