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Growth Factors

Children who first encounter changes of scale in the lower grades will recognize much later, when they learn about exponential notation, that doubling the size in dimension three leads to an increase in the volume of a factor of 23, whereas doubling the size of a two-dimensional square increases its area by 22. Whatever it might mean to have a box in four dimensions, exponents make very clear a pattern of doubling that predicts its size will increase by 24.

Each dimension, therefore, corresponds to its own growth exponent. A surprising fact is that there are geometric objects whose growth exponents are not whole numbers. These strange objects, which have a kind of "fractional dimension," are examples of a fascinating collection of geometric patterns known as "fractals." Since the creation of fractals usually requires a process that is applied an infinite number of times, it is only with the advent of modern computer graphics that it has been possible to carry out the experiments necessary to explore them effectively.

One of the earliest examples of a fractal was invented long before computers by the Polish mathematician Waclaw Sierpinski. The first step in creating Sierpinski's figure is to remove a small triangle from the middle of a large one. The second step is the same as the first: remove the middle of each of the remaining triangles. Repeat this over and over again to obtain what is known as the "Sierpinski gasket" (Figure 18).

Figure 18. This infinitely punctured triangle, known as Sierpinski's gasket, comprises three half-size copies of itself-not two or four as one would expect if its dimension were one or two. Hence it has a fractional dimension in between one and two.

What's remarkable about Sierpinski's gasket is that doubling its size produces a figure that is composed of three copies of the original figure. This is very strange, because our experiments with tiles and cubes show that doubling factors are always powers of 2: if we double the size of something of dimension one, we get two copies of the original, whereas if we double the size of something of dimension two, we get four copies of the original. The Sierpinski gasket, therefore, must have a dimension somewhere between one and two-hence a fractional dimension. (Specifically, its dimension is the number d with the property that 2d = 3; this number d is the logarithm of three to the base two, namely 1.5849 .... )

Fractals can be used to motivate a large number of mathematical discussions. Since they arise as a result of an infinite process, they can be discussed in relation to geometric series or repeating decimals. The unusual doubling properties of fractals give a geometric interpretation for the logarithm to base two. Other fractal processes lead to figures like the Mandelbrot set, including some of the most striking examples of mathematical art.5 [an error occurred while processing this directive]