Dimension by Box-Counting

Grid Placement Sample

Pictured below are two coverings of the Koch curve with boxes of size 16. The offsets are

top: x = 0, y = 0

bottom: x = 8, y = 12

In the top picture, 60 boxes are occupied by parts of the Koch curve.

In the bottom picture, 73 boxes are occupied.

If only a small range of box sizes is used, the grid placement can have a significant effect on the estimated dimension.

Here are complete calculations for two offsets. The size = 4 grids are displayed

top: x = 0, y = 0

bottom: x = 3, y = 3

Here are the box counts and the corresponding dimension calculations.

offset = 0, 0 offset = 3,3

box size count

64 9

32 27

16 57

8 162

4 396

box size count

64 9

32 27

16 61

8 163

4 383

dimension = 1.350 dimension = 1.342

Offset can alter the count, but both these estimates are too high. This is a cautionary note about how many sizes are needed to accurately estimage dimension. The range from size 4 to size 64 is not much over an order of magnitude. Evidently, more is needed.

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