fractal urban areas

int. j. geographical information science, 2002vol. 16, no. 5, 419 437

Research Article

Fractal dimension and fractal growth of urbanized areas

GUOQIANG SHEN

Division of City and Regional Planning, College of Architecture, University ofOklahoma, Norman, OK 73019, USA; e-mail: [email protected]

(Received 26 November 1999; accepted 3 October 2001 )

Abstract. Based on a box-accounting fractal dimension algorithm (BCFD) and aunique procedure of data processing, this paper computes planar fractal dimensionsof 20 large US cities along with their surrounding urbanized areas. The results showthat the value range of planar urban fractal dimension (D) is 1

G. Shen420

the metes and bounds of each parcel must be precisely determined for legal andengineering purposes. The fractal structure of urban form becomes more apparentwhen the urbanized areas of a city, metropolis, or urban system are viewed as awhole. In addition, a complex urban form, as evolved from its very beginning as ahandful of developed land parcels, shows an organic growth process in which moreurbanized areas are added over time. This kind of urban growth is similar to thefractal growth in Sander (1987) and the fractal urban growth in Batty (1991) andBatty and Longley (1994). Thus, the overall form of the urbanized areas of a citycan be treated as a fractal and described by fractal geometry.

This paper examines fractal dimensions of urban forms based on urbanized areasof 20 cities in the United States. The relationships between fractal dimension, urban-ized areas, and urban population are explored. Fractal urban growth of City ofBaltimore, MD is studied through 12 time periods from 1792 to 1992. Speci cally,the research focuses on, rst, whether diVerent urban forms (e.g., planar urbanizedarea, urban population) can have virtually the same fractal dimensions; secondly,how urban and population sizes are statistically related to fractal dimension; andthirdly, whether fractal dimension is a stable measure of urban population density.

The paper is organized as follows. Section 2 brie y reviews the literature onfractal dimension and its applications to urban form and growth research. Section 3presents a box-counting methodology and a log-linear function for tting urbanpopulation and fractal dimension. Section 4 describes data processing procedures.Section 5 discusses computational results of fractal dimension and fractal urbangrowth. Conclusions and remarks are included in section 6.

2. An overview of existing researchOne of the important aspects of fractal geometry is fractal dimension. Although

many researchers had contributed to the development and formalization of fractaldimension (HausdorV 1919, Richardson 1961), it was not until Mandelbrot (1967)that the fractal dimension concept was rmly established. Mandelbrot argued thatif a straight line or a plane is absolute with Euclidean dimension 1 or 2, respectively,then spatial objects such as coastlines which twist in the plane must intuitively havea fractal dimension between 1 and 2.

In urban analysis, Batty and Longley (1987a, 1987b, 1994) and Batty and Xie(1996) studied fractal dimensions of planar urban form and urban growth. Theycalculated three types of urban fractal dimensions based on city size, shape, andscale using a method similar to the box-accounting algorithm. The fractal dimensionsof US cities (e.g. BuValo, NY, Columbus, OH, and Pittsburgh, PA) and internationalcities (e.g. Seoul, CardiV, London, and Paris) were obtained with values ranging from1.312 to 1.862. Fractal dimensions representing urban growth of London between18201962 were also calculated in Batty and Longley (1994) with values rangingfrom 1.322 to 1.791. These fractal dimensions are consistent with their analysis ofthe fractal growth of CardiV, England in the sense that urban fractal growth isessentially a space lling process, that is, the larger the fractal dimension value, themore lled a planar city becomes. Batty and Xie (1999) further examined the fractalspace lling process by using the concept of self-organized criticality and BuValosurban development over the period of 17501989. The images used in Batty andLongley (1994) came from Abercrombie (1945) and Doxiadis (1968) for the case ofLondon, England. The detailed data sets used for BuValo and other US cities wereprimarily based on 100m 100m grid images derived from a number of sources,including the 1990 TIGER les.

