The Fractal Approach. a New Tool for the Spatial Analysis of Urban

7/30/2019 The Fractal Approach. a New Tool for the Spatial Analysis of Urban

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Pierre Frankhauser

The fractal approach. A new tool for the spatial analysis of urban

agglomerationsIn: Population, 10e anne, n1, 1998 pp. 205-240.

Citer ce document / Cite this document :

Frankhauser Pierre. The fractal approach. A new tool for the spatial analysis of urban agglomerations. In: Population, 10e

anne, n1, 1998 pp. 205-240.

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Abstract

Frankhauser (Pierre).- The fractal approach. A new tool for the spatial analysis of urban agglomerations.

Fractal geometry is a new approach for the study of spatial distributions. The basic model is a law of

hierarchical distribution corresponding to Pareto's law which is familiar to urban geographers and

demographers. The methods of fractal analysis can be used to study the spatial organization of human

activities across scales. The regularities and the discontinuities in the distributions can then be

identified. These discontinuities can be spatially situated. Applying this concept to urbanized areas has

shown that districts can be defined and classified according to their scaling relations, thereby allowing

development of a typology of locational patterns. This observation reveals the existence of a principle of

self-similarity in land-use patterns. An examination of time series shows that despite the apparent

fragmentation of these urban tissues, urbanization is often accompanied by self- structuring

development. Subsequent research will need to employ complementary morphological measures, such

as measures of space filling and of population distribution, which could be used to validate the

simulation models based on fractal geometry.

Rsum

Frankhauser (Pierre).- L'approche fractale. Un outil de rflexion dans l'analyse spatiale des

agglomrations urbaines. La gomtrie fractale est une nouvelle approche pour tudier des rpartitions

spatiales. Le modle de rfrence est une loi de distribution hirarchique qui correspond la loi de

Pareto, bien connue en gographie urbaine et en dmographie. L'utilisation des mthodes d'analyse

fractale permet d'tudier l'organisation spatiale des activits humaines travers les chelles. Il est ainsi

possible de dcouvrir aussi bien des hirarchies rgulires que des ruptures. Ces ruptures peuvent tre

identifies dans l'espace. L'application de ce concept aux tissus urbains a montr qu'il est possible de

distinguer et de classifier des quartiers selon leur comportement sealant, et de dvelopper une

typologie des tissus urbains. Cette observation met en vidence l'existence d'un principe d'auto-

similarit dans les tissus urbains. L'tude de sries temporelles montre que l'urbanisation est souvent

accompagne de phnomnes de structuration, en dpit de la fragmentation apparente de ces tissus.

Les futures recherches devraient utiliser des mesures morphologiques complmentaires: mesures de

lacunarit, mesures concernant la rpartition de la population. Ces mesures pourront servir valider

des modles de simulation bass sur la gomtrie fractale.

Resumen

Frankhauser (Pierre). - El anlisis fractal. Un nuevo instrumente de reflexion en el anlisis espacial de

las aglomeraciones urbanas. La geometria fractal es un nuevo mtodo de estudio de reparticiones

espaciales. El modelo de referencia es una ley de distribucin jerrquica que corresponde a la ley de

Pareto, bien conocida en geografia urbana y en demografia. La utilizacin de los mtodos de anlisis

fractal permite estudiar la organizacin espacial de las actividades humanas a tra- vs de escalas.

Estas escalas permiten establecer tanto jerarquias regulares como rupturas, que se pueden identificar y

situar en el espacio. La aplicacin de este concepto a los tejidos urbanos ha abierto la posibilidad dedistinguir y clasificar gradualmente barrios segn su comportamiento, y de esta forma desarrollar

tipologias urbanas. Este mtodo muestra la ex- istencia de un principio de auto-similaridad en los

tejidos urbanos. El estudio de series temporales muestra que el proceso de urbanizacin va

frecuentemente acompafiado de un proceso de estructuracin, a pesar de la fragmentacin aparente

de taies tejidos. En investigaciones futuras deberian utilizarse medidas morfologicas complementa-

rias: medidas de lagunaridad, medidas referentes a la reparticin de la poblacin. Estas medidas

podrian utilizarse tambin para validar los modelos de simulacin basados en la geometria fractal.


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THE FRACTAL APPROACH.A new tool for the spatial analysis

of urban agglomerations

Pierre FRANKHAUSER*

IntroductionA fundamental question when examining a territory's social and economic functioning concerns the spatial distribution of population. This phenomenon has been studied in different contexts and on different scales bya number of disciplines: demographers and geographers but also town planners and economists. Whatever the scale of observation used, the resultsobtained all show the spatial distribution of human activities to be essentially non-homogeneous. Various reasons can be given for this: first, someplaces are more propitious than others to human activity, thereby influencinghe areal distribution of settlement; second, urbanization has long been

accompanied by a hierarchical organization of towns and cities. There havebeen many attempts to produce a theoretical formalization of this hierarchical structure, either from a demographic and descriptive angle as in therank-size rule, or in the explanatory approach of central place theory.On the larger scale of agglomerations, Clark introduced a mathematicalormulation to model the decrease in population as distance from acity centre increases; the same phenomenon has been studied by Bussireand Stovall using a different formalization. It might be thought that theprocess of periurbanization or population decentralization has helped toreduce the difference in the density of occupation between urban coresand their suburbs, a development which could be accentuated by the growthof tertiary activities in the central districts at the expense of residentialhousing. However, this purely demographic perspective is in fact misleading.hat is observed here is simply a segregation of functions, and landuse in the city centres often actually intensifies due to the new demand.This results in a high day-time concentration of non-residential population.Universit de Franche-Comt, Besanon.

Population: An English Selection, special issue New Methodological Approachesin the Social Sciences, 1998, 205-240.


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206 P. FRANKHAUSERBy contrast, densities remain low in the residential districts of the outersuburbs. In addition, the importance of transport in the suburbs encouragesa tentacular growth along the axes of circulation, thereby adding to thedisparity in the distribution of the built-up areas.Yet if the existence of a heterogeneous distribution seems omnipresentand features in theoretical analyses of the functioning of the settlementsystem, the measures employed are always based on the paradigm of auniform space: the geometrical reference remains homogeneity. The mostcommonly used measure is density, which indicates the mean distributionof population in a given space and thus assumes a proportionality betweenpopulation and surface occupied. The shortcomings of this approach arewell known. Population density is high if a small administrative unit ischosen and lower if a larger area is considered. Without proportionality,it is hard to move from one scale to another. The value obtained for thedensity is thus dependent on the size of the reference surface, and thuson the scale at which we operate.Other methods for measurement besides density have been suggested,such as the 'nearest neighbour' method and the 'quadrats' method. Butthese methods also fail to take into account the variation in a phenomenondepending on the scale of reference (Frankhauser, 1997a).A disparity exists therefore between, on the one hand, the observationsnd theories - whether descriptive or explanatory - and, on the other,the geometrical paradigm which underpins our spatial measures. And thesemeasures simply reflect our approach to space, which remains based onthe tradition of Euclidean geometry.The same geometrical approach is present in the theories of locationof the urban economy which take as their reference the linear or circularurban settlement. Yet with Euclidean geometry it has not been possible toproduce a vision able to accommodate the complex forms found in thespatial distribution of human activities. Town planners, for example, oftendescribe contemporary urban settlements as amorphous and irregular, andlacking any discernible internal organization. The same limits are foundin every discipline. For example, the physical sciences long gave little atten ti on to forms that were perceived to be irregular, and Euclidean geometryappeared to be adequate for the study of crystals and planetary paths. In-depth study of more complex, nonlinear phenomena has established theshortcomings of these traditional concepts. Examples that can be mentionedare the discovery of chaotic attractors, the in-depth study of phase transitionphenomena and the increased understanding of materials, notably their surfaces and textures. These investigations have discovered a close relationshipbetween the development of complex forms and the existence of self-organization phenomena (Pumain, 1989; Schroeder, 1994).

The only alternative approach of an authentically geometrical character to date is that of fractal geometry. The physicist Nicholis has summedup the particular value of this approach: "Fractal objects are a new model


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THE FRACTAL APPROACH 207for structures which although complex, originate in relatively simple mechanisms" (Nicholis, 1985). Applied in many scientific fields and at differentscales, this approach has made it possible to identify hitherto unknownprinciples of internal order, and to incorporate these results in explanatorytheories. The fractal approach has also proved a valuable instrument inimage analysis, for the segmentation and classification of objects in a set.The use of fractal geometry for the study of a spatial phenomenonis thus not simply about introducing some new measures but involves applying a new geometrical frame of reference."* However, there are morespecific reasons favouring the adoption of this approach in demographyand urban geography. Fractal geometry reflects the existence of a hierarchical organization within a spatial system which obeys a particular logic,that of the Pareto-Zipf distribution. Such a logic is known to operate in thesystem of settlement, so it would seem that fractal geometry can be usedto verify this law of distribution, and to identify the deviations from it . Thefact of being able to study a phenomenon across different scales also offersthe possibility of identifying critical thresholds in spatial organization.

