• 研究专题：京津冀协同发展 •

### 城市地理研究中的单分形、多分形和自仿射分形

1. 北京大学城市与环境学院城市与经济地理系,北京 100871
• 收稿日期:2018-02-25 修回日期:2018-07-06 出版日期:2019-01-28 发布日期:2019-01-22
• 作者简介:

作者简介：陈彦光(1965— ),男,河南罗山人,教授,博士,从事城市和理论地理学研究。E-mail: chenyg@pku.edu.cn

• 基金资助:
国家自然科学基金项目(41671167)

### Monofractal, multifractals, and self-affine fractals in urban studies

Yanguang CHEN()

1. Department of Urban and Economic Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China
• Received:2018-02-25 Revised:2018-07-06 Online:2019-01-28 Published:2019-01-22
• Supported by:
National Natural Science Foundation of China, No.41671167.

Abstract:

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.