int. j. geographical information science, 2002
vol. 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 of
Oklahoma, Norman, OK 73019, USA; e-mail: [email protected]
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(Received 26 November 1999; accepted 3 October 2001)
Abstract. Based on a box-accounting fractal dimension algorithm (BCFD) and a
unique procedure of data processing, this paper computes planar fractal dimensions
of 20 large US cities along with their surrounding urbanized areas. The results show
that the value range of planar urban fractal dimension (D) is 1 <D <2, with D for
the largest city, New York City, and the smallest city, Omaha being 1.7014 and
1.2778 respectively. The estimated urban fractal dimensions are then regressed to
the total urbanized areas, L og(C), and total urban population, L og(POP ), with loglinear functions. In general, the linear functions can produce good-Ž ts for L og(C )
vs. D and L og(POP ) vs. D in terms of R2 values. The observation that cities may
have virtually the same D or L og(C) value but quite disparate population sizes
indicates that D itself says little about the speciŽ c orientation and conŽ guration of
an urban form and is not a good measure of urban population density. This paper
also explores fractal dimension and fractal growth of Baltimore, MD for the 200-year
span from 1792 –1992. The results show that Baltimore’s D also satisŽ es the inequality
1 <D <2, with D 5 1.0157 in 1822 and D 5 1.7221 in 1992. D5 0.6641 for Baltimore
in 1792 is an exception due mainly to its relatively small urban image with respect
to pixel size. While D always increases with L og(C) over the years, it is not always
positively correlated to urban population, L og(POP ).
1.
Introduction
Fractals are tenuous spatial objects whose geometric characteristics include
irregularity, scale-independence, and self-similarity. While natural spatial objects
such as coastlines, plants, and clouds have long been treated as fractals of various
dimensions (Mandelbrot 1967, 1983, Peitgen and Saupe 1985, Orbach 1986, Falconer
1990, Takayasu 1990, Porter and Gleick 1990, Lam and De Cola 1993), recent
research on spatial analysis has concluded that artiŽ cially planned and designed
spatial objects such as urban forms and transportatio n networks can also be treated
as fractals (Fotheringham et al. 1989, Batty and Longley 1987a, 1987b, 1994,
Arlinghaus 1985, 1990, Goodchild and Mark 1987, Arlinghaus and Nystaen 1990,
Benguigui and Daoud 1991, Frankhauser 1990, 1992, Batty and Xie 1996, Shen
1997, Batty and Xie 1999). Spatial objects with Euclidean geometric regularity are
regarded as special fractals with an integer fractal dimension of 1, 2, or 3.
Research on fractal nature of urban form has been inhibited since our conventional attention to urban forms has been on their regularity of Euclidean geometry.
This is especially true in the land subdivision planning and design process in which
Internationa l Journal of Geographica l Informatio n Science
ISSN 1365-881 6 print/ISSN 1362-308 7 online © 2002 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/13658810210137013
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420
G. Shen
the metes and bounds of each parcel must be precisely determined for legal and
engineering purposes. The fractal structure of urban form becomes more apparent
when the urbanized areas of a city, metropolis, or urban system are viewed as a
whole. In addition, a complex urban form, as evolved from its very beginning as a
handful of developed land parcels, shows an organic growth process in which more
urbanized areas are added over time. This kind of urban growth is similar to the
fractal growth in Sander (1987) and the fractal urban growth in Batty (1991) and
Batty and Longley (1994). Thus, the overall form of the urbanized areas of a city
can be treated as a fractal and described by fractal geometry.
This paper examines fractal dimensions of urban forms based on urbanized areas
of 20 cities in the United States. The relationships between fractal dimension, urbanized areas, and urban population are explored. Fractal urban growth of City of
Baltimore, MD is studied through 12 time periods from 1792 to 1992. SpeciŽ cally,
the research focuses on, Ž rst, whether diVerent urban forms (e.g., planar urbanized
area, urban population) can have virtually the same fractal dimensions; secondly,
how urban and population sizes are statistically related to fractal dimension; and
thirdly, whether fractal dimension is a stable measure of urban population density.
The paper is organized as follows. Section 2 brie y reviews the literature on
fractal dimension and its applications to urban form and growth research. Section 3
presents a box-counting methodology and a log-linear function for Ž tting urban
population and fractal dimension. Section 4 describes data processing procedures.