As regards urban agglomerations, recent studies have established thevalue of a fractal formalization both for developing new morphologicalmeasures and for studying at a more conceptual level the spatial organizationnd dynamic associated with them. The built-up component has beenanalyzed by M. Batty and P. Longley (1994), R. White, (1993b) andP. Frankhauser (1994b, 1995)). These studies were initially conducted ata regional scale, but more recent efforts have been directed to the studyof urban agglomerations (Frankhauser, 1997; Batty, 1996). These worksapply a binary logic inasmuch as they distinguish simply between built-upand non-built up areas. 2) Batty and Kim introduced the intensity of landuse by making population density the third dimension (Batty and Kim,1992). In this approach the decline of population density towards the periphery is modelled by a hyperbolic function, as has been discussed byseveral authors, for example Bussire and Stovall. A general and consistentformalization of population distribution requires more complex conceptssuch as the multifractal approach. H. Le Bras has employed such a logicto model population distribution (Le Bras, 1993).

This paper starts with an introduction to fractal geometry. Theexamples have been chosen to illustrate the application of this approachto the description of urban agglomerations. We use here mainly the binaryapproach, which can add significantly to our understanding of urban spatialorganization. There follows a discussion of the techniques of fractal measurement and the results produced by analysis of actual urbanized areas.This analysis is conducted using analysis programs applied to digitized maps.

(" Methods of measurement introduced or used in other contexts can be used to studyfractal behaviour, for example the variogram or mathematical morphology (cf. below).(2' The concept used can be extended so as to examine the spatial distribution of severaltypes of land use together (Frankhauser, 1994b).


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208 P. FRANKHAUSERIn a final section we present the example of an extension of the concept to include not only a description of built-up and non-built-up areasbut population density.I. - The fractal formalization of urban agglomerations

The Sierpinski carpetand the fractal dimension The essential characteristic of fractal geometry is that the same type of geometricallements are found at an infinite numberof scales. Fractal objects do not belong to any particular scale. The presenceof the same element across many scales is reflected in the existence of ahierarchical structure within the fractal object, a property of recurrence whichis often referred to as self-similarity. We illustrate this phenomenon with aspecial class of fractal which is particularly adapted to the study of spatialdistribution in a settlement system: the Sierpinski carpet (figure 1).

Figure 1. - First iteration steps fo r the construction of two fractalstructures. Figure (a) shows the first iteration steps in theconstruction of a Sierpinski carpet. The curve in (b) forms theborder of each side of the Sierpinski carpetTo produce this fractal we select an initial figure, which in this caseis a square of length L. A geometrical operation known as the generatoris then defined which transforms the initial figure. In our example it isreduced by a factor r = ^ , and N,= N = 5 of these squares of length /, = r Lare arranged to form a chess-board as appears in figure la, left. This operationis repeated for each of the five squares (figure la, middle). The figure


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THE FRACTAL APPROACH 209thus now comprises N2 = N2 = 25 squares of length I2 = r2 L = ^L . It canbe seen that the chess-board appearance has been lost but that a spatialhierarchy has appeared in the free spaces. Repeating this operation produces an additional hierarchical level of free spaces, as in Figure la, right.If this iteration procedure is continued to step n, the number of grey squareswill be Nn - Nn and their length will be reduced to l = rn L, so that thesurface of each stippled square will then be an-ri L2. For the total surfaceA of the grey squares which form the fractal we get:

= {N-r2)n-L2 (1)

(2)Since | < 0, the surface diminishes at each step, and by repeatingthis operation to infinity we get a set of separate points whose surfacetends to zero.The hierarchical distribution of the free spaces means that these pointsare not distributed homogeneously in the surface but produce clusters.Figure lb shows how an iteration can be used to construct a fractalstructure which reproduces at each step the border of the Sierpinski carpet.In this case the initial figure used was the segment of line of length L.

We confirm that the total length of this object increases at each step andthat it tends to infinity. The fact that the length of a curve increases towardsinfinity suggests that it is of a dimension greater than that of a normalcurve such as a circle, i.e. one. Yet it remains topologically linear. Thisbehaviour is inconsistent with Euclidean geometry, even more so since theobject's surface seems to tend to zero. To define such sets, measure theoryhas introduced fractional value dimensions. A general measure, L isdefined, that is required to remain constant during the iteration:L const =Nn-{ln)D

= LD (3)whereNn = const- l;D (4)

Converting to logarithms, we get a linear relation:(5)

where D is the slope of a line, defined by the points xn = log / , yn = log N.The condition that L remain constant is satisfied by an appropriate choiceof D which is determined by relation (5). By introducing Nn =N" and / = f1 L,


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210 P. FRANKHAUSERthis new parameter D is obtained which does not depend on the step nand which is referred to as the fractal dimension.

(6)Since the parameters have the same values for both the constructions

described, the same fractal dimension D - -^ ~ 1 .47 is obtained for theperimeter and for the surface, a result which is inconsistent with normalgeometry, in which curves have the dimension one and surfaces have thedimension two.

This result reflects the fact that during iteration the length of theperimeter increases disproportionately compared with a normal geometricalobject, whereas the surface is increasingly dominated by free spaces. Indeed, in the limit case n every element of the surface is also an elementof the perimeter (Frankhauser, 1994b).We verify that by the same calculation the dimension D = 1 is obtained for a line and D = 2 for a totally occupied surface, that is, valueswhich are consistent with Euclidean geometry. Thus it is established thatfractal geometry contains conventional geometry as a limiting case.In the Sierpinski carpet, the iteration was interpreted as a gradualreduction in the surface occupied by the object, such as it appears in relation (2). A different interpretation is also possible (see for exampleGouyet, 1992): it is imagined that at each step a given mass or populationis concentrated on the remaining surfaces of the object. The density oneach element is then calculated. If the initial population is P, we get, atstep n, for density gn = p/an on each of the Nn squares occupied:

By introducing new constants c = p/l2 and/=(Vr)2 > 1 we get:gn = c-r (8)

Since / > 1 the density increases towards infinity when n . Inthis interpretation of a fractal structure, density becomes a measure thatis subject to large fluctuations:

gn = 0 at the empty places,gn at the occupied places.

Density is thus an inappropriate indicator for the description of astructure presenting a fractal morphology.In this case a better representation of the spatial distribution of theelements is obtained by using the fractal dimension and related supplementaryeasures.


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THE FRACTAL APPROACH 211Hierarchical propertiesand multifractality The particularity of Sierpinski carpets istheir hierarchical structure which is seen inthe distribution of free spaces. The numberof non-occupied areas Nauc)(kn) of size Xn is in general:-s (9)where 5 is an exponent which may differ from D. In the present case, thefree spaces are of size \\ = ^ L, \2 = (\) L, etc., and it is verified that

(10)While iteration necessarily produces a hierarchy in the distribution ofthe free spaces, the elements are of the same size even though they form

clusters. However, generators can be selected which produce hierarchies ofclusters. Such a fractal appears in figure 2. In this case the generator doesnot distribute the elements homogeneously as in the case of the chess-board(figure la, left) but concentrates N} =9 elements in a cluster while puttingN2 =4 outside this cluster. The repetition of the operation therefore produces a hierarchical structure. During the iteration there emerges a singlelarge cluster, composed of increasingly slender branches. On the other hand,in each of the N = N} + N2 elements at each step is generated a clustercomposed of N} squares reduced by the factor r = ^. It is confirmed thatat step n we have obtained one cluster composed of m{ni)=gn elements,four of mi2) = 9"-' squares, and thereafter the following series:

Sizem(3) = 9n~2m{t] = 9"-3 N{4)N(5)

= N2= N2= N2

= N2

NumberNN2N3

= 4-= 4= 4

= 4

13132133

i3*-2

The number W = > (m $) and the size m {nk) can be connected bythe following relation:~8 (11)


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212 P. FRANKHAUSER

Figure 2. - A fractal at iterationstep 2 with / = 1 3 and r= . Thearrangement of the elements inthe generator produces ahierarchy of clusters

Figure 3. - A multifractal atiteration step 3. For some elementsthe factors of reduction are indicated.They are made up of two factors ofreduction h=\ and r2 = | eachraised to a power. The numbers ofelements here are /Vi = 1 andN2 = 4 (see text).The distribution obtained is thus hyperbolic or paretian, which ischaracterized by the exponent 5 = og . For each size m# of clusters, the

number N(k) of elements of this size depends on the step n of iteration bythe same factor cn = N2 TVf"-''8.So far we have considered only fractals generated by a single factor ofreduction. The name we use for such structures is monofractals.{3) More comp

licated hierarchies can be constructed if the generator is constituted by severalfactors of reduction, for example if it produces N\ elements of size r, L andN2 of size r2 L. Such a structure exhibits a multifractal logic. In the examplein figure 3 we selected N\ - 1 , r\ = j and N2 4 , r\ - j In the subsequentiterations the same logic is applied to each element generated previously.Elements are thus produced whose factors of reduction are composed of combinations of multiples of r, and r2 . Iteration 3 in the first ring produces, atthe centre, an element of size r\ surrounded by four elements of size r\ r2 . Thesecond ring contains four clusters in the same arrangement but whose sizeis reduced by the factor r2 : the centres were generated by the sequence of

f3> In preference to the name homogeneous fractal, which is often used but which inour view can lead to confusion with the usual notion of homogeneity.


