Fractal systems of central places based on intermittency of space-filling

Abstract The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d  = 2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D d  = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places.

[1]  Michael Batty,et al.  URBAN SHAPES AS FRACTALS , 1987 .

[2]  M. Goodchild,et al.  The Fractal Nature of Geographic Phenomena , 1987 .

[3]  A. Lösch.  The economics of location , 1954 .

[4]  Wendong Xu,et al.  Symmetry distribution of cities in China , 2001 .

[5]  S. Albeverio.  The dynamics of complex urban systems : an interdisciplinary approach , 2008 .

[6]  Pierre Frankhauser,et al.  The fractal approach. A new tool for the spatial analysis of urban agglomerations , 1998, Population.

[7]  Feasible optimality implies Hack's Law , 1998 .

[8]  Michael Batty,et al.  Fractal Cities: A Geometry of Form and Function , 1996 .

[9]  Reginald G. Golledge,et al.  Cities, Space and Behavior: The Elements of Urban Geography , 1978 .

[10]  Yanguang Chen,et al.  Analogies between urban hierarchies and river networks: Fractals, symmetry, and self-organized criticality , 2009 .

[11]  Nina S. N. Lam,et al.  Fractals in Geography , 1993 .

[12]  S. Havlin,et al.  Scaling laws of human interaction activity , 2009, Proceedings of the National Academy of Sciences.

[13]  B. Basu,et al.  Zipf’s law and distribution of population in Indian cities , 2009 .

[14]  P. Haggett,et al.  Integrated Models in Geography , 2014 .

[15]  Alan W. Evans.  Urban Economics: An Introduction , 1985 .

[16]  Michael Batty,et al.  Hierarchy in Cities and City Systems , 2006 .

[17]  S. Appleby.  Multifractal Characterization of the Distribution Pattern of the Human Population , 2010 .

[18]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[19]  M. Batty,et al.  The size, shape and dimension of urban settlements , 1991 .

[20]  Roger White,et al.  Cellular Automata and Fractal Urban Form: A Cellular Modelling Approach to the Evolution of Urban Land-Use Patterns , 1993 .

[21]  Yanguang Chen,et al.  Reinterpreting Central Place Networks Using Ideas from Fractals and Self-Organized Criticality , 2006 .

[22]  R. Botet,et al.  Aggregation and Fractal Aggregates , 1987 .

[23]  I. Prigogine,et al.  Order out of chaos , 1984 .

[24]  I. Rodríguez‐Iturbe,et al.  The fractal nature of river networks , 1988 .

[25]  Yanguang Chen,et al.  Multi-fractal measures of city-size distributions based on the three-parameter Zipf model , 2004 .

[26]  S. Arlinghaus.  Fractals Take a Central Place , 1985 .

[27]  Susanna C. Manrubia,et al.  Intermittency model for urban development , 1998 .

[28]  S. Arlinghaus,et al.  The Fractal Theory of Central Place Geometry: A Diophantine Analysis of Fractal Generators for Arbitrary Löschian Numbers , 2010 .

[29]  D. Zanette,et al.  ROLE OF INTERMITTENCY IN URBAN DEVELOPMENT : A MODEL OF LARGE-SCALE CITY FORMATION , 1997 .

[30]  Yanguang Chen,et al.  Urban gravity model based on cross-correlation function and Fourier analyses of spatio-temporal process , 2009 .

[31]  D. Pumain.  Hierarchy in natural and social sciences , 2006 .

[32]  P. Allen.  Cities and Regions as Self-Organizing Systems: Models of Complexity , 1997 .

[33]  Brian J. L. Berry,et al.  RIVERS AND CENTRAL PLACES: ANALOGOUS SYSTEMS? , 1967 .

[34]  Roger White,et al.  Urban systems dynamics and cellular automata: Fractal structures between order and chaos , 1994 .

[35]  Gerard Rushton,et al.  Locational Analysis in Human Geography , 1967 .

[36]  Central place fractals: Theoretical geography in an urban setting , 1989 .

[37]  Per Bak,et al.  How Nature Works: The Science of Self‐Organized Criticality , 1997 .

[38]  Yanguang Chen,et al.  Scaling laws and indications of self-organized criticality in urban systems , 2008 .

[39]  K. Davis,et al.  World urbanization, 1950-1970 , 1969 .

[40]  A. Rinaldo,et al.  Fractal River Basins: Chance and Self-Organization , 1997 .

[41]  Isao Orishimo,et al.  Systems of Cities , 1982 .

[42]  B. Berry,et al.  Central places in Southern Germany , 1967 .

[43]  B. H. Kaye.  A random walk through fractal dimensions , 1989 .

[44]  Lucien Benguigui,et al.  When and Where is a City Fractal? , 2000 .

[45]  Günter Haag,et al.  The rank-size distribution of settlements as a dynamic multifractal phenomenon , 1994 .

[46]  Hernán D. Rozenfeld,et al.  Laws of population growth , 2008, Proceedings of the National Academy of Sciences.

[47]  R. Rosso,et al.  On the fractal dimension of stream networks , 1989 .

[48]  B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

[49]  Martin J. Beckmann,et al.  City Hierarchies and the Distribution of City Size , 1958, Economic Development and Cultural Change.

[50]  L. King.  Central-Place Theory , 2021, Encyclopedic Dictionary of Archaeology.