Box-Counting Dimension of a Line Segment

Covering a line segment with smaller and smaller boxes we see the pattern in the table on the right.

N(1) = 1

N(1/2) = 2 = 1/(1/2)

N(1/4) = 4 = 1/(1/4)

N(1/8) = 8 = 1/(1/8)

and so in general

N(r) = 1/r.