G. ERDOGAN, MCP *
K. M. CUBUKCU, PhD **
(*) Instructor, Department of City and Regional Planning Department, Pamukkale University, Denizli, Turkey
(**) Associate Professor, Department of City and Regional Planning, Dokuz Eylul University, Izmir, Turkey
Introduction
A fractal is a rough or fragmented geometric shape that can be split into parts,
each of which is (at least approximately) a reduced-size copy of the whole, a
property called self-similarity (Mendelbrot, 1983).
Fractals are spatial objects whose geometric characteristics include scale
dependence, irregularity, and self-similarity (Shen, 2002).
The Koch curve
The Mandelbrot set illustrates selfsimilarity.
Introduction
Recent research has demonstrated that the urban form can not be fully described
by Euclidean geometry, but rather be treated as fractals (Batty and Longley,
1987; Benguigui and Daoud, 1991; Batty and Xie, 1996; 1999; Shen 1997; 2002).
Hausdorff and Besicovitch define
fractal dimension, D, as a statistical
magnitude measuring space-filling
efficiency.
Fractal dimension is a quantitative
measure of the efficiency of spacefilling.
It is a real number, often between 1 and
2, which implies that fractal objects
occupy irregularly shaped spaces
(Ball, 2004).
Fractal dimensions an efficient gateway
for describing the urban spatial system
and the urban morphology.
Fractals preferred by scholars for two main advantages:
1.
Storing data pertaining to urban boundaries at different scales is
time and money consuming. Fractal dimension can avoid
disadvantage of scale.
2.
Second, Euclidean geometry is not satisfying to explain complex
spatial forms.
Aim
This study examines the urban space-filling efficiency measured by
fractal dimension pertaining to randomly selected 20 high populated
world cities using urban explanatory variables.
DATA PROCESSİNG
City_Name
Country
Population
Mexico City
Mexico
23,200,000
Delhi
India
22,900,000
Cairo
Egypt
15,600,000
Bangkok
Thailand
13,700,000
Tehran
Iran
13,500,000
Istanbul
Turkey
13,300,000
Tientsin
9,800,000
Chicago
China
United States of
America
Kinshasa
Congo (Dem. Rep.)
9,550,000
Nagoya
Japan
8,400,000
Saigon
Vietnam
7,750,000
Kuala Lumpur
Malaysia
6,450,000
Santiago
Chile
6,100,000
Bandung
Indonesia
5,600,000
Khartoum
Sudan
5,050,000
Luanda
Angola
5,050,000
Saint Petersburg
Russia
5,050,000
Barcelona
Spain
4,575,000
Riyadh
Saudi Arabia
5,800,000
9,750,000
The sample includes randomly selected 20 highly populated world
cities.
DATA PROCESSİNG
The data were derived from satellite images available from the Google Earth
(2012).
DATA PROCESSİNG
The urban forms were first refined using image processor software, Photohop
CS2, and then scaled, registered, and vectorized using GIS software, ArcGIS 10.
The fractal dimensions are then calculated for each city using fractal analysis
software,Fractalyse.
ANALYSIS
Box-counting method is used to compute fractal dimensions. The method is based
on a repeat the process of changing the box size to determine the presence of builtup land.
The estimated fractal dimension D is derived by estimated slope of the
log(n(s)) and log(1/s) graph (Shen, 2002):
log(n(s)) = log(U) + Dlog(1/s) + εs
D : fractal dimension
log(U) : constant (U is builted area)
s : the box size
εs : the error term
ANALYSIS
ANALYSIS
The fractal dimensions for the sample
varies between 1.28 and 1.64.
RESULTS
The selected model is a log-log linear model.
The depended variable is the linear fractal dimension (in log form)
The explanatory (independent) variables in the selected model includes;
urban population (in log form),
population density (in log form),
development level of the country,
forest area variance
national income per capita.
RESULTS
Urban space-filling efficiency measured by fractal dimension increases:
when urban population size (-) or population density degreases (-).
when the development level of the country increases (+).
Coefficientsa
Standardized
Unstandardized Coefficients
Model
1
B
(Constant)
Coefficients
Std. Error
Beta
.636
.136
log_Pop_Dens
-.038
.019
Ulke_Gel_Duz
.000
OrDeg2008
GSMH_2009
log_kntsl_nfs_2011
t
Sig.
4.686
.000
-.382
-2.013
.065
.000
-1.099
-2.357
.035
-2.004E-8
.000
-.119
-.622
.545
-6.421E-7
.000
-.216
-.503
.623
-.137
.056
-.654
-2.443
.030
a. Dependent Variable: log_F_L_DIMENT
variables forest area variance and national income per capita are not
statistically significant at the 0.10 level, but they contribute to the
model statistically.
CAVEATS
However,
The selected model explains %55 of the variance in fractal dimension of
the selected sample.
This finding indicates that more explanatory variables are needed to
explain the complex nature of urban space-filling in populated cities.
Model Summary
Model
R
R Square
Adjusted R Square
Std. Error of the Estimate
1
.739a
.546
.372
.025228277281682
a. Predictors: (Constant), log_kntsl_nfs_2011, log_Pop_Dens, OrDeg2008, GSMH_2009,
Ulke_Gel_Duz
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