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Fractal dimension and f ractal growth of urbanized areas 421

Frankhauser (1990, 1992, 1994) conducted extensive fractal dimension studies forUS (e.g. Los Angeles), European (e.g. Rome), and international cities (e.g. MexicoCity). The fractal dimensions for these cities ranged from 1.39 for Taipei to 1.93 forBeijing. Frankhauser (1990, 1991, 1994) also reported the fractal dimensions showingthe growth of Berlin in 1875, 1920, and 1945 to be 1.43, 1.54, and 1.69, respectively.

Similar work on urban fractal dimensions and growth can also be found inThibault and Marchand (1987), Batty et al. (1989), Wong and Fortheringham (1990),Smith (1991), Batty (1991), Benguigui and Daoud (1991), Batty and Howes (1996),and Shen (1997). These studies focus more on the fractal nature of a speci c elementof urban environment (e.g. urban transportation network, drainage utility network),a new technique (e.g. visualization, diVusion-limited aggregation) , or an importantissue (e.g. scale) in modelling fractal urban development.

While these studies have provided some interesting theoretical formulations andempirical results revealing the fractal nature of urban form and growth, they are notsystematic in the sense that cities were not selected according to a spatial scheme(e.g. city or population size hierarchy) and a common set of parameters (i.e. mapcoverage, resolution, scale). Thus, the results are incomplete and less useful for thepurpose of inter-city comparison from the urban system perspective. Also, the analysisof fractal urban growth and the linkage to urban population in these studies aremainly based on model simulation rather than referenced to the observed urbangrowth or population. Therefore, leaving the fractal dimension and growth simulationdisconnected from the actual urban growth of land and population.

In this study, a systematic analysis of planar urban fractal dimensions of 20 USurbanized areas, including their central cities and surrounding urban places, ispresented. These urbanized areas, identi ed with their central cities, were selectedfrom the top 40 cities ranked by 1992 population. The relationship between fractaldimensions of these cities and their urban population is examined through a log-linear function. Fractal urban growth is examined for Baltimore, MD, whose digitaldata of urbanized areas and population can be found in Clark et al. (1996) and USBureau of the Census (1999a) .

3. Methodology3.1. T he Box-Counting Fractal Dimension (BCFD) algorithm

There are a variety of fractal dimensions, including HausdorV-Besicovitch dimen-sion, Minkowski-Boulingand dimension, the capacity dimension, and the similaritydimension (Barnsley, 1988). Fractal dimensions also can be calculated in a numberof ways, including the Calliper method, which is based on linear measurement sizesand steps, the box-counting method, which uses a set of meshes laid over an image,the pixel-dilation method, which calculates the Minkowski-Boulingand dimensionbased on a set of in nitive small circles, and the mass-radius method, which is basedon the image portion found within a set of concentric rings covering the image(Mandelbrot 1983, Peitgen and Saupe 1985). Each of these methods can be usedto analyse spatial objects ranging from strictly self-similar to non-self-similar fora range of scales. For strictly self-similar mathematical fractals the mass-radiusdimension, the capacity dimension, and the similarity dimension are the same asthe HausdorV-Besicovitch dimension and the Minkowski-Boulingand dimension(Falconer, 1990). However, for non-self-similar fractals, these methods would yieldslightly diVerent dimension values.

In urban and spatial analysis, fractal dimensions are mainly computed using the

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box-counting method (i.e. Batty and Longley 1994, Shen 1997) and the mass-radiusmethod (i.e. Batty and Longley 1987a, Benguigui and Daoud, 1991, Frankhauser1994, Batty and Xie 1996). Given that urban forms or urbanized areas are notstrictly self-similar and that the scales, image resolutions, and coverage sizes used inthese studies are not the same, the reported fractal dimensions from these studieswould necessarily vary, though in many cases the diVerences are fairly small. Inthis study, the box-counting algorithm fractal dimension (BCFD) algorithm isbased upon the HausdorV-Besicovitch dimension. The mathematical description ofbox-counting and its speci c treatment in BCFD are brie y given below.

