Koch Curve

The Koch curve (also known as the von Koch curve) is a type of fractal. It was first described in a 1904 paper by Swedish mathematician Helge von Koch, making it one of the first fractal curves to be described.

It's pretty simple to create:

• Start with a straight line.
• Break the line into three equal line segments.
• Using the middle line segment, create an equilateral triangle with that segment as the base, and then delete the original line segment.
• Repeat the above process indefinitely.

A Koch snowflake.

Repeating the process an infinite amount of times creates the Koch curve. The first seven iterations of this process are illustrated at the top of the page.

A similar curve is the Koch snowflake, which is the same as a Koch curve, except the starting point is a triangle, which results in the shape on the right, one that looks a little bit like a snowflake. Even though the length of the curve is infinite, the area enclosed by the curve is finite; it converges on an area 8/5 of the area of the original triangle.

Here are some activities to try:

• If you have experience with infinite series, prove that the area of the Koch snowflake is 8/5 of the area of the original triangle.
• Try constructing an anti-snowflake by drawing the triangles inward instead of outward. What is the limiting area of this curve?
• What would happen if you were to construct the curve in a different manner, such as by using other polygons instead of triangles, or by constructing multiple polygons on each side?

Sources used (see bibliography page for titles corresponding to numbers): 17, 56.