The Koch Curve is a very simple and very well known fractal. It is a member of the L-system of fractals. Its formual is:
F - F + + F - F.
The first 2 steps of the fractal look thus:

0) --------

     /\
1)  /  \
  --    --

For each iteration, on each line, a triangle is drawn in the middle. This causes the length of the line to explode. The length of the line is as follows:

iteration length
0             1
1            4/3
2           16/9
3           64/27
4          256/81
5         1024/243
n           4n/3n

As part of a triangle (called the Koch Snowflake) inscribed in a circle, the Koch Curve has the interesting properties of having infinite perimeter but finite surface area.