Let the diameter of the smallest circle covering all areas in set V be diam (V ). Ifa given set Q is contained in the union of sets {V

i} and each V

ihas diameter less

than s, then {Vi} is an s-cover of Q. The HausdorV d-dimensional measure of Q is:

L im

s 0H

d(s) 5 L im

s 0[Min

Vi |Q 0diam(V

i)d] (1)

The result of this limiting process may be in nite since it is a non-increasingfunction of s (Falconer 1990).

The fractal dimension D, as given by HausdorV and studied by Besicovitch, isthe limiting value:

L im

s 0H

d(s) 5 G2 0


estimated fractal dimension D of the true fractal dimension d is given by the bestslope of the L og(N(s)) vs. L og(1/s) graph. Thus, fractal dimension values of the 20urbanized areas are actually least-square estimates of their true fractal dimensions.Equation (4 ) can be rewritten as:

L og (N(s)) 5 L og (C )1 DL og (1/s) 1 Es

(5)

where Esis the error term, L og(C ) the regression constant with C representing the

size of urbanized areas, D the estimated fractal dimension.This estimation produces some ambiguity for D due mainly to choices of s. Pruess

(1995) pointed out that sampling N at only a nite number of s values may producepoor estimates for D. He suggested that to produce reasonable estimates for D mayrequire s to be very small. However, box size smaller than image resolution (pixelsize) is not necessary for using BCFD. On the other hand, a box size larger than thecity image size is also not used since N(s) is always equal to 1. In this study, themaximum box size is set to one half of the image size. Speci cally, the maximumbox size for the 20 cities is 500 pixels and for the City of Baltimore is 60 pixels.

3.2. L inkage between urban size, population and fractal dimensionDue to the small sample size of 20 cities, only log-linear functions (6) and (7)

were used to t urbanized areas and population sizes by fractal dimensions (L og(C )vs. D and L og(POP ) vs. D. The best- t functions, along with their constants,coeYcients, R-square values, and charts are reported in 5.

L og(C ) 5 a1 1 a2D 1 ec(6)

L og(POP ) 5 b1 1 b2D 1 epop(7)

where a1 , a2 , b1 , b2 , ec, e

popare constants or coeYcients or error terms to be

estimated.

4. Data processingIn line with the de nition by the US Bureau of the Census(1999a) , urbanized

areas were regarded as developed areas in central cities with 50 000 or moreinhabitants and their surrounding densely settled urban places, whether or notincorporated. These urbanized areas, identi ed by their central cities, were selectedas follows. First, 1992 US urban population was ranked for all cities. Then, the top40 cities were selected and numbered. The 20 cities with odd numbers ranging from1 to 39 were selected for this study.

The BCFD algorithm requires a city image in PICT format as input. The imagesof the 20 urbanized areas were obtained from TIGER Map Service (TMS) providedby US Bureau of the Census (1999b) . TMS is an Internet-based public domainservice that provides on-line high quality image maps in GIF format of anywherein the United States. Colour maps of urbanized areas can be created on-the- y froma special binary version of TIGER92 at customized scales.

The necessary input to TMS to create a city map showing urbanized areasincludes city centre coordinates, map sizes, and image resolution, etc. The scale ofthe city maps is about 1:1 463 500. The map projection type is Albers Equal-AreasConic (Conterminous US). These input parameters and their relationships are illus-trated in gure 1 for the case of Chicago, IL. Clearly, each pixel area in the maprepresents an area of 178m 178m on the ground.

The city centres longitudes and latitudes were obtained from the city/town searchfunction within TMS. The city maps generated by TMS in GIF format were converted

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Figure 1. Map size, coordinate, and resolution parameters for Chicago.

into black and white PICT images in Adobe PhotoShop. The black areas in eachcity map represent the urbanized areas in and around the city. Each city image inPICT format was then used as an input city image le to the BCFD algorithmdeveloped in this study.

Each city map represents a rectangular area bounded by latitude and longitudelines 0.8 decimal degrees away from the city centre. For example, the map of Chicagoin gure 1 represents the area bounded by 41.05 and 42.65 latitudes and 86.85 and 88.45 longitudes. The image size (width by height) in pixels is 1000 1000. Sucha map or image size was selected to ensure that the central city be included completelytogether with nearby minor cities or towns. Although the 20 central cities are ofdiVerent sizes, only New Yorks urbanized areas are slightly out of its image boundary.

