Recursive Subdivision of Urban Space and Zipf's law
  Zipf's law can be used to describe the rank-size distribution of cities in a region. It was seldom employed to research urban internal structure. In this paper, we demonstrate that the space-filling process within a city follows Zipf's law and can be characterized with the rank-size rule. A model of spatial disaggregation of urban space is presented to depict the spatial regularity of urban growth. By recursive subdivision of space, an urban region can be geometrically divided into two parts, four parts, eight parts, and so on, and form a hierarchy with cascade structure. If we rank these parts by size, the portions will conform to the Zipf distribution. By means of GIS technique and remote sensing data, the model of recursive subdivision of urban space is applied to three cities of China. The results show that the intra-urban hierarchy complies with Zipf's law, and the values of the rank-size scaling exponent are very close to 1. The significance of this study lies in three aspects. First, it shows that the strict subdivision of space is an efficient approach to revealing spatial order of urban form. Second, it discloses the relationships between urban space-filling process and the rank-size rule. Third, it suggests a new way of understanding fractals, Zipf's law, and spatial organization of urban evolution. 
Yanguang Chen, Jiejing Wang
physics.soc-ph