Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities. 1. Introduction Dimension is a measurement of space, and measurement is the basic link between mathematical models and empirical research. So, dimension is a necessary measurement for spatial analysis. Studying geographical spatial phenomena of scaling invariance such as cities and systems of cities has highlighted the value of fractal dimension [1–6]. However, there are two problems in practical work. On the one hand, sometimes it is difficult for us to determine the numerical value of fractal dimension for some realistic systems, but it is fairly easy to calculate the ratio of different fractal parameters. On the other hand, in many cases, it is enough to reveal the system’s information by the fractal-dimension ratios and it is unnecessary to compute fractal dimension further [7]. The ratio of different dimensions of a fractal can constitute an allometric coefficient under certain conditions. As a parameter of scale-free systems, the allometric coefficient is in fact a scaling exponent, and the fractal-dimension ratio can be called allometric scaling exponent (ASE). The use of ASEs for the simple regular fractals in the mathematical world is not very noticeable. But for the quasifractals or random fractals in the real world, the function of ASEs should be viewed with special respect. A city can be regarded as an evolutive fractal. The land-use
References
[1]
M. Batty, Cities and Complexity: Understanding Cities with Cellular Automata, MIT Press, Cambridge, Mass, USA, 2005.
[2]
M. Batty and P. A. Longley, Fractal Cities: A Geometry of Form and Function, Academic Press, London, UK, 1994.
[3]
P. Frankhauser, La Fractalité des Structures Urbaines, Economica, Paris, France, 1994.
[4]
G. Haag, “The rank-size distribution of settlements as a dynamic multifractal phenomenon,” Chaos, Solitons and Fractals, vol. 4, no. 4, pp. 519–534, 1994.
[5]
B. B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, San Francisco, Calif, USA, 1982.
[6]
R. White and G. Engelen, “Urban systems dynamics and cellular automata: fractal structures between order and chaos,” Chaos, Solitons and Fractals, vol. 4, no. 4, pp. 563–583, 1994.
[7]
Y. Chen and S. Jiang, “An analytical process of the spatio-temporal evolution of urban systems based on allometric and fractal ideas,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 49–64, 2009.
[8]
M. Batty, “Generating urban forms from diffusive growth,” Environment & Planning A, vol. 23, no. 4, pp. 511–544, 1991.
[9]
M. Batty, “Cities as fractals: simulating growth and form,” in Fractals and Chaos, A. J. Crilly, R. A. Earnshaw, and H. Jones, Eds., pp. 43–69, Springer, New York, NY, USA, 1991.
[10]
L. Benguigui, D. Czamanski, M. Marinov, and Y. Portugali, “When and where is a city fractal?” Environment and Planning B, vol. 27, no. 4, pp. 507–519, 2000.
[11]
L. Benguigui, E. Blumenfeld-Lieberthal, and D. Czamanski, “The dynamics of the Tel Aviv morphology,” Environment and Planning B, vol. 33, no. 2, pp. 269–284, 2006.
[12]
M.-L. De Keersmaecker, P. Frankhauser, and I. Thomas, “Using fractal dimensions for characterizing intra-urban diversity: the example of Brussels,” Geographical Analysis, vol. 35, no. 4, pp. 310–328, 2003.
[13]
I. Thomas, P. Frankhauser, and M.-L. De Keersmaecker, “Fractal dimension versus density of built-up surfaces in the periphery of Brussels,” Papers in Regional Science, vol. 86, no. 2, pp. 287–308, 2007.
[14]
I. Thomas, P. Frankhauser, and C. Biernacki, “The morphology of built-up landscapes in Wallonia (Belgium): a classification using fractal indices,” Landscape and Urban Planning, vol. 84, no. 2, pp. 99–115, 2008.
[15]
J. Gayon, “History of the concept of allometry,” American Zoologist, vol. 40, no. 5, pp. 748–758, 2000.
[16]
Y. Lee, “An allometric analysis of the US urban system: 1960–80,” Environment & Planning A, vol. 21, no. 4, pp. 463–476, 1989.
[17]
R. S. Naroll and L. von Bertalanffy, “The principle of allometry in biology and social sciences,” General Systems Yearbook, vol. 1, Part II, pp. 76–89, 1956.
[18]
R. S. Naroll and L. von Bertalanffy, “The principle of allometry in biology and social sciences,” Ekistics, vol. 36, no. 215, pp. 244–252, 1973.
