Fractal Systems of Central Places Based on Intermittency of. Space-filling

Transcription

1 Fractal Systes of Central Places Based on Interittency of Space-filling Yanguang Chen (Departent of Geography, College of Urban and Environental Sciences, Peking University, Beijing 0087, P.R.China. E-ail: Abstract: The central place odels are fundaentally iportant in theoretical geography and city planning theory. The texture and structure of central place networks have been deonstrated to be self-siilar in both theoretical and epirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the echaniss by which the fractal patterns can be generated fro central place systes. The structural diension of the traditional central place odels is d=2 indicating no interittency in the spatial distribution of huan settleents. This diension value is inconsistent with epirical observations. Substituting the coplete space filling with the incoplete space filling, we can obtain central place odels with fractional diension D<d=2 indicative of spatial interittency. Thus the conventional central place odels are converted into fractal central place odels. If we further integrate the chance factors into the iproved central place fractals, the theory will be able to well explain the real patterns of urban places. As epirical analyses, the US cities and towns are eployed to verify the fractal-based odels of central places. Key words: central place network; fractals; hierarchical scaling law; interittency; spatial scaling law; urban hierarchy; the US cities. Introduction Central place theory seeks to explain the relative size and spacing of huan settleents, including cities and towns, as a function of people s econoic activities, especially shopping behavior. In fact, geographers have long recognized that the functions of cities and towns as arket centers, traffic centers, or adinistrative centers result in a hierarchical syste of

2 settleents. A central place can be defined as a settleent at the center of a region, in which certain types of products and services are available to consuers (King and Golledge, 978; Knox and Marston, 2007). In other words, the doinant function of a central place is to provide arket and supply services for the region. The tendency for central places to be organized in hierarchical systes and network structure was first explored by Christaller (933/966), and his ideas led to central place theory, which was consolidated and developed by Lösch (940/954). Fro then on, this theory went gradually beyond geography and influenced any related fields, including econoics, sociology, city planning theory, and even physical geography. The theory on central places concerns with the way that huan settleents evolve, and are spaced out and organized regularly. Three concepts are basic for us to understand this theory, that is, order, range, and threshold. The order of a central place is deterined by the size of the region which is served and in turn reacts on it. A higher-order central place serves a larger region, while a lower-order central place serves a saller region. Thus we need another concept, range, indicating the spacing of settleents. The range of a central place function denotes the axiu distance which consuers will norally travel to obtain a particular product or service. Both order and range can be related to the third concept, threshold of central place function, which suggests the iniu arket size required to ake the sale of particular product or service profitable (Knox and Marston, 2007). The three concepts can be atheatically easured with population size, spatial distance (service area), and nuber of central place, by which we can bring to light the latent scaling relations of central place systes. The work of central place studies was involved with a series of concepts of coplexity theory such as self-siilarity, scaling laws, and self-organization. A central place syste can be seen as both a hierarchy with cascade structure and a network with self-siilar properties. A hierarchy and a network actually represent different sides of the sae coin. Hierarchical structure is a very significant notion for us to understand urban fractals (Batty, 2006; Batty and Longley, 994; Frankhauser, 998). On the other hand, the hierarchies of central places bear an analogy with the network of rivers (Chen, 2009; Woldenberg and Berry, 967), and river networks have been deonstrated to take on fractal nature (LaBarbera and Rosso, 989; Rodriguez-Iturbe and Rinaldo, 200; Tarboton et al, 988). So, central place networks ay be of self-siilarity. Theoretically, the texture of central place network odels can be interpreted with fractal geoetry (Arlinghaus, 2

3 985). Epirically, the central places of southern Gerany do follow the scaling laws indicative of fractal structure (Chen and Zhou, 2006). The fractal texture of central place odels have been expounded by Arlinghaus (993) and Arlinghaus and Arlinghaus (989). However, the generation echanis of fractal structure of real central place systes is still an outstanding proble reaining to be resolved. The central place theory is based on the postulates such as an unbounded isotropic plain with a hoogeneous distribution of the purchasing power. The regular hexagonal foration has long been criticized on both description and explanation in geography because we cannot find this kind of patterns fro the systes of settleents in the real world. However, the essence of this theory rests with its prediction on hierarchical scaling law and the average coordination nuber of cities around an urban place (six). Both population growth and huan interaction activity follow the scaling laws (Rozenfeld et al, 2008; Rybski et al, 2009). The average coordination nuber was frequently deonstrated to be close to 6 (Niu, 992; Haggett, 969; Ye et al, 2000). More and ore facts and epirical observations lend further support to the predictions fro central place theory (this paper will present several evidences). In fact, fractal central place theory ay be one of the channels for us to coprehend the potential links between huan systes and physical systes (e.g. rivers). However, the classical central place odels reflect the interittency-free systes of huan settleents, which predict a Euclidean diension for urban space in a region (d=2). This is in contradiction to the fractal patterns of settleents in the real world (Batty and Longley, 994; Chen, 2008; Frankhauser, 994). This paper is devoted to probing into interittency of space-filling of central place systes. Interittency is a significant concept to understand urban developent, especially at large scale (Manrubia and Zanette, 998; Zanette and Manrubia, 997). The rest of the article is structured as follows. Section 2 gives the growing fractal odel of central places by introducing the ideas of incoplete space-filling and interittency, and section 3 provides epirical evidences by applying the fractal scaling relations to the systes of urban places in the United States. Related questions are discussed in section 4, and finally, the writing will be concluded by suarizing this study. The ain novel points of this paper are as follows. First, the incoplete space-filling process and interittent structure predicting fractal patterns are introduced into the classical central place odels. Second, a new fractal odel of cities, Koch snowflake odel, is derived 3

