Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.

Dimension is a measurement of space, and measurement is the basic link between mathematical models and empirical research. So, dimension is a necessary measurement for spatial analysis. Studying geographical spatial phenomena of scaling invariance such as cities and systems of cities has highlighted the value of fractal dimension [

The use of ASEs for the simple regular fractals in the mathematical world is not very noticeable. But for the quasifractals or random fractals in the real world, the function of ASEs should be viewed with special respect. A city can be regarded as an evolutive fractal. The land-use patterns, spatial form, and internal structure of a city can be modeled and simulated with ideas from fractal geometry [

This work is devoted to discussing how to characterize growth and form of fractal cities with ASEs. The remaining part of this paper is structured as follows. Section

The law of allometric growth originated from biological sciences [

The allometric equation usually takes the form of power law. Let

If we study

The three kinds of allometric relations can be applied to urban studies. If

The allometric relations between urban area and population are more familiar to geographers, but this paper attaches more importance to the scaling relationship between urban area and perimeter. A problem arises about defining urban boundary. Since urban form can be treated as a fractal body like Fournier dusts [

Scaling and dimensional analysis actually proceeded from physics, starting with Newton. Nowadays, allometric analysis, scaling analysis, and dimensional analysis have reached the same goal in theoretical exploration by different routes of development [

Allometric growth, including longitudinal allometry and cross allometry, can be divided into five types in terms of

In the next section we will reveal the allometric scaling relations in fractal bodies. To give a better explanation to allometric scaling of fractals, we should draw a mathematical analogy between self-similar structure and urban cascade structure. For an urban hierarchy with cascade structure, we can use three exponential functions based on hierarchical structure to characterize it. These three exponential functions can be changed into a set of power functions [

The relation between fractal geometry and Euclidean geometry is of “duality”. For a Euclidean geometrical body, the dimension is known (0, 1, 2, or 3), but its direct measures such as length, area, and volume are unknown without calculation. On the contrary, for a fractal geometrical body, the direct measures, including length, area, and volume, are always known, but its dimension required to be calculated. For example, the dimension of a rectangle is

However, there are no any real fractals in the real world. The so-called fractals in natural and human systems are of scaling invariance only within certain scale range. They are actually prefractals or quasifractals in the statistical sense. Therefore, we need the concepts of length, area, or even volume to build the allometric models although a fractal body has no length (e.g., it is 0 for the Cantor set, or infinite for the Koch curve) or area (it equals 0) in theory. A solution to this problem is to rely on the scale concept: we examine fractals within certain scaling range. For a fractal hierarchy with infinite classes (

A fractal usually suggests the allometric scaling relations in the broad sense, and this can be illustrated with several classical fractal patterns. Figure

A growing fractal and its self-similar boundary line (the first four steps are obtained by Vicsek [

Fractal growth

Subsequent division

Fracal line of boundary

where

The boundary of the growing fractal is a kind of the quadratic Koch curve (Figure

in which

For the growing fractal of infinite accumulative process displayed in Figure

For the fractal growth of continuous subdivision process illustrated by Figure

Where the growing process is concerned, the scaling relations between fractal-part number and boundary length can be derived by taking the logarithm of (

Next, we should investigate the cross allometry. In fact, the cross allometry based on Figure

Another well-known example of fractals is the Sierpinski gasket which is displayed in Figure

The Sierpinski gasket and its interior boundary lines (the first four steps).

Sierpinski gasket

Boundary of Sierpinski gasket

where

It is easy to understand the cross allometry of the Sierpinski gasket. For simplicity, let us consider the result of step

The longitudinal allometry of the gasket is somewhat different. In the

Generally speaking, we can describe a fractal in two ways based on allometric relations: one is longitudinal allometry, and the other is cross allometry. For the longitudinal allometry indicative of growing process, the measures of number, area, and length can be expressed with three exponential equations as follows:

represent different ASEs.

