Third Homework Set Answers

4(c) On the left is the completed table, on the right is the graph.

r N(r) 1/r Log(1/r) Log(N(r))

.25 6 4 0.602 0.778

.25² = .0625 36 16 1.204 1.556

.25³ = .0156 6³ = 216 4³ = 64 1.806 2.334

.25⁴ = .0039 6⁴ = 1296 4⁴ = 256 2.408 3.112

.25⁵ = .0010 6⁵ = 7776 4⁵ = 1024 3.010 3.891

.25⁶ = .0002 6⁶ = 46656 4⁶ = 4096 3.612 4.669

The points appear to fall along a straight line; the slope of this line is the box-counting dimension. We measure dim = (4.669 - 0.778)/(3.612 - 0.602) = 1.293.

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