Abstract:

Fractal cities and city fractals have been studied for about 30 years, but many basic problems have not yet been fully solved. Among the various basic fractal problems about cities, the most important are how to estimate fractal dimension and how to identify fractal nature of a city as a system or a system of cities in an effective way. Based on urban form and the box-counting method, this article discusses the approaches to calculating fractal dimension and determining fractal property of cities. The theoretical foundation of fractal dimension computation is the concept of perfect covering. In other words, in the process of fractal dimension measurement, a fractal object should be covered with boxes in the best way: nothing more, nothing less. In practice, it is hard to find the optimal way of box covering. Therefore, three rules should be followed. First, the sequence of measuring scales should be consistent with the cascade structure of a fractal city so that the fractal details can be captured in a reasonable way. Second, the operation of obtaining observational data should be simple and convenient to use so that the method can be applied by beginners. Third, the size of the dataset for fractal dimension estimation should be large enough so that the result of fractal parameters is stable. A conclusion can be reached that the geometric scale rather than the arithmetic scale should be employed to make a measurement because the fractal structure can be abstracted as geometric sequences instead of arithmetic sequences, and the measuring scale sequence should comply with the 1/2ⁿ rule (that is, 1, 1/2, 1/4, …). After estimating the fractal dimension of a city, the fractal property can be identified by the statistics from fractal dimension calculation. The good way of identifying the fractal nature of urban form is to use confidence statement, which consists of fractal dimension values, margin of error, and level of confidence. Given a level of significance (for example, α=0.05), we can draw an inference that a city's form is of fractal structure because it has a fractal dimension with a determinate level of confidence (for example, (1-α)×100%=95%). Using statistical analysis, however, one can never have full (100%) assurance that a city has a fractal form.

Key words: urban form, fractals, fractal dimension, self-affinity, box-counting method, multifractals, scaling range