Figure 1

Snowflakes are amazing creations of nature. They seem
to have intricate detail no matter how closely you look
at them. One way to model a snowflake is to use a
fractal which is any mathematical object showing
"selfsimilarity" at all levels.
The Koch snowflake is constructed as follows.
Start with a line segment. Divide it into 3
equal parts. Erase the middle part and substitute
it by the top part of an equilateral triangle.
Now, repeat this procedure for each of the 4
segments of this second stage.
See Figure 1. If you continue repeating this
procedure, the curve will never selfintersect,
and in the limit you get a shape known as the
Koch snowflake.
Amazingly, the Koch snowflake is a curve of
infinite length!
And, if you start with an equilateral triangle
and do this procedure to each side, you will
get a snowflake, which has finite area, though
infinite boundary!
Presentation Suggestions:
Draw pictures. If they like this Fun Fact, ask them:
can you figure out how to construct a 3dimensional
example? [Hint: start with a regular tetrahedron.
See Koch Tetrahedron for what happens.]
The Math Behind the Fact:
You can see that the boundary of the snowflake has infinite
length by looking at the lengths at each stage of
the process, which grows by 4/3 each time the process
is repeated. On the other hand, the area inside the
snowflake grows like an infinite series,
which is geometric and converges to a finite area!
You can learn about fractals in a course on
dynamical systems.
How to Cite this Page:
Su, Francis E., et al. "Koch Snowflake."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