For the cross allometry indicating hierarchical structure, we can derive an area-length scaling relation. If

For the same scaling relation between area and length, there exists a difference between the longitudinal allometry and the cross allometry. Sometimes the longitudinal allometry is an inverse allometry, while the cross allometry is always a positive allometry or a negative allometry. The allometric scaling relations are very useful for us to characterize the random fractals. In fractal theory, if

In geographical studies, the scaling exponents have been associated with allometry and dimension in the context of fractal properties of cities for many years. A comparison of relations between urban area and border length, between urban population and radius, between urban border length and radius, and so forth was discussed by Longley et al. [

The self-affine fractals can also be described with allometric scaling equations. If the way of isotropic growth is turned into the way of anisotropic growth, then the self-similar growing fractal in Figure

A sketch map of the self-affine growing fractal (the first four steps are obtained by Vicsek [

From (

The allometric scaling relations between fractal-part area and boundary length can be characterized from the perspective of fractal growth. The area (

The relation between area and perimeter of the self-affine fractal can be expressed by a linear equation. Based on the fractal structure in Figure

The scaling relation between urban area and perimeter bears an analogy with the allometric relation of fractal growth. Let us take China’s cities as an example to illuminate this question. The processed data came from the study by Wang et al. [

In the digital map, we can use a set of grid consisting of uniform squares to cover the urban figure. Changing the side length of squares of the grid

The fractal dimension values of urban boundaries and ASEs of the area-perimeter relations of 31 megacities in China in 1990 and 2000.

City | 1990 | 2000 | ||||
---|---|---|---|---|---|---|

Scaling exponent ( | Boundary dimension 1 ( | Boundary dimension 2 ( | Scaling exponent ( | Boundary dimension 1 ( | Boundary dimension 2 ( | |

