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
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.
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Here is the data for this box-count.
| rn | N(rn) | 1/rn | Log(1/rn) | Log(N(rn)) |
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| 1 | 17 | 1 | 0 | 1.230 |
| 1/2 | 53 | 2 | .301 | 1.724 |
| 1/4 | 183 | 4 | .602 | 2.262 |
Plot the points. They appear to fall pretty close to a straight line.
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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.