Section 5 discusses computational results of fractal dimension and fractal urban
growth. Conclusions and remarks are included in section 6.
2.
An overview of existing research
One of the important aspects of fractal geometry is fractal dimension. Although
many researchers had contributed to the development and formalization of fractal
dimension (HausdorV 1919, Richardson 1961), it was not until Mandelbrot (1967)
that the fractal dimension concept was Ž rmly established. Mandelbrot argued that
if 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 have
a 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. They
calculated three types of urban fractal dimensions based on city size, shape, and
scale using a method similar to the box-accountin g algorithm. The fractal dimensions
of US cities (e.g. BuValo, NY, Columbus, OH, and Pittsburgh, PA) and international
cities (e.g. Seoul, CardiV, London, and Paris) were obtained with values ranging from
1.312 to 1.862. Fractal dimensions representing urban growth of London between
1820 –1962 were also calculated in Batty and Longley (1994) with values ranging
from 1.322 to 1.791. These fractal dimensions are consistent with their analysis of
the fractal growth of CardiV, England in the sense that urban fractal growth is
essentially a space Ž lling process, that is, the larger the fractal dimension value, the
more ‘Ž lled’ a planar city becomes. Batty and Xie (1999) further examined the fractal
space Ž lling process by using the concept of self-organized criticality and BuValo’s
urban development over the period of 1750 –1989. The images used in Batty and
Longley (1994) came from Abercrombie (1945) and Doxiadis (1968) for the case of
London, England. The detailed data sets used for BuValo and other US cities were
primarily based on 100 m Ö 100 m grid images derived from a number of sources,
including the 1990 TIGER Ž les.
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Fractal dimension and f ractal growth of urbanized areas
421
Frankhauser (1990, 1992, 1994) conducted extensive fractal dimension studies for
US (e.g. Los Angeles), European (e.g. Rome), and international cities (e.g. Mexico
City). The fractal dimensions for these cities ranged from 1.39 for Taipei to 1.93 for
Beijing. Frankhauser (1990, 1991, 1994) also reported the fractal dimensions showing
the 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 in
Thibault 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 speciŽ c element
of urban environment (e.g. urban transportatio n network, drainage utility network),
a new technique (e.g. visualization, diVusion-limited aggregation) , or an important
issue (e.g. scale) in modelling fractal urban development.
While these studies have provided some interesting theoretical formulations and
empirical results revealing the fractal nature of urban form and growth, they are not
systematic 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. map
coverage, resolution, scale). Thus, the results are incomplete and less useful for the
purpose of inter-city comparison from the urban system perspective. Also, the analysis
of fractal urban growth and the linkage to urban population in these studies are
mainly based on model simulation rather than referenced to the observed urban
growth or population. Therefore, leaving the fractal dimension and growth simulation
disconnected from the actual urban growth of land and population.
In this study, a systematic analysis of planar urban fractal dimensions of 20 US
urbanized areas, including their central cities and surrounding urban places, is
presented. These urbanized areas, identiŽ ed with their central cities, were selected
from the top 40 cities ranked by 1992 population. The relationship between fractal
dimensions of these cities and their urban population is examined through a loglinear function. Fractal urban growth is examined for Baltimore, MD, whose digital
data of urbanized areas and population can be found in Clark et al. (1996) and US
Bureau of the Census (1999a) .
3. Methodology
3.1. T he Box-Counting Fractal Dimension (BCFD) algorithm
There are a variety of fractal dimensions, including HausdorV-Besicovitch dimension, Minkowski-Boulingand dimension, the capacity dimension, and the similarity
dimension (Barnsley, 1988). Fractal dimensions also can be calculated in a number
of ways, including the Calliper method, which is based on linear measurement sizes
and steps, the box-counting method, which uses a set of meshes laid over an image,
the pixel-dilation method, which calculates the Minkowski-Boulingand dimension
based on a set of inŽ nitive small circles, and the mass-radius method, which is based
on 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 used
to analyse spatial objects ranging from strictly self-similar to non-self-similar for
a range of scales. For strictly self-similar mathematical fractals the mass-radius
dimension, the capacity dimension, and the similarity dimension are the same as
the HausdorV-Besicovitch dimension and the Minkowski-Boulingand dimension
(Falconer, 1990). However, for non-self-similar fractals, these methods would yield
slightly diVerent dimension values.