Anshan | 0.735 | 1.469 | 1.250 | 0.690 | 1.380 | 1.174 |

Beijing | 0.751 | 1.502 | 1.277 | 0.722 | 1.444 | 1.228 |

Changchun | 0.702 | 1.404 | 1.194 | 0.701 | 1.401 | 1.192 |

Changsha | 0.766 | 1.532 | 1.303 | 0.763 | 1.526 | 1.298 |

Chengdu | 0.838 | 1.676 | 1.425 | 0.837 | 1.674 | 1.424 |

Chongqing | 0.753 | 1.505 | 1.281 | 0.723 | 1.446 | 1.230 |

Dalian | 0.745 | 1.489 | 1.267 | 0.737 | 1.474 | 1.254 |

Fushun | 0.706 | 1.411 | 1.201 | 0.683 | 1.366 | 1.162 |

Guangzhou | 0.702 | 1.403 | 1.194 | 0.772 | 1.544 | 1.313 |

Guiyang | 0.874 | 1.748 | 1.487 | 0.871 | 1.742 | 1.482 |

Hangzhou | 0.800 | 1.599 | 1.361 | 0.783 | 1.565 | 1.332 |

Harbin | 0.685 | 1.369 | 1.165 | 0.654 | 1.307 | 1.112 |

Jilin | 0.712 | 1.424 | 1.211 | 0.716 | 1.432 | 1.218 |

Jinan | 0.717 | 1.433 | 1.220 | 0.732 | 1.463 | 1.245 |

Kunming | 0.794 | 1.588 | 1.351 | 0.736 | 1.472 | 1.252 |

Lanzhou | 0.741 | 1.482 | 1.260 | 0.736 | 1.471 | 1.252 |

Nanchang | 0.727 | 1.454 | 1.237 | 0.751 | 1.502 | 1.277 |

Nanjing | 0.785 | 1.569 | 1.335 | 0.747 | 1.494 | 1.271 |

Qingdao | 0.689 | 1.377 | 1.172 | 0.653 | 1.305 | 1.111 |

Qiqihaer | 0.678 | 1.355 | 1.153 | 0.670 | 1.340 | 1.140 |

Shanghai | 0.741 | 1.481 | 1.260 | 0.711 | 1.422 | 1.209 |

Shenyang | 0.650 | 1.300 | 1.106 | 0.639 | 1.278 | 1.087 |

Shijiazhuang | 0.786 | 1.571 | 1.337 | 0.733 | 1.466 | 1.247 |

Taiyuan | 0.777 | 1.554 | 1.322 | 0.769 | 1.538 | 1.308 |

Tangshan | 0.750 | 1.500 | 1.276 | 0.728 | 1.456 | 1.238 |

Tianjin | 0.688 | 1.376 | 1.170 | 0.678 | 1.356 | 1.153 |

Urumchi | 0.724 | 1.447 | 1.232 | 0.721 | 1.441 | 1.226 |

Wuhan | 0.738 | 1.475 | 1.255 | 0.747 | 1.494 | 1.271 |

Xian | 0.731 | 1.461 | 1.243 | 0.683 | 1.366 | 1.162 |

Zhengzhou | 0.753 | 1.506 | 1.281 | 0.713 | 1.426 | 1.213 |

Zibo | 0.763 | 1.525 | 1.298 | 0.747 | 1.493 | 1.271 |

Average | 0.742 | 1.483 | 1.262 | 0.727 | 1.454 | 1.237 |

The true dimension of urban form, however, is expected to be less than 2; namely

A question may be put as follows: what are the real values of the boundary dimension and form dimension of China’s cities? We cannot know them by the data from the study by Wang et al. [

The allometric scaling analysis of urban form can be generalized to the relation between urban area and population. The allometric model of urban area-population scaling can be used to predict regional population growth [

First of all, we carry out a longitudinal allometric analysis of Beijing’s growth. The data are urban population and built-up area from 1991 to 2005 (Table

The urban area and population of Beijing from 1991 to 2005.

Year | Original data | Logarithmic value | Standardization | |||
---|---|---|---|---|---|---|

Urban population | Built-up area ^{2}) | |||||

1991 | 629.6 | 397.4 | 6.445 | 5.985 | ||

1992 | 634.7 | 429.4 | 6.453 | 6.062 | ||

1993 | 640.8 | 454.1 | 6.463 | 6.118 | ||

1994 | 649.6 | 467.0 | 6.476 | 6.146 | ||

1995 | 656.1 | 476.8 | 6.486 | 6.167 | ||

1996 | 662.8 | 476.8 | 6.496 | 6.167 | ||

1997 | 670.0 | 488.1 | 6.507 | 6.191 | ||

1998 | 675.3 | 488.3 | 6.515 | 6.191 | ||

1999 | 682.4 | 488.3 | 6.526 | 6.191 | ||

2000 | 690.9 | 490.1 | 6.538 | 6.195 | ||

2001 | 861.4 | 747.8 | 6.759 | 6.617 | 0.387 | 0.485 |

2002 | 949.7 | 1043.5 | 6.856 | 6.950 | 0.748 | 1.282 |

2003 | 962.7 | 1180.1 | 6.870 | 7.073 | 0.799 | 1.577 |

2004 | 1187.0 | 1182.3 | 7.079 | 7.075 | 1.573 | 1.581 |

2005 | 1538.0 | 1200.0 | 7.338 | 7.090 | 1.617 | |

Average | 806.1 | 667.3 | 6.654 | 6.415 | 0.000 | 0.000 |

Stdev | 260.3 | 313.4 | 0.270 | 0.418 | 1.000 | 1.000 |

The allometric scaling relation between the built-up area and urban population of Beijing (1991–2004).

Excluding the data of 2005 from our consideration according to the value of double standard error, we can make a regression analysis by using the data from 1991 to 2004. On the whole, the process of urban growth conforms to the allometric scaling law to some extent (Figure

It will not be surprising if some readers doubt the result from the Beijing case. The effect of fitting Beijing’s dataset to (

Next, let us make a cross-sectional allometric analysis based on the rank-size relationships. For simplicity, only the allometric pattern in 2005 is shown (Figure

The allometric scaling relation between built-up area and urban population of China’s cities (2005).

Finally, we can make an allometric analysis based on hierarchy of cities with cascade structure. Putting the 660 cities in order by population size, we can classify them in a top-down way in terms of the

The allometric scaling relation between the average built-up area and urban population of China’s hierarchy of cities (2005).

