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

If the object is 1dimensional,
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 will be needed to cover the object.) 

If the object is 2dimensional,
such as the (filledin) 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 boxcounting
dimension. 

To show the boxcounting dimension agrees with the standard dimension in
familiar cases, consider the filledin triangle. 

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

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

Now we compute the boxcounting 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 boxcounting dimension. 

Here are some practice problems. 

Finally, here is a common mistake
in computing boxcounting dimensions. 