In urban and spatial analysis, fractal dimensions are mainly computed using the
G. Shen
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422
box-counting method (i.e. Batty and Longley 1994, Shen 1997) and the mass-radius
method (i.e. Batty and Longley 1987a, Benguigui and Daoud, 1991, Frankhause r
1994, Batty and Xie 1996). Given that urban forms or urbanized areas are not
strictly self-similar and that the scales, image resolutions, and coverage sizes used in
these studies are not the same, the reported fractal dimensions from these studies
would necessarily vary, though in many cases the diVerences are fairly small. In
this study, the box-counting algorithm fractal dimension (BCFD) algorithm is
based upon the HausdorV-Besicovitch dimension. The mathematical description of
box-counting and its speciŽ 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 ). If
a given set Q is contained in the union of sets {V } and each V has diameter less
i
i
than s, then {V } is an s-cover of Q. The HausdorV d-dimensional measure of Q is:
i
L im H (s) 5 L im [Min ž
diam(V )d]
(1)
d
i
s 0
s 0
Vi |Q¡ 0
The result of this limiting process may be inŽ nite since it is a non-increasing
function of s (Falconer 1990).
The fractal dimension D, as given by HausdorV and studied by Besicovitch, is
the limiting value:
L im H (s) 5
d
s 0
G
2
0 <d<D
0
D <d < 2
h
d5 D
(2)
When d 5 D, h is a Ž nite positive number and is known as the HausdorV measure,
which numerically characterizes the size of Q. The computations of equation (1 ) and
(2) must be approximated due to the limiting process, hence errors are introduced
that may be signiŽ cant to the true values of D.
To minimize over all s-covers, the BCFD algorithm uses a uniform Ž xed-size
square covering for V to approximate circle covering deŽ ned in (1). First the set Q
i
is embedded in some larger squares, which are then divided into smaller sub-squares
of constant diagonal s. This simple choice of V would produce the correct D as long
i
as the minimum and limit as s  0 are carefully chosen.
With a uniform Ž xed-size square covering, equation (1) becomes:
L im [H (s)] 5 L im [Min ž
diam(V )d] 5 L im [MinN (s)sd]
(3)
d
i
s 0
s 0
Vi |Q¡ 0
s 0
where N(s) is the number of boxes that cover areas of Q. Note that the minimization
is still present, since the location of Q in the plane is not speciŽ ed and mathematically
it is necessary to minimize over all possible box orientations by rotations and shifts.
This, unfortunately, is impossible with available computing capability except for only
a few orientations.
The approximation procedure in the BCFD algorithm is based upon a description
by Mandelbrot (1983) with the following basic relationship:
N(s) Ö sd 5 C
(4)
where N(s) is the number of boxes containing the urbanized areas C, d the true
fractal dimension. A square mesh of various sizes s is laid over the city image. The
mesh boxes N(s) that contain every part of the image are counted and a Ž nite number
of value points (s, natural logarithms L og(N(s)), and L og(1/s)) are generated. The
Fractal dimension and f ractal growth of urbanized areas
423
estimated fractal dimension D of the true fractal dimension d is given by the best
slope of the L og(N(s)) vs. L og(1/s) graph. Thus, fractal dimension values of the 20
urbanized areas are actually least-square estimates of their true fractal dimensions.
Equation (4 ) can be rewritten as:
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L og(N(s)) 5 L og(C )1
DL og(1/s)1
E
(5)
s
where E is the error term, L og(C ) the regression constant with C representing the
s
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 Ž nite number of s values may produce
poor estimates for D. He suggested that to produce reasonable estimates for D may
require s to be very small. However, box size smaller than image resolution (pixel
size) is not necessary for using BCFD. On the other hand, a box size larger than the
city image size is also not used since N(s) is always equal to 1. In this study, the
maximum box size is set to one half of the image size. SpeciŽ cally, the maximum
box 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 dimension
Due to the small sample size of 20 cities, only log-linear functions (6) and (7)
were used to Ž t urbanized areas and population sizes by fractal dimensions (L og(C )
vs. D and L og(POP) vs. D. The best-Ž t functions, along with their constants,
coeYcients, R-square values, and charts are reported in §5.
