Coastline of a fractal island.

Imagine you start out with a square island 27 units long on a side. Then it is divided into 9 equal blocks, and the middle blocks on each side are removed, leaving the corner and very middle block. In the next reiteration of creating this island, you repeat this same rule for each of the remaining 5 blocks. Each block will end up creating 5 blocks, each with a length one third of the original block. One can compute the aggregate 'coastline' length for the assemblage as given in the table below. If we now think of using successively smaller rulers, each one the size of the side of the square element, then we can see how this coast line length changes as one uses smaller and smaller rulers to measure its length.

Fractal island as seen through reiteration.

reiteration length of side = l, ruler size # of boxes or elements length of outer perimeter = p log(l) log(p)
0 27 1 108 1.43136376415899 0
1 9 5 180 0.954242509439325 0.698970004336019
2 3 25 300 0.477121254719662 1.39794000867204
3 1 125 500 0 2.09691001300806
4 0.333333333333333 625 833.333333333333 -0.477121254719662 2.79588001734408
5 0.111111111111111 3125 1388.88888888889 -0.954242509439325 3.49485002168009
6 0.037037037037037 15625 2314.81481481481 -1.43136376415899 4.19382002601611
7 0.0123456790123457 78125 3858.02469135802 -1.90848501887865 4.89279003035213

The graph for this is below, and we once again see how as we look closer and closer the log of the length increases linearly.


Copyright by Harmon D. Maher Jr.. This material may be used for non-profit educational purposes if proper attribution is given. Otherwise please contact Harmon D. Maher Jr.