2. B. Box-Counting Dimension

Measuring in the Wrong Dimension

The reason for the useless results we got when we tried to measure the area and length of the Koch curve is this.

Trying to measure in a dimension lower than an object gives infinity (Imagine the length of infinitely thin thread needed to cover up a filled-in square.) and

trying to measure in a dimension higher than an object gives zero. (Think of the volume of a filled-in square. How much water can it hold?)

For the Koch curve, the dimension lies between 1 and 2, so we should not be surprised that its length is infinite and its area zero.

To emphasize this point, recall that using boxes of side length r_n -> 0 as n -> infinity,

we measure the length of a shape by N(r_n) r_n as n -> infinity, and

we measure the area by N(r_n) r_n² as n -> infinity.

So to measure a shape in dimension d, we might expect to use

N(r_n) r_n^d as n -> infinity.

Here are some graphs, using the Koch curve data, for different values of d.

The curves support the fact that the Koch curve dimension lies betwen 1.2 and 1.3.

Note: being an exponent, the dimension can be found with fairly coarse calculations, such as covering the shape with boxes of a limited collection of sizes.

Determining the measure in a particular dimension is a much more subtle problem: N(r_n) r_n^d is not adequate to find the measure.

Return to Box-Counting Dimension.