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THE FRACTAL APPROACH 213factors ) r2 r\ = r\ r2 L . They are identical in size to those of the periphery of the first ring. The size of the peripheral elements is r, r\ .

Thus it can be seen that the different rings contain squares that areof the same size but of a different 'function': the peripheral elements ofthe first ring are of the same size as the central elements of the second,and so forth. For iteration n, we thus obtain a series of different factorswith: [ , rf1 r2 ,..., = f\~k A >> A.- The number of elements correspondingo size r"fk A is:N (rnrk-rk)= N"-kNk (12)' 2 (n-k)lk\ ' 2

where the binomial prefactor expresses the phenomenon whereby elementsof the same size are found in different types of clusters.When dealing with towns and cities it is useful not to put all theelements of the same size together but to respect their place in a particularring and thus to respect their position (central or peripheral) in the sub-clusters. This position can be identified by respecting the order of the factors of reduction as they were gradually added together during the iteration.A distinction is thus made between the elements of r, r2 r{ and those ofthe factor r] r2 . It is verified that the corresponding numbers of elementsare then for each arrangement: 4)

If r2 < r\ and N2 > N\ there is an increasing number of units of smallerand smaller size and which are in more and more distant rings. This corresponds to the sharp decrease in mass towards the exterior of the occupiedsurface in the fractal of figure 3, compared with the Sierpinski carpet. Byselecting different ratios between |2, A^andTV, it is possible to representa lesser decrease or the coexistence of several large clusters of the samesize and in equivalent position. This illustrates the rich potential of thisapproach.When the number of elements Nn {r1k r|) with the same mass isknown it is possible to introduce a relation that corresponds to (3):

= (N]r-N2r)n (14)and with an appropriate choice of D it is possible to satisfy the conditionL = const. The values for r, and r2 selected in figure 3 can be used to makean explicit calculation of the fractal dimension: D = 1.36 (Gouyet, 1992).

(4) Since the indices are no longer to be mixed, the binomial prefactor disappears.


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214 P. FRANKHAUSERThe fractal dimension can usually only be determined by a numerical calculation. It is only a global measure since in a multifractal the fractal behaviour varies according to one's position. Information about this localbehaviour can be obtained by introducing the mass exponent or Lipschitz-Hb'lder exponent o*(LH) . Each point is surrounded by an interval , and to we link the mass |i() present in this interval by the relation (Vicsek,1989, Feder, 1988):

(15)For a monofractal this exponent is the same at each point and identicalto the fractal dimension. By contrast, in a multifractal structure it variesat every point. It is then possible to identify all the places for which thereis the same value a(Lff> . A multifractal can be broken down into sub-setseach of which has its own dimension, such that a whole spectrum of dimensions s obtained.

Random fractals The extremely regular form of constructed fractalsmight be considered an obstacle to an applicationto actual systems. However, fractal behaviour is not limited to objects witha regular morphology. A random element can be introduced into the iteration without affecting the fractal organization of the structure obtained:the position of the elements can be varied at each step of the iteration. Inthis way the free spaces already created are not affected but the distributionof the elements is different to that observed earlier. The parameters N andr do not change, so the fractal dimension stays the same.(5)If some range is also established within which the number of elements TVis varied at random during the iteration, the fractal behaviour remains statistically valid. However, the fact of having many numbers N has the sameeffect as the presence of several factors of reduction, and the structurebecomes multifractal (Gouyet, 1992). These variations can produce changesof fractal behaviour at some scales, as is observed in empirical structures.To describe this limited self-similarity, which reflects a hierarchical organization that is present at only a small number of scales, we prefer to adoptthe terminology of physics and talk of scaling behaviour.

II. - The methods of fractal analysisAn iterative mapping procedure cannot be used directly to measurethe fractality of empirical structures, most of which present an irregularmorphology. Methods of measurement have thus been elaborated whichapply a logic equivalent to that of iteration: a series of measures of variable H. Le Bras used this method to simulate the distribution of the population at aregional and national scale (Le Bras, 1993).


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THE FRACTAL APPROACH 215size e is introduced, and for each value e we determine the number ofelements N(e) needed to cover the structure:

Nn-(ln)D = L -> N(e) (e) D = const (16)The scaling behaviour is usually analyzed using the bi-logarithmicrepresentation of the function N(e):log N(e) = log const- D log 8 (17)

In the case of a fractal behaviour, this curve is a line whose slopevalue corresponds to the fractal dimension.Several methods have been elaborated for transcribing this logic intoconcrete algorithms with which to analyse the fractal behaviour of structures. In urban analysis, four main methods have been used: grid analysis,dilation analysis, radial analysis, and correlation analysis (Battey, Longley,1994; White, Engelen, 1993a; Frankhauser, 1994b, 1997b).For simple fractal structures, like monofractals, all these methodsshould give identical results, but empirical analysis has shown that this isnot necessarily the case. This must be interpreted as a sign that the structurefollows a multifractal logic. In these cases the information obtained byusing more than one method is found to be complementary. Some methodsproduce general information about the spatial distribution of the elements,such as the built-upon surface, in a zone. These methods we refer to asglobal analyses. Information of this type can be treated as equivalent to

the fractal dimension such as it is given by relation (14). In contrast, thereis a different approach which measures the distribution of the elements inthe vicinity of a selected point and which provides more detailed information about the spatial organization of the phenomenon in question. Werefer to this type of analysis as local analysis.Global informationThe grid method

The zone to be studied is defined and covered by a quadratical grid,and the grid distance is then varied. Following the logic described earlier,for each value of e, the number of squares N(z) containing any occupiedpoint is counted. The fractal relation corresponds to (17) and is used todetermine the grid-dimension Dg (Frankhauser, 1994b).This method has affinities with the procedure used in spatial statistics forstudying the distribution of a phenomenon. On the other hand it shows up theambiguity of the results obtained by counting in a grid if we use only a singlescale and thus a single value of e. This can be seen by applying the method tothe Sierpinski carpet in figure la. We vary the grid distance according to theiteration e = e = rn L . For each step n, we consider that part of the occupied


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216 P. FRANKHAUSERsurface vn or , what is equivalent, the density of the population that we assumeto be concentrated on this surface (see above). For vn we get:

v - N>in which we have replaced TV according to the fractal relation N = r-D andused the relation r" = en/L and where Nitot> is the number of squares neededto cover the whole surface. The proportion of the surface on which themass is concentrated thus depends directly on the size of the mesh: thedistribution of the elements cannot be determined by a single mesh sizeand it is parameter D which measures the variation in v, which is consistentwith our remark about density.

The results obtained by the grid method can be affected by the location and size of the selected zone. This is not surprising if the structurepresents multifractal behaviour. In some cases, however, the results varyaccording to the position and size of the window chosen for analysis eventhough the surface occupied inside the window remains the same. Thisambiguity can be avoided by means of dilation analysis.The dilation method

This method is based on the algorithm introduced by Minkowski andBouligand to establish the dimension of an object using the measure theoryapproach/6' In this analysis each occupied point is surrounded by a squareof size e, the surface of which is considered to be completely occupied(see figure 5). The size of these squares is then gradually enlarged, andwe measure the total surface covered A(e) at each stage. As the squaresare enlarged, any details smaller than e are overlooked and we graduallyobtain an approximation of the original form.(7) Because more and moresquares overlap, the total occupied surface A^dl> (e) for a particularvalue e is less than what it would be if the same number of occupied pointsthat make up the original form were surrounded individually. By dividingthis total surface by the surface a^dll> (e) = e2 of a test square, we get thenumber of elements N(e) necessary to cover the whole and we get a relationconsistent with (17). The corresponding fractal dimension DM is known asthe Minkowski dimension or dilation dimension.The correlation method and the dimensions series

A third method, spatial correlation analysis (see Batty, Longley, 1994;Frankhauser, 1994b), has sometimes been used. This involves counting the(6> This approach has also recently been discussed in the context of 'mathematicalmorphology'.(7> This approach can be compared to the gradual change in the degree of cartographicaldetail in drawing.