L og(C ) 5 a1 1 a2 D1 e
(6)
c
L og(POP) 5 b1 1 b2 D1 e
(7)
pop
where a1 , a2 , b1 , b2 , e , e
are constants or coeYcients or error terms to be
c pop
estimated.
4.
Data processing
In line with the deŽ nition by the US Bureau of the Census(1999a) , urbanized
areas were regarded as developed areas in central cities with 50 000 or more
inhabitants and their surrounding densely settled urban places, whether or not
incorporated. These urbanized areas, identiŽ ed by their central cities, were selected
as follows. First, 1992 US urban population was ranked for all cities. Then, the top
40 cities were selected and numbered. The 20 cities with odd numbers ranging from
1 to 39 were selected for this study.
The BCFD algorithm requires a city image in PICT format as input. The images
of the 20 urbanized areas were obtained from TIGER Map Service (TMS) provided
by US Bureau of the Census (1999b) . TMS is an Internet-based public domain
service that provides on-line high quality image maps in GIF format of anywhere
in the United States. Colour maps of urbanized areas can be created on-the- y from
a special binary version of TIGER’92 at customized scales.
The necessary input to TMS to create a city map showing urbanized areas
includes city centre coordinates, map sizes, and image resolution, etc. The scale of
the city maps is about 1:1 463 500. The map projection type is Albers Equal-Areas
Conic (Conterminous US). These input parameters and their relationships are illustrated in Ž gure 1 for the case of Chicago, IL. Clearly, each pixel area in the map
represents an area of 178 m Ö 178 m on the ground.
The city centres’ longitudes and latitudes were obtained from the city/town search
function within TMS. The city maps generated by TMS in GIF format were converted
G. Shen
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424
Figure 1.
Map size, coordinate, and resolution parameters for Chicago.
into black and white PICT images in Adobe PhotoShop. The black areas in each
city map represent the urbanized areas in and around the city. Each city image in
PICT format was then used as an input city image Ž le to the BCFD algorithm
developed in this study.
Each city map represents a rectangular area bounded by latitude and longitude
lines 0.8 decimal degrees away from the city centre. For example, the map of Chicago
in Ž gure 1 represents the area bounded by 41.05 and 42.65 latitudes and Õ 86.85 and
Õ 88.45 longitudes. The image size (width by height) in pixels is 1000 Ö 1000. Such
a map or image size was selected to ensure that the central city be included completely
together with nearby minor cities or towns. Although the 20 central cities are of
diVerent sizes, only New York’s urbanized areas are slightly out of its image boundary.
The images of the 20 cities are shown in Ž gure 2. County boundaries are included
for reference purposes (e.g. location and scope). The city images actually used in
BCFD for computing fractal dimension do not include county boundaries.
Given that each city map typically contains urbanized areas within one central
city and the urbanized areas surrounding the city, the population data must correspond to the population in the central city as well as its surrounding urbanized
areas. This was accomplished by using the ArcView GIS 3.2 software and the ArcUSA database. Each city image map was Ž rst imported to ArcView and rectiŽ ed
with ArcView’s Image Analyst Extension with the city image’s projection and longitude and latitude coordinates. The geo-referenced city image was then overlaid with
places and counties layers available in the Arc-USA database. The populations of
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Fractal dimension and f ractal growth of urbanized areas
Figure 2.
Urbanized areas of 20 US cities with county boundaries.
425
G. Shen
426
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the places inside the city image boundary were added and the total was used as the
population for the city and its urbanized areas.
5. Fractal dimension computation
5.1. Fractal dimension
The BCFD algorithm calculates the ‘box dimension’ of 2-D urbanized areas. For
each 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 values
were then imported into SAS for computing the fractal dimension using the leastsquare estimation. The fractal dimension D was regarded as the best regression slope
of 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 not
speciŽ ed, multiple measures of N(s) were computed for diVerent mesh origins. The
graphed 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 are
listed 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 Ž gure 3 and Ž gure 4. The fractal dimension estimates for Omaha,
NE and New York City, NY are 1.2778 and 1.7014, while the intercepts are 10.33
and 12.408, and the associated values are 0.9932 and 0.9993 respectively. Thus,
the linear regression functions are L og(N(s)) 5 10.331 1.2778L og(1/s) for Omaha
and L og(N(s)) 5 12.4081 1.7014L og(1/s) for New York City. The fractal dimension
estimates and urban populations for the 20 US cities are listed in the table 2.
