2. B. Box-Counting Dimension

Box-Counting Dimension of the Product of a Sierpinski Gasket and a Line Segment

Arrange the gasket and line segment side-by-side. What is the box-counting dimension of this shape? To compute the box-counting dimension, cover this shape with smaller and smaller boxes. We find the pattern of boxes shown in the table below the picture.

N(1) = 2 = 1 + 1

N(1/2) = 5 = 31 + 21

N(1/4) = 13 = 9 + 4 = 32 + 22

and in general

N((1/2)n) = 3n + 2n.

Here we compute the exact value of the box-counting dimension.

Return to Box-Counting Dimension.