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THE FRACTALAPPROACH 217number of occupied points which lie at a certain distance from each occupied point, and thus the number of correlations. Fractality can also beanalyzed using a bi-logarithmic representation. In a multifractal dimension,this correlation dimension Dc is the first of a series of dimensions whichcharacterize the proximity of three, four etc. points (Feder, 1988). Thisgives more detailed results than grid analysis about the distribution of occupied points and can also be interpreted as the mean of the radial dimensionhat is examined in the next section (Frankhauser, 1994b).Local informationThe radial method

Information of a different type is obtained by using radial analysis.This method refers to a specific point known as the counting centre andgives the law of distribution of the occupied sites around this point. Acircle is drawn around this point, and the radius r is gradually increased.At each step the total number of occupied points 7V(p) inside the circle iscounted. The fractal law takes the following form:

=> logtf(p)=Dr-logp (19)With linear relation (19) it is possible to calculate the radial dimension r . When analyzing digitized patterns it is usual to surround the count

ingentre with squares. This avoids the rounding errors that result fromthe quadratical grid of the pixels. As a reminder that the distances aremeasured in pixels, and thus in discrete terms, we prefer to write p, .If we consider this type of analysis from the angle of multifractaltheory, the radial dimension has a logic similar to that of the Lipschitz-Hlder exponent mentioned earlier. For if the mass (x(e) is identified withthe number of occupied points Nit), and the interval with p, relation (15)is identical to relation (19).The local character of this type of analysis means it can be used fordetailed investigation of urban land-use patterns by comparing the results obtained for different counting centres. On the other hand, information aboutthe space surrounding the counting centre is aggregated. Counting the occupiedpoints inside a circle whose radius is gradually enlarged, implies an integrationover the whole range of 2 of the angles and, simultaneously, an integration on the radius from p' = 0 up to the present distance p' = p. We haveseen that the notion of density is ill-suited to a fractal logic. However, radialanalysis also measures the radial decline in mean density with distance fromthe counting centre. This is clear if we consider the proportion of the occupiedsurface at a distance p from the centre, and thus the mean density g inside


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218 P. FRANKHAUSERthis zone. Since the number Nm(p) of pixels needed to cover the wholesurface is proportional to p2, for g we get:

D.-2 (20)P 'Since Dr < 2, the exponent is negative and so the loss of density inthe occupied surface with radial distance from the selected centre followsa hyperbolique function. The mean density thus varies continuously fromthe centre (Batty, Kim, 1992).

The curves of scaling behaviourIf the aim is to compare the radial loss of density at different distances

from the centre and above all to identify the changes in fractal behaviour,the bi-logarithmic distortion can also be problematic. To avoid this effect,we have calculated for each distance p, the local value of the slope in thebi-logarithmic representation produced by the radial analysis. These slopevalues are represented for the range of distance p; (Batty, Longley, 1994;Frankhauser, 1997b):logN-logN (21)a = log p, - log p;_,In what follows the function oc,(p,) is referred to as the curve of scaling

behaviour.This mode of representation is especially useful for identifying andmeasuring changes in the spatial organization of urban structures. Somecare is needed when interpreting the results, however. As an example wepresent a theoretical pattern formed by a Sierpinski carpet surrounded bya black frame (figure 4a). In the central part we expect to observe a constant slope value equal to the fractal dimension. But contrary to this hypothesis, radial analysis (figure 4b) show sharp variations which appear asfluctuations in the curve of scaling behaviour (figure 4c). This is due tothe fact that the fractal law is only valid at distances from the centre thatare multiples of three of the length of the square, particular to that step,which is consistent with the logic of iteration. Radial analysis does notrespect this logic and deviations are observed for the intermediary distances.In particular, local values of oc, may greatly exceed the value of two.

This local variation of the slope is a well-known phenomenon in fractal eometry: free spaces in particular are responsible for local distortionsin the scaling behaviour, even if their presence is consistent with the fractallogic of a structure, for example of a constructed fractal. This phenomenoncan be accounted for by introducing a generalized fractal law:,)-? (22)


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THE FRACTAL APPROACH 219

0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 2,4 2,7 3logo

15 29 43 57 71 85 99 113Figure 4. - Figure (a) shows a theoretical reference structurecomposed of the Sierpinski carpet, surrounded by a black zone.The radial analysis and the scaling behaviour analysis are shown

in figures (b) and (c) (see text)where a(p,) indicates the local variations of the slope. 8) These local deviations from the fractal law should therefore not be over-estimated.If the local variations generate slope values that can exceed two, evenlarger variations are observed beyond the distance where the scaling behaviour changes. Beyond this distance the values fall and approach asymptotically the value of two which corresponds to the structure. This is atransient phenomenon: if the structure was also of dimension two prior tothe rupture, the number of occupied points at this critical distance will behigher than in the case of a fractal structure. So in relation to the externalpart of dimension two there is a shortfall of occupied points. As distancefrom the centre is increased this phenomenon declines in size: the proportionf missing points in relation to two becomes less and less significant.This example shows that the value a, does not correspond directlyto the fractal dimension but is subject to different types of perturbation.A relation has been established which identifies the different phenomena


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220 P. FRANKHAUSERinfluencing the value a,. So as to be able to allow for any change in thefractal dimension, we have generalized relation (22) so that the dimensioncan also vary with distance: D = D(p) . In addition, we have treated distance p as a continuous variable, which makes it possible to interpret theslope a as the first derivative in the bi-logarithmic representation:

. . dlogN .. (23)a(p) = ,. & = lim a.dlogp ]Ogp^iogPilBy means of law (22) generalized, it is verified that for the slope oc(p)we finally get the relation:d\ogN dloga . dD _ (24)e = 6 + logp + D = a.logp /logp dlogp

Thus it can be seen that three different terms influence the slope value a: the first term represents the local change in the prefactor a whichcharacterizes the local deviations. These perturbations appear onthe curve of scaling behaviour as fluctuations around a constantmean value; the second term measures the change in the fractal dimension D,and thus authentic ruptures in the scaling behaviour. The latterappear as changes in the mean behaviour of the curve a; ; the third term is actually the fractal dimension itself. Because it

is now assumed that D varies with distance p, we prefer to speakof the local value of dimension D.The local values a, of the slopes must not be taken as the fractaldimension. In particular, because the first two terms measure variations,it is possible for the total of the three terms to exceed value two.i9)In order to distinguish local fluctuations from structural changes, itis useful to isolate the changes in the mean scaling behaviour. Gaussiansmoothing has been found an effective tool for eliminating the fluctuationsin a gradual and well controlled manner. (10)III. - Fractal investigation of urban patterns

Fractals, hierarchies and urbanizationFractal analysis combines several aspects. It is: a method of spatial analysis. In particular, it can be used to examinethe law of distribution by moving across scales, as was seen in the(9) A formal demonstration is given in Frankhauser, 1996a.(10) Comparisons have shown that the most suitable smoothing parameter for identifyingthe structural aspects of the curves of scaling behaviour has to be in a range of between 12and 18 pixels. Use of the same value means that the results can be compared.


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THE FRACTAL APPROACH 22 1grid analysis. The reference model is a Pareto distribution which ischaracteristic of a hierarchical organization. It is also possible tomeasure the deviations from this type of distribution and identify theruptures present in the spatial organization. The case of a homogeneous distribution is also included by the value D = 2;

a geometrical approach. Geometric structures can be generated whichrespect a defined law of distribution. This makes it possible to createmodel urban patterns with which to illustrate certain types of spatialorganization. A multifractal approach offers many possibilities forgenerating complex urban patterns. It may even be possible to modelthe ruptures at some scales by introducing a variation of the generator; an instrument for research. It is possible to compare empirical structures, even when these appear irregular, with constructed structures

that follow the same law of distribution (see figure 7). These constructed patterns can then be used as reference models by town planners and allow the calculation of spatial measures of value for planningpurposes. Such a paradigmatic transcription remains difficult to realizeat present. The fractal dimension is a fairly global parameter whichis not affected by the presence of free spaces/1 ')The link between fractal geometry and the Pareto distribution appearsto offer particularly rich possibilities in the field of urban theory. The existence of such a law of distribution is a well-known phenomenon in systems of settlement and at different scales: at the scale of urban networks, this distribution has been used to formalize the rank-size distribution of towns (see for example Pumain,1982; Gurin-Pace, 1990), and Bussire and Stovall (1981) have foundthe same law by comparing the surfaces of cities by size class. In centralplace theory, economics provides the basis for deducing a similar lawof distribution which is reflected in the presence of numerous smallurban settlements and progressively fewer large urban settlements. Thishierarchy is already visible at the scale of metropolitan areas, where asystem of sub-centres can often be identified; in urban agglomerations, a radial decrease is frequently observedin the proportion of occupied surface in the agglomeration with

distance from the centre, and a hyperbolic law has been proposedto describe this phenomenon, thus adopting an approach that corresponds to fractal geometry. A more general formalization of thisradial decrease is possible with a multifractal approach.From the angle of the decline in population density S. Korzibski hasalready analyzed this phenomenon by comparing the evolution of the urbanpopulation in London and Paris. Also in relation to population density,Batty (1992) discusses the similarities between the exponential and(11) A different organization of the elements in a generator can change the aspect of

a structure without the fractal dimension being altered.


