The relationship between the number of pieces and the scaling factor for the Koch curve does not produce an integer. In fact, the answer depends on the length of the Ruler you use for the measurements. A shorter Ruler measures more of the sinuosity of bays and inlets than a larger one, so the estimated length continues to increase as the Ruler length decreases. I propose that curves for which the fractal dimension exceeds the topological dimension 1 be called fractal curves. And within the scale of interest to a geographer, coastlines can be modeled by fractal curves.
Coastlines are fractal patterns.
The coast line length turns out to be an elusive notion that slips between the fingers of one who wants to grasp it. All measurements methods ultimately lead to the conclusion that the typical coast line s length is very large and so ill determined that it is best considered infinite. Hence, if one wishes to compare different coastlines from the viewpoint of their extent, length is an inadequate concept. As you can see each leaf of the fern is in fact the same as the full fern. This infinite level of self-similarity is what gives this image its fractal dimension.