The Fractal Dimension

Abstract

How long is the coast of Norway? Take a look at figure 2.1. On the scale of the map the deep fjords on the western coast show up clearly. The details encountered moving northeast along the coast from the southern tip are more difficult to resolve, but I can assure you that the maps I use when sailing in that area show structures quite similar to those of the west coast. In fact when sailing there you find rocks, islands, bays, faults and gorges that look much the same but do not show up even on my detailed maps. Before answering the question we have to decide on whether the coast of the islands should be included. And what about the rivers? Where does the fjord stop being a fjord and become a river? Sometimes this is an easy question and sometimes not. However, even after we have decided on all these questions we have a difficulty. I could walk a divider with an opening corresponding to δ km along the coastline on the map and count the number of steps, N(δ), needed to move from one end of the map to the other. Being in a hurry I would choose such a large opening of the divider that I would not have to bother about even the deepest fjords and estimate the length to be L = N(δ) · δ. If objections were raised I would use a somewhat smaller opening δ and try again. This time the large fjords would contribute to the measured length but the southeastern coast would still be taken in a few steps. For a really serious discussion I would have to get the kind of maps neighbors use when they settle questions of where the fence should go, or how far up the river the fishing rights extend.