You should see a triangle.
- Press 'z', make the order '1', 'enter'
- Press 'z', make the order '2', 'enter'
- Repeat with some other orders. (Orders 8 and higher will take a LONG time, so don't choose numbers that high unless you want to wait.)
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Fractint repeats a simple rule many times to generate
Sierpinski's Triangle. The repetition of a rule is called an
iteration. The order 5 image, for example, is created by
performing 5 iterations. The Triangle is only complete after an
infinite number of iterations of the rule. Can you figure out what the
rule is?
One of the characteristics of fractals is that they exhibit
self-similarity on different scales. For example, consider one of the
filled triangles inside of Sierpinski's Triangle. If you zoomed in on
this triangle it would look identical to the entire Triangle. You could
then find a smaller triangle inside this triangle that would also look
identical to the whole fractal. Sierpinski imagined the original
triangle like a piece of paper. At each step, he cut out the middle
triangle. How much of the piece of paper would be left if Sierpinski
repeated this procedure an infinite number of times?
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- Press 't', choose 'lsystem' from the menu, choose 'koch1'
- Make the order '0', press 'enter'
- Press 'z', make the order '1', 'enter'
- Repeat for order 2
- Look at some other orders that are less than 8 (8 and higher orders take more time)
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What is the rule that generates the von Koch snowflake?
Is the von Koch snowflake self-similar in any way? Describe how.
Since it is possible to draw a circle around the von Koch
snowflake, it has a finite area. Keeping in mind that the von Koch
snowflake is created by an infinite number of iterations, what can you
say about the perimeter (the distance around the outside) of the
snowflake?
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- Press 't', choose 'lsystem', choose 'bush'
- Make the order '0', press 'enter'
- Press 'z', make the order '1', 'enter'
- Look at orders 2, 3, 4, and 5 (Orders 6 and higher take a long time)
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Can you describe the rule that generates the bush?
In what way is the fractal bush self-similar?
Name some other objects or processes in nature which exhibit
self-similarity on different scales. Can think of any pattern that
might be formed by the repetition of a rule?
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Sierpinski's Triangle, the von Koch Snowflake, and the fractal bush
are all generated by a geometric replacement rule. During each
iteration, one type of geometric figure is replaced by another shape.
There are other fractals that are generated by numerical replacement
rules. Julia sets are examples of this type of fractal. Julia sets
are much more colorful and intricate than the fractals you have seen
so far. You need to be familiar with two pieces of mathematics to
understand how these beautiful pictures are created.
'Squaring' a number simply means multiplying it by itself. So 5
squared is 25 (5 x 5 = 25). This can also be written 52 =
25. Notice that -5 squared is also 25. (A negative times a negative
is always positive.)
Consider the following numerical
replacement rule: during each iteration, the old number is replaced by
the square of itself. Since z is the first letter in the word
Number, we will write our rule in symbol form this way:
new z = z2 .
- Let's use this rule starting with the number 1.1.
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1.12 = 1.21 1.212 = 1.4641 1.46412 = 2.14358881 2.143588812 = 4.594972986 4.5949729862 = 21.11377675 21.113776752 = 445.7915685 |
445.79156852 = 198730.1225 198730.12252 = 3.949366159 x 1010 (3.949366159 x 1010)2 = 1.559749306 x 1021 (1.559749306 x 1021)2 = 2.432817897 x 1042 (2.432817897 x 1042)2 = 5.918602921 x 1084 |
You can see that the numbers became big very quickly. The iterates are "escaping to infinity."
- Now let's repeat this procedure with 0.9
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0.92 = 0.81 0.812 = 0.6561 0.65612 = 0.43046721 0.430467212 = 0.185302018 0.1853020182 = 0.034336838 0.0343368382 = 0.001179018 |
0.0011790182 = 0.00000139 0.000001392 = 1.932334983 x 10-12 (1.932334983 x 10-12)2 = 3.733918487 x 10-24 (3.733918487 x 10-24)2 = 1.394214727 x 10-47 (1.394214727 x 10-47)2 = 1.943834705 x 10-94 |
This time, the numbers became smaller. The iterates are
"bounded" - they do not escape to
infinity. Even though 1.1 and 0.9 are very close together, the replacement rule sends iterates of
1.1 towards infinity and sends iterates of 0.9 towards zero.
