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    2Size n Shape Scale nDimension

    We are here face to face with the crucial paradox of knowledge. Year by year wedevise more precise instruments with which to observe nature with more fineness.nd when we look t the observations we are discomforted to see that they areall still fuzzy nd we feel that they are as uncertain as ever. We seem to be runningafter a goal which lurches away from us to infinity every time we come withinsight of it. Bronowski 1973 p. 256.

    2.1 Scale Hierarchy and Self SimilarityWe have already seen in Chapter 1 that cities are organized hierarchicallyinto distinct neighborhoods their spatial extent depending upon the econ-omic functions which they offer to their surrounding population. This hier-archy of functions exists throughout the city with the more specializedserving larger areas of the city than those which meet more immediate ndlocal needs. The centers nd their hinterlands which form this hierarchyhave many elements in common in functional terms which are repeatedacross several spatial scales and in this sense districts of different sizesat different levels in the hierarchy have a similar structure. Moreover thehierarchy of functions exists for economic necessity nd the growth of citiesnot only occurs through the addition of units of development at the mostbasic scale bu t through increasing specialization of key centers thus raisingtheir importance in the hierarchy. These mechanisms of urban growth alsoensure that the city is stable in the sense portrayed in the previous chapterwhere hierarchical differentiation was associated with the process ofbuilding resilient systems Simon 1969 .Cities are primarily vehicles for bringing people together to engage inthe exchange of ideas nd material goods nd city size depends upon thelevel at which the city exists in the entire hierarchy of size from the smallesthamlet to the most global city. But large cities grow from the tiniest seedsnd the nature of economic production nd consumption which are relatedto each other in the market is directly based on the level of populationthe market can support and vice versa. Perhaps incredibly the w y spatialmarkets are organized across the range of spatial scales is virtually ident-

    ical. When consumers purchase goods in retail outlets the same structuresnd mechanisms are used at whatever level of the hierarchy such

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    iz and Shape Scale and Dimension 9

    transactions take place. Such structures which repeat themselves at differentlevels of the hierarchy nd which in turn are associated with different scalesnd sizes are said to be self similar Moreover, i t is this property of selfsimilarity that is writ large in the shape nd form of cities, nd provides therationale for a new geometry of cities which is to be elaborated in this book.

    To make progress in tracing the link between urb n form nd function,we must now embark upon a more structured analysis of city systems. Inthis chapter, we will outline the rudiments of this new geometry of formnd function which has been developed over the last two decades, ndwhich we anticipated in our introduction nd in the previous chapter. Thisgeometry has been christened by its greatest advocate, Benoit Mandelbrot(1983), as a geometry of nature , nd although its most graphic examplesexist in nature, it is increasingly being used to explore the w ys in whichartificial or man-made systems develop nd are organized. In this book,we will speculate on how this geometry can be applied to cities, nd inthis chapter we will present its rudiments, concentrating on natural forms,but gradually introducing nd demonstrating man-made forms which have

    s ~ m i l r properties.When we talk of geometry, we usually talk of a geometry based on thestraight line, the geometry of Euclid upon which our concept of dimensionis based. Although most natural shapes that we can imagine are clearly notcomposed of straight lines, we are able to approximate any object to thedesired degree by representing it as straight line segments. However, wecan only make formal sense of such objects if we can represent the entireform of the object in w ys in which we might pply the calculus of ewtonnd Leibnitz, nd invariably with real objects this is not p o s ~ i b l e is sometimes possible to make progress by studying gross simplifications of naturalobjects in which form is continuous nd differentiable, but we are so accustomed to assuming that our understanding of natural objects must be basedon atomistic principles that we often assume w y any pattern nd orderwhich does not fit our Euclidean-Newtonian methods of analysis. In short,our understanding of natural form nd how it relates to function has beenwoefully limited, usually lying beyond analysis.During the last 20 years, there has emerged a geometry of nature basedon the very assumption th t objects whose structure is irregular in Euclide n terms, often display patterns within this irregularity which are asordered as those in simpler objects composed of straight lines. Objects composed of a multitude of lines which are nowhere smoothm y well manifestorder in more aggregative terms than the sorts of simple objects whichare dealt with in mathematics. Such objects which show the same kind ofirregularity t m ny scales have been called fractals (from the Latin adjectivefractus meaning broken ) by their inventor nd popularizer Mandelbrot(1983, 1990) nd the geometry which has emerged in their study is calledfractal geometry . In essence, a fractal is n object whose irregularity as anon-smooth form, is repeated acrossm ny scales, nd in this sense, systemssuch as cities which manifest discrete self-similarity are ideal candidatesfor such study. A somewhat looser definition is given by Lauwerier 1991who says that A fractal is a geometrical figure that consists of n identicalmotif repeating itself on n ever-reduced scale . Cities with their manifestself-similarity of market area nd repeating orders of centers nd neighbor-

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    ractal ities

    hoods meet these criteria and form the set of fractal objects which we willexplore in this book.The best way to begin describing fractals is by example. A coastline anda mountain are examples of natural fractals, a crumpled piece of paper anexample of an artificial one. However, such irregularity which characterizesthese objects is not entirely without order and this order is to be found infractals in terms of the following three principles. First, fractals are alwaysself similar at least in some general sense. On whatever scale, and withina given range you examine a fractal, i t will always appear to have the sameshape or same degree of irregularity. The whole will always be manifestin the parts ; look at a piece of rock broken off a mountain and you cansee the mountain in the part. Look at the twigs on the branches of a treeand you can see the whole tree in these, albeit at a much reduced scale.Second, fractals can always be described in terms of a hierarchy of selfsimilar components. Fractals are ordered hierarchically across many scalesand the tree is the classic example. In fact, the tree is a literal interpretationof the term hierarchy and as such, it represents the most fundamental offractals. There are many other examples of hierarchy: as we indicated inthe last chapter, the organization and spacing of cities as central places issuch an order while the configuration of districts and neighborhoods, andspatial distribution of roads and other communications are hierarchicallystructured. The third principle relates to the irregularity of form. Here byirregularity we mean forms which are continuous but nowhere smooth,hence non-differentiable in terms of the calculus. This point is so importantthat we mus t elaborate upon it further.