4 fro the central place network. Third, the average nearest distance ethod is proposed to testify the central place scaling in the real world. Last but not least, the hierarchical scaling based on central place fractals can be eployed to estiate the friction coefficient of distance decay. 2. Model odification 2. Coplete space-filling and Euclidean plane The basic central place systes can be divided into three types: the k=3 systes indicting a arketing-optiizing case, the k=4 systes suggesting a traffic-optiizing situation, and the k=7 systes indicative of an adinistrative-optiizing situation (Figure ). A central place network is also a hierarchy consisting of 7 levels of settleents (L, P, G, B, K, A, M), including cities and towns. Each central place possesses 6 coordination locations for other central places. Suppose that the central place classes are nubered =, 2,, M in a top-town order for siplicity (generally M=7±2 in practice). This differs fro the traditional odels of central place hierarchies which are nubered in the botto-up order. Three necessary easures of huan settleent are sybolized as follows: N and P denote respectively the nuber and the average population size of the central places in the th class, and correspondingly, L denotes the average distance between adjacent central places of order. We can exaine and bring to light the scaling relations aong the nuber N, size P, and distance L. a. k=3 b. k=4 c. k=7 Figure Three interittency-free central place networks representing different arrangeents of huan settleents The nearest neighboring distance L should be explained as it is the first basic easure to 4

5 define the fractal diension of central place networks. In the k=3 networks, a central place of order have 3 central places of the sae order as the nearest neighbors (Figure a). In the k=4 networks, for each central place in the th level, the nuber of the ost proxiate central places of order is 4 (Figure b). In the k=7 networks, the nuber of iediate central places of each central place of order equals 5 (Figure c). The easure L can be defined as the distance between a central place of order and the nearest neighboring central places in the sae order. Based on the nuber N and distance L, the fractal diension of central place networks can be defined atheatically (Chen and Zhou, 2004; Chen and Zhou, 2008). For siplicity, let x=l, y=l +. For the k=3 network (Figure a), we have which is equivalent to x y ( ) y ( ) =, () x = 2 y cos30 o = 3 2 y. (2) Thus we get 2 2 x = 3y, (3) which iplies a scaling relation For k=4 network (Figure b), obviously we have x = 3y. (4) x = 2 y = 4y. (5) As for k=7 network (Figure c), according to the cosine theore, we get o 2 x = y + ( 2y) 2y(2y) cos20 = 7y. (6) This suggests x = 7 y. (7) To su up, a general scaling forula can be derived fro the central place odel (King and Golledge, 978), and the result is x = k y. (8) So the length ratio of the average distance of order to that of order (+) is such as 5

6 r x L = y L l = = + k. (9) The length ratio can also be called distance ratio in our context. On the other hand, according to Christaller (933/966), the nuber ratio of the (+)th-order central places to that of the th-order central places is as follows (Chen and Zhou, 2006) r N N + n = = k. (0) Then the siilarity diension of central place systes can be defined by the nuber ratio and the distance ratio in the for ln( N D = ln( L + / N) ln( rn ) ln k = = / L ) ln( r ) ln k + = Equation () suggests the below spatial scaling relation such as l 2. () N AL d =, (2) where A denotes the proportionality coefficient, and the subscript can be oitted for siplicity. The paraeter d=2 refers to a Euclidean diension, this iplies that the geographical space is coplete filling with huan activities (Table, Figure 2). Therefore, the conventional central place odels are based on a Euclidean plane despite the fact that the texture of the network coprises fractal lines (Arlinghaus, 985). Actually, Christaller s central place theory provided an equilibriu solution based on the thinking of classical econoics to the proble that huan settleents are organized and distributed spatially (King and Golledge, 978). The network is regular and the distribution of the urban places in the sae layer is hoogeneous, thus the hexagonal arket areas are the basic feature of central place distribution. As Prigogine and Stengers (984, page 97) once observed: Obviously, in actual case, such a regular hierarchical distribution is very infrequent: historical, political, and geographical factors abound, disrupting the spatial syetry. The real systes of central places ay be far fro equilibriu (Allen, 997). However, the network structure of regular hexagons ay be broken down, but the spatial and hierarchical scaling relations hidden in the hierarchies are constantly held by the self-organized evolveent of settleents. 6

