this point rather clearly. Consider a Euclidean line segment. Bisecting that line produces two line segments, each with half the length of the original segment. Bisecting the segments again produces four segments, each with one-quarter the length of the original segment. Next, consider a square on a Euclidean plane. Bisecting each side of the square results in four copies, each one-quarter the size of the original square. Bisecting each side of the new squares will result in 16 squares, each a quarter the size of the squares in the second step. Finally, consider a cube. Bisecting each face of the cube will yield eight one-eighth sized copies of the original cube.
These cases provide an illustration of the general idea behind fractal dimension. Very roughly, fractal dimension is a measure of the relationship between how many copies of a figure are present at different levels of magnification and how much the size of those copies changes between levels of magnification. In fact, we can think of it as a ratio between these two quantities. The fractal dimension d of an object is equal to log(a)/log(b), where a = the number of new copies present at each level, and b is the factor by each piece must be magnified in order to have the same size as the original. This definition tells us that a line is one-dimensional: it can be broken into n pieces, each of which is n-times smaller than the original. If we let n = 2, as in our bisection case, then we can see easily that log(2)/log(2) = 1. Likewise, it tells us that a square is two-dimensional: a square can be broken into n2 pieces, each of which must be
- In the illustration here, we had to build in the presence of “copies” by hand, since a featureless line (or square or cube) has no self-similarity at all. That’s OK: the action of bisecting the figure is, in a sense, a purely abstract operation: we’re not changing anything about the topology of the figures in question by supposing that they’re being altered in this way. In figures with actual self-similarity (like fractals), we won’t have to appeal to this somewhat arbitrary-seeming procedure.
somewhat more technically precise than the discussion there; Mitchell hides the mathematics behind the discussion, and fails to make the connection between fractal dimension and topological dimension explicit, resulting in a somewhat confusing discussion as she equivocates between the two senses of "dimension." For a more formal definition of fractal dimensionality (especially in the case of Pythagoras Tree-like figures), see Lofstedt (2008).