    you try to describe a coastline, you will encounter the following problem. you measure its length from a map, the map will have been constructed at a scale which omits lower level detail. you actually measurethe length by walking along the beach, you will face a problem of knowingwhat scale or yardstick to use and deciding whether to measure aroundevery rock and pebble. In essence, what you will get will be a length whichis dependent on the scale you use, and as you use finer and finer scalesdown to microscopic levels even, the length of the coastline will continueto increase. We are forced to conclude that the coastline s length is infinitely long or rather, that its absolute length has no meaning and the lengthgiven is always relative to the scale of measurement.This conundrum has been known for a very long time. Richardson 1961wrote about it for coastlines and national boundaries, while the geographerAndreas Penck 1894 alluded to it (Nysteun, 1966; Perkal, 1958a). There issome evidence that Leonardo da Vinci knew about it (Stevens, 1974) and Leonardo knew about it, so probably did the Greeks. But it was not untilthe mid-1960s that the problem was raised formally and explicitly by Mandelbrot (1967). Building on Richardson s 1961 paper, Mandelbrot in an article entitled How Long is the Coast of Britain? argued that if the lengthof a straight line is absolute with Euclidean dimension I, and the area ofa plane is absolute with dimension 2, then something like a coastline whichtwists about in the plane must intuitively have a dimension between 1 and2; in short a fractional or fractal dimension. Thus Mandelbrot was arguingthat fractals were not only irregular lines like coastlines with self-similarityacross a range of scales or orders in a hierarchy, bu t were also characterized

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    iz and Shape cale and imension 6

    by fractional dimension. Mountains would thus have fractal dimensionsbetween 2 and 3, as if they were sculpted out of a solid block, more thanthe plane but less than the volume cube . Fractals, however, would not berestricted to simply the dimensionswe can visualize between the point andthe volume but could exist between any adjacent pairs of higher dimensions. Indeed, as Mandelbrot 1967 argued, such objects are more likely tobe the rule than the exception with Euclidean being a special case of fractalgeometry. Most objects would thus have fractional, not integral dimension.In this chapter, we will illustrate this geometry using idealized forms. Inother words, we will present fractal geometry in terms of objects which arewell-specified and manifest similarity across scales which we can modelexactly. This is in contrast to most of the fractals illustrated in the rest of thisbook which will not be exact in terms of their self-similarity, but manifestsimilarities across scales which are ordered only in terms of their statisticaldistribution. nthe sequel, we will begin by exploring the simplest of deterministic fractals - the Koch curve - and then we will use this to derive thebasic mathematics used to describe fractal forms, in particular, emphasizingthe meaning of fractal dimension. We will explore one-dimensional curveswhich fill two-dimensional space, hierarchies and tree structures, and wewill then outline a rather different approach to fractals based upon repeatedtransformations used to generate their form at every scale. This approachwhich is largely due to Barnsley 1988a is called Iterated unction ystemsIFS and it provides a powerful way of illustrating the critical propertiesof fractals.In this book, we will be speculating on the measurement of urban formin two ways: first in terms of boundaries around and within cities, andsecond, in terms of the way cities grow and fill the space available to them. urideas will be largely restricted to those fractals which exist betweenthe one dimension of the line and the two dimensions of the plane. urmathematics will be elementary, requiring no more than high schoolalgebra and calculus, and when we introduce some trickier developmentwe will explicitly present our algebra through all the needed steps. Anotherfeature of our approach and indeed of the development of fractals generallyis that it is easiest to work with them using computer graphics. Indeedsome say that without computer graphics, fractals would certainly not havecome alive in the last 20 years. We will thus present many computer graphics to illustrate these ideas, some of which will be in gray tones or blackand white, and others in color see color plate section .

    The eometry of the Koch urveTo construct the simplest fractals we follow Mandelbrot 1983 in startingwith a geometric object which we call an initiator To this we apply a motifwhich repeats itself at every scale calling this the generator We constructthe fractal by applying the generator to the initiator, deriving a geometricobject which can be considered to be composed of several initiators at thenext level of hierarchy or scale down. Applying the generator once again

