hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su

The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

# Koch Snowflake

 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 "self-similarity" 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 self-intersect, 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 3-dimensional 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.

Su, Francis E., et al. "Koch Snowflake." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
K. Falconer, Fractal Geometry.

Keywords:    real analysis
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Jorge Aarao
Suggestions? Use this form.
4.29
current
rating
Click to rate this Fun Fact...
*   Awesome! I totally dig it!
*   Fun enough to tell a friend!
*   Mildly interesting
*   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!