The Koch Snowflake

Why use this problem?

This problem is an introduction to fractals, and also uses limits and geometrical sequences.  It can be quite surprising to find that there is a theoretical shape with infinite perimeter but finite area!

Here are printable word and pdf versions of the problem.

Key Questions

• How is each iteration related to the one before?
• How does the number of sides change with each iteration?  How does the length of each side change?
• How many "new triangles" are added at each stage?  How does the size of one of these relate to the size of the original triangle?

Possible Extension

The problems Squareflake and Sierpinski Triangle explore other fractals, including the dimensions of these shapes.

In this video Claire and Charlie discuss complex numbers and how these are related to the Mandelbrot Set.

Students might like to investigate other fractals, such as the Cantor Set and the Menger Sponge.