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    Fractal Cities

    at this new scale results in a further elaboration of the object s geometry atyet a finer scale, and the process is thus continued indefinitely towards thelimit. In practice, the iteration or recursion is stopped at a level below whichfurther scaled copies of the original object are no longer visible in terms ofthe scale at which the fractal is being viewed. In essence, however, thetrue fractal only exists in the limit, and thus what we see is simply anapproximation to it.We illustrate this process in Figure 2 1 for the non-rectifiable curve introduced by Helge von Koch i n 1904 where we show t he initiator - a str aightline, and the generator which replaces the line by four copies of itselfarranged as a continuous line but scaled so that each copy is one third thelength of t he initiator. The recursion which defines the process is shownas a hierarchy, or cascade as we will term it, in Figure 2 1 The tree whichdefines this cascade is indicative of the generative process which at eachlevel replaces each part of the object by four smaller parts. As we havealready implied, the tree which defines the cascade is itself a fractal, andin the rest of this chapter we will define all our fractals in terms of initiators,generators and the cascade which forms the process of application. In somefractals, we will see their geometry in terms of the cascading tree m uc hmore clearly than in others.The Koch curve is an excellent example of a line whi ch is scaled up inlength at each iteration t hr ough replacing each st raight line acting as itsinitiator by a line four thirds the length of the initiator, ordered in fOUfcontinuous straight line segments. An even better illustration of this processfor our pur poses is given by the Koch isl and whose construction and thefirst three levels of its cascade is illustrated in Figure 2 2 I n this case, theinitiator is an equilateral triangle - the island, and the generator consistsof scaling the triangle to one which is one third the length of each of theinitiator s sides, and then gluing each smaller copy of the triangle to eachof the initiator s sides. Each side of the Koch island is t hu s a Koch curvewhich in the limit defines a fractal. It is easy to see that the Koch island iscomposed of smaller and smaller Koch islands which are identical motifsscaled successively by the same ratios. I t is i n this sense t hen t hat we speak

    Cascade

    nd level

    0 3rd level

    @ s t level

    Generator . - - -__e

    Initiator @ @N==4r n = 1/30 1.2618

    igur 2 The construction of the Koch curve.

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    Size and Shape Scale and Dimension 6

    Generator Initiator

    ascade

    2.2

    igure The Koch island

    of self-similarity . We also illustrate the island after four cascades in Plate2.1. In the Koch curve, the endlessly repeating motifs are strictly self-similarin that the scaling imposed by the generator does not stretch the object inone direction over another. In fact, objects which are stretched or distortedand scaled in the manner of fractals at successive scales are still fractals inour u se of the term, a lt ho ug h their scaling is s ai d to embrace the propertyof self-affinity rather than self-similarity. We will encounter examples ofthese later in this chapter; in practice, most real fractals in nature and inthe man-made world display self-affinity rather than strict self-similarity.

    The Koch island represents one of the best fractals with which to illustratethe various conundrums which throw into doubt our Euclidean conceptionsof space and dimension. Figure 2.1 suggests that the length of the Kochcurve like the coastline of Britain is infinitely long, whereas Figure 2.2 andPlate 2.1 suggest that the area o f th e Koch i sla nd co mp ose d of three Kochcurves is in fact b ou nd ed . We ca n illustrate these intuitions formally asfollows. Let the length of the initial Koch curve which is each side of theoriginal Koch island be defined as As Figure 2.1 implies, each side of thegenerator has length f /3 and consists of four copies of the initiator. Thenwe can define the increasing length of the Koch curve as follows. The lengthof t he original line is

    Lo =f 2.1Applying the generator to the initiator results in a line L which is 4/3 thelength of o

    4 4L1 = Lo = fand subsequent recursion gives

    2.3)

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    Fractal Cities

    2.4t is clear that as k 00 then

    lim Lk= i kr 00 2.5k oo To all intents and purposes, the line is of infinite length like the coastlineof Britain in Mandelbrot s early writing on fractals. In fact, the Koch curveis a somewhat serendipitous choice for as Mandelbrot 1967 shows, it ha sa fractal dimension close to the coastline of Britain, thus providing a graphicexample of this conundrum concerning length.

    we now examine the Koch island in Figure 2.2, then it is clear that theperim eter of the island is three times the length of a single Koch curve andthus we must replace the length r with in equations 2.1 to 2.5 above.We will now examine the area of this island and show t ha t despite itsp er imete r be in g o f infinite length, its area converges to a finite value. Toshow this, first let us define the area of the initiating equilateral triangle asAo and with each s id e gi ven as r, the area is

    rZ 2.64 .On the first iteration, three equilateral triangles are added to each of thethree sides of the initiator and the area of each defined as At is

    A = r)Zt 4 3and the area of the three triangles is given as

    1=3 4 rZClearly < Ao and t hu s t he sum of areas so far given as At

    A, =Ao A = r

    2.7)

    2.8)

    2.9)A t each stage of th e recursion 3 x 4k t triangles are added and the cumulat-ive total area to k is th usk= Ao 4Az ... 4kt Ak.

    Ak is defi ned as1 1 2 Ak 4 0,

    and this simplifies to

    2.10)

    2.11

    2.12)

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    Size and Shape Scale and Dimension

    The cumulative total area on iteration k can now be written as

    Ak = r2 r2 {I ... ~ }Equation 2.13 can be writ ten more compactly as

    l 4 jA = 2 r ~ k 4 12 L.J 9 .j

    2.13

    2.14The summation term in equation 2.14 is a converging geometric serieswhich can easily be shown to sum to 9/5 as k - 00 and thus the limit ofequation 2.10 is given as

    _ 9A =lim Ak = - r2 - r 2 k-+oo 4 12 58=5 r2 = Ao 2.15

    Equivalent summations are given in Woronow 1981 and by Peitgen, Jurgens and Saupe 1992 .

