分形几何学在城市地理研究中具有广泛的应用,然而很多基本概念却让初学者感到迷惑。如何区分单分形、自仿射分形与多分形,是一个基本而重要的问题。简单分形容易理解,而真实的地理现象很少是单分形的。城市生长过程具有自仿射特征,而城市空间格局却具有多分形性质。作者发现,各种分形的共性在于三个方面：标度律、分数维和熵守恒。论文基于标度、分维和熵守恒公式,借助隐喻城市生长的规则分形来区分单分形、多分形和自仿射分形,讨论分形系统演化的机理、分形与空间自相关和空间异质性的联系,同时澄清一些在地理分形研究中的常见错误概念。最后以城市位序-规模分布为例,说明并对比单分形和多分形在城市地理研究中的建模与应用思路。

Fractal geometry provides a powerful tool for scale-free spatial modeling and analyses in geography. However, a number of basic concepts are puzzling. The three common fractals, that is, monofractal (unifractal), multifractals, and self-affine fractal, are often misunderstood by students of geography. This article clarifies some confusing fractal concepts for urban fractal modeling and fractal dimension analysis. Using simple mathematical models based on three growing fractals that bear an analogy to urban growth, we can distinguish the three types of common fractal structure. The similarities and differences between monofractal, multifractals, and self-affine fractal are as follows: 1) A monofractal is a simple self-similar fractal that bears only one scaling factor (scaling ratio), and a multifractal object is a complex fractal system that bears at least two scaling factors for different parts. Each scaling factor dominates all different scales and is independent of directions and levels. 2) A self-affine fractal bears different scaling factors in different directions of growth or at different levels of scales. The basic feature of self-affine growing fractal is anisotropy, which differs from the isotropic self-similar growing fractals. 3) Both self-affine fractal and multifractals may possess two scaling factors, but there are essential differences between self-affine fractals and multifractals. A self-affine fractal often takes on the form of bi-fractals, which can be reflected by two scaling ranges on a log-log plot. However, there is only one scaling range for a multifractal pattern. As an example, two-scaling fractal modeling is applied to the rank-size distributions of cities to illustrate the concept of urban multifractals. By comparison with these multifractal models, we can better understand monofractals and self-affine fractals in geographical research.

The authors have declared that no competing interests exist.