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222 P. FRANKHAUSERhyperbolic approaches in empirical urban patterns with reference to thework of Zielinski (1980).

at the intra-urban scale, hierarchy appears rather at the level of thefree spaces: on the one hand there is a large number of smallresidential streets and internal courtyards, and on the other a smallnumber of main avenues and large squares; from a morphological standpoint, the fragmented aspect of built-upareas is also reminiscent of fractal geometry: irregularities are observed at very different scales and, in particular, the distributionof the built-up zones is non-homogeneous. This is reflected in aconsiderable lengthening of the perimeter of cities in relation totheir built-up surface, which appears to be consistent with the geometry of Sierpinski carpets: their border line p is the same as

their surface a, giving the proportionality between p and a whichis in contradiction with Euclidean geometry.So we should not expect to find a fractal law that is valid for all thescales: the proportion of the surface occupied by man in space is not thesame if we are studying settlement patterns at a regional scale or if weare considering the interior of an urbanized zone. Consequently we areunlikely to find a single type of behaviour for all the geographical scales.Instead, it is more reasonable to assume that there are ruptures in the hierarchical organization at some scales. We have referred to different typesof fractals according to the context under discussion.Even at a single scale, fractal behaviour may vary according to the objectunder review: topography or the morphology of the transportation networkcan influence the morphogenesis of an urban pattern in different ways.It can also be noted that some algorithms are better suited than othersto the study of a particular phenomenon. Thus radial analysis appears tobe especially valuable in measuring the micro-structure of intra-urban patterns. By contrast, global analysis proves more useful at a regional scale,where attention is on the spatial distribution of settlements and where settlements are separated by extensive unoccupied spaces.

An established field of application: Fractal analysis has proved a po-the structure of urban patterns werful tool for studying the spatialorganization of urban patterns. Ment

ion can be made of the analyses by M. Batty and P. Longley (1994), M. Battyand Y. Xie (1996), P. Frankhauser (1994b, 1997b) and R. White (1991, 1993a).Here we shall limit ourselves to examining different aspects of theinformation collected using examples from actual research.Some general findings

We begin with a number of results which confirm the assumptionspresented above. From a theoretical standpoint, it has been possible to construct


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THE FRACTAL APPROACH 223a fractal which obeys a logic similar to that which underpins central placetheory (Frankhauser, 1994b). (12) For the French urban system, N. Franoishas demonstrated that such a fractal hierarchy can be observed using aradial analysis that takes Paris as its centre (Franois, 1995). An analysisin the same vein concerns a number of metropolitan areas. We defined anumber of surface size classes and counted the number of cities whosebuilt-up surface corresponded to these (Frankhauser, 1994a). The metropolitan areas which present a high degree of fragmentation, such as Stuttgart, Moscow, Berlin and Pittsburgh, confirm the existence of a hyperbolicdistribution that is consistent with fractal geometry. Urbanization in thesecases has been influenced either by the presence of valleys or by the importance of a public transport network akin to the RER (suburban railway)around Paris. By contrast, no such hierarchy is discernible for a numberof North American and Australian cities whose development has been determined by the private motor car from a very early date. Figure 5 shows

1 1 n i i0 0,5 1 1,522,53 4 567

Figure 5. - (a) The built-up surfaces of Pittsburgh, (b) the thirdstep of dilation, (c) the curve of the dilation analysis, and (d) thecurve of the clusters(l2) N. Franois has recently developed a fractal formalization of the distribution ofpopulation in a central place system (Franois, 1997a).


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224 P. FRANKHAUSERthe curve of this analysis for Pittsburgh, that we have compared with thedilation curve. These two curves indicate the existence of a spatial hierarchy. It may be noted, however, that fractal analysis also translates thenon-homogeneous distribution of human activities within this space.A different analysis has examined the relationship between the built-up surface a and the perimeter p in a sample of sixty urban areas. Theresult confirms a close proportionality between the values of a and p(Frankhauser, 1994b). The fractal dimension of the perimeter can thereforebe used as an indication of the degree of fragmentation of an urban pattern.A typology of metropolitan areas

Many fractal investigations of urban patterns have dealt with the analysis of metropolitan areas and are thus conducted at a regional scale. Detailed discussion of these results is in M. Batty and P. Longley (1994)Fractal Cities, a Geometry of Form and Function{l3) and P. Frankhauser(1994) La fractalit des structures urbaines.

These analyses are often conducted using large-scale cartographic representations from which the details of intra-urban structures are absent. Ageneral information is thus obtained about the spatial organization of thecities. In the sample of thirty cities that we analyzed, it was found thatEuropean cities and some old-established cities in North America, such asPittsburgh and Boston, are different from the cities of North America andAustralia whose growth has been strongly marked by the spread of privatecar ownership, such as Los Angeles and Melbourne.04 In the first group,the fractal dimensions are lower, notably the global dimension (grid andcorrelation). The difference between the three dimensions (radial, grid andcorrelation) is greater in this group. Their behaviour is thus more consistentwith a multifractal logic. The values obtained are direct reflections of theurban morphology: cities like Berlin and Moscow have low values of Dr,which is explained by their axial development along the lines of transportnetworks. The degree of urbanization in the urban hinterland is indicatedby the global dimensions. Highly fragmented patterns like that in the Stuttgart region, for example, give low values. In this way several types ofurban agglomeration can thus be identified.Analysis of urban agglomerations - some comments on methodology

For the urban and intra-urban scale we draw on analyses we haveconducted for several urban agglomerations in the Franche-Comt regionof France, in particular Besanon, the Montbliard urban area, the regionsof Dole and of Lons-le-Saunier.(l5)(13' This book assembles the results obtained by other authors. The Third World cities we have analyzed do not form a homogeneous group. Theresults reflect their disparate historical evolution and an incomplete urban development.(15) These analyses are included in a number of masters theses in Geography (S. Trin-

cat, Besanon; L. Goguel, Montbliard; S. Lhomme, Dole; J. Prost, Lons-le-Saunier).


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THE FRACTAL APPROACH 225A cartographical representation as close as possible to reality is required for this scale of observation. We used topographical maps on the1: 50 000 and 1: 25 000 scales, completed in the case of Besanon by a

database compiled from the GIS of the local authority.(16)Each of these studies focused on the radial analysis completed bythe representation of the scaling behaviour. Local information relative toa specific point in a town and the ability to identify directly the rupturesin urban patterns are particularly well-adapted to this scale of study. Thevalue of radial analysis also stems from the fact that this analysis has affinities with human perception of the urban space (Franois, 1995). Studying spatial patterns by gradually increasing the area around a fixed pointcorresponds to the situation of an observer at a certain point in a townwho gradually expands his field of vision. This aspect is further reinforcedby the logarithmic representation which gives greater emphasis to the phenomen a near the counting centre while aggregating the information relativeto the more distant zones. 17) Studying the form of the curves has usuallybeen found to be more revealing than the actual values of the dimensions.Intra-urban ruptures in agglomerations

The curves for urban patterns present the same characteristics alreadydiscussed in the analysis of theoretical reference structures: the scaling behaviour fluctuates locally around a dominant mean behaviour. However,these fluctuations are usually less than in the theoretical reference structures, since the random phenomena present in urban patterns tend to makethe curves smoother. Thus although the bi-logarithmic curves are in placessurprisingly regular they also contain ruptures. By selecting different countingentres near to each other, it is found that the scaling behaviour variesonly for the points in the immediate vicinity of the counting centre. Thisis due to local deviations such as the presence of large buildings or vacantlots; these are in fact the effects of free spaces. In addition, it is importantto remember the in-built limits to an analysis of urban patterns: resultsthat relate to distances smaller than the average size of small buildingsgive unreliable information/18'However, the character of the curves may vary depending on thecounting centre selected when we take a position close to the points wherethe scaling behaviour changes. The fluctuations in these zones are alsogreater, which is consistent with our theoretical propositions: change inthe fractal dimension contributes to larger deviations.Figure 6 shows the built-up area in the north of the outer suburbanring of Besanon, plus the smoothed curves of scaling behaviour for two

(l6) The author would like to thank the municipal authorities of Besanon for makingthis data base available.(17> Mention can be made in this context of the work by several authors on the non-Euclidean, heterogenous and anistropic nature of geographical space (Reymond, 1981; Brunet,1990). (18) For districts composed of blocks of flats this threshold may be at a greater distance,as is confirmed by an analysis of the district of Besanon-Planoise.