    3 Length rea and Fractal DimensionThe mathematical argument presented above in equations 2.1 to 2.15 isa formal statement of the coastline conundrum Although the length of thecurve which bounds the Koch island increases without bound as the scaleis reduced - becomes finer, the area of the island converges to a finite valuewhich is 0 693r2 the length of the curve converged, then its dimensionwould clearly be 1 and the area it enclosed 2 but in the case of the Kochcurve, it is apparent that length measured in the conventional Euclideansense is unbounded In an attempt to unravel the paradox of infinite lengthand finite area, let us restate the generation of the Koch curve in the following way. We will first repeat equation 2.4 as

    2.4Now this length in 2.4 is made up of the number of copies of the initiatorused in generating the curve which we call N k and the scaling ratio applied to the original l ine length r which gives the length of each line inthe Nk copies. Therefore length Lk is defined as

    2.16from which it is clear that Nk =4k and = r 3kIn Table 2.1, we show the increase in Nk in comparison to as well asthe length Lk for both the Koch curve in equation 2.16 and the straightline, where Nk is divided into the same number of parts as the inverse of

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    the individual length of each part r,;l. t is clear that for the Koch curve,the number of copies k increases at a much faster rate than the inverser,;l. This can be seen by examining krk which for the straight line is aconstant, whereas for the Koch curve is increasing. we were to predictthe number of copiesk generated from the number of parts r,;l the initiatoris divided into, then it is clear that r,;l would need to be raised to a powerD greater than unity. That is

    2.17For a straight line, D = while for the Koch curve the similarity factor rkmust scale as a power D > 1. owi we substitute k in equation 2.17into equation 2.16 , the equation for the length of the line becomesL N r r l -D . 2.18the parameter 0 is equal to then equation 2.18 gives a constant unitlength, while if D > 1, k is unbounded. Moreover, it is intuitively obviousthat 0 plays the role of dimension in these equations and as such, we have

    now demonstrated that the Koch curve has a dimension which is greaterthan 1, hence must be fractional, not integral in value.In strictly self-similar curves, the dimension D can be calculated exactly,for the recursion which generates the curve is itself identical at each levelor scale. Then taking the log of equation 2.17 , the dimension D at anylevel k called k is given aso - log k _ log k 219k - - log rk logl rkr .

    In the case of the Koch curve where we assume without loss of generalitythat r = 1, then Nk=4k and l/rk =3k, and equation 2.19 reduces tolog 4 o =k = log 3 = 1.262,

    which is the value of the fractal dimension. This bears out our intuitionthat the Koch curve has a dimension nearer 1 than 2 in that the curvedeparts from the straight line significantly but does not. fill very much ofthe two-dimensional space.

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    Size and Shape Scale and Dimension

    There are many di ff eren t t yp es of fractal d im en si on Falconer, 1990;Takayasu, 1990). The one we ha ve d er iv ed is often called t he similarityd im en si on w hi ch is o nl y d ef in ed for strictly self-similar objects. I n thiscontext, we will u se t he n ot io n o f t he fractal d im en si on in its generic sensefor we wi ll defi ne suc h d im en si on s in a va ri et y of contexts, and as ourpurpose is with appl icat ions, it will suffice t o t hi nk of suc h d im en si on s asa measure of the extent to which space is filled. In fact, our concern herewill be almost exclusively with fractal dimensions which vary b et we en 1and 2. Our focus is on the spatial structure of cities that exist in t he p la neand although there are many studies of urban structure which stress theirthree-dimensional form, we will be mainly dealing with cities as the y areexpressed through two-dimensional maps. However, the fractal dimensionswe will develop will depend upon the particular aspect of urban form weare measuring; t he b ou nd ar ie s of cities, for example, will have differentd im en si on s fr om th e d en si ty of d ev el op me nt , w hi le t he a ct ua l v al ue o f t hedimensions computed will inevitably depend upon the me thods us ed intheir measurement and calculation.From equation 2.17), it is clear that th e number of copies of the initiatorgenerated at any iteration k N k varies inversely with r and their productwill b e c onst ant, t ha t is

    2.20)As illustrated in Table 2.1, thi s r el at io n h ol ds for t he str ai gh t l ine when D= 1. Where the object in q ue st io n fills t he p la ne as in the case of a square,then the number of copies generated varies with t he sq uar e of t he scalingfactor 1/ as 11/ where D 2. Clearly for a fractal object with dimensionbetween 1 and 2 then equation 2.20) is only satisfied when the value ofD ensures it is constant.However, consider the case where k varies as r where 8 is not equalto D. Then we c an w ri te e qu at io n 2.20) as

    2.21)

    2.22)

    f 8 is less than D, the actual fractal dimension, then N will diverge towardsinfinity. This would be the case where we assumed that the object w er e astraight line, but in fact i t is a Koch curve. On the other hand if we assumedthat 8 w er e greater than D, then equation 2.21) would converge towardszero. Thus the fractal di men si on D is th e on ly v alue which would ensurethat e qu at io n 2.20) is satisfied. For mal ly this c an b e w ri tt en a s

    [

    -- if 8 < D,lim Nkrf = rf-8 = 1 if 8 = D,k-- oo

    -- 0 if 8 > D.t is possible to vi su ali ze t he v al ue o f 0 co nv er gi ng toward D from aboveor below and only when it is exactly e qu al to D will e qu at io ns 2.20) and2.22) b e e qu al to 1 Feder, 1988).W e h av e a lr ea dy i nt ro du ce d t he n ot io n that fractals exist which are self-

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    Fractal Cities

    affine in th t their scaling ratios {r m y differ. Such a fractal would eself-similar in that there would be copies of the object t every scale utthat these copies would e distorted in terms of the original initiator insome way. There is a straightforward generalization of equation 2.20 forthis case. We now have m scaling factors which we call r nd i f we associate a distinct scaling factor with each copy of the object, then the fractaldimension must satisfy

    L = 1.j

    2.23In this case, we can find D y solving the equation for ny level k whichis based on the fact that the fractal never changes its scaling factors overthe range of levels nd scales for which the object is observed Barnsley,1988a; Feder, 1988 .We can examine this best y example. In Figure 2.3 nd in Plate 2.2, theKoch curve h s been regularly distorted in that the two base pieces of thegenerator have quite different scaling nd the two perturbed pieces whichare equal in length form a spike rather th n a pyr mid to the curve. Thisgenerates wh t Mandelbrot 1983 calls a Koch forest. We can easily compute the dimension from this figure, given the scaling factors. Then in thecase where =4 for the Koch forest, equation 2.23 simplifies to

    0.30D 0.42D 0.42D O 6 = 1,nd the dimension D which solves this equation is 1.750, considerablylarger, as expected, than the regular Koch curve whose dimension is 1.262.