The images of the 20 cities are shown in gure 2. County boundaries are includedfor reference purposes (e.g. location and scope). The city images actually used inBCFD for computing fractal dimension do not include county boundaries.

Given that each city map typically contains urbanized areas within one centralcity and the urbanized areas surrounding the city, the population data must corre-spond to the population in the central city as well as its surrounding urbanizedareas. This was accomplished by using the ArcView GIS 3.2 software and the Arc-USA database. Each city image map was rst imported to ArcView and recti edwith ArcViews Image Analyst Extension with the city images projection and longit-ude and latitude coordinates. The geo-referenced city image was then overlaid withplaces and counties layers available in the Arc-USA database. The populations of

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Figure 2. Urbanized areas of 20 US cities with county boundaries.

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the places inside the city image boundary were added and the total was used as thepopulation for the city and its urbanized areas.

5. Fractal dimension computation5.1. Fractal dimension

The BCFD algorithm calculates the box dimension of 2-D urbanized areas. Foreach city image, BCFD outputs a set of values, including box sizes (s), box counts(N(s)), and their appropriate logarithmic values L og(N(s)) and L og(1/s). These valueswere then imported into SAS for computing the fractal dimension using the least-square estimation. The fractal dimension D was regarded as the best regression slopeof a L og(N(s)) vs. L og(1/s) graph.

Because the origin of the boxes with respect to the pixels in the image was notspeci ed, multiple measures of N(s) were computed for diVerent mesh origins. Thegraphed value of N(s) actually was the average of N(s) from diVerent mesh origins.For example, the box-counts (N(s)), box sizes (s), and logarithmic values L og(1/s),L og(N(s)) generated by the BCFD algorithm for New York City and Omaha arelisted in table 1.

The L og(N(s)) vs. L og(1/s) graphs for the New York City, NY and Omaha,NE are given in gure 3 and gure 4. The fractal dimension estimates for Omaha,NE and New York City, NY are 1.2778 and 1.7014, while the intercepts are 10.33and 12.408, and the associated values are 0.9932 and 0.9993 respectively. Thus,the linear regression functions are L og(N(s)) 5 10.33 1 1.2778L og(1/s) for Omahaand L og(N(s)) 5 12.408 1 1.7014L og (1/s) for New York City. The fractal dimensionestimates and urban populations for the 20 US cities are listed in the table 2.

Figure 3 and gure 4 show a strong linear association between L og(N(s)) andL og(1/s), with the slope values of the regression lines between 1 and 2. This observa-tion, as can be seen from table 2, is true for all the 20 cities. This observation canalso be regarded as proof that the urbanized areas are indeed fractals. Also fromtable 2 we can see that New York City has the highest fractal dimension value of1.7014, while Omaha has the lowest value of 1.2778.

With the same map scale, resolution, projection, and map coverage, this fractaldimension value diVerence can be visually associated with New York City, which

Table 1. BCFD outputs for New York City, NY and Omaha, NE.

New York City, NY Omaha, NE

s L og(1/s) N(s) L og(N(s)) s L og(1/s) N(s) L og(N(s))

1 0 283 175 12.5538 1 0 38 840 10.56722 0.69315 76 591 11.2462 2 0.693147 11 142 9.318484 1.38629 21 087 9.95641 4 1.38629 3501 8.160833 3.49651 581 6.36475 33 3.49651 356 5.8749362 4.12713 201 5.3033 62 4.12713 213 5.3612993 4.5326 103 4.63473 93 4.5326 123 4.81218125 4.82831 61 4.11087 125 4.82831 73 4.29046156 5.0.986 44 3.78419 156 5.04986 49 3.89182187 5.23111 33 3.49651 187 5.23111 41 3.71357218 5.3845 30 3.4012 218 5.3845 25 3.21888250 5.52146 21 3.04452 250 5.52146 24 3.17805500 6.21461 7 1.94591 500 6.21461 9 2.19722

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Table 2. Fractal dimension estimates and populations for 20 US cities.