7 Table Christaller s odels for the hierarchies of central places in Southern Gerany Type of Class k=3 k=4 k=7 place L N L N L N L 08 () 256 () 372 () P G B K A M Note: The data coe fro Christaller (966), see also King (984). N refers to the nuber of tributary areas, and L to the distance between places (ks). 000 N = 7776L N = 4952L N = L -2 R 2 = R 2 = R 2 = N N 00 N L L L a. k=3 b. k=4 c. k=7 Figure 2 The scaling relations between the nuber of tributary areas and the distance between places in the odels of Southern Gerany (Note: The first class is special so that it always goes beyond the scaling range. Therefore, the L type of central places is excluded fro the scaling relations as outliers. See Chen and Zhou, 2006) 2.2 Incoplete space-filling and fractal systes In the classical theory, one of the basic ais of central place systes is to ake the best of geographical space. A network of settleents in a region can be grouped into several layers/levels in light of size and function of central places. Theoretically, the area served by a central place is a round field (Figure 3a). The areas served by different central places in the sae layer are equal to one another. Because of copeting with each other for trade/service area, round fields doinated 7

8 by lots and lots of central places crowd and change to the close-packed hexagonal nets with cells of fixed diaeter within a layer (Figure ). The hexagonal network structure was assued to avoid gaps or overlapping arket area. In short, the traditional central place odels indicate the interittency-free systes of settleents aking use of every bit of geographical space. There is no roo for physical systes, and huan activities are onipresent in a territory. a. Coplete space-filling b. Incoplete space-filling Figure 3 The sketch aps of the coplete space-filling and the incoplete space-filling (k=4) (Note: The hexagons represent the networks of central places, while the circles indicate the space-filling extent of huan activities.) However, this pattern of coplete space-filling fro spatial copetition is only of ideality but never accord with the reality. The real systes of central places in Southern Gerany, studied by Christaller (933/966) are in fact fractal systes with fractional diension, and the fractal diension values ranges fro D.48 to D.84 (Chen and Zhou, 2006). In order to interpret this phenoenon, we need new assuption of incoplete space-filling associated with interittency. Figure 3a refers to a schee of coplete space-filling process, while Figure 3b to a schee of incoplete space-filling. The siilarities and differences between the two schees are very clear. The forer suggests close-packed systes of service areas, while the latter suggests interspace representing the place which cannot or should not be occupied by huan beings despite the copact distribution of urban places. Therefore, the revised odel indicates an interittent syste of urban settleents with fractional diension (see Appendix ). 8

9 a. b. c. d. Figure 4 A growing fractal of central places based on incoplete space-filling and interittency (the first four steps) Owing to the sall but pivotal revision of the postulate on space-filling, the odels of central places are iproved. The central place networks can be transfored into a Koch snowflake pattern of systes of cities and towns, which is tered urban snowflake (Figure 5). Fro the standard central place odels, we can naturally derive a Koch snowflake odel (Chen, 2008; Chen and Zhou, 2006). The urban snowflake odel (USM) is siilar to a growing fractal presented by Jullien and Botet, (987). The siilarity diension of the spatial for of the revised central place network is ln N ln( N+ / N) ln(7) D n = = =.772, ln L ln( L / L ) ln(3) + where N refers to the nuber of central places of order, and L to the distance between two adjacent central places of the sae order. The boundary of an urban fractal is possibly a fractal line, which contains useful geographical spatial inforation (Batty and Longley, 987; Frankhauser, 998). The perieter of USM is a Koch curve with a diension such as ln B ln( B+ / B ) ln(4) D b = = =.269, ln L ln( L / L ) ln(3) + where B denotes the boundary length of the urban snowflake of order. The relation between the nuber of places and the boundary length of the urban snowflake follow the alloetric scaling law 9

10 N B α = B, (3) D n / D b where α refers to the alloetric scaling exponent, which is given by Dn ln N ln( N+ / N) ln(7) α = = = = D ln B ln( B / B ) ln(4) b + Actually, there exist various alloetric relations in the real systes of central places. Now, equation (2) can be generalized to the following for N AL D = n, (4) where D n indicates the fractal diension of central place network. Equation (4) represents the spatial scaling law of central places. As soon as the interittency is introduced to the central place systes, ore than one kind of fractal structure can be derived fro the coon odels by using the idea of fractals. No atter what kind of odel it is, the urban place will follow the spatial scaling law. a. Central place network (CPN) b. Koch snowflake odel (KSM) Figure 5 A sketch ap of derivation of the Koch snowflake fro central place network (k=3) (the first four levels, see Chen and Zhou, 2006) 3. Epirical evidences 3. Cases of large scale The central place odels of Christaller (933/966) are abstracted fro the real networks of huan settleents. The fractal diension of standard central place odels can be divided into 0