    4 The asic Mathematical Relations of ractalGeometry

    So far we have assumed that we are measuring the geometric propertiesof a single object nd we have shown how we might do this for strictlyniti tor

    Generator

    Cascade

    igure 3 The self affine Koch forest.

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    Size and Shape Scale and Dimension 9

    self-similar fractals. In essence, we compute the fractal dimension by examining the object at different scales, taking the ratio of the number of itsrepetitions to its scaling factors. There are, however, other ways of lookingat the form of such objects and we will use two such ways here. First, wemay have a set of objects which we know are of the same type and whichare all measured at the same scale. In such a case, we can develop methodsfor computing the fractal dimension of the set by examining changes inform at the fixed scale of measurement. Second, we may have an objectwhich is constructed or measured at the same scale but whose size changesin some regular way which we might associate with growth or decline. Inshort, its mass or the number of its parts increases or decreases as the objectgrows or dies. In one sense these differences in measurement are stronglyrelated to the fact that the object s in question changes its size or scale,and i t is such changes that are essential in computing its fractal dimension.We will begin by treating single objects for which we are able to controlthe scale of measurement as in the cases already introduced for the Kochcurve. We will now use the variable r to measure continuous scale and wewill drop the explicit reference to iterations of fractal construction. Thengeneralizing equations 2.17 and 2.18 , we get

    N r =and

    L r N r r Kr 1-D .

    2.24

    2.25N r and L r are the number of parts and the length of the object respectively where we are implying that we are dealing with fractal lines, and Kis a constant of proportionality. we have a series of observations of N rand L r at different scales then we can derive the fractal dimension bytaking log transforms of each of these equations and performing aregression of these variables in cases where the variation is stochastic, hencestatistical not deterministic. Because we will be mainly concerned withregressing length on scale, then the log transform of equation 2.25 yields

    log L r log K I-D log r, 2.26

    2.27

    where it is clear that the slope of the regression line is I-D from whichthe dimension can be derived directly. We will say much more about thisin Chapters 5 and 6 where we will deal with statistical variations, but notehere that if we apply equations 2.24 and 2.25 to, say, the Koch curve,then these equations collapse back to those from which they have been generalized.To derive the dimension from a set of objects all of different size bu tmeasured at the same scale r, we now need to define the scale moreexplicitly and for this we assume that the scaling ratio r is applieddirectly to a size R which is the size of the object in question. Equation2.25 can be written as

    L r =K ~ =KRD r

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    Fractal Cities

    varies. Incorporating the various constants into G, dropping the scale randsubscripting the variables L and Rj where i indicates a particular object,equation 2.27 can be written as2.28

    The size Rj is clearly related to the area of the object A j , and if we considerRj to be equal to the square root of A j , thenRj A ~ / 2

    Substituting for Rj in equation 2.28), we obtain a relationship betweenlength and area, the so-called perimeter-area relation,

    A log transform of equation 2.29 givesDlog L =log G log Ai

    2.29

    2.30from which it is clear t ha t the slope of any regression line estimated usingequation 2.30 is D 2 This equation has been widely used to me asu refractal dimensions in sets of physical objects such as clouds Lovejoy, 1982),moon craters Woronow, 1981 and islands Goodchild, 1980).The third and last m ethod of m easuring dimension is based on a singleobject where the scale change is implied by the object increasing or decreasing in size. Here we will be concerned with the mass of t he object whichwe consider is measured by the number of parts of the object at scale N r . Using R again as the size of the object, equation 2.24 can be w ritten as

    N r K ~ r 2.31Then at a fixed scale t he mass or number of p ar ts of th e object scaleswith R as

    2.32Note that we now use R instead of r for the ind ex of scale change w hi chis based on the change in the size R of the object. f we have the area ofthe object A R , then we can normalize equation 2.32 to obtain a density relation

    N R RDp R) = =z - RD A r TrR2 2.33We can obtain the fractal dimension by taking a log transform of equations2.32 or 2.33 and we will use these equations extensively when we discussurban growth models in Ch apt er s 7 to 10. No te also t ha t in the sequel wewill refer to the measure of mass N R as a mea su re of p op ul at io n size.In this section we have examined three ways of computing fractal dimension and in Table 2.2 we summarize these methods in terms of their emphasis on size and scale. In fact we do n ot h av e a ny me th od s for estimatingdimension where there are several objects which vary across several scales.

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    Size and Shape, Scale and Dimension

    ble Equations associated with measuring fractal dimensionNumber o objectsOne objectMore than one

    Varying scale2.24 , 2.25

    Varying size2.32 , 2.332.28 , 2.29

    In this case, some selection from the three methods introduced here wouldbe necessary. For example, it would be possible to treat every object separately nd to estimate a set of fractal dimensions using equations 2.24 nd2.25 . a single fractal dimension were required, then all the scale changesfor all the objects could be combined into one set nd a single regressioncarried out. However, in the case of a set of objects whe re their scale canbe varied, t he n it is likely t ha t t he empha sis would be on estimating a setof dimensions nd making comparisons between objects. We will developthese ideas further in C hapter 6 when we deal with the fractal dimensionsof different land uses in a town.

