2. B. Box-Counting Dimension

A Common Mistake

Incorrect computations of dimension abound, so some care is needed when referring to examples, especially on websites. Here is an incorrect calculation, similar to a web example frequently found by students over the last few years.
We want to show the side-elevation view of the building (in blue) is a fractal.
To do this, we
cover the image with boxes of several sizes,
count the boxes, and
do a log-log plot of the number of boxes.
If the points are close to a straight line, the slope of the line is the box-counting dimension. If the dimension is not a whole number, the shape must be a fractal, because only fractals have non-integer dimensions.
Here is the data for this box-count. Plot the points. They appear to fall pretty close to a straight line.
rn N(rn) 1/rn Log(1/rn) Log(N(rn))
1 17 1 0 1.230
1/2 53 2 .301 1.724
1/4 183 4 .602 2.262
We could compute the slope by finding the best-fitting line, but some examples (incorrectly) compute slopes for each pair of points.

first and second(1.724 - 1.230)/(.301 - 0) = 1.641
second and third(2.262 - 1.724)/(.602 - .301) = 1.787
first and third(2.262 - 1.230)/(.602 - 0) = 1.714
These numbers are not close to 2, so the shape must be a fractal.
WRONG WRONG WRONG As we saw in the example of the gasket and line segment example, if a shape consists of several pieces, the dimension of the shape is the largest of the dimensions of the pieces.
This shape contains filled-in rectangles, having dimension 2, so the whole shape has dimension 2.
What went wrong with the calculation? Not nearly small enough boxes were used. Box-counting ratios can be very slow to converge to the dimension.
When you look at a website, the first question you should ask is "Based on what I know, does this make sense?" Not nearly everything posted on the web is correct.

Return to Box-Counting Dimension.