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.
How to Cite this Page:
Su, Francis E., et al. “Koch Snowflake.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
K. Falconer, Fractal Geometry.
Fun Fact suggested by:
Jorge Aarao