    5 ore dealizedGeometries Space FillingCurves and Fractal DustsThe last section w as s ome th in g o f a digression in th t the equations wepresented, although derived from the geometry of the Koch curve nd ofextremely general import throughout this book, will be used mainly for themeasurement of fractals using statistical methods. ow we will return toour discussion of methods applicable to exact fractals nd provide somemo re examples to impre ss the ide a t hat a large number of objects can existwith fractal dimensions between 0 nd 2. Per ha ps t he b es t examples arethose continuous lines which have a Euclidean or topological dimension of1 bu t a fractal dimension of 2 nd are called space-filling curves, for reasonsw hich will become obvious. W e will begin with the curve first introducedby Peano in 1890 Mandelbrot, 1983 nd whose construction is shown inFigure 2.4.This curve is formed by applying a generator which spans a squarewh ose initiator is a diag onal line across the square. I n Figure 2.4, t eachlevel of recursion, the generator replaces the straight line by nine copies ofitself, each scaled to one-third the length of the line. To present the curve,we cut off each co me r of the right-angled twists in the line to show thatthe line is continuous. Four generations of the line are shown starting withthe originator t = the recursion is continued beyond = 4 the curvefalls below the resolution of the computer screen on which it was generatednd for all intents nd purposes t the scale of the picture, the curve literallyfills the space. However, because it is a fractal, the line hasno width, ndas we continue to z oo m into the picture, the curve continues to generatee ver mor e detail on its p th towa rds infinity. In short, it never fills the

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    Fractal Cities

    Initiator

    a

    Generator

    b

    ascade

    c

    aigure 4 Peano s space-filling curve.

    space because we can never reach the limit, thus illustrating the paradoxwhich Bronowski 1973 referred to in the quote which introduces this chapter. In generating the curve at any level it is clear that the number ofparts k are k and the similarity ratio is 1/3 k. From equation 2.19 , thefractal dimension is computed as D = log 9 / log 3 , and our intuition thatthis curve fills all the two-dimensional space available to it is rewarded inthat the fractal dimension is indeed 2.There are several other curves which we can generate which have a fractal dimension equal to 2. we take a straight line as init iator and generatetwo parts to the line forming a right angled triangle resting on the initiatorwhich in turn is the hypotenuse of this triangle shown in Figure 2.5 , then

    Initiator

    Generator

    ascade

    nigure 5 The e curve.

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    Size and Shape Scale and Dimension

    Initiator enerator l

    ascade Aigur 6 The Dragon curve.we generate a closed curve which is called the c curve. Assuming thatthe length of each of the two ne w lines is I, then the original initiator hasa length V2 an d the similarity ratio is 1/ V2 Using these values in equation2.19 also gives this curve a fractal dimension of D = log 2)/ log J2 =2For the C curve, we generate ne w lines always outwards from the initiator,whereas i f we replace the two lines which are generated with one outwardsan d one inwards as s ho wn in Figure 2.6, the construction which we generate is called the dragon curve which also ha s a fractal dimension of 2 Tofurther impress this not ion of space-filling, examine the double dragoncurve in Plate 2 3 which suggests that infinite space can be perfectly tiledan d entirely filled with such constructions.Figures 2.4 to 2.6 il lustrate that the actual shape of an object does no tnecessarily influence the value of its fractal dimension which is, in a sense,obvious in that there is nothing that we have introduced so far which relatesdimension to geometric shape. In fact, we can change the fractal dimensionby virtually keeping the same shape. Figure 2.7 illustrates ho w w e mig ht

    Initiator

    enerator

    ascade

    \JLAigur 7 The Peano-Cesaro triangle sweep.

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    Fractal Cities

    sweep ou t a d esig n in the p lan e u sin g a generator similar to those u sed i nthe C an d dragon curves. In this case, the generator is the same rightangled triangle as u se d in th e dragon curve but with both triangles turnedinwards at each level of recursion. To actually show the curve, we have se tthe similarity ratio at 0.673 which is a little less than 1/ ./2 ( = 0.707) an dthis leads to a dimension of 1.750. The design we have gener at ed i n Figure2.7 is called by Mandelbrot (1983) the Peano-Cesaro triangle sweep an d itsself-similarity is sometimes reminiscent of a fern, although as we shall seelater, much more realistic designs can be generated as the stricturesimposed by exact self-similarity are relaxed.

    Another design we must introduce is one which can be interpreted assculpting out scaled versions of th e object in the two-dimensional plane,t hu s sh ow in g th at fractals ca n be generated by taking a wa y r at he r thanadding to th e initiator. In Figure 2.8, we begin with an initiator which is asolid equilateral triangle, and take ou t a scaled copy of the original, positioned centrally in the object at each level of recursion. Another way oflooking at th e generation process which we will invoke la ter is to see th escaling as taking three copies of the triangle, an d scaling these in such away as to tile th e original triangle. A final way of seeing this which isalso s ho wn i n Figure 2.8 is as a continuous curve w hi ch s pa ns t he triangleand it is this which we can us e to calculate th e fractal dimension. In essence,th e scaling is 1/2 an d the number of pieces of scaled line or triangle generated at each generat ion is 3. The fractal dimension of th e resulting latticelike structure, called the Sierpinski gasket, after its originator, is D = log 3)/log 2 = 1.585. We will return to this construction w hen we introduceBarnsley s (1988a) IFS approach below.There is one last construction we must mention before we change tackan d examine branching structures, an d this is a fractal whose dimensionlies between 0 an d 1. We refer to such fractals as dusts an d the best oneto illustrate this is named after th e mathematician George Cantor, t he Canto r Set . This is based on a straight line initiator which is replaced at eachiteration by tw o copies of itself, but these copies are scaled by 1/3 of th e

    Initiator

    enerator

    Cascade

    igure 8 Sierpinski s gasket: sculpting and tiling space

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    Koch Island

    late 3 Twin Dragon Curves

    late 5 Tiling the Landscape with Trees

    late Koch Forest

    late 4 BinaryTree in Full Foliage

    late 6 Savannah a Minimalist Fractal Landscape

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    Plate left SimpleRendition of Mandelbrot s lanetrise

    3 2lpine Scene

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    he Hierarchy of the lanetrisefour levels of construction are shown with the two tricks to render planetrise the circular sculpting n the light source

    ding illustrated in the last two subplates.

