Box-Counting Dimension of a Filled-in Triangle

Covering a filled-in triangle with smaller and smaller boxes

we see

N(1) = 1

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

N(1/4) = 10 = 1 + 2 + 3 + 4

N(1/8) = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

and in general

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

The pattern may not be as obvious as those of the covering of the line or square. We expect the filled-in triangle is 2-dimensional, so we should get a box-counting dimension of 2. Before trying to develop an analytical argument, let's do the Log-Log plot.

Although not transparent, the pattern is simple enough that we can prove the filled-in triangle has box-counting dimension 2.

Return to Box-Counting Dimension.