Taking a square root, 
, is the opposite of squaring a number. If you take the square root of 25,
you will get 5. We saw in the last section that there is no way to square a real number and get a
negative number as a result. Therefore, there is no real answer to the square root of -25. (If you
try this on your calculator, you will get an error message.) There is no real number
which will be equal to -25 when it is squared, but there is an imaginary number that will work!
We designate imaginary numbers by the letter i, so the square root of -25 is 5i, and the square
root of -16 would be 4i.
Julia sets are created using complex numbers. Complex numbers have a real part and an
imaginary part added together. Some examples of complex numbers are:
| A) -1.2 + 1.01i | B) 1 + 2i | C) -1.3 - 0.8i | D) 0.43 - 0.12i |
We can plot these complex numbers on a graph by letting the
horizontal direction show the real part and the vertical direction
show the imaginary part.
- Press 'x' - the basic options screen should appear (If it doesn't, press 'escape' to return to the main menu and then press 'x')
- Change 'Inside Color' to '0' (this will make the inside of the fractals black), press 'enter'
- Press 't', choose 'julia'
- Change the real parameter to '0.32' and the imaginary parameter to '0.04' and press 'enter'
The replacement rule which generates this Julia set is new z = z2 + 0.32 + 0.04i. We usually call
the complex number at the end c for short.
The computer begins with the point in the upper left corner of the screen. It uses the complex number corresponding to that point (might be z = -2 + 2i, for example).
The computer squares this number and adds 0.32 + 0.04i to get the new z.
On the next iteration it takes the new z, squares it, and adds c.
The computer repeats this process until the iterates either become "large" (bigger than 2), which means they are escaping to infinity, or until the operation has been performed a predetermined number of times.
The computer assigns a color to the point based on how many iterations it takes for the number to escape.
The computer then proceeds to the next point on the screen and begins again.
- Blue background points escaped after the first 2 iterations.
- Points in the green band escape after 3 iterations.
- The light blue band shows the points that escape after 4 iterations.
- Points that are bounded, that never escape, are colored black.
There are as many color bands as there are iterations. Fractint's default number of iterations for Julia sets is 150.
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Let
c = 0.32 + 0.04i and let the first z
be -2 + 2i. Using the 'FOIL' method (also known as
double distribution), calculate the next z value by squaring
z and adding c. (You should get a new z of 0.32 -
7.96i if you performed this caluclation correctly.)
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- Press 'z' to change the c-value of the Julia set
- This time set the real part equal to 0.3 and the imaginary part to 0.6
- Press 'enter'
Notice that this fractal has broken up into isolated "island" color bands. This happens because
there is no connected black region around which they are centered. To see this better, we need to
look at the fractal in black and white.
- Press 'x'
- Change 'Outside Color' to '1', Change 'Maximum Iterations' to '5', press 'enter'
- Press 'F8' to switch to black and white mode
All of the escape bands are now white. The black region represents the points that do not escape
within 5 iterations. After 5 iterations, the black region is still connected.
- Press 'x', change 'Maximum Iterations' to '10', press 'enter'
- Repeat this procedure for 20 and 30 iterations
Notice that the black region is no longer connected after 20 iterations. After 30 iterations, the fractal begins to look like dust specks. Eventually, all of the black region will blow away.
Julia sets like new z = z2 + 0.3 + 0.6i are called "dust sets."
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What is the highest number of iterations that you can choose so that the black region is still
connected?
Give an example of a maximum iteration value that appears to remove all of the black points
leaving only the white background.
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- Press 'z' to change the c-value of the Julia set
- This time set the real part equal to -0.122 and the imaginary part to 0.745, press 'enter'
- Press 'x', set 'Maximum Iterations' to '5'
- Look at other maximum iteration settings
Notice that Douady's Rabbit stays connected no matter how many iterations are performed.
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Do you think that this Julia set, new z = z2 + -0.122 + 0.745i, looks like a rabbit?
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- Press 'z' to generate a Julia set with a new c-value
- Set the real part equal to any number between -2 and 1.5
- Set the imaginary part equal to any number between -1.25 and 1.25
Is your fractal connected for all maximum iteration settings?
