Click on the graphic above to generate a new image.

**Sierpinski's Gasket** is an example of a fractal. A
fractal is an object that is self-similar across multiple scales. If
you were to zoom in on an area of the image above, it would not look
considerably different. Fractals have amazing properties: some
fractals are infinite in area, and some are infinite in perimeter.
This isn't necessarily exceptional, until you realize that fractals
are finitely bounded. What this means is you can have a circle of
known area and known perimeter that contains a fractal of infinite
area and/or infinite perimeter inside itself! This is possible
because fractals are not entirely one-dimensional, but rather they
vary between 1 and 2 dimensions.

**Sierpinski's Gasket** is especially cool because it
produces a pattern familiar to some students of algebra. Remember
polynomial expansion from school?

`
(1 + x) ^{0} = 1
(1 + x)^{1} = 1 + x
(1 + x)^{2} = 1 + 2x + x^{2}
(1 + x)^{3} = 1 + 3x + 3x^{2} + x^{3}
(1 + x)^{4} = 1 + 4x + 6x^{2} + 4x^{3} +
x^{4}
...
(1 ± x)^{n} = 1 ± nx⁄1! +
n(n - 1)x^{2}⁄2! + ... (x^{2}
< 1)
`

If you disregard the `x`'s, and focus on the coefficients, you
can create a diagram known as **Pascal's Triangle**.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1

**Pascal's triangle** is interesting for a number of
reasons, one of which involves combinatorial mathematics. How many
ways are there to choose two objects from a collection of five unique
objects? Let's say the objects are the letters `A`,`B`,
`C`, `D`, and `E`. The possible combinations of
two of these letters (where order does not matter) are `AB`,
`AC`, `AD`, `AE`, `BC`, `BD`,
`BE`, `CD`, `CE`, and `DE`. There are 10
total combinations. Did you notice that by counting five down and two
over in Pascal's Triangle you find a 10?

What do you get if you color the odd numbers in Pascal's Triangle black?

111121133114 6 411510 10511615201561172135352171

A little hard to see in the text, so I'll draw it with symbols:

• • • • • • • • • • • • • • • • • • • • • • • • • • •

Yes, there it is again! At least to me, that's unbelievable. Using a very simple pattern you can create Sierpinski's Triangle, which relates to some fundmental sets of numbers in mathematics.

If you're interested in this kind of thing, a neat book (and a quick
read) is *Introducing Fractal Geometry* by Nigel Lesmoir-Gordon,
Will Rood and Ralph Edney.

If you're interested in the source code that generates the fractal, it's here. The applet source code is here.