Figure 3 and Ž gure 4 show a strong linear association between L og(N(s)) and
L og(1/s), with the slope values of the regression lines between 1 and 2. This observation, as can be seen from table 2, is true for all the 20 cities. This observation can
also be regarded as proof that the urbanized areas are indeed fractals. Also from
table 2 we can see that New York City has the highest fractal dimension value of
1.7014, while Omaha has the lowest value of 1.2778.
With the same map scale, resolution, projection, and map coverage, this fractal
dimension 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
s
1
2
4
33
62
93
125
156
187
218
250
500
L og(1/s)
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
0
0.69315
1.38629
3.49651
4.12713
4.5326
4.82831
5.0.986
5.23111
5.3845
5.52146
6.21461
Omaha, NE
N(s)
L og(N(s))
s
283 175
76 591
21 087
581
201
103
61
44
33
30
21
7
12.5538
11.2462
9.95641
6.36475
5.3033
4.63473
4.11087
3.78419
3.49651
3.4012
3.04452
1.94591
1
2
4
33
62
93
125
156
187
218
250
500
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
Õ
L og(1/s)
0
0.693147
1.38629
3.49651
4.12713
4.5326
4.82831
5.04986
5.23111
5.3845
5.52146
6.21461
N(s)
L og(N(s))
38 840
11 142
3501
356
213
123
73
49
41
25
24
9
10.5672
9.31848
8.1608
5.87493
5.36129
4.81218
4.29046
3.89182
3.71357
3.21888
3.17805
2.19722
Fractal dimension and f ractal growth of urbanized areas
Table 2.
Fractal dimension estimates and populations for 20 US cities.
City
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427
New York City, NY
Dallas-FW, TX
Chicago, IL
Phoenix, AZ
San Francisco, CA
Boston, MA
Cleveland, OH
Oklahoma City, OK
Seattle, WA
Denver, CO
Pittsburgh, PA
Nashville, TN
Atlanta, GA
New Orleans, LA
Cincinnati, OH
Charlotte, NC
Albuquerque, NM
Tulsa, OK
Indianapolis, IN
Omaha, NE
Figure 3.
D
L og(C )
R2
Population
L og(POP )
1.7014
1.6439
1.6437
1.6388
1.6285
1.6022
1.5869
1.5660
1.5473
1.5114
1.4981
1.4973
1.4950
1.4745
1.4666
1.4643
1.4294
1.4250
1.4129
1.2778
12.408
12.202
12.067
11.683
11.757
11.892
11.613
11.851
11.339
11.228
11.502
11.337
11.477
11.068
11.315
11.290
10.490
11.036
11.032
10.033
0.999
0.999
0.998
0.995
0.998
0.998
0.999
0.998
0.996
0.996
0.999
0.998
0.999
0.997
0.998
0.998
0.993
0.998
0.998
0.993
16 413 024
3 805 838
7 642 330
2 095 926
6 397 514
4 814 438
3 093 683
1 069 034
2 497 675
1 837 507
2 596 305
1 036 950
1 602 367
1 368 778
2 276 239
1 204 531
641 363
771 529
1 674 032
861 495
16.614
15.152
15.849
14.556
15.671
15.387
14.945
13.882
14.731
14.424
14.770
13.852
14.287
14.129
14.638
14.002
13.371
13.556
14.331
13.666
L og (N(s)) vs. L og (1/s) for Omaha, NE.
G. Shen
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428
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 total
urbanized area, as shown in Ž gure 2. This numeric disparity indicates that the fractal
dimension can be thought of as a space-Ž lling measure—the measure of urbanized
areas Ž lling the city map coverage. The fractal dimension of urbanized areas can
also be thought of as an indicator of the complexity or dispersion of urban form. In
general, the higher the value of a city’s fractal dimension, the more complex or
disperse the city becomes. In this sense, the urban form of New York City is the
most complex or disperse while the urban form of Omaha is the least complex or
disperse.
