Fractal dimension and fractal growth of urbanized areas

Based on a box-accounting fractal dimension algorithm (BCFD) and a unique procedure of data processing, this paper computes planar fractal dimensions of 20 large US cities along with their surrounding urbanized areas. The results show that the value range of planar urban fractal dimension (D) is 1< D <2, with D for the largest city, New York City, and the smallest city, Omaha being 1.7014 and 1.2778 respectively. The estimated urban fractal dimensions are then regressed to the total urbanized areas, Log (C), and total urban population, Log (POP), with log-linear functions. In general, the linear functions can produce good-fits for Log (C) vs. D and Log (POP) vs. D in terms of R² values. The observation that cities may have virtually the same D or Log(C) value but quite disparate population sizes indicates that D itself says little about the specific orientation and configuration of an urban form and is not a good measure of urban population density. This paper also explores fractal dimension and fractal growth of Baltimore, MD for the 200-year span from 1792-1992. The results show that Baltimore's D also satisfies the inequality 1< D <2, with D =1.0157 in 1822 and D =1.7221 in 1992. D =0.6641 for Baltimore in 1792 is an exception due mainly to its relatively small urban image with respect to pixel size. While D always increases with Log (C) over the years, it is not always positively correlated to urban population, Log(POP).