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    l te 4 The Hierarchy of the Fractal ityBlue is commercial industrial land use red is residential ngreen is open space recreational.

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    Plate 4 Sampling o the Space o ll ities

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    l te 4 Realistic ypothetical Urban imulations

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    Plate 4 right The Graphical Data aseg of Housing in Londonverage age of housing: 8 years white , 26 years

    light blue , 48 years magenta , 78 years darkblue , 110 years yellow , 150 years green , and175 years red .

    late 4 4 above Deterministic SimulationsHouse Type in Londonverted flats red , purpose-built flats yellow ,

    housing green , and detached semihousing blue .

    ate 5 right Urban Growth of Cardiff s

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    5 Structured Walks long the Urban Boundary

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    5 Cell Counts the Space Along the Urban Boundary

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    6 Land s in Swindon

    7 Fractal London Employment Densities

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    l te 7 Urban Employment ensity in England and Wales

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    Plate 7 Dendritic Growth from the Diffusion-Limited ggregationDLA Modelb sed on a 500 X 500 lattice with 000 particles

    Plate 8 Dendritic Growth from the Dielectric Breakdown Model DBM .b sed on a 150 X 150 lattice with 1856 particles

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    l te 8 hysically onstrained M Simulations

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    l te 8 left TheBaseline Simulation

    l te S below TheUrban Area Cardiff

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    l te 8 Urban Forms enerated by Systematic istortions to the M Field

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    l te 8 Simulating t Urban rowth ardiff

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    Size and Shape, Scale and Dimension

    previous line. This construction is shown in Figure 2.9 where it is clear thatthe object is being systematically reduced from the one-dimensional linewhich is its initiator y removing the middle third of each line. Using equation 2.19), the dimension is D = log 2)/log 3 =0.631. In one sense, wec an see oth the 5ierpinski gasket nd the Cantor dust as objects whichbegin with two- nd one-dimensional shapes respectively nd graduallyreduce the dimension of the shape as pieces of it are removed. In this sense,then we m ig ht th in k of oth of these as dusts .Finally in this section we will anticipate later chapters of this book ygeneralizing these results. In Figure 2.10, we show the sorts of objects whichexist across a co nti nuu m of dimensions from points to lines to planes tovolumes, in Euclidean terms from zero to three dimensions. We also showthree typical fractals which exist between zero nd one, one nd two, ndtw o nd three dimensions, these being dusts, trees nd surfaces respectively. In fact as we have already implied, we will mainly concentrate uponobjects with a fractal dimension between 1 nd 2 i n this b ook because ourpredominant w y of representing cities will e through maps. So far in ourdiscussion of fractals we have only dealt with lines nd points of the simplest kind, nd insofar as we have dealt with trees or de nd ri te s it h as b eenthrough ideas a bo ut hierarchy, n ot with any more substantial representation of reality. In the next section we will remedy this nd the n be in aposition to introduce a som ew hat different approach to fractals whichenables us to round off the elementary insights we are attempting to presentin this chapter.

    6 r s and ierarchiesWe have already noted that the tree or cascade structure used to show howthe generator relates to the initiator in deterministic fractals is itself a fractal,

    Cascade

    Initiator 0 0N I II Ir n) = 1/3 I II ID 0 6309 I II II II II II II II II II II IGenerator 0 0 0 0

    I II II II II II I+ +

    igure 9 The Cantor curve: fractals as dusts .

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    Fractal Cities

    C\J

    Qc:0 iic:JEC

    vic:0.... .iiic:QE :0 -0c: 0 n0 .....0 E ::::::c: 0uQ 0 ce.C lI4f:::

    u

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    ize and Shape Scale and Dimension

    for the tree shows how m ny self-similar copies of the object are generatedbut not their scaling. In fact we will deal with a much reduced set of treestructures here nd restrict our attention to structures which branch intotwo copies of the object t each level. In fact we know that this yields onlya subset of all possible trees but it is sufficient for our purposes which issimply to establish the background. There h s already been substantialwork on the morphology of trees nd we will not attempt to summarizethis work here. Readers who wish to follow up these ideas are referred toMacDonald 1983 in the biological literature to Aono nd Kunii 1984 incomputer graphics nd to Prusinkiewicz nd Lindenmayer 1990 for atreatment in terms of fractals.Firstwe will state some of the obvious relationships governing branchingstructures of the binary or dichotomous kind. The number of branches ofny tree which are generated t a given level of recursion or hierarchy k isgiven as

    2.34where {} is the bifurcation ratio equal to 2 for binary branching 3 for ternarynd so on. The number of branches t ny level of the hierarchy is the sumof these numbers over k defined as

    2.35Equations 2.34 nd 2.35 can be used to compute the number of elementary operations in any recursive scheme t nd down to any level k Withrespect to botanical trees several relationships have been establishedbetween branch lengths nd widths angles their scaling or contractionnd their symmetry but we will only state one which was first articulatedby Leonardo d Vinci in 16th century Italy. This is based on relating thewidth of ny stem in a tree to the two stems which branch from thisbetween levels 1 nd k Then

