when and where a city is fractal

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Introduction

Ever since the concept of `fractals' was proposed by Mandelbrot (1983) it has been

applied extensively to describe a variety of physical phenomena (for example, see

Fleischmann et al, 1989). In recent years the concept has been applied in the context

of human geography as well. In two recent books, by Batty and Longley (1994) and byFrankhauser (1994), the claim was made that urban morphology is fractal. A number

of studies have reported estimates of the fractal dimension of several large cities. The

estimated values reported are around 1.7, and sometimes larger (see Anas et al, 1998).

There are a number of issues concerning urban morphology that have not received

sufficient attention. First, it is not self-evident that all cities are fractal, or that fractality

is a general property of all cities. Should it become evident that some urban structures

are not fractal, it will become interesting to classify cities according to the type of

morphology that they exhibit. It will be a challenge to explain why the development

of some towns has a fractal character and why in some cases such characteristics areabsent. Second, Batty and Longley (1994) have reported that the fractal dimension (D)

of London changes over time. A similar result for Berlin was reported by Frankhauser

(1994). These results suggest that it is likely that a city becomes fractal only at some late

stage of its development.

The third issue concerns the boundaries of urban phenomena. It is well known that

it is difficult to choose the outer boundaries of a metropolis and there is no agreement

on a theoretically correct definition.1 There are two common approaches. Function-

alists define urban boundaries in terms of geographic homogeneity of patterns of

consumption and other social characteristics and in terms of volumes of commutingto work. Others define them in terms of contiguity of built areas. Studies that rely on

When and where is a city fractal?

Lucien BenguiguiDepartment of Physics and Solid State Institute

Daniel Czamanski, Maria Marinov Center for Urban and Regional Studies, Faculty of Architecture and Town Planning,Technion ^ Israel Institute of Technology, Haifa 32000, Israel;e-mails: [email protected]; [email protected];

[email protected] PortugaliDepartment of Geography, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel;e-mail: [email protected] 9th March 1999; in revised form 27th November 1999

Environment and Planning B: Planning and Design 2000, volume 27, pages 507 ^ 519

Abstract. We present here an analysis of the development of the Tel Aviv metropolis by using theconcept of fractals. The fractal dimension of the entire metropolis, and of its parts, was estimated as a

function of time, from 1935 onwards. The central part and the northern tier are fractal at all times.Their fractal dimension increased with time. However, the metropolis as a whole can be said to befractal only after 1985. There is a general tendency towards fractality, in the sense that the fractaldimension of the different parts converge towards the same value.

1The early literature concerning urban boundaries has mapped out very clearly the issues butwithout forming a clear consensus (see Berry et al, 1968).

DOI:10.1068/b2617

mailto:[email protected]












(a) 1935 (b) 1942

(c) 1952 (d) 1962

10 km 10 km

10 km 10 km

Figure 1. Maps of the Tel Aviv Metropolis (built area).

508 L Benguigui, D Czamanski, M Marinov, Y Portugali



10 km 10 km

10 km 10 km

(e) 1971 (f ) 1978

(g) 1985 (h) 1991

When and where is a city fractal? 509



informal methods are not uncommon. Recently, Martin (1998) suggested the use of an

experimental, computationally intensive method that relies on a variety of population

and socioeconomic geographically disaggregate data as inputs.It is not clear whether assuming different boundaries would yield different estimates

of fractal dimensions. As an example, one can consider the case of London. If one takes

the values of D as reported by Batty and Longley, it is noteworthy that D does not

increase regularly but exhibits an isolated maximum in 1840. This does not seem very

reasonable. The source of this strange result may be in the choice of city limits. It is

possible that different boundaries would have yielded a different, and perhaps more

reasonable, result.

The study reported on herein is related to the above issues and, in particular, for

the morphology of the Tel Aviv metropolis from 1935 to 1991. Our definition of Tel Avivboundaries follows a relatively restricted definition of the metropolis and is consistent

with the contiguity approach to defining urban areas. It includes only the towns that

are in close physical proximity one to another. The sea coast forms a natural boundary

in the west. In the east there is no issue as regards the boundary. There is again a

`natural' boundary because urban expansion is limited by the frontier between Israel

and the occupied territories. The boundaries in the north and in the south were chosen

such that beyond the chosen boundaries the main land use is agricultural.