11 two types: one is textural diension characterizing the boundary lines of service areas; the other is structural diension characterizing spatial distribution of urban places. Where the odels are concerned, the boundary lines of the traditional central place systes are fractal lines, and the diensions are D=ln(4)/ln(3).269 for the k=3 syste, D=ln(3)/ln(2).585 for the k=4 syste, and D=ln(9)/ln(7).292 for the k=7 syste (Arlinghaus, 985). All these can be regarded as textural diension value. The structural diension of traditional odels is defined by equation (2), which suggests a Euclidean diension d=2. However, the real networks of central places are of fractional diension. Using the original data given by Christaller (933) and the later suppleental data, Chen and Zhou (2006) estiated different fractal diension values for central places in Southern Gerany, that is, D n.7328 for Munich region, D n.6853 for Nureberg region, D n.8370 for Stuttgart region, and D n.482 for Frankfort region. The diension of urban population distribution in Southern Gerany in 933 is about D p.859. Now, we can apply the odel of fractal central places to other countries such as Aerica. In order to avoid the subjectivity of data processing, we can use the data processed by other scholars who know little about fractals. For aking a ultisided investigation, we should introduce the scaling relation between the nuber and average size of urban places of the sae class. Fro the size distribution of the central place population forulated by Beckann (958), we can derived the size ratio of the urban population of order to that of order +, that is r p = P P = + s, (5) h where s refers to the nuber of equivalent urban places of the (+)th level that are served by the th level city, and h to a proportionality factor that relates the urban population to the total population served by that city. Both s and h are assued to be constant over the levels of the hierarchy. Please note the botto-up order in the classical central place odels is replaced by the top-down order in this paper for the purpose of siplicity of atheatical transforation, and this substitution does not influence the analytical conclusions. A spatial and hierarchical scaling relations predicted by central place theory can be derived fro equations (9), (0), and (5), and the results are as follows D P p = CL, (6)

12 N. (7) α = KP where D p denotes the fractal diension of central place population, and α is the scaling exponent of the power-law relation between city nuber and average population size of urban places in the th class. Both C and K are proportionality constants. Coparing equations (4) and (6) with (7) shows that D = D n α. (8) p This suggests that b is the ratio of the network diension to the population diension. In theory, the α value denotes the fractal diension of city size distribution. It is equivalent in nuerical value to the Pareto exponent and equals the reciprocal of Zipf s rank-size scaling exponent. Now, let s exaine the hierarchy of urban places in the United States. The data for the period 900 to 980 were processed by King (984) who specialized in central places theory but sees to know little about fractals then (Table 2). Owing to absence of the distance easure, L, we can only probe the scaling relation between the city nuber and population size of urban places by fitting the data in Table 2 to equation (7). The results show that the US cities follow the scaling law on the whole. The last class indicating sall urban places in 960 and 980 slop over the scaling range due to undergrowth of huan settleents. As is often the case, the power-law relations break down when the scale are too large or too sall (Bak, 996). The urban centers with population size under 2000 can be treated as the so-called lae-duck class (Davis, 978). The lease squares calculations yield the four atheatical odels taking on power-law relations. The equations, estiated paraeter values, and related statistic quantities are listed in Table 3 for coparison. The effect of data points atching with the trend lines are displayed in Figure 6. Fro 900 to 980, the scaling exponent α=d n /D p varied around. This suggests that, at the large scale, the diension of networks of urban centers is very close to that of the population distribution all over the central place systes. The sae scaling analysis can be applied to Indian cities based on the census data fro 98 to 200, and the effect is ore satisfying (Appendix 2). Table 2 The size scale and nuber of urban places in the United States, Class Population size Lower liit of size P Nuber of places N

13 Over Under * * Source: The United States Bureau of the Census (960, 970, 980). The Data is processed by King (984). * Data unavailable. The last class, i.e., the 0 th class, is a lae-duck class owing to undergrowth of cities. Table 3 The scaling odels, scaling exponents, and corresponding goodness of fit for the US urban places, Year Matheatical odel Scaling exponent b Goodness of fit R N = P N = P N = P N = P N N 00 0 N = P R 2 = N = P R 2 = P P a.940 b

