2. B. Box-Counting Dimension

Covering with boxes

Instead of approximating the Koch cuve with line segments or triangles, we could cover it with squares
Certainly, smaller squares will pick up more detail of the Koch curve, and will give a better approximation of the curve. Suppose we need
N(r) squares of side length r
to cover the curve. Then
N(r)⋅r approximates the length of the curve, and
N(r)⋅r2 approximates the area of the curve.
For several examples, we shall find the pattern of how N(r) changes with r. This will tell us something about the complexity of the shape.

Return to Box-Counting Dimension.