2. B. Box-Counting Dimension

We have seen that trying to measure the length of the Koch curve gives infinity, while trying to measure the area of the Koch curve gives zero.

Neither is a useful result. Here we shall introdce a more general measure that leads to the idea of box-counting dimension.

We cover a shape with boxes and find how the number of boxes changes with the size of the boxes.

If the object is 1-dimensional, such as the unit line segment, we expect N(r) = 1/r. (It's 1/r instead of r because as the squares get smaller, more are needed to cover the object.)

If the object is 2-dimensional, such as the (filled-in) unit square, we expect N(r) = (1/r)².

For more complicated shapes, the relation between N(r) and 1/r may be a power law, N(r) = k⋅(1/r)^d.

This leads to the definition of the box-counting dimension.

For the Sierpinski gasket we obtain d_b = Log(3)/Log(2) = 1.58996 ... . The gasket is more than 1-dimensional, but less than 2-dimensional.

For the Koch curve we obtain d_b = Log(4)/Log(3) = 1.26186 ... . The Koch curve is more than 1-dimensional, but less than 2-dimensional.

Now we compute the box-counting dimension of the Cantor Middle Thirds Set.

What happens when we measure an object in the wrong dimension?

and of a combination of the Cantor set and line segment.

and of a combination of the Gasket and line segment.

Here is some Java software to investiate properties of the box-counting dimension, and an application to estimating the dimenson of a deer skull suture.

Here is a common mistake in computing box-counting dimensions.

Finally, here are some practice problems.