Introduction to Fractals

Date: 09/02/98 at 20:39:00
From: Shannon Donaldson
Subject: Formulas for Fractals

Dear Doctor Math,

I have been trying to do a math project in fractals. However, every 
time I try to learn about them, I end up getting lost. I really want 
to know a basic formula for the fractal, and how each part of the 
formula works. What numbers do I use, and where do I insert those 
numbers in the formula?

Thank you,

Shannon Donaldson

Date: 09/03/98 at 07:08:48
From: Doctor Mitteldorf
Subject: Re: Formulas for Fractals

Dear Shannon,

There is no one formula for a fractal, but many different procedures 
for drawing different fractals.

A fractal is a shape with a lot of detail to it. "A lot" is actually 
an understatement. The more closely you look at a fractal, the more 
the lines appear jagged and convoluted. This is the basic property 
that distinguishes a fractal from any other curve you might draw. If 
you draw an ordinary curve and look at the curve with a magnifying 
glass (figuratively) then any part of the curve will look smoother and 
smoother the closer you look, until you really can't tell it from a 
straight line. But not a fractal. The closer you look, the more detail 
shows up.

All fractals require lots and lots of computation to draw. There is no 
formula like y = a*x^2 + b or y = 1/(x-2) where you can just plot 
(x,y) and the answer has the fractal property. Even the simplest ideas 
for drawing fractals are impractical to execute by hand, and require a 
good deal of time even on a fast computer.

Here's a prescription for drawing a fractal. First, pick two numbers 
a and b, which we'll keep the same for the rest of this computation.  
Then draw your x and y axes. Every point on the plane will be colored 
either black or white. Now, of course, there is an infinite number of 
points because they are infinitely close together. So we must tell the 
computer to take samples of the plane that are a finite distance 
apart.  

For every point (x,y), here's how to decide whether to color it black 
or white:  

   Calculate a new point by the prescription:

      new x = x*x - y*y + a
      new y = 2*x*y + b

Now use your new x and y to do the same thing over again, and 
calculate another x and y. Feed those two numbers back to do the same 
thing over and over and over again.  

After a while, one of two things will happen. Either you'll see that 
the numbers get bigger and bigger (either positive or negative) and 
that looks like it's going to go on without end; if this happens color 
the point white. Or else, the numbers jump about randomly, but don't 
appear to be getting any bigger on the whole; in this case, color the 
point black. You can usually tell within 20 or so repetitions (called 
"iterations") whether the point is sailing off into the infinite or 
remaining close to home.

If you're doing this on a computer screen, you might take the dots on 
the screen as points. Say you have 640 points from left to right, and 
480 from top to bottom. The top left corner might be x = -3.20, 
y = 2.40. The next point to the right would be x = -3.19, y = 2.40, 
continuing across....

Remember that you have to repeat your computation many times for 
every single point on the screen before you decide whether each point 
is to be colored white or black. That's what makes the procedure so 
long that you need a computer. But as you can see, it's not terribly 
complicated.

Wait for the computer to finish and look at the pattern on the screen.  
You can ask the computer to "magnify" a little portion of the image, 
by selecting any point for the upper left corner. Maybe this time 
each dot on the screen will represent a jump of 0.001 instead of 0.01 
like last time. You should see new details, new curliques in the 
picture when you do this.

You don't need to be an experienced computer programmer to do this, 
but if it's your first program, you'll want to get some help.

For a good online introduction to fractals, please see:

   http://www.angelfire.com/art2/fractals/learning.htm   

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/