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 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 n$2$ pieces, each of which must be

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