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		1.5.4— Example Where the Average Does Not Exist: Diffusion Limited Aggregation (DLA)

		1— The Average Density of a Non-Fractal Checkerboard

		We want to determine the average density of the pixels in a black and white checkerboard. To find the average density within a circle of radius r, we count the number of black pixels within the circle and divide that by the total number of pixels. We repeat this measurement for circles of larger radii. As the radius increases, we find that there are some fluctuations in these averages. However, we also find that as the radius increases, these average densities approach a finite, limiting value that we therefore identify as the "real" average density of the checkerboard.

		2— There Is No Average Density of a Fractal DLA

		We now want to determine the average density of a fractal object called a diffusion limited aggregate (DLA). It is formed by particles that are released one at a time from far away and randomly walk until they hit and stick to the growing structure. The DLA is self-similar. It has little spaces between the small arms and bigger spaces between the larger arms. We measure the average density within a circle of radius r. As the radius of the circle increases, we catch more of the ever bigger spaces between the ever larger arms. Thus, as the radius r of the circle increases, the average density within radius r decreases. The average density does not reach a finite, limiting value. It approaches zero. Therefore, the "real" average density for the DLA does not exist.

 

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