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226 P. FRANKHAUSER

321

point 1

21 4

250 500 750 1 000 1250p (in metres)point 2

250 500 750 1 000 1 250 1 500 1 750 (in metres)Figure 6. - The smoothed curves of scaling behaviour fo r twocounting centres located in the northern part of Besanon. Thesquares correspond to the ruptures marked in the curves. The factthat the squares corresponding to different counting points overlapin places indicates that the ruptures are important


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THE FRACTAL APPROACH 227counting centres. The squares marked on the map correspond to the rupturesin the curves. It can be seen that some squares corresponding to differentcounting centres partially overlap. This indicates that in this sector the rupture is large enough to dominate the result obtained for the whole of theperimeter of this square.By choosing a succession of positions for the counting centre it isthus possible to obtain detailed information on the aspect of the urbanpattern as it appears from these points. Such information is more than asimple division into zones: it is a descriptor of the quarters in question.

Fractal measures can thus be used to make a morphological comparison and classification across different scales.A division into zones according to scaling behaviour can also be doneusing other methods of analysis. We measured the size of the grid by gradually moving for a fixed distance the window in which the counting isdone, over the total surface of the urban pattern. The values obtained forthe fractal dimension were then mapped: each position selected is surrounded by a square whose degree of shading corresponds to the value ofthe dimension observed. For example, the colour black corresponds todimension two, while the colour white indicates dimension zero. In thisway an information is obtained about the distribution of dimensions andthus about the ruptures in the urban patterns.

Additional information is supplied by certain ruptures in the actualcurves obtained by the grid method. In particular, a rupture often occurswhich relates to a small grid distance. This rupture corresponds to the minimum distance between buildings: when the usual size of backyards isreached, a shortfall of occupied squares will be noted for a smaller mesh,indicating that this is the lower limit at which structural phenomena canstill be observed.The information obtained in this way about the spatial organizationof urban districts can be used to construct theoretical fractal patterns equivalent in their internal organization to the observed patterns. Figure 7 showsan attempt to represent the aspect of the centre of Besanon in the formof a Sierpinski carpet by respecting approximately the fractal dimensionand the distribution of free spaces in the urban pattern on the basis of aqualitative resemblance.' l9)Urban cores: centres of symmetry in urban patterns?Characteristics of urban cores

The aspect of the curves obtained for the outer suburban districtsvaries depending on the spatial organization of the districts. On the other


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228 P. FRANKHAUSER

Besanon town centrer = 1,81VMM:: VAVAVAVA VA VA VA VA VAVAVAVAAVAVAAVAVA

VA .: s.sVAVAVAVAVAVAAVAVAAVAVA MB VAVAVA constructed fractalD=1,73

Figure 7. - The centre of Besanon (Dr= 1.81) compared toa constructed fractal (D = 1.73). The distribution of free spaceshas been chosen to be close to actual conditions, which seemto be characterized by the presence of large blocks aroundinternal court yardshand, if counting points at the centre of an agglomeration are selected,the curves obtained present a similar form, even when the towns and citiesare of very different sizes. Figure 8 shows this result for the agglomerationof Moscow based on a 1:500 000 map. Curves of the same type are obtainedfor other major European agglomerations such as Munich, Berlin and Stuttgart, but also for smaller towns like Besanon, Dole and Lons.(20) In thecase of villages, the curves observed are either very irregular, with no discernible pattern, or correspond to the pattern for towns.

This schema can be characterized as follows: as distance from the centre increases, scaling behaviour remainsquite constant at a fairly high value of approximately 1.8 if topographical maps are used, and closer to 2 for less detailed maps.This zone, which varies in size, corresponds to the urban core; when the edge of this central zone is reached, the values of a, fall.This transitional zone varies in extent according to the morphologyof the agglomeration. It indicates a change in spatial organization at(2) The fact of having used different cartographical bases for the large cities as forthe towns obviously makes comparison of the results problematic. However, as comparativestudies have established, such a comparison appears to be acceptable if attention is confinedto the general aspect of the curves.


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THE FRACTAL APPROACH 229this distance from the centre. This lower limit is marked on figure 8,and it can be seen that it does indeed correspond to the end of thezone of dense urbanization which precedes the outer suburban ringdominated by strong axial growth;next there comes a fairly extensive zone that corresponds to theperipheral ring of the urbanized area. The curve here often presentsan impressive degree of regularity, so we observe a constant scalingbehaviour, though the value of exponent a is lower (see figure 8);

Figure 8. - The Moscow agglomeration and the curve of scalingbehaviour obtained by a radial analysis from the city centre. Thetwo distances for which ruptures are observed have beenindicated on the curve by lines and on the diagram by squares:one corresponds to the urban core, the other is at the boundary


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230 P. FRANKHAUSER continuing towards the exterior another transitional zone appears,where the slope frequently declines gradually and which marks thestart of a zone as yet largely unaffected by the growth of the central

core. In figure 8, this second inflexion is indicated on both thecurve and the map.In many cases, therefore, the spatial organization of urban patternsappears to follow the same principle of internal order at different scales,so this is a phenomenon of self-similarity.'21' An explanation can be foundin the dendritic morphology of urbanized zones: beyond a certain distancefrom the centre, large unoccupied areas appear between the urbanized zonessituated close to the axes of transportation, as can be seen in the exampleof Moscow. The same also applies to the villages which present this typeof scaling behaviour.The strikingly regular form and the similarity of the curves obtainedfor the urban cores shows that these play the role of centres for the radialdecay of built-up areas. In a certain sense they represent centres of symmetry in the morphology of urban patterns. (22)The schema described above is sometimes modified according to thespecific features of the urban pattern. Smaller scale ruptures often occur,which correspond to sub-centres present some distance from the main core.

Some specific examplesThere are also cases which diverge considerably from the schema out

lined above, but from what has been said it is possible to provide an interpretation of such results. A first example is that of Los Angeles whichwe studied at the scale of 1:500 000. Although the urbanized zone in thisagglomeration is very extensive there exists no hierarchy of towns in thehinterland. In this case the a, values decline over a much longer range ofdistance and the stage of the peripheral ring is missing.Another case that has proved interesting is that of the Montbliardconurbation, a medium-sized agglomeration. The curve obtained for thehistoric centre of Montbliard is more like that obtained for an outer suburban district. By contrast, if the Peugeot car factory is taken as the countingentre, the form of the curve is like that observed for city centres.The Peugeot factory thus plays the role of a centre of symmetry for theradial decay in this urban pattern. On the other hand, for Audincourt situated at the centre of the conurbation, the curve presents merely a slightinflexion as distance from the centre increases. Thereafter it rises and aninflexion occurs only when the edge of the conurbation is reached. This

(21) At the scale of metropolitan areas, the difference in scaling behaviour betweenthe urban core and the suburban periphery has been noted by other authors, notably byR. White who refers in this context to a bifractal behaviour (White and Engelen, 1994a;1994b).(22) It may be added that recent analysis conducted at a national scale for the urbannetwork by N. Franois has shown that Paris similarly plays the role of a "natural centre ofradial decay" in the system of settlement (Franois et al., 1995).


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THE FRACTAL APPROACH 23 1indicates that the town belongs to a network of towns which together forma larger unit. The fact that no major change is observed in the scalingbehaviour shows that this set of towns together has the same spatial organization as a central core. Thus the conurbation as a whole forms a structure organized according to a common internal hierarchy. The relativeregularity of the curve for Audincourt indicates that this town is the geographical centre of the conurbation.Spatial analysis of the urbanization processThe degree of spatial absorption

Urbanized zones are usually composed of a number of sub-centresgrouped around a large urban centre. It is common for the centres of settlements closest to the central cluster to be absorbed by it in the processof urbanization, thereby losing their independent character and becoming partof the enlarged central cluster - even if administrative boundaries do notalways keep up with such changes. The value of radial analysis as a sensitive instrument for distinguishing the degree of absorption of a centre ofsettlement has been demonstrated by the analyses we have conducted for theperiurban towns of Munich (Frankhauser, 1994a) and Berlin, as well as atthe scale of medium- and small-sized towns in the Franche-Comt region.When other clusters are situated near to the main cluster, the smoothedcurve of scaling behaviour is seen to have a weak inflexion before climbingagain. On the other hand the same behaviour is observed when the positionof reference is the centre of a commune (local district) situated on the periphery. For a periurban zone that has not yet been absorbed into the centralcluster, however, a curve with a clear break is observed.By comparing the degree of absorption for the same agglomerationat different periods we can retrace the successive stages of urbanizationin a periurban zone. Figure 9 shows the urban pattern of Lons-le-Saunierin 1970 and in 1985, as well as the smoothed curves of scaling behaviourfor Montmorot, a commune close to Lons. A sharp break is observed in1970 but has completely disappeared in 1985. The scaling behaviour hasthus become more regular, and the urbanization process is found to act asa self-structuring process.Change and continuity in urban morphogenesis