14 N 00 N 00 0 N = P R 2 = N = P R 2 = P P c.960 d. 980 Figure 6 The scaling relations between the population size and nuber of urban places in the United States, (Note: In 960 and 980, the last class is treated as an outlier because of undergrowth of cities. The circles indicate the exceptional data points out of the scaling ranges) 3.2 Cases of ediu and sall scales More evidences can be found to support the scaling relation of fractal central places, equation (7), including the US cities in 2000 (Chen and Zhou, 2004), Chinese cities in 2000 (Chen and Zhou, 2008), India cities in 98, 99, and 200(Basu and Bandyapadhyay, 2009), and the classical exaple of central places in southern Gerany (Christaller, 933). Next, let s turn to the urban places in the US sub-regions. Rayner et al (97) once grouped the huan settleents in the two US states, Iowa and North Dakota, into 7 or 9 classes in the botto-up order in ters of the idea fro central place theory. The distance refers to the ean distance to first nearest neighbor in iles. It is easy to rearrange the results in the top-down order according to our usage of classification (Table 4). The botto level of the population size ranging fro 0 to 99 is a lae-duck class due to undergrowth of urban places. These data can be eployed to testify the fractal central place odels. Table 4 The nearest-neighbor distance for different-sized urban places in Iowa and North Dakota Class Population size Lower liit Iowa North Dakota of size P Nuber N Distance L Nuber N Distance L 4

15 Over Source: Rayner JN, et al. 97. The least squares coputations by fitting the data in Table 4 to equation (4), (6), and (7) yield two sets of results, which are listed in Table 5. The scaling relations are visually displayed in Figure 7. The fractal diension of the network of huan settleents, D n, is about.8763 for Iowa and.7555 for North Dakota. These results are noral. However, the fractal diension of the population distribution corresponding to these networks, D p, are abnoral. The results are for Iowa and for North Dakota. The expected results are that the fractal diension coes between and 2. As siilarity diension, the values which are greater than the diension of ebedding space d E =2 are understandable and acceptable (Chen, 2009). These suggest that the huan settleents of sall scale are concentrative, or the gap between the average distance of one class and that of its iediate class is not wide enough. The scaling exponent, α, is supposed to be close to. But the value is less than what is expected because the population diension D p exceeds the proper upper liit. Anyway, urban fractals are evolutive processes rather than deterinistic patterns. The fractal landscape of cities as systes and systes of cities often evolves fro the nontypical into the typical for, and fro the siple into the coplex patterns (Benguigui et al, 2000; Chen, 2008). After all, huan geographical systes differ fro the classical physical systes. The odels of huan systes are dynaical odes instead of the static relations. Just because of this, we can optiize the spatial structure of huan systes by using the ideas fro fractals and related theories. Table 5 Matheatical odels, scaling exponents, and goodness of fit for systes of urban places in Iowa and North Dakota (97) 5

16 Region Matheatical odel Scaling exponent D or b Goodness of fit R 2 N D n = L Iowa P D p = L N α = P N D n = L North Dakota P D p = L N α = P 000 N = 2708L R 2 = N = 9878L R 2 = N 00 N L L a. Iowa b. North Dakota P = L R 2 = P = L R 2 = P 000 P L L c. Iowa d. North Dakota 6

17 000 N = 092P R 2 = N = P R 2 = N 00 N P P e. Iowa f. North Dakota Figure 7 The scaling relations between nubers, population sizes, and ean distances to the first nearest neighbors of urban places, Iowa and North Dakota, USA (97) (Note: In figures c, d, e, and f, the last class is an exceptional value owing to undergrowth of urban places. The circles indicate the outliers out of the scaling ranges) 4. Questions and Discussion Central place theory is one of the cornerstones of huan geography, and our understanding of the growth and evolution of urban settleent systes largely rests upon the edifice of central place theory and its elaboration and epirical testing through spatial statistics (Longley et al, 99). Intuitively, the hierarchical structure of central place odels is consistent with fractal geoetry. However, the classical central place odels cannot entirely explain the fractional diension of systes of urban places in the real world. As soon as the idea of spatial interittency is taken into account, the fractal structure of central place systes can be generated by recursive subdivision of space (Batty and Longley, 994; Chen, 2008; Goodchild and Mark, 987). A key question is what is the doinative variable or paraeter of the fractal diension of central place systes. In order to answer this inquiry, let s ake a siple atheatical transforation. Equation (9) can be generalized to the following expression r l = L L + = k / w, (9) where w is a positive nuber equal to or less than 2. Thus equation () changes to 7