The purpose of the study was to determine the fractal dimension as a function

of time for the entire metropolis and for different parts of it. Previous studies of Tel Aviv have suggested that there are differences in the behaviour of the central

part and the northern and southern tiers of the metropolis (Krakover, 1983; Shachar,

1975). The important conclusion of this work is that only the northern tier of the

metropolis is fractal throughout its history, whereas the entire metropolis appears to

have become fractal only recently. However, there is a clear tendency towards fractality

of the entire metropolis. We conclude that the concept of fractal must be used with

caution and that a critical examination is necessary before deciding whether or not a

city is indeed fractal.

The method

The basis for our estimates of the fractality of the Tel Aviv metropolis is a series of

maps of the Tel Aviv metropolis that were prepared by Portugali as part of a study

of urban growth. The series of maps from 1935 to 1991 is displayed in figure 1.

There are three main methods to determine the fractal dimension of an object

(see Vicsek, 1989). One method is particularly suited to the case of growth from a

definite center. As this is not the case in Tel Aviv, this method was rejected as

unsuitable. Another method was rejected because it requires a complicated computation

of a correlation function and significant computer time. In our analysis we utilized thethird alternative, the box counting method. The object (embedded in a two-dimensional

space) is covered by a grid made of squares of size l . The grid that covers the entire

object is assigned size L. Next, the number N of squares in which a part of the object

appears is counted. If the object is compact, a part of the object appears in each

square. The number, corresponding to the surface of the square, N is equal to the ratio

of the total surface L2 divided by l 2, or N (Lal )2. If the object turns out to be fractal,

then N (Lal )D, where D is its fractal dimension. Thus, a logarithmic plot of ln N

versus ln l yields a straight line with a slope equal to ÀD [for an illustration of this

point, see Vicsek (1989) and Batty and Longley (1994)].From a practical point of view, the lowest value of l must be larger than the smallest

details in the map, such as isolated points or the thickness of a line. Our choice of the

series of values of l was as follows. The first value, l 1, is the size of the square in which




the entire object is contained (see figure 2), That is L. Thus l 1 L. The next value, l 2,

is equal to La2. The i th value of l is l i , l i La(2i ). In our case, the highest value of

index i is 9. As 29 512 the value of L is chosen equal to 512, and l 9 1.

The maps of Tel Aviv were digitized by means of a scanner. In order to prevent the

introduction of errors in the scanning process, we verified that the smallest l value islarger than the smallest `detail' in the map. We wrote a special computer program that

yields the series of values of N i corresponding to the series of values of l i from the

scanned image of the map. In the ideal case, the fractal dimension is estimated by a

regression fit of a straight line that best represents the distribution of the individual

point estimates. Modern cities, however, have time-limited histories and thus a limited

number of observations are possible. As a result, rigorous rests of misspecification

error as to the linearity of the results are not possible.

We consider here two extreme examples: the results (ln N versus ln l ) for the

whole ensemble in 1952 and for the central part of the metropolis in 1991 (below weshall also present the limits of the central part). Figure 3(a) (see over) presents data for

the central part in 1991. The points are well aligned and it is easy to conclude that the

object is fractal. However, in the case of the entire metropolis, presented in figure 3(b),

the situation is far less clear. Close inspection of the points suggests that the curve that

will yield the best fit is a parabola and not a straight line. We estimated two different

functions, a straight line ( y aÀDx) and a parabola ( y a bx cx 2 ). The results

are presented in figure 3(b). By direct inspection, it is evident that the parabola yields a

far better fit. The adjusted R 2 value is 0.991 for the straight line and 0.998 for the

parabola. Although a misspecification error test is not possible it seems difficult toconclude that the entire metropolis is fractal. In fact, a downward curvature of the

curve for ln N versus ln l is ever present. In some cases it is very small and we can

conclude that the object is fractal. In other cases it seems difficult to neglect.

We are faced with a need to define a criterion for distinguishing between fractal

and nonfractal cases. When a straight line is fitted to data for the entire metropolis, the

standard error d decreases with an increase in time. This means that there is a tendency

towards fractality when the time elapses. Our intention is to avoid characterization of

objects that are more and less fractal. We prefer to define a clear-cut criterion that

permits a direct characterization of objects as fractal or not fractal.We did not find a rigorous rule to define the largest acceptable value, dm, of the

error. But from visual inspection it seems fair to assign dm the value of 0.040. This is

equivalent to using R 2 0X996 as a criterion. For R 2 greater than this value we can say

l

Figure 2. Grid on a two-dimensional object of irregularshape.




