On the Spaces and Dimensions of Geographical Systems
  To remove the confusion of concepts about different sorts of geographical space and dimension, a new framework of space theory is proposed in this paper. Based on three sets of fractal dimensions, the geographical space is divided into three types: real space (R-space), phase space (P-space), and order space (O-space). The real space is concrete or visual space, the fractal dimension of which can be evaluated through digital maps or remotely sensed images. The phase space and order space are both abstract space, the fractal dimension values of which cannot be estimated with one or two maps or images. The dimension of phase space can be computed by using time series, and that of order space can be determined with cross-sectional data in certain time. Three examples are offered to illustrate the three types of spaces and fractal dimension of geographical systems. The new space theory can be employed to explain the parameters of geographical scaling laws, such as the scaling exponent of the allometric growth law of cities and the fractal dimension based on Horton's laws of rivers. 
Yanguang Chen
physics.soc-ph