City D L og(C ) R2 Population L og(POP )

New York City, NY 1.7014 12.408 0.999 16 413 024 16.614Dallas-FW, TX 1.6439 12.202 0.999 3 805 838 15.152Chicago, IL 1.6437 12.067 0.998 7 642 330 15.849Phoenix, AZ 1.6388 11.683 0.995 2 095 926 14.556San Francisco, CA 1.6285 11.757 0.998 6 397 514 15.671Boston, MA 1.6022 11.892 0.998 4 814 438 15.387Cleveland, OH 1.5869 11.613 0.999 3 093 683 14.945Oklahoma City, OK 1.5660 11.851 0.998 1 069 034 13.882Seattle, WA 1.5473 11.339 0.996 2 497 675 14.731Denver, CO 1.5114 11.228 0.996 1 837 507 14.424Pittsburgh, PA 1.4981 11.502 0.999 2 596 305 14.770Nashville, TN 1.4973 11.337 0.998 1 036 950 13.852Atlanta, GA 1.4950 11.477 0.999 1 602 367 14.287New Orleans, LA 1.4745 11.068 0.997 1 368 778 14.129Cincinnati, OH 1.4666 11.315 0.998 2 276 239 14.638Charlotte, NC 1.4643 11.290 0.998 1 204 531 14.002Albuquerque, NM 1.4294 10.490 0.993 641 363 13.371Tulsa, OK 1.4250 11.036 0.998 771 529 13.556Indianapolis, IN 1.4129 11.032 0.998 1 674 032 14.331Omaha, NE 1.2778 10.033 0.993 861 495 13.666

Figure 3. L og (N (s)) vs. L og (1/s) for Omaha, NE.

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Figure 4. L og (N (s)) vs. L og (1/s) for New York City, NY.

has a larger total urbanized area, and Omaha, which has a much smaller totalurbanized area, as shown in gure 2. This numeric disparity indicates that the fractaldimension can be thought of as a space- lling measurethe measure of urbanizedareas lling the city map coverage. The fractal dimension of urbanized areas canalso be thought of as an indicator of the complexity or dispersion of urban form. Ingeneral, the higher the value of a citys fractal dimension, the more complex ordisperse the city becomes. In this sense, the urban form of New York City is themost complex or disperse while the urban form of Omaha is the least complex ordisperse.

5.2. Urban size and population as functions of fractal dimensionSince the size, distribution, and complexity of urbanized areas are in uenced by

many other city parameters, such as population, fractal dimension of urbanized areasmay be used as an important parameter for urban form and growth modelling, andthis, indeed, has been manifested in the well-known work reviewed in 2. For the 20US cities, the best- t log-linear functions are displayed in gure 5 and gure 6. Sinceonly 20 observations are used and the values (0.6404 and 0.8761) are quite high,log-linear functions of urbanized areas or population sizes over fractal dimensionscan generate reasonably good estimates.

5.3. Fractal dimension, f ractal growth, and urban population in Baltimore, MDThe linkage between fractal dimension and urban growth was studied for the

case of Baltimore, MD. Speci cally, the urbanized areas of Baltimore for 12 timeperiods from 1792 to 1992 and the corresponding urban population were used. Theurbanized areas, shown in gure 7, were obtained from Clark et al. (1996 ) and

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Figure 5. L og (POP ) as a linear fraction of D for 20 US cities.

Figure 6. L og (C ) as a linear function of D for 20 US cities.

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Figure 7. Urbanized areas in Baltimore, MD in 12 selected years.

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transformed into black and white PICT les as required by the BCFD algorithmfor fractal dimension computation. These urban images were synthesized from mul-tiple resources, including historic maps, topographic maps, commercial road maps,remotely sensed data, existing digital land use data, and Digital Line Graphs (DLG)with scales ranging from 1:12 000 to 1:250 000. The nal images were set withUniversal Transverse Mercator (UTM) projection and scaled at 1:100 000 after theywere referenced to the 1:100 000-scale DLG road network with an acceptable accu-racy. The urban population data, shown in table 3, were obtained from populationcensuses from 1790 to 1990 and interpolated to the 12 speci c years. The log-linearfunctions (6) and (7) were used to t fractal dimensions to urban population sizesand urbanized areas.