In the previous example, urbanization is accompanied by a phenomenonf self-structuring which is reflected in a greater regularity of thecurves, despite the fragmented morphology of the urban patterns. Thereforeit seems interesting to study the process of urbanization by comparing theurban patterns of a town across time. We have conducted such investigationsfor Munich, Berlin, the Montbliard region, and Lons-le-Saunier. The resultsconfirm that fluctuations diminish greatly in the course of urbanization:


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232 P. FRANKHAUSERLons-le-Saunier 1970

Montmorot .,:Lons-le-Saunier 1985

500 1000 1500p (in metres) 500 1000 1500p (in metres)Figure 9. - The absorption of Montmorot, a village onthe outskirts of Lons-le-Saunier, between 1970 and 1985.The same counting centre has been used fo r both periods.In 1970, the smoothed curve of scaling behaviour is seen tocontain several ruptures which were then situated between the two

agglomerations. In 1985 there remains only a slight inflexion.The distance of 60 pixels corresponds to 1 km for large agglomerations the principal observation is a decline inthe fluctuations in the suburbs and beyond.This indicates that the urbanized space is increasingly dominatedby the central cluster; for some periurban villages no structure can be identified to beginwith, whereas in the course of urbanization the curves get closerto the pattern described for the urban cores.Urbanization should thus be interpreted as a self- structuring process.Figure 10 presents the comparison of two chronological sequencesof smoothed curves of scaling behaviour relative to periurban developmentin the region of Berlin and of Lons. Although these two agglomerationsare not comparable by size, in qualitative terms the two curves are similar.In both cases we observe, at the first date, that the centre is still fairlylimited in extent.(23) For distances beyond the centre, the curve for Berlin(23> By isolating the break between the urban core and the peripheral ring at eachperiod it is possible to measure the growth of the space affected by urbanization. This informat ion can be used to study and model urban growth (Frankhauser, 1994a).


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THE FRACTAL APPROACH 233

a3,

a3-,

Berlin1875

10 15 20 25 30 35p (in kilometres)1910

10 15 20 25 30 35p (in kilometres)1920

5 10 15 20 25 30 35p (in kilometres)1945

0 5 10 15 20 25 30 35p (in kilometres)

Lons-le-Saunier1913

50 0 1 000 1 500 2000 2500 3000 3500p (in metres)1947

i 1 ~r50 0 100015002000250030003500p (in metres)1970

500 1000 1500 2000 25003000p (in metres)1985

0 50 0 1 00 0 1 50 0 2000 2500 3000p (in metres)Figure 10. - Comparison of the growth of Berlinand Lons-le-Saunier as transcribed bythe smoothed curves of scaling behaviour.It will be noted that the form of both curves changesin the same way (see text)


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234 P. FRANKHAUSERlevels out at a lower value before going on to reach higher values, withina limited range. In the case of Lons, it is seen that the curve falls afterreaching the edge of the core, and that a second break is observed, corresponding to the suburban ring which surrounds this core. Only at a greaterdistance from the centre do there exist also higher values. In both thesecases this increase in values is due to the presence of other agglomerationssituated at this distance. A comparison with the curves for the subsequentperiods shows that the inflexions in the curves are reduced and that weobserve a more extensive transitional zone. The fluctuations caused by distant agglomerations gradually give way under the influence of a periurbanring whose presence is indicated by a stage of constant behaviour as discussed earlier.If the growth of the central cluster has a considerable influence onthe spatial organization of the suburbs, it is found that the values of theparameter a, scarcely increase inside the urban core which thus does notreach the dimension of two that is indicative of a homogeneous occupationof space. The values of a, in the peripheral zones are observed to movegradually closer to those of the core. It follows that the additional built-upsurface is thus distributed according to the same principle as inside thecluster, with the result that larger free spaces continue to exist inside theurbanized zone. This form of growth corresponds to the allometric principle, often observed in biological systems (Frankhauser, 1994b) and isconsistent with the logic discussed above of a gradual fragmentation ofthe built-up area.

This mode of growth is not the result of a planning policy but tendsin fact to run counter to such policies, which means that it is a phenomenonof self-organization. In fact, the growing influence of the central clusteron the periurban area is reminiscent of the slaving principle elaborated insynergetics to explain phenomena of self-organization. One of the fundamental principles of this approach is based on the idea that the competition etween sub-systems culminates in the dominance of a single elementwhich achieves preeminence through a process of natural selection. In thecase of a settlement system it can be imagined that one particular agglomerationenefits from an advantageous situation, in the form of political oreconomic advantages, or by its position. The neighbouring agglomerationsthen risk losing their independence. 24)These observations point to the use of the fractal approach to modelurban morphogenesis, and several authors have suggested growth modelsbased on fractal rules (Frankhauser, 1994b; Batty, Longley, 1994; White,Engelen, 1994a; Makse, Halvin, Stanley, 1995). Particular mention can be

(24) w. Weidlich and G. Haag (1988) have elaborated a synergetic model of a networkof towns to show the spontaneous growth of one town to the detriment of the others. Themulti-agents model, elaborated by the P.A.R.I.S. team (Gurin-Pace, 1995), also reveals thedevelopment of an urban hierarchy. In both cases, however, these are demographic modelsrather than a morphogenetic approach. It may be added that the socio-economic aspects ofthe fragmentation of the urban space are discussed in Frankhauser, 1994b.


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THE FRACTAL APPROACH 235made of the attempts by Batty et al. and more recently A. Makse et al. tomodel urban growth by applying methods used in physics to describe phenomen a of diffusion and of electrical discharges. These results are interesting in that what is being modelled is the peripheral growth of cities.In addition, Makse et al. also consider the influence that the surroundingsof a place have on its probability or not of being urbanized. However,such an approach remains essentially descriptive, since it is still hard tojustify this method on the basis of micro-economic processes, as forexample, decisions over residential location. In this respect, the approachadopted by R. White and G. Engelen represents a different logic: they introduce the laws of spatial interaction between different land-uses. Forexample, the proximity of an industrial zone is unfavourable to the development of a residential zone. With this model is it also possible to takeinto account the influence of supplementary models. On the other hand,no allowance is made for changes in the behaviour of the human agents,for example a greater sensitivity to increasing density of the built-up space.Yet such phenomena appear to have an important role in explaining thereasons which contribute to the fragmentation of the periurban space. Theuse of spatial models to explore urban growth does seem nonetheless apromising field, since space is explicitly present in this approach, in contrast to the traditional models whose approach is essentially economic. Amodelling concept has recently been proposed which attempts to integratethe reaction of the human agents to the spatial transformation provokedby the process of urbanization (Frankhauser, 1996b).

IV. - Towards an analysis of population distributionOur discussion so far has been limited to studying the distributionof built-up surfaces. In terms of demography, this point of view corresponds to a homogeneous distribution of the population across the built-upsurface. At the level of a more detailed analysis of urban patterns, it wouldbe desirable to also take into account other aspects of urban spatial or

ganization, such as the density of built-up surfaces and the distributionof the population. Reference was made earlier to the work of H. Le Braswho has used a multifractal approach to analyse the non-homogeneousdistribution of the population. This approach is based on an iterationwhereby at each step a part of the population is allocated to a part ofthe available surface. A similar approach has recently also been used byS. Appelby to analyse the distribution of population at the scale of theurban network (Appelby, 1996).

Application of such an approach at the scale of agglomerations requires introduction of a bivalent logic with which to model both the distribution of the built-up surface and the density of land use. We need tomeasure the radial decay of the occupied surface from a city centre and


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236 P. FRANKHAUSERa distribution of the population which may be different/25' The aim is thusto combine two different logics of distribution.

The generator

Iteration step 2

Figure 11. - The generator discussed in the text and the iterationstep 2. The percentages of population present in each square arerepresented by different degrees of shading: the greyer the scale,the higher the percentage(25) It is known, for example, that the residential population is concentrated in theperipheral zones even though densities of built-up surfaces in the urban cores reach veryhigh levels.


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THE FRACTAL APPROACH 237Figure 11 gives the example of a fractal which corresponds to sucha logic: the initial figure is a square containing the entire population. Agenerator is then introduced comprising nine squares reduced to one fifth.

