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.

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 |