Corresponding to urban population, the total urbanized area, L og(C ), increasesfrom 3.3722 in 1792 to 9.6215 in 1992, while the fractal dimension, D, increases from0.6641 in 1792 to 1.7211 in 1992. Given the fact that from 1792 to 1992, City ofBaltimores urbanized areas grew substantially, the positive relationship betweenurbanized areas and fractal dimension indicates that fractal dimension is a reasonablygood measure of urban spatial growth.

Interestingly, the goodness-of- t in terms of R2 values improves consistently from0.8549 in 1792 to 0.9990 in 1992. This goodness-of- t disparity can be seen in gure 8and gure 9. While inaccuracy of box-counting estimation is well-known, the abovedisparity indicates that the box-counting algorithm in general and the BCFD inparticularly may be more accurate for computing fractal dimensions of well developedurbanized areas.

Figure 10 shows that the urbanized areas in Baltimore increased consistently.The growth rate was higher during 17921990 and again during 19381953 thanthat during 19251938. Slower growth occurred around 1953 and continued through1992. This process is also clearly re ected by the Baltimores population chroniclefor the period 17921992, with the population growing from 16105 in 1792 to itspeak 946 530 in 1953 and then declining to 726 096 in 1992.

The positive correlation between fractal dimension and urbanized areas for theperiod of 17921992 can be seen clearly in gure 10. However, given that Baltimoresurbanized areas grew slowly ( gure 10) while the population increased dramaticallyduring 17921953 and dropped steadily during 19531992 ( gure 11), it can beinferred that the growth of urbanized areas may not necessarily lead to the growth

Table 3. Population, fractal dimension, and urbanized areas for Baltimore.

Year D L og(C ) R2 Population L og(POP )

1792 0.6641 3.3722 0.8549 16 105 9.6871822 1.0157 4.3981 0.9251 66 314 11.1021851 1.1544 5.4106 0.9376 173 390 12.0631878 1.2059 6.1801 0.9553 319 321 12.6741900 1.3024 7.4534 0.9980 508 957 13.1401925 1.3836 7.9415 0.9968 769 350 13.5531938 1.4374 8.0559 0.9971 848 255 13.6511953 1.5953 9.0426 0.9975 946 503 13.7611966 1.6450 9.3018 0.9980 919 065 13.7311972 1.6822 9.5833 0.9986 881 962 13.6901982 1.7163 9.5947 0.9988 776 623 13.5631992 1.7211 9.6215 0.9990 726 096 13.495

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Figure 8. Baltimore, MD 1851.

of population, that is, the fractal dimension does not always linearly and positivelycorrespond to population growth or decline.

For the 12 time periods of Baltimore, the best- t log-linear functions are displayedin gure 12 and gure 13. Since only 12 observations are used and the R2 values(0.8590 and 0.9672) are quite high, log-linear functions of urbanized areas orpopulation sizes over fractal dimensions can provide fairly good estimates.

6. Conclusions and RemarksIndividual parcels of urbanized areas can be geometrically planned and designed

using Euclidean geometry. However, the planar urban form as a whole system cannotbe fully described by Euclidean geometry. This is because the urban form and itsdevelopment demonstrate the distinct nature of fractals, namely, irregularity, scale-independence, and self-similarity at least for a range of scales. Thus, it is appropriateto regard the urbanized areas as fractals and study their spatial forms in fractalgeometry.

Fractal dimension, a necessary dimensional measure of fractal geometry, can becalculated for urbanized areas. Since a fractal dimension computation is essentiallya limiting process and requires good algorithms to approximate for values, it isinevitably associated with errors. In this study, the BCFD algorithm and the least-square estimation technique were presented and discussed. The fractal dimensionsof 20 large US cities were computed and linked to urban population and urbanizedareas. The urban form and population growth of Baltimore was examined for theperiod 17921992 and linked to fractal dimensions.