Five of these squares are arranged in the initial square in the form of achess-board, and another is added in each corner. This construction is notmuch more complicated than the carpet in figure 1. Unlike the Sierpinskicarpet, however, the population is distributed over the occupied squares ina non-homogenous way. We allocate only /?i=| to the central square,whereas each of the squares in the first suburban ring have p2 = | of thepopulation, and we assume that the furthest squares, situated in the corners,have only = -^ of the population. This operation is repeated in the nextstep. The logic is thus like that of figure 3: also produced are factors ofreduction which contain the factors pu p2 and p3. A hierarchy of sub-centresis obtained which obey a logic equivalent to that of the central cluster.The proportion of the built-up surface does indeed fall as distance from thecentre increases. However, the distribution of the population is consistent withthe idea that the largest share of the population lives in the inner suburbs,while the centre and outer suburbs are less densely populated.A fractal analysis based on this approach could be used to study thisphenomenon and to divide the zones by their type of demographic distribution. This requires the adaptation of multifractal methods of measuremento this phenomenon of distribution.Conclusion

Application of fractal theory is not simply about the introduction ofalternative methods of spatial measurement. It in fact involves a new geometr ica l approach for understanding the organization of humanized space:the paradigm referred to is no longer that of homogeneity as in the Euclidean perception but that of the hierarchical distribution of elements inspace. This logic appears to be consistent with many observations and witha number of theoretical formalizations such as the rank-size law and centralplace theory. By using the fractal approach it is possible to establish theexistence of such an organization within a spatial system, even when it isnot directly visible because masked by random phenomena. Regular andconstructed geometrical structures can thus be compared with structureswhich though they appear to be irregular follow the same principle of internalorder. In addition, the possibility of studying a phenomenon across scales alsoallows the identification of critical thresholds in a spatial system.The geometrical character of the fractal formalization makes it particularly suited to exploration of the morphology and genesis of urban patterns, and in a way which goes beyond a simple topological approach. Forexample, the curves of fractal analyses obtained for different periods are


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238 P. FRANKHAUSERfound to become more regular which reflects an underlying phenomenonof self-organization. The same type of evolution is also observed for smalltowns as for large agglomerations. Yet it must not be thought that this isa trivial phenomenon, as topographical, socio-economic conditions and accessibility can all influence morphogenesis and hence the fractal indices.It must therefore be concluded that the fractal indices characterize themorphology of urban structures. Fractal dimensions have a concrete meaning:they measure the degree of non-homogeneity in a spatial distribution. A fractaldimension close to two expresses an almost uniform distribution of the elements, whereas a dimension close to zero is indicative of a strong concentration.ompared with some traditional measures which relate to topology, suchas measures of connectivity, fractal analysis provides supplementary information,ince it measures the non-topological properties of these structures. Inother cases the fractal indices indicate the unsuitability of traditional measures:if density varies greatly with the scale of observation, fractal analysis caneliminate this ambiguity, for example.Fractal analysis is primarily a descriptive tool: we may observe astructure which obeys a fractal law or identify the existence of a threshold,but these facts are not explained. However, identifying a certain type ofspatial organization provides a tool with which to test theories or explanatory models from a morphological standpoint: first, a model of urbanmorphogenesis has to respect the results obtained by this approach, whilesecond, as in the physical sciences, the discovery of a certain type of structure can contribute to reflections about explanatory theories which reproduce the observed spatial logic.Although the fractal measures currently in use enable us to compareand classify, it can be seen that they are sometimes inadequate. Futureresearch should thus be directed at developing supplementary measures butwhich are consistent with the logic of spatial hierarchies. As has beenshown, a promising direction for conceptual extension appears to be thatof the notion of multifractality. This should facilitate the interpretation ofcertain results obtained by earlier analyses.

Pierre Frankhauser

AcknowledgementsThe author would like to thank Denise Pumain, Lena Sanders and FranceGurin-Pace of the P.A.R.I.S. group of the CNRS, fo r many fruitful discussions,and my colleagues at the IRADES (CNRS THEMA group) for the interest theyhave shown in this work. Thanks also to Jolle Maillardet and Thomas Theveninfor the diagrams, and to Anne Marie Odouze for her critical comments on the text.


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THE FRACTAL APPROACH 239REFERENCES

Appelby S. (1996), "Multifractal characterization of the distribution pattern of the humanpopulation", Geographical Analysis, (2).Batty M., Kim K.-S. (1992), "Form follows function: reformulating urban population densityfunction", Urban Studies, 29(7).Batty M., Longley P. (1994), Fractal Cities, a Geometry of Form and Function, AcademicPress.Batty M., Xie Y. (1996), "Preliminary evidence for a theory of fractal cities", Environmentand Planning, A, 28.Le Bras H. (1993), La plante au village, DATAR, ditions de l'Aube.Brunet R. (1990), "Le dchiffrement du monde", in: Gographie Universelle, volume 1, Ha-chette-RECLUS.BussiERE R., Stovall T. (1981), Systmes volutifs urbains et rgionaux l'tat d'quilibre, C.R.V.Feder J. (1988), Fractals, Plenum Press.Franois N., Frankhauser P., Pumain D. (1995), "Villes, densit et fractalit", Les Annalesde la Recherche Urbaine, 67(6).Franois N. (1997), Mthode d'application de la gomtrie f aciale la description dessystmes de villes et des rseaux de transport, Doctoral Thesis, Universit de Paris I.Frankhauser P. (1994a), "Fractales, tissus urbains et rseaux de transport", Revue d'conomiepolitique, La dynamique chaotique, (2/3).Frankhauser P. (1994b), La fractalit des structures urbaines, Anthropos.Frankhauser P. (1996a), Echelles, hirarchies et fait urbain, Universit de Franche-Comt.Frankhauser P., Gillon P., Moine A., Tannier (1996b), "Vers une nouvelle approchepour simuler l'influence du comportement des agents sur l'volution du tissu urbain",Actes du XXIIe Colloque de l'ASRDLF, Berlin.Frankhauser P. (1997), "L'analyse fractale, un nouvel outil pour l'analyse spatiale de s tissuqurbains", in J.-P. Bocquet-Appel, D. Courgeau, D. Pumain (eds), Spatial Analysis of iode graphie Data, Congresses & Colloquia n16, John Libbey/Ined.Gouyet J.-F. (1992), Physique et structures fractales, Masson.Gurin-Pace F. (1990), La dynamique d'un systme de peuplement - volution de la population des villes franaises de 1831 1982, Doctoral Thesis, Universit Paris VII.Gurin-Pace F., Mathian, H., Pumain D., Sanders L., Bura S. (1995), "SIMPOP, un modlemulti-agents pour l'volution du systme de peuplement", in Actes des premires rencontres Tho. Quant., pp . 157-1654.Makse A., Havlin S., Stanley H.E. (1995), "Modelling urban growth pattern", Nature, 37 7 .Nicholis G. (1985), "Symmetriebruche und die Perzeption von Formen", in: M. Baudson(d.), Zeit - die vierte Kategorie in der Kunst.Pumain D. (1982), La dynamique des villes, Economica.Pumain D., Sanders L. , Saint-Julien Th . (1989), Villes et autoroganisation, Economica.Reymond H. (1981), "Pour une problmatique thorique", in: H. Isnard, J.B. Racine, H. Rey-mond (eds), Problmatique de la gographie, PUFSchroeder M. (1994), Fraktle, Chaos und Selbstandigkeit, Spektrum des Wissenschaft -Akademischer VerlagVicsek T. (1989), Fractal Growth Phenomena, World Scientific.Weidlich W., Haag G. ed s (1988), Interregional Migration - Dynamic Theory and ComparativeAnalysis, Springer.White R. (1991), "A cellular automata approach to the evolution of urban land use patterns",in: 7th European Colloquium on Theoretical and Quantitative Geography, Stockholm.White R., Engelen G. (1993a), "Cellular automata and fractal urban form: a cellular modelling approach to the evolution of urban land use patterns", Environment and Planning, A, 25.White R., Engelen G. (1993b), "Cellular automata modelling of fractal urban land-use patterns: forcasting change for planning applications", 8th European Colloquium on Theoretical and Quantitative Geography, Stockholm.White R., Engelen G. (1994a), "Cellular dynamics and GIS modelling spatial complexity",Geographical Systems (1).White R., Engelen G. (1994b), "Urban systems dynamics and cellular automata: Fractalstructures between order and chaos", Chaos, Solitons and Fractals, 4(4).Zielinski K. (1980), "The modelling of urban population density: a survey", Environmentand Planning, A, 12 .


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240 P. FRANKHAUSERFrankhauser (Pierre).- The fractal approach. A new tool for the spatial analysis of urban agglomerations.

Fractal geometry is a new approach for the study of spatial distributions. The basicmodel is a law of hierarchical distribution corresponding to Pareto's law which is familiarto urban geographers and demographers.

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The Fractal Approach. a New Tool for the Spatial Analysis of Urban