Sierpinski Triangle

Date: 01/15/97 at 10:28:11
From: Anonymous
Subject: Sierpinski Triangle

Hi,

My name is Ryan and I would like to ask you a question.  What is a 
Sierpinski Triangle?

Date: 01/20/97 at 11:26:36
From: Doctor Toby
Subject: Re: Sierpinski Triangle

Waclaw Sierpinski invented the triangle (or gasket) named after him in 
1916. Here's how to make one:

Draw an equilateral triangle on a piece of paper, and shade the area 
outside the triangle.  Now divide the triangle into four equilateral 
triangles like this:

                 /\
                /  \
               /    \
              /______\
             /\      /\
            /  \    /  \
           /    \  /    \
          /______\/______\

Shade the area in the center triangle (the one that's upside down).  
Now you have three equilateral triangles left.  Divide each of these 
triangles into four triangles, and shade each of the center triangles 
again. Now you have nine triangles.  Divide them and shade their 
centers.  Repeat this process forever.

Sierpinski's gasket is the *unshaded* part.  The lines you drew are 
also part of the gasket.

Here's an amazing fact:  During the course of your drawing lines and 
shading triangles, every single point on the paper either eventually 
gets shaded or eventually becomes part of of one of the lines.  What's 
so amazing about this?  It means there are two ways to think about 
making Sierpinski's gasket, and these two ways, as you will see, seem 
to contradict each other.

Here's one way:

At each step in constructing the gasket, the area left unshaded 
(together with the lines, which form its boundary) is a two-
dimensional figure.  After you've performed an infinite number of 
steps, this figure becomes Sierpinski's gasket.  Since each figure 
along the way is two-dimensional, you might think that the gasket 
should be two-dimensional as well.

Here's another way:

At each step in constructing the gasket, the lines themselves, alone, 
form a one-dimensional figure.  After you've performed an infinite 
number of steps, this figure also becomes Sierpinski's gasket.
So now you might think the gasket is really only one-dimensional.

In fact, the gasket is neither one-dimensional nor two-dimensional.
Its dimension is a number somewhere between 1 and 2.  A figure whose 
dimension is not a whole number is called a `fractal';  Sierpinski's 
gasket is one of the oldest known fractals.  Calculating the actual 
dimension of a fractal is a tricky matter; if I didn't make any 
mistakes, the dimension of Sierpinski's gasket is about 1.58.

-Doctor Toby,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/