18 ln k D = = w 2. (20) / w ln k This suggests that, in the siplest case, the scaling exponent of the ratio of the distance between the adjoining urban centers in one class to that in the iediate class controls the diension of central place network. Further, suppose that equations (9) and (0) can be replaced with r l = L L + = k / u, (2) r n N N + = = k / v, (22) where u and v are two paraeters greater than zero. Thus the siilarity diension is such as ln r D = ln r n = l u v. (23) This suggests that the diension can be doinated by both the scaling exponent of nubers of urban places and that of distances between urban centers in different classes. In short, the paraeters controlling the fractal diension of networks are relative to the paraeters of hierarchy of urban places. The structural diension of central place systes is independent of the paraeter k. This differs fro the textural diension, which depends on k values of central place odels (Arlinghaus, 985). Another question is why the average nearest-neighbor distance rather than the average distance is eployed to estiate the fractal diension of central place systes. As we know, to judge whether or not a geoetric body is a fractal, we should drawn an analogy between the scaling exponent of the geoetric body with Hausdorff s diension. Hausdorff s diension in pure atheatics can be replaced by box diension in technique or technology. If we use the box-counting ethod to estiate the diension of the geoetric body, we should use the inial/least box instead of the larger one to cover the body. Generally speaking, for diension estiation, we should eploy the boundary values rather the average value of scales to easure a geoetric body. To estiate fractal diension, we can adopt the upper liits of scales (the largest one) to ake a easureent for the positive power law (PLR) relation or the area-radius scaling, or, the lower liits of scales (the sallest one) for the negative power law (NPL) relation or the box-counting scaling. If we substitute the scale ean for the scale boundary to ake a 8

19 easureent and thus to build a scaling relation, we ay get a fractal rabbit (Kaye, 989), rather than a valid fractal (see Appendix 3 for a siple exaple). The hierarchical scaling laws derived fro central place theory are very iportant for spatial analysis of cities. For exaple, we can use the scaling law to estiate the distance friction coefficient (DFC) of the urban gravity odel (UGM). The principal paraeter of UGM, actually a scaling exponent, is the DFC, which is hard to estiate in practice (Haggett et al, 977). By eans of the spatial and hierarchical scaling laws, we can derived a forula such as (the detailed derivation process will be given in a copanion paper) b = qd n, (24) where b refers to DFC, q to the scaling exponent of Zipf s rank-size distribution, and D n, the fractal diension of spatial distance of cities defined above (Chen, 2008; Chen and Zhou, 2006). In ters of equation (8), we have q = = α D D p n. (25) Substituting equation (25) into equation (24) yields D b = n = D. (26) α p This suggests that if we easure the urban gravity with city population size, the average DFC value for the cities in a region is just the fractal diension of spatial distribution of urban population. For exaple, this ethod can be applied to the urban places of Iowa and North Dakota discussed in Subsection 3.2. For the urban settleents in Iowa, the DFC value can be directly estiated as d=d p 2.782, or indirectly estiated as d=d/b.8763/ ; for the urban settleents in North Dakota, the DFC value can be directly estiated as d=d p , or indirectly estiated as d.7555/ The third question is the type of central place fractals which is put forward preliinarily by Chen and Zhou (2006). If we consider the shadow effect of urban developent, a variant of USM can be derived fro the standard USM displayed in Figure 4. The shadow effect of regional science is proposed by Evans (985, pages 98-99), who said: In earlier years one would have expected hinterland effects to have doinated as the growth of the larger cities during the industrial revolution occurred at the expense of the saller towns. In effect the large city cast it 9

20 shadow over the surrounding area depriving the saller towns of growth as a larger tree prevents the growth of others by depriving the of light. But the extent of the shadow and its effect will change as technical, social and econoic factors change. Owing to the shadow effect, the sall interittency changes to large interittency, and we have a hollow snowflake odel (Figure 8a), which present a contrast to the solid snowflake odels (Figure 8b). The network diension (D n ), boundary diension (D b ), and the alloetric scaling exponent (α) indicating the ratio of the two diensions of the forer are as follows ln(6) ln(4) ln(6) D n =.6309, D b =. 269, α = ln(3) ln(3) ln(4) As a atter of fact, a ajority of diension values of central place networks in the real world vary fro D n.63 for the hollow snowflake to D n.77 for the solid snowflake (Chen, 2008). a. With shadow effect b. No shadow effect Figure 8 Two kinds of urban snowflake odels for central place networks with interittency Anyway, central place theory is very iportant in theoretical geography, urban and regional econoics, and city planning theory. One of the great triuphs of central place theory is that it iplies the scaling law of spatial and hierarchical distribution of urban places, and the other, it predicts the city coordination nuber equal to six. What is ore, it can be associated with fractals indicating spatial optiization. However, few systes of cities and towns in the real world can be fitted to the regular hexagonal patterns. Just because of this, central place theory has long been denounced on both institutional and ethodological grounds. Fortunately, the proble used to 20

