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 onequarter 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 onequarter 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 oneeighth 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^{[1]}. 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 onedimensional: it can be broken into n pieces, each of which is ntimes 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 twodimensional: a square can be broken into n^{2} pieces, each of which must be

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 Treelike figures), see Lofstedt (2008).
 ↑ 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 selfsimilarity 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 selfsimilarity (like fractals), we won’t have to appeal to this somewhat arbitraryseeming procedure.
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