105

104

103

102

101

100

1 10 100 1000

l n N

ln l (a)

10

8

6

4

2

0

0 1 2 3 4 5 6

ln l

l n N

(b)

10

8

6

4

2

0

0 1 2 3 4 5 6

ln l

l n N

(c)

Figure 3. (a) LnN versus ln l for region 1 in 1991. LnN versus ln l for the entire ensemble(region 3) in 1952: (b) linear fit; (c) polynomial fit.




that the object is fractal. There is some arbitrariness in choosing this value because it is

based on a visual inspection. This choice does not modify seriously our conclusions

because in the extreme cases there is no ambiguity. There is room for hesitation only insome intermediate cases. We believe that this is not a serious problem.

The maps and the divisions

A map of the Tel Aviv metropolis is presented in figure 4. The structure of the entire

metropolis is visible clearly. The sea coast limits the development of the town on the

west. The municipality of Tel Aviv can be seen at the center of the metropolis. More

than twenty smaller municipalities (towns) are dispersed around the center.

The maps used for the estimation of the fractal dimension are shown in figure 1.

These indicate in black the ground surface occupied by buildings only. Markings of land occupied by major roads have been removed.

Three regions were chosen (see figure 5, over). For each of these, the series of values

of the quantities ln N i and ln l i were determined. The values of D and of d were estimated

0 10 20 km

Figure 4. Map of Tel Aviv metropolis with municipal jurisdictions.




by means of the above-described fitting program. The first region is the central part

(region 1). It contains Tel Aviv and the five towns that are contiguous to it (Givataim,

Ramat Gan, Bnei Brak, Holon, Bat Yam). The common characteristic of these towns

is that from an economic base perspective they were always urban centers, whereas the

majority of the other towns in the metropolis began as agricultural settlements. This is

evident from an examination of the evolution of their population over time (see

figure 8 later). For the towns belonging to the central part, the curves do not exhibit

any anomaly at the time of creation of the State of Israel in 1948; other towns display

an upward discontinuity at this time.

10 km

Figure 5. Map of the entire Tel Aviv metropolis with three study regions: 1, central part; 2, north-

ern part; 3 entire ensemble.




The second region includes the central part and the northeast part of the entire

metropolis. It was chosen because visually the built area in this region appears to be

more developed than in the south.The third region is the entire metropolis. It is at this geographic scale that the

problem of the region's boundaries becomes apparent, especially to the east. The

boundaries in the north and in the south were chosen such that beyond the bounda-

ries the main land use is agriculture. On the east, there exist urban activities beyond

the boundary that was chosen. Our choice was governed by the existence of a `natural'

boundary, because urban expansion is limited by the frontier between Israel and the

occupied territories. Indeed the dynamics of the urban phenomenon beyond the chosen

boundary is different to the east and to the west of the chosen line. It is a rather loose

definition. Yet it should not give rise to an important error in the resulting conclusions.

The results

Results for the three regions are presented in two forms; in table 1 and in figure 6.

Inspection of the table suggests that region 1 is always fractal. It displays a small error,

always smaller than or equal to 0.025. As was already evident from inspection of

figure 3, the quality of the fit for the curve ln N versus ln l is very good. Thus, without

hesitation, it can be concluded that the central part of the Tel Aviv metropolis is

fractal. It is also evident that over time D increases from 1.533 to 1.809, with a small

discontinuity (or rapid increase) over the period 1971 ^ 78.

Table 1. Evolution of the fractal dimensions growth of the study regions in Tel Aviv.

Year Region 1 Std error, d Region 2 Std error, d Region 3 Std error, d

1935 1.533 0.021 1.387 0.031 1.312 0.0411942 1.607 0.018 1.434 0.027 1.341 0.0441952 1.644 0.017 1.546 0.038 1.477 0.0501962 1.672 0.0251971 1.695 0.025 1.621 0.038 1.565 0.0521978 1.773 0.014 1.679 0.026 1.600 0.041

1985 1.787 0.014 1.707 0.026 1.632 0.0371991 1.809 0.014 1.733 0.025 1.667 0.037

1.85

1.80

1.75

1.70

1.65

1.60

1.55

1.501.45

1.40

1.35

1.30

1930 1940 1950 1960 1970 1980 1990 2000 2010

Year

F r a c t

a l d i m e n s i o n D

D1

D2

D3

Figure 6. Evolution of the fractal dimension of the study regions in Tel Aviv.




