Fractal dimension accounts for allometric scaling exponents of cities

Fractal theory has long been used to explain the scaling exponent of allometric growth models. However, the relationship between fractals and allometry is still questioned by some scholars. One viewpoint is that fractals involve scaling laws, but scaling laws do not necessarily mean fractals. The reason for this misunderstanding may lie in that the knowledge linking fractals and allometric scaling laws has not yet been effectively clarified. By means of mathematical reasoning and empirical analysis, this paper is devoted to explaining allometric scaling exponents of cities using fractal dimensions. As power laws, allometric relationships are geometric measure relation, which must adhere to the principle of dimensional consistency. Thus, the allometric scaling exponent can be proved to equal the ratio of two values of dimensions. If both dimensions are Euclidean dimensions, then the allometric scaling exponent must be an integer or a ratio of integers. However, a large number of computational results based on urban observation data for allometric scaling exponents deviate from the theoretical expectations based on Euclidean geometry. The conclusion is that a scaling exponent of urban allometry is at least made of one fractal dimension. The fractal dimension of urban form may changes due to urban growth, but the ratio of two correlated fractal dimensions keeps constant in a certain period.