Box-Counting Dimension of a Gasket

Plotting the points in the table below, we obtain the graph on the right. The points lie on a straight line of slope about 1.59. (The graph shows a greater range, and more widely spaced, points than these.)

(Log(1/r₀), Log(N(r₀))) = (Log(1), Log(1)) = (0, 0)

(Log(1/r₁), Log(N(r₁))) = (Log(2), Log(3)) = (0.301, 0.477)

(Log(1/r₂), Log(N(r₂))) = (Log(4), Log(9)) = (0.602, 0.954)

(Log(1/r₃), Log(N(r₃))) = (Log(8), Log(27)) = (0.903, 1.432)

(Log(1/r₄), Log(N(r₄))) = (Log(16), Log(81)) = (1.204, 1.908)

... ... ...

So the box-counting dimension of the gasket is about 1.59.

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