Panorama of Fractals and Their Uses

Rivers

Box-counting is a third approach to computing dimensions of rivers. Below we see a trace of part of the Mississippi river, covered by a sequence of smaller boxes. Notice that smaller boxes pick up more detail of the river. Denoting by

N(r) the number of boxes of side length r needed to cover the river trace,

we expect

N(r) = kr^-D.

Plotting Log(N(s)) versus Log(1/s) should give points lying approximately on a straight line of slope D.

Here is a table of the data in these three pictures.

side length r	number of boxes N(r)
1/4	52
1/8	115
1/16	275

The plot of Log(N(r) vs Log(1/r) shows the points lie very nicely along a straight line of slope 1.2. So at least in this range of scales, this part of the Mississippi River has dimension D = 1.2.

Return to Rivers.