A Set of Formulae on Fractal Dimension Relations and Its Application to Urban Form

TL;DR

This paper derives new mathematical formulae relating fractal dimensions of urban boundaries and forms, providing more accurate tools for urban analysis and planning based on fractal geometry.

Contribution

It introduces a corrected formula for calculating urban boundary dimensions and establishes a hyperbolic relation between boundary and form dimensions.

Findings

Derived a new boundary dimension formula based on box-counting and dimensional consistency.

Established the hyperbolic relation between boundary and form dimensions.

Validated the formulae with data from Chinese and UK cities.

Abstract

The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is about 1.7071. The fractal dimension relation formulae are applied to China's cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.