5.2. Urban size and population as functions of fractal dimension
Since the size, distribution, and complexity of urbanized areas are in uenced by
many other city parameters, such as population, fractal dimension of urbanized areas
may be used as an important parameter for urban form and growth modelling, and
this, indeed, has been manifested in the well-known work reviewed in §2. For the 20
US cities, the best-Ž t log-linear functions are displayed in Ž gure 5 and Ž gure 6. Since
only 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 dimensions
can generate reasonably good estimates.
5.3. Fractal dimension, f ractal growth, and urban population in Baltimore, MD
The linkage between fractal dimension and urban growth was studied for the
case of Baltimore, MD. SpeciŽ cally, the urbanized areas of Baltimore for 12 time
periods from 1792 to 1992 and the corresponding urban population were used. The
urbanized areas, shown in Ž gure 7, were obtained from Clark et al. (1996 ) and
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Fractal dimension and f ractal growth of urbanized areas
Figure 5.
Figure 6.
L og (POP ) as a linear fraction of D for 20 US cities.
L og (C) as a linear function of D for 20 US cities.
429
G. Shen
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430
Figure 7.
Urbanized areas in Baltimore, MD in 12 selected years.
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Fractal dimension and f ractal growth of urbanized areas
431
transformed into black and white PICT Ž les as required by the BCFD algorithm
for fractal dimension computation. These urban images were synthesized from multiple 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 Ž nal images were set with
Universal Transverse Mercator (UTM) projection and scaled at 1:100 000 after they
were referenced to the 1:100 000-scale DLG road network with an acceptable accuracy. The urban population data, shown in table 3, were obtained from population
censuses from 1790 to 1990 and interpolated to the 12 speciŽ c years. The log-linear
functions (6) and (7) were used to Ž t fractal dimensions to urban population sizes
and urbanized areas.
Corresponding to urban population, the total urbanized area, L og(C ), increases
from 3.3722 in 1792 to 9.6215 in 1992, while the fractal dimension, D, increases from
0.6641 in 1792 to 1.7211 in 1992. Given the fact that from 1792 to 1992, City of
Baltimore’s urbanized areas grew substantially, the positive relationship between
urbanized areas and fractal dimension indicates that fractal dimension is a reasonably
good measure of urban spatial growth.
Interestingly, the goodness-of-Ž t in terms of R2 values improves consistently from
0.8549 in 1792 to 0.9990 in 1992. This goodness-of-Ž t disparity can be seen in Ž gure 8
and Ž gure 9. While inaccuracy of box-counting estimation is well-known, the above
disparity indicates that the box-counting algorithm in general and the BCFD in
particularly may be more accurate for computing fractal dimensions of well developed
urbanized areas.
Figure 10 shows that the urbanized areas in Baltimore increased consistently.
The growth rate was higher during 1792–1990 and again during 1938–1953 than
that during 1925–1938. Slower growth occurred around 1953 and continued through
1992. This process is also clearly re ected by the Baltimore’s population chronicle
for the period 1792 –1992, with the population growing from 16105 in 1792 to its
peak 946 530 in 1953 and then declining to 726 096 in 1992.
The positive correlation between fractal dimension and urbanized areas for the
period of 1792 –1992 can be seen clearly in Ž gure 10. However, given that Baltimore’s
urbanized areas grew slowly (Ž gure 10) while the population increased dramatically
during 1792–1953 and dropped steadily during 1953 –1992 (Ž gure 11 ), it can be
inferred 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
1822
1851
1878
1900
1925
1938
1953
1966
1972
1982
1992
0.6641
1.0157
1.1544
1.2059
1.3024
1.3836
1.4374
1.5953
1.6450
1.6822
1.7163
1.7211
3.3722
4.3981
5.4106
6.1801
7.4534
7.9415
8.0559
9.0426
9.3018
9.5833
9.5947
9.6215
0.8549
0.9251
0.9376
0.9553
0.9980
0.9968
0.9971
0.9975
0.9980
0.9986
0.9988
0.9990
16 105
66 314
173 390
319 321
508 957
769 350
848 255
946 503
919 065
881 962
776 623
726 096
9.687
11.102
12.063
12.674
13.140
13.553
13.651
13.761
13.731
13.690
13.563
13.495
G. Shen
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432
Figure 8.
