1.C. Initiators and Generators

The Koch curve

Take as initiator the line segment of length 1, and as generator the shape on the right.

Click the initiator on the left to see successive iterates of the rule. This gives a sequence of shapes converging to the Koch curve, not named after the mayor of New York. (This is still true, but no longer as funny as it was once.)

Though its construction is so simple, the Koch curve has some properties that appear counterintuitive.

For example, we shall see that it is infinitely long, and that every piece of it, no matter how small it appears, also is infinitely long.

Click the picture to see four copies, indicated by colors.

Using the shape of the generator as a guide, we see the Koch curve is made of four copies of itself, each scaled by a factor of 1/3 horizontally and vertically.

Challenge The Koch curve is made of two copies of itself. Do you see them? What is the scaling factor?

Return to Initiators and Generators.