21 puzzle geographers can nowadays be resolved to the core. First, by eans of the hierarchical scaling law, we can iprove central place theory so that it is based on fractal geoetry rather than Euclidean geoetry. Copared with Euclidean geoetry, fractal geoetry can go beyond graphics (figures) and yield ore abstract atheatical equations. Second, if we introduce the chance factors into the regular central place fractals, we will obtain irregular central place fractal odels, which look like the real settleent patterns. As a result, as done in Section 3, we can fit the observational data of urban settleents to the atheatical odels rather than fitting the spatial distribution of urban places to the regular hexagonal network. In order to develop central place theory, we should consider various factors affecting urban evolution such as chance, interittency, and ulti-scales. A fractal is a phenoenon of scaling syetry consisting of for, chance, and diension (Mandelbrot, 983). Chance factor has been introduced to central place theory by Allen (997), and interittency factor is considered in this paper. As soon as both the chance for processes and interittency for patterns are integrated into central place fractals, we will have brand-new central place odels with irregular fors, the core of which is the spatial and hierarchical scaling and fractal diension rather than hexagonal networks. The hexagonal structure can be treated as a postulate instead of the theoretical odel itself. Thus the distance-based urban space concept will be replaced by the diension-based urban space notion. In next step, USM should be generalized to ultifractals since that central place hierarchies are associated with size distributions of huan settleent and population distribution (Appleby, 996; Chen and Zhou, 2006; Frankhauser, 2008), and the rank-size rule is related to ultifractal phenoena (Chen and Zhou, 2004; Haag, 994). As space is liited, the ultifractal odels of central place systes reain to be discussed in future studies. 5. Conclusions Fractals suggest the optiized structure of systes in nature. A fractal body can occupy its space in the ost efficient way (Chen, 2009; Rigon et al, 998). Using ideas fro fractals to plan cities and systes of cities will help to iproving huan environent and guaranteeing sustainable developent of huan society. Central place theory is one of the basic theories available for city planning. However, the theory is based on the concept of coplete space-filling. 2

22 No buffer space, no vacant space, no interittency for physical phenoena. In light of the traditional notion of econoics, aking the best of geographical space iplies aking use of geographical space in the best way. However, this notion ay be old-fashioned. Going too far is as bad as not going far enough, and things will develop in the opposite direction when they becoe extree. Coplete space-filling suggests a siple Euclidean plane with little vital force and profound order. In contrast, fractal structure suggests the order behind chaos of cities (White and Engelen, 993; White and Engelen, 994). The unity of opposites of chaos and order ay indicate driving force of urban evolveent. The ain points of this paper can be suarized as follows. First, the real systes of central places are of interittency indicating incoplete space-filling of huan activities. This is different fro the classical odels of central places based on coplete space-filling and interittency-free patterns. The central place systes are actually fractal systes taking on self-siilar network and hierarchy with cascade process. Second, the regular hexagonal landscapes of the traditional odels in ideality are broken down in reality, but the scaling relations behind presentational fors will keep and the spatial and hierarchical scaling laws doinate the spatio-teporal evolution of settleents. Third, the fractal landscapes of urban places are ore acceptable for urban theory, resting with two aspects: one is that, by introducing the chance and interittency factors into the hexagonal network, the fractal central place odels accord with the real syste of urban places and correspond to physical phenoena such as rivers, and the other is that fractals suggest optial structure for huan systes and provide potential application to planning systes of cities and towns in the future. Since fractal geoetry can go beyond the liit of graphics, we can fit the observational data of urban settleents to the scaling laws, instead of fitting the real settleent pattern to the ideal hexagonal hierarchical systes, to test central place theory in the future. Acknowledgeents This research was sponsored by the National Natural Science Foundation of China (Grant No ). 22

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25 Knox PL, Marston SA (2007). Places and Regions in Global Context: Huan Geography (4th Edition). Upper Saddle River, NJ: Prentice Hall LaBarbera P, Rosso R (989). On the fractal diension of strea networks. Water Resources Research, 25(4): Longley PA, Batty M, Shepherd J (99). The size, shape and diension of urban settleents. Transactions of the Institute of British Geographers (New Series), 6(): Lösch A (940). The Econoics of Location (tran. By W.H. Woglo and W.R. Stolper in 954). New Haven: Yale University Press Mandelbrot BB (983). The Fractal Geoetry of Nature. New York: W. H. Freean and Copany Manrubia S, Zanette D (998). Interittency odel for urban developent. Physical Review E, 58(): Niu W-Y (992). Theoretical Geography. Beijing: The Coercial Press Prigogine I, Stengers I (984). Order out of Chaos: Man's New Dialogue with Nature. New York: Banta Book Rayner JN, Golledge RG, Collins Jr. SS (97). Spectral analysis of settleent patterns. In: Final Report, NSF Grant No. GS-278. Ohio State University Research Foundation, Colubus, Ohio, pp Rigon R, Rodriguez-Iturbe I, Rinaldo A (998). Feasible optiality iplies Hack's law. Water Resources Research, 34 (): Rodriguez-Iturbe I, Rinaldo A (200). Fractal River Basins: Chance and Self-Organization. Cabridge: Cabridge University Press Rozenfeld HD, Rybski D, Andrade Jr. JS, Batty M, Stanley HE, Makse HA (2008). Laws of population growth. PNAS, 05(48): Rybski D, Buldyrev SV, Havlin S, Liljeros F, Makse HA (2009). Scaling laws of huan interaction activity. PNAS, 06(3): Tarboton DG, Bras RL, Rodriguez-Iturbe I (988). The fractal nature of river networks. Water Resources Research, 24: White R, Engelen G (993). Cellular autoata and fractal urban for: a cellular odeling approach to the evolution of urban land-use patterns. Environent and Planning A, 25(8): White R, Engelen G (993). Urban systes dynaics and cellular autoata: fractal structures between 25