Baltimore, MD 1851.
of population, that is, the fractal dimension does not always linearly and positively
correspond to population growth or decline.
For the 12 time periods of Baltimore, the best-Ž t log-linear functions are displayed
in Ž gure 12 and Ž 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 or
population sizes over fractal dimensions can provide fairly good estimates.
6.
Conclusions and Remarks
Individual parcels of urbanized areas can be geometrically planned and designed
using Euclidean geometry. However, the planar urban form as a whole system cannot
be fully described by Euclidean geometry. This is because the urban form and its
development demonstrate the distinct nature of fractals, namely, irregularity, scaleindependence, and self-similarity at least for a range of scales. Thus, it is appropriate
to regard the urbanized areas as fractals and study their spatial forms in fractal
geometry.
Fractal dimension, a necessary dimensional measure of fractal geometry, can be
calculated for urbanized areas. Since a fractal dimension computation is essentially
a limiting process and requires good algorithms to approximate for values, it is
inevitably associated with errors. In this study, the BCFD algorithm and the leastsquare estimation technique were presented and discussed. The fractal dimensions
of 20 large US cities were computed and linked to urban population and urbanized
areas. The urban form and population growth of Baltimore was examined for the
period 1792 –1992 and linked to fractal dimensions.
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Fractal dimension and f ractal growth of urbanized areas
Figure 9.
Figure 10.
Baltimore, MD, 1992.
Growth of urbanized areas and fractal dimensions, Baltimore, MD.
433
G. Shen
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434
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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Fractal dimension and f ractal growth of urbanized areas
Figure 13.
435
L og (C) as a linear function of D, Baltimore, MD.
Since the box-counting algorithm developed in this study requires a speciŽ c
image format, the black and white PICT format, as input, the preparation and
processing of urban image data were discussed. The 20 city images were obtained
from Tiger Map Service (TMS), a free public service on the Internet through the
WWW. To make the fractal dimensions comparable, the map coverage and
boundary, 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 fractal
dimension variances would be expected if a slightly enlarged map boundary for the
20 US cities or Baltimore were used.
The linkages between fractal dimension and urban population and urbanized
areas, as Ž tted with log-linear functions for the 20 US cities in general and for
Baltimore in particular, were established. In general, log-linear functions of fractal
dimensions can yield satisfactory Ž ts for population sizes and urbanized areas in
terms of R2 values even though some graphs (e.g. Ž gure 12) suggest a curvilinear
relationship.
DiVerent urban forms can have virtually the same fractal dimension value. For
example, from table 2 we can see that the fractal dimensions for inland city DallasFort 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 fractal
dimension values of 1.4981 and 1.4973 respectively. This observation, as pointed out
in Shen (1997), indicates again that fractal dimension alone says little about speciŽ c
orientation and conŽ guration of a physical urban form. The usefulness of fractal
dimension lies primarily in its aggregate measure of overall urban form as a fractal.
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436
G. Shen
Cities with virtually the same fractal dimension values and urbanized areas may
have quite diVerent population sizes. For example, the population of Chicago was
slightly over twice the population of Dallas ((7,642,330 vs. 3,805,838) and Pittsburgh’s
population was 2.5 time Nashville’s population (2,596,305 vs. 1,036,950) in 1992.
This observation implies that fractal dimension alone is a fair indicator of the total
of 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 positively
correlated 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 while
the 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 a
good direct measure of urban population density. The indirect use of fractal dimension for urban density modelling, as reported in some previous studies (e.g. Batty
and Longley 1994), also warrants further justiŽ 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 of
1981 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 the
fractal dimension of 1.4981 for Pittsburgh and 1.5869 for Cleveland of 1992 in this
study. In addition to computing-metho d variations, disparities in image size, map
coverage 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 a
more uniŽ 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 Ž lling process,
planar fractal dimension can certainly be used to modelling 2-D urban growth and
urban form. It would be interesting to see a more complete set of fractal dimensions
for 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 spatial
dimension measure and its role in urban modelling and spatial analysis.
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