| 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. |