Geographical space based on urban allometry and fractal dimension

TL;DR

This paper introduces a new conceptualization of urban geographical space based on fractal dimensions and allometry, linking real, phase, and order spaces to better understand city structures.

Contribution

It proposes a novel framework connecting fractal dimensions with urban allometry types, expanding the traditional concept of geographical space to include generalized spaces for geospatial analysis.

Findings

Urban allometry types correspond to different fractal dimensions.

Three categories of geographical space are defined: real, phase, and order space.

The generalized space framework enhances understanding of city fractal structures.

Abstract

The conventional concept of geographical space is mainly referred to actual space based on landscape, maps, and remote sensing images. However, this notion of space is not enough to interpret different types of fractal dimension of cities. The fractal dimensions derived from Zipf's law and time series analysis do not belong to the traditional geographical space. Based on the nature of the datasets, the urban allometry can be divided into three types: longitudinal allometry indicating time, transversal allometry indicating hierarchy, and isoline allometry indicating space. According to the principle of dimension consistency, an allometric scaling exponent must be a ratio of one fractal dimension to another. From abovementioned three allometric models, we can derive three sets of fractal dimension. In light of the three sets of fractal dimension and the principle of dimension uniqueness, urban geographical space falls into three categories, including the real space based on isoline allometry and spatial distribution, the phase space based on longitudinal allometry and time series, and order space based on transversal allometry and rank-size distribution. The generalized space not only helps to explain various fractal dimensions of cities, but also can be used to develop new theory and methods of geospatial analysis.