[19]
S. Nordbeck, “Urban allometric growth,” Annaler B, vol. 53, no. 1, pp. 54–67, 1971.
[20]
W. R. Tobler, “Satellite confirmation of settlement size coefficients,” Area, vol. 1, no. 3, pp. 30–34, 1969.
[21]
M. J. Beckmann, “City hierarchies and distribution of city sizes,” Economic Development and Cultural Change, vol. 6, no. 3, pp. 243–248, 1958.
[22]
Y. Chen and Y. Zhou, “Multi-fractal measures of city-size distributions based on the three-parameter Zipf model,” Chaos, Solitons and Fractals, vol. 22, no. 4, pp. 793–805, 2004.
[23]
P. A. Longley, M. Batty, and J. Shepherd, “The size, shape and dimension of urban settlements,” Transactions of the Institute of British Geographers, vol. 16, no. 1, pp. 75–94, 1991.
[24]
P. A. Longley and M. Batty, “On the fractal measurement of geographical boundaries,” Geographical Analysis, vol. 21, no. 1, pp. 47–67, 1989.
[25]
P. A. Longley and M. Batty, “Fractal measurement and line generalization,” Computers and Geosciences, vol. 15, no. 2, pp. 167–183, 1989.
[26]
J.-H. He and J.-F. Liu, “Allometric scaling laws in biology and physics,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 1836–1838, 2009.
[27]
B. J. West, “Comments on the renormalization group, scaling and measures of complexity,” Chaos, Solitons and Fractals, vol. 20, no. 1, pp. 33–44, 2004.
[28]
Y. Chen and Y. Zhou, “Reinterpreting central place networks using ideas from fractals and self-organized criticality,” Environment and Planning B, vol. 33, no. 3, pp. 345–364, 2006.
[29]
Y. Chen and Y. Zhou, “Scaling laws and indications of self-organized criticality in urban systems,” Chaos, Solitons and Fractals, vol. 35, no. 1, pp. 85–98, 2008.
[30]
R. Jullien and R. Botet, Aggregation and Fractal Aggregates, World Scientific, Teaneck, NJ, USA, 1987.
[31]
T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 1989.
[32]
R. White and G. Engelen, “Cellular automata and fractal urban form: a cellular modelling approach to the evolution of urban land-use patterns,” Environment & Planning A, vol. 25, no. 8, pp. 1175–1199, 1993.
[33]
P. Frankhauser, “The fractal approach: a new tool for the spatial analysis of urban agglomerations,” Population, vol. 10, no. 1, pp. 205–240, 1998.
[34]
P. Longley, M. Batty, J. Shepherd, and G. Sadler, “Do green belts change the shape of urban areas? A preliminary analysis of the settlement geography of South East England,” Regional Studies, vol. 26, no. 5, pp. 437–452, 1992.
[35]
M. Batty and K. S. Kim, “Form follows function: reformulating urban population density functions,” Urban Studies, vol. 29, no. 7, pp. 1043–1069, 1992.
[36]
A. R. Imre and J. Bogaert, “The fractal dimension as a measure of the quality of habitats,” Acta Biotheoretica, vol. 52, no. 1, pp. 41–56, 2004.
[37]
Y. Chen and J. Lin, “Modeling the self-affine structure and optimization conditions of city systems using the idea from fractals,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 615–629, 2009.
[38]
X. Wang, J. Liu, D. Zhuang, and L. Wang, “Spatial-temporal changes of urban spatial morphology in China,” Acta Geographica Sinica, vol. 60, no. 3, pp. 392–400, 2005.
[39]
M. Batty and P. A. Longley, “The morphology of urban land use,” Environment & Planning B, vol. 15, no. 4, pp. 461–488, 1988.
[40]
C. P. Lo and R. Welch, “Chinese urban population estimates,” Annals of the Association of American Geographers, vol. 67, no. 2, pp. 246–253, 1977.
[41]
Y. G. Chen, Fractal Urban Systems: Scaling, Symmetry, and Spatial Complexity, Scientific Press, Beijing, China, 2008.
[42]
K. Davis, “World urbanization: 1950–1970,” in Systems of Cities, I. S. Bourne and J. W. Simons, Eds., pp. 92–100, Oxford University Press, New York, NY, USA, 1978.
[43]
M. Batty, “The size, scale, and shape of cities,” Science, vol. 319, no. 5864, pp. 769–771, 2008.