Region 2 is fractal over the entire study period. This is because at all times the

error is smaller than the maximum value we defined above. However, the values of D

are much smaller than those for region 1 (central part). The rapid increase in D isclearly evident between the years 1941 and 1952, a period during which these settle-

ments experienced intensive population growth. The difference in the values of D for

regions 1 and 2 decreased regularly. In 1935 the difference was 0.221 and in 1991 it

became equal to 0.142.

For region 3, the entire metropolis, things are different. The error is relatively large

and one can say that the Tel Aviv metropolis is fractal only after 1985. To support

this conclusion, we display in figure 7 the curve lnN versus ln l for the year 1991.

Effectively, a straight line is a good fit. Even the anomaly between 1942 and 1952

is evident.

A comparison of data and the three curves presented in table 1 and in figure 6

suggests a tendency towards fractality. It can be observed that there is a general trend of

increase in the values of D. Assuming a linear increase from 1978 on, it is evident that

the slopes of the three regions are increasing, from region 3 to region 1. From a linear

extrapolation of the three curves it seems that they will converge around year 2010.

To conclude this section, a comparison between regions 1 and 3 is presented.

Region 1, which was always an urban metropolis, is fractal with an increasing fractal

dimension D. In 1991 the value was similar to that of other large cities, such as Paris

and London. There is also a rapid increase between 1971 and 1978, although thereseems to have been no marked event during this period (see below).

Region 3 exhibits an important change around 1950 when there as a large wave of

immigration immediately after the creation of the state in 1948. This is expressed by the

rapid increase in D during this period. Later on, a regular increase in D is observed.

But the metropolis becomes fractal only after 1985. This is a consequence of the choice

of dm. As d decreases from 1971 onwards, one expects that after 1991 this decrease will

continue. Therefore, even with a different choice of dm, it is likely that region 3 would

become fractal, but at a somewhat later time.

105

104

103

102

101

100

l n N

1 10 100 1000

ln l

Figure 7. LnN versus ln l for region 3 in 1991.




Discussion

The first and most important conclusion from the above is that regions 1 and 2 of the

Tel Aviv urban metropolis display a spatial evolution that follows a fractal pattern.Three periods in the development of the center are distinguishable: before 1940,

between 1942 and 1971, and after 1971. Only one value of D was estimated for the first

period. In the second and third periods, D increased linearly with a smaller slope in the

third period. It is interesting to compare the evolution of the population during these

periods (see figure 8). From 1950 to 1975 the population increases linearly and there is

no sign of a rapid variation in D values between 1971 and 1978. After 1975 and until

1990, the population of region 1 is practically constant while D increases slightly. One

can conclude that this last increase in D is due to building for other land uses, rather

than for residential housing. In fact it is well known that an important business centerhas been developing during the 1980s in the core of the metropolis. During the same

time period many of its residents moved to various more remote suburban locations.

So far as the entire metropolis is concerned, the discontinuity in D between 1942

and 1952 indicates the birth of the region as a large urban metropolis. Before the

creation of the state and the waves of immigration that followed it, the urban metro-

polis was in its central part only. Very shortly thereafter, numerous settlements were

progressively transformed into secondary urban centers.

Because the northern part is fractal, but the whole metropolis is not, it must be

concluded that the south part of the metropolis was not fractal until 1985. Thisconclusion was verified directly. It is noteworthy that nothing special occurred in the

Tel Aviv metropolis, nor in Israel, in 1985. Yet the resulting finding suggests that

the structure and the growth mode of the north and the south (excluding the central

part) were completely different. In other words, the Tel Aviv metropolis is hetero-

geneous. Here the observer is confronted with an unexpected situation. On the one

hand, it must be concluded that the Tel Aviv metropolis is not fractal before 1985,

implying different development patterns for different parts of the metropolis. But, on

the other hand, it can be concluded that before 1985 the south part is not an organic

part of the metropolis. However, an intermediate point of view is possible. Accordingto this view, the south belongs to the metropolis, and its fractal nature has appeared

only progressively.