What is the expected value of the scaling exponent of urban area and population relation? Through spectral analysis, we can learn that the dimension of the urban population is

The allometric scaling pattern based on the rank-size distribution is equivalent in theory to the allometric scaling relation based on hierarchical structure. The former differs from the latter to some extent in empirical analysis, but the difference between the two is not very significant. One gives the scaling exponent

Further, by using the least squares calculation, we can easily fit the data to the scaling relation between the urban number and population as well as the scaling relation between the urban number and area. Of course, the lame-duck class should be removed as an outlier from the regression analysis because the number of cities in this class is too few to support the scaling relation. Based on the hierarchy of cities without the tenth class, the modeling results are as follows:

In light of the nature of allometric growth, we can classify the geographical space into three types. The allometric patterns reflected by Figures

In addition, there are some inherent relationships among fractal structure, allometric relation and self-organized networks. In the process of measuring the fractal dimension of the form displayed in Figures

A fractal growth is actually an allometric growth; the process of allometric growth is always involved with a number of fractal dimension relations. For a simple regular fractal, say, the Koch curve, one fractal dimension is enough to characterize its geometrical property. However, for a random fractal, especially, for a prefractal phenomenon, for example, a city, it is not sufficient to characterize its form and structure with only one fractional dimension. We should employ a set of fractal parameters including various fractal dimensions, the ratios of fractal dimensions, among others, to describe the complicated systems of scaling invariance. The ratio of one fractal dimension to the other related fractal dimension can constitute an ASE discussed above. Now, the main conclusions in this paper can be drawn and summarized as follows.

First, if the form of the growing phenomena such as cities is self-similar, then the boundaries of the phenomena will be of self-similarity also. The geometric relationship between the boundary length and the whole form is always an allometric scaling relation. The allometric relation can be described from two angles of view. One is the longitudinal allometry, and the other is the cross allometry. The former reflects the progressively evolutive process of the fractal growing from an initiator, while the latter reflects the hierarchy with cascade structure corresponding to the growing process. The method of allometric scaling analysis can be applied not only to the isotropic growing phenomena indicating self-similar fractals, but also to the anisotropic growths indicative of self-affine fractals. For the self-affine patterns, there exists an allometric scaling relation between the parts in different directions.

Second, the mathematical description of the allometric growth rests with two aspects: one is various fractal dimensions, and the other is ASEs. The fractal characterization is a static method, laying emphasis on the best result by assuming the linear size of fractal elements to approach to zero. Theoretically, if the linear size of fractal measure becomes infinitesimal, then the fractal dimension value will approach a real constant. By contrast with fractal dimension, ASE lays stress on an evolutive process or a kind of spatial relations. The linear size of yard measure for estimating ASEs does not necessarily tend towards infinitesimal. As long as the scaling range is long enough, the result will be satisfying. This is significant for urban studies because the self-similarity of cities is valid only within certain scale limits. By means of allometric analysis we can reveal the regularity and complexity of urban evolvement and structure efficiently.

Third, the fractal studies can be generalized from real space to the abstract space in terms of allometric growth. All of the fractals that can be directly exhibited by maps or pictures are fractal in actual space. However, there are lots of fractals which cannot be immediately represented by graphics. The scaling invariance of this kind of fractals can be indirectly revealed with mathematical transformation and log-log plots. The majority of these fractals often comes from the generalized space. Urban form and boundaries belongs to the real space, but the scaling relation between urban area and population belong to the generalized space. It is difficult to evaluate the dimensions of the fractals in the abstract space, but it is easy to estimate the ratio of different fractal dimensions based on the generalized space. In many cases, what we want to know is just the fractal dimension ratios rather than the fractal dimensions themselves. Through allometric analyses we can directly calculate the ratio of fractal dimensions and thus obtain ASEs; thereby we can research the structure and functions of fractal systems.

The city of Beijing is taken as an example to show how to reveal the outliers of an allometric scaling relation. The allometric growth is in fact based on exponential growth theoretically. Suppose that both urban area and population increase exponentially with the passage of time. Taking the logarithm of urban population

This research was sponsored by the National Natural Science Foundation of China (Grant no. 40771061) and Beijing Natural Science Foundation (Grant no. 8093033). Many thanks are offerred to two anonymous referees who provided interesting suggestions. The errors and omissions which remain are all mine.