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### 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.

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
making Sierpinski's gasket, and these two ways, as you will see, seem

Here's one way:

(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

-Doctor Toby,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Fractals

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