Box-Counting Dimension of a Deer Skull Suture

Here we see two pictures of a deer skull suture, photos by Donna Laine.

To estimate the box-counting dimension of part of the suture, we cover the image with a collection of boxes, each smaller box has 1/2 the side length of larger box. Five levels of box sizes gives an overall side length reduction by a factor of 16, hardly more than one decade of data, so this example is meant only to illustrate the process and should not be taken as a serious demonstration of fractality. Typically, at least two decades (the smallest boxes have side length 1/100 that of the largest boxes) are required for a plausible claim of fractality.

Pictured here are the largest and smallest boxes. Certainly, smaller boxes pick up more detail than larger. The dimension measures this increase, under the assumption - validated if in a log-log plot the data points lie nearly on a straight line - that this increase is according to a power law.

The points do appear to lie close to a straight line. The slope of this line approximates the dimension of the suture.

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