Irrespective of which interpretation is adopted, the clear tendency of the metro-

politan region towards fractality is interesting. It is expected that in less than fifteen

2000

1800

1600

1400

1200

1000

800

600

400

200

01920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Year

P o p u l a t i o n ( t h o u s a n d s )

Region 3Region 2

Region 1

Figure 8. Population of the Tel Aviv metropolis (source: Israel Central Bureau of Statistics, 1997).




years the metropolis will have a uniform fractal dimension (around 1.8). It is entirely

plausible that at that time the boundaries of the metropolis will be different. This then

is a special feature of urban growth: fractality appears progressively. In other words,it is probable that a large town is not always fractal but becomes fractal at some stage

of its development. It should be very interesting to find the phenomenon associated

with the appearance of fractality. This would also be important for understanding the

concept itself.

The patterns of spatial evolution described above, namely the transformation of the

urban structure of region 3 around 1985 from nonfractal to fractal and the rapid

change in dimension D of the central part between 1971 and 1978, require an explana-

tion. Several aspects of regional growth and development were considered.

The rate of population growth of the entire region during the 1970s does notindicate any significant changes that might explain the essential change in the values

of the fractal dimension D. On the other hand, an examination of Israel's real

per capita GDP provides convincing evidence of rapid economic growth in the late

1960s and early 1970s, as compared with the uniform and almost linear growth since

1950 and until today (figure 9). This change in growth patterns is attributed to the

influence of the Six-Day War on the Israeli economy. The increase in the national

product resulted, among other things, in an increase in the volume of business activity

and in an improvement in the population's standard of living.

One of the expected byproducts of economic prosperity was improvement inthe housing stock of the population of big cities in the central part of the country. To

achieve higher housing standards residents of cities migrated to locations with newer

and more spacious housing (see Carmon and Czamanski, 1992). Generally, the intra-

urban migration was away from the central city of the metropolis and towards suburban

communities in a rural environment. Hence, following the rapid economic growth

between 1967 and 1971, the population of the older urban centers spread out to the

countryside. Generally the new housing developments were around existing small towns

within the Tel Aviv metropolis. This leap-frogging process (see Benguigui et al, 1998)

increased the homogeneity of the built area and the extent to which the developmentfilled the open space of the metropolis.

The spread of urban activities around main urban poles was responsible for the

observed change in the fractal dimension in region 3. It is likely that this `1960s ^ 1970s'

effect first gave an impetus to the development of the business sector, and the associated

50 000

40 000

30 000

20 000

10 000

1950 1960 1970 1980 1990 2000

G D P p e r c a p i t a

Year

Figure 9. Per capita GDP at constant prices (source: Israel Central Bureau of Statistics, 1997).




expansion of commercial building stock in the central part of the metropolis, and only

afterwards to the growth in housing stock. There seems to be no doubt that it gave rise

to a discontinuity in D between 1971 and 1978 and to fractality of the whole metropolis.Most importantly, there is no interpretation, or indeed a model, that provides an

explanation of the increase in the fractal dimension as growth takes place (Vicsek,

Mandelbrot). Dimension D is kept constant in models of fractal growth. This can be

understood very easily in the case of deterministic fractals, where a fractal is created by

self-similarity (Mandelbrot). It seems that this is a particular property of town growth

and this point necessitates further investigation.

Conclusions

From our study of the Tel Aviv metropolis two important conclusions are suggested.First, fractality appears in some areas of the urban metropolis at early stages of its

development, such as in the central part and the north-east region, and in other areas it

evolves progressively, as in the case of the entire metropolis. This finding suggests that in

the same metropolis different parts have their own mode of development. The second

finding is that the fractal dimension increases over time. This clearly means that the

density also increases with time. But the structure of a fractal is conserved, that is it

has the same structure at different scales. It is not clear why a town is indeed fractal. A

significant effort has to be made in order to relate fractality to the other properties of

town growth. The tentative interpretation of the sudden variation of D of the centralpart between 1971 and 1978 and the appearance of a fractal city after 1985 is suggestive

of the type of explanation needed.

Acknowledgements. This research was supported in part by the Technion VPR Fund and by P andE Nathan research fund for the promotion of research at the Technion Institute.

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