Putting the colors back:
- Press 'x'
- Change 'Outside Color' to 'iter', Change 'Maximum Iterations' to at least 150, press 'enter'
- Press 'F4' to switch to color mode
To zoom in on your fractal:
- Press 'page up' until the box is the right size ('page down' makes the box bigger)
- Use the mouse to move the box to the right place and press 'enter'
Fractint makes three passes over the fractal making it less fuzzy each time. It beeps when it is
done. The farther you zoom in, the longer this takes. You can either wait for the image to
become clear, or you can press 'page up' and zoom in farther before it is done.
To zoom back out, press 'h' until you are back where you want to be. 'Ctrl-h' takes you back in
if you zoom out too far.
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Is the Julia set that you chose self-similar? That is, do you find the same types of structures as
you zoom in? Do you find different structures if you zoom in different places?
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We have seen that all Julia sets are generated by a rule of the
form new z = z2 + c where c is a complex
number. Some values of c generate connected Julia sets (like
Douady's Rabbit) while others generate dust
sets. The Mandelbrot set is a fractal that serves as a sort of
"dictionary" for the Julia sets. Among other things, it shows which
values of c yield connected Julia sets.
- Begin with a Julia set on the screen
- Press the space bar to toggle to the Mandelbrot set
- After the beep, press the space bar again to see where in the Mandelbrot set your Julia set came from (the cursor shows this). The outline of your Julia set appears in the little box.
- Move the cursor to the middle of the black region, press the space bar and look at the Julia set
- Press the space bar once to return to the Mandelbrot set
- After the beep, press the space bar again to display the Julia inset
- Put the cursor on the blue background and look at the Julia set
- Put the cursor near the boundary of the black region and exmine that Julia set
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Another important property of fractal processes is that the end result can be 'sensitive to initial conditions'. This means that a small change in a parameter value can make a big difference in the type of picture that you end up with. Julia sets are sensitive to initial conditions for parameter values near the boundary of the Mandelbrot set. Find an example of two different values for c which are very close to each other but which produce Julia sets that are different in an important way. Move the cursor slowly around the Mandelbrot set, stopping the
mouse every so often to look at the shape of the Julia set. Press the
space bar to look at the Julia sets more closely. Find your favorite
Julia set. Give it a name and record its formula. To find the
c-value, press 'z' while the Julia is on the screen.
Remember that Douady's Rabbit, new z = z2 + -0.122 +
0.745i, already has a name!
Name:
Formula:
Try changing the number of iterations to
see how your Julia set looks best. Press 'x' for basic options, and
change maximum iterations.
Zoom in on
your Julia set. Record your favorite window. (Hit tab and copy the
numbers of the corners.)
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Experiment with changing the inside and outside colors. (Under
basic options. Type a number or one of the other options it gives
you). Other things to play with are log palette, biomorph color, and
decomp option.
For a thrill press 'c' to start and stop
color cycling ('+' or '-' changes direction)
Press 'ctrl-a' to have your fractal eaten by the ant automaton.
To Save and Print Your Work
Printing directly from Fractint doesn't work very well, so you will need to save your image first.
- Press "s" to save your image (Fractint automatically names it "fract###.gif" (### is some number 001, 002 ...)
- Open your picture in an image editor or word processor
For Word Perfect 7
- Click "Graphics" then "Image"
- Your File should be in "c:\fractals\" and will be named "fract###.gif"
- Use the "Edit Image" command to modify settings and add text to your fractal
- Print your fractal!!
Installing Fractint From a Disk
Fractint works best if it is run off of the hard drive. To copy files to the hard drive (Win95):
- Click the "Start" button, and select "Programs" then "Windows Explorer"
- Insert disk into drive
- In the Left window, scroll and select "3
Floppy (A:)" - In the Right window, use the mouse to highlight all of the files (or left click on the right window and press 'Ctrl-a'
- Right Click and select "Copy"
- In the Left window, scroll and select "(C:)"
- In the Right window, Right click on any blank spot
- Select "New" and then "Folder"
- type "fractals" and press Enter twice
- In the Right window, right click on any blank spot
- Select "Paste"
- Double click on "
"
Now you are ready to explore the other fractals and Fractint features on your own!
Fractint is a freeware program created by the Stone Soup Group. More advanced versions of the
program can be found on the Internet as they come out.
Fractint's homepage is at http://spanky.triumf.ca/www/fractint/fractint.html
The Fractint web site has a Fractal Screen-Saver, and several add-ons for Fractint.
It also has information for creating your own formulas, and customizing your copy of Fractint.