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Figure 9. Baltimore, MD, 1992.

Figure 10. Growth of urbanized areas and fractal dimensions, Baltimore, MD.

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Figure 11. Population growth by year, Baltimore, MD.

Figure 12. L og (POP ) as a linear function of D, Baltimore, MD.

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Figure 13. L og (C ) as a linear function of D, Baltimore, MD.

Since the box-counting algorithm developed in this study requires a speci cimage format, the black and white PICT format, as input, the preparation andprocessing of urban image data were discussed. The 20 city images were obtainedfrom Tiger Map Service (TMS), a free public service on the Internet through theWWW. To make the fractal dimensions comparable, the map coverage andboundary, image scale, and resolution were set the same for all the 20 city maps.Similar treatment was also adopted for the Baltimore data set. Clearly, small fractaldimension variances would be expected if a slightly enlarged map boundary for the20 US cities or Baltimore were used.

The linkages between fractal dimension and urban population and urbanizedareas, as tted with log-linear functions for the 20 US cities in general and forBaltimore in particular, were established. In general, log-linear functions of fractaldimensions can yield satisfactory ts for population sizes and urbanized areas interms of R2 values even though some graphs (e.g. gure 12) suggest a curvilinearrelationship.

DiVerent urban forms can have virtually the same fractal dimension value. Forexample, from table 2 we can see that the fractal dimensions for inland city Dallas-Fort Worth, TX and waterfront city Chicago, IL are 1.6439 and 1.6437 respectively.The two inland cities Pittsburgh, PA and Nashville, TN also have similar fractaldimension values of 1.4981 and 1.4973 respectively. This observation, as pointed outin Shen (1997), indicates again that fractal dimension alone says little about speci corientation and con guration of a physical urban form. The usefulness of fractaldimension lies primarily in its aggregate measure of overall urban form as a fractal.

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Cities with virtually the same fractal dimension values and urbanized areas mayhave quite diVerent population sizes. For example, the population of Chicago wasslightly over twice the population of Dallas ((7,642,330 vs. 3,805,838) and Pittsburghs

population was 2.5 time Nashvilles population (2,596,305 vs. 1,036,950) in 1992.This observation implies that fractal dimension alone is a fair indicator of the totalof urbanized areas but not a good measure of urban population density. The data

and results for Baltimore also show that while fractal dimension is always positivelycorrelated to urban size and growth, it does not display such a monotonic relation

with urban population size and change. The underlying cause for this is that whilethe population of a city may have sizable changes (e.g. growth or decline) over time,its urbanized areas usually increase at various paces. In fact, the total physical size

of urbanized areas of a livable city rarely decrease. Thus, fractal dimension is not agood direct measure of urban population density. The indirect use of fractal dimen-

sion for urban density modelling, as reported in some previous studies (e.g. Battyand Longley 1994), also warrants further justi cation.

The fact that previous literature on urban fractal research (e.g. Frankhauser 1994,Batty and Longley 1994) and this study use slightly diVerent methods to compute

the fractal dimension inevitably generates some diVerences in results. For example,Frankhauser (1994) reported a fractal dimension of 1.59 and 1.775 for Pittsburgh of1981 and 1990 respectively. Batty and Longley (1994) showed a fractal dimension

of 1.732 for Cleveland of 1990. These results are similar but still diVerent from thefractal dimension of 1.4981 for Pittsburgh and 1.5869 for Cleveland of 1992 in this

study. In addition to computing-method variations, disparities in image size, mapcoverage and boundary, image resolution, data accuracy, time period, box-size, and

scale may also contribute to diVerences in results. It would be interesting to see amore uni ed method, database, and a set of modelling parameters to be adopted in

future endeavours.Given that planar fractal growth can be regarded as a 2-D space lling process,

planar fractal dimension can certainly be used to modelling 2-D urban growth andurban form. It would be interesting to see a more complete set of fractal dimensionsfor major urbanized areas in the US or other countries. Such a fractal dimension

set may shed more light on the intriguing nature of fractal dimension as a spatialdimension measure and its role in urban modelling and spatial analysis.

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