26 order and chaos. Chaos, Solitons & Fractals, 994, 4(4): Woldenberg MJ, Berry BJL (967). Rivers and central places: analogous systes? Journal of Regional Science, 7: Ye D-N, Xu W-D, He W, Li Z (200). Syetry distribution of cities in China. Science in China D: Earth Sciences, 44(8): Zanette D, Manrubia S (997). Role of interittency in urban developent: a odel of large-scale city foration. Physical Review Letters, 79(3): Appendices (The first and third appendixes can be deleted after review) A A sketch ap of spatial interittency The spatial disaggregation is an iportant concept in theoretical geography. Spatial disaggreation includes recursive subdivision of space, hierarchy and network structure (Batty and Longley, 994). The spatial subdivision can be divided into two types: interittency-free subdivision (Figure A(a)) and interittent subdivision (Figure A(b)). The forer is associated with the Euclidean space while the latter with the fractal space. The classical central place odels are based on the interittency-free recursive subdivision of geographical space. This paper tries to introduce the idea of recursive subdivision of space with interittency into central place theory and thus yield fractal central place systes. 26

27 a b c a. Spatial subdivision process without interittency a b c b. Spatial subdivision process with interittency Figure A Sketch aps for two kinds of recursive subdivision of geographical space (Note: Figure A (a) displays a Euclidean process with diension d=ln(4)/ln(2)=2; Figure A (b) shows a interittent process with a fractal diension D f =ln(3)/ln(2).585) A2 Another case of large scale based on Indian cities The sae ethod of scaling analysis as that in Subsection 3. can be easily applied to Indian censuses of 98, 99 and 200. The results are very satisfying (Tables A and A2, Figure A2). The case of India lends further support to the hierarchical scaling relation between city nuber and size predicted by central place theory. Table A The population size and nuber of Indian urban places: Order Population size City nuber (N ) () (P ) * * 3 27

28 Source: The Indian censuses of 98, 99 and 200. Cited fro: Basu and Bandyapadhyay, * Data unavailable. The last class, i.e., the 9 th class, is a lae-duck class owing to undergrowth of cities. Table A2 The scaling odels, scaling exponents, and corresponding goodness of fit for Indian urban places, Year Matheatical odel Scaling exponent b Goodness of fit R N = P N = P N = P 000 N = P R 2 = N = P R 2 = N = P R 2 = N N N P P P a. 98 b. 99 c. 200 Figure A2 The scaling relations between the lower liit of population size and nuber of urban places in India, A3 Why to use the least average distance instead of the average distance In order to calculate the diension of a geoetrical body, we should eploy the boundary value instead of the ean value of scale to ake a easureent. Concretely speaking, we use the upper liits of scales for positive power law (PLR) relation or the area-radius scaling to avoid iproper scale translation, and use the lower liits of scales for negative power law (NPL) relation or the box-counting scaling to avoid box overlapping. The scaling exponent iplies the 28

29 diension. For exaple, if we want to copute the diension of a 2D plane, we can ake use of the PLR-based area-radius scaling. Drawing a syste of concentric circles yields a set of data (Figure A3, Table A3). Based on the upper liits of radius, a least squares coputation gives the following result A( r) r 2 = π r d = 3.46, where r refers to the upper liit of a radius for a ring, and A(r) to the area of the corresponding ring, which is defined as the geoetric region between two iediate circles. As for the paraeters, π denotes the ratio of the circuference of a circle to its diaeter, and d=2 is just the Euclidean diension of the geoetric plane. This is a perfect fit (R 2 =). However, if we substitute the average radius for the upper liit of the radius, the result is as below.5935 A ( r) = 5.775r. This is a typical fractal rabbit! The goodness of fit is about R 2 =0.9928, and scaling exponent is about.5935, not close to 2. Besides, the proportionality coefficient is not close to the pi value. Table A3 The scaling odels, scaling exponents, and corresponding goodness of fit for Indian cities Nuber Lower liit of radius Average radius Ring area