    2.36where W is the width t the relevant level is a parameter of the relationnd nd L indicate the right nd left branches respectively. LeonardoStevens 1974 suggested that the parameter in equation 2.36 be equalto 2 nd in this case the tree could be called Pythagorean in th t the widthof the branch stem k would be the hypotenuse of a right angled trianglewhose two sides are R nd LWk Examples of such trees are given in thebook by Lauwerier 1991 . McMahon 1975 suggests th t the width of nybranch Wk should be proportional to its length as

    2.37In fact strict Pythagorean trees have the length of their branches identicalto their width although this leads to somewhat squ t looking trees. AsMcMahon 1975 nd others imply the parameter c;, which makes therelation in equation 2.37 realistic is unlikely to be as low as 2 but nevermore th n 3. However the biggest single factor affecting the shape of treesconcerns their branching angles but these do not affect our computation of

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    Fractal Cities

    fractal dimension which only depends upon the rate at which the branchescontract or scale nd the num er of branches which are associated witheach stem or trunk.Computing the fractal dimension of trees illustrates the existence of several dimensions which depend upon the particular aspect of the tree s formwhich is measured. Examining the branch tips of a bifurcating binary treesuggests that the canopy of the tree which contains the branch tips is akind of dust. the width of the t wo branches of the s te m are less th n thewidth of the stem itself, then the canopy formed is a Cantor set with dimension between a nd 1. However, this dimension takes no account of t helength of the branches. A more obvious dimension is based on the fact thatt he initiator is a stem nd the t wo branches are scaled copies of the stemas in the C nd dr gon curves shown earlier in Figures 2.5 nd 2.6. Oftenthe bra nch angles are chosen so th t the tree is self-avoiding in that thebranch tips do not touch or t least just touch ut do not overlap. In Figure2.11, we show a tree in w hi ch the b ran ch angles a re chosen so th t the tipsof the branches just touch one another nd the contraction or scaling ratiofor each b ra nc h is 0.6. The fractal dimension using equation 2.19 is computed as -log 2) /log 0.6) = 1.357.In contrast in Figure 2.12, we show two more realistic looking trees. Thefirst is symmetric ut with the branches overlapping with a contractionratio of 0.8, hence a fractal dimension of 3.106 which implies that the overlap more th n covers three dimensions, while the second tree is asymmetricwith contraction factors of 0.8 nd 0.7 for the left nd right branches respectively; hence using equation 2.23), the dimension is 2.435 covering moreth n the plane. This tree has been computed to a depth of 10 branches, thusillustrating how the branches contract to the canopy in contrast to the othertree in this figure which is only plotted to a depth of five branches. Noteth t in these cases where the branches are not self-avoiding, our equationsfor fractal dimension gives values greater than 2, thus indicating that our

    igur 2 A self avoiding symmetric tree

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    ize and hape Scale and Dimension 79

    igur 2 2 Fractal trees with overlapping branches.pi ctures of these trees ar e not en tir ely a de qu at e i n v is ua li zi ng th ei r geometric form. The full tree is also pictured in Plate 2.4, while its use in tilingspace to form landscapes is illustrated in Plates 2.5 nd 2.6.The est example of a tree which fills the pl ne is provided y the Htree which is shown in F ig ur e 2.13 a). This tre e is symm etri c, it is selfavoiding in th t its r nch angles are chosen to e 9 nd the rate t whichoth its branches contract is 0.707. This gives a fractal dimension of 2 whichbears out our intuition. A slight variation of this contraction ratio down to0.7 nd a slight decrease in the r nch angles from 9 to 8 p ro du ce s aslightly more realistic structure with a dimension of 1.943, ut this remainsstrictly self-similar. This is also shown a s Fi gu re 2.13 b). These forms a re

    a

    igur 2 3 trees as plan forms.

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    Fractal Cities

    classic space-filling curves. They are reminiscent of traffic systems in residential areas of towns. In fact, one of the features of these binary treeswhich is brought out by this analogy is that it is possible to visit everybranch of the tree without crossing any other branch. As we noted in thefirst chapter, this form of layout plan was suggested by Clarence Stein asbeing an ideal layout for a residential housing area in that its residentscould walk around the layout without crossing any of the roads. This wasadopted quite widely as a model for pedestrian segregation from vehiculartraffic and it was widely implemented in the design of residential areas inthe British New Towns, illustrated earlier in Figure 1.19. Another featureof such trees is that they are a minimal form of strongly connected graphwhere every branch is connected directly or indirectly to every other, butwhich will break into two parts if any branch in the structure is severed.This model has also been used to show how trees can grow in a constrained space, and the example which best illustrates this is the humanlung. Analogies between trees and human lungs as well as rivers, citiesand electric breakdown were made almost thirty years ago by Woldenberg1968 and Woldenberg and Berry 1967 . More recently the analogies havebeen derived and extended to make the tree model more realistic. Figures2.14 a and b show how bifurcating trees can fill a circular space and beself- avoiding with suitably chosen contraction ratios Nelson and Manchester, 1988 . These trees like the H trees, are reminiscent of views of the treefrom directly above, plans of the tree rather than end or side views. Inanother context, they could be seen as cross sections of the growth of thetree above and below ground showing its roots as well as its foliage in themanner illustrated by Doxiadis 1968 . fact, Nelson and Manchester1988 use this type of spatial constraint as a model of the growth of thehuman lung although these models go back to the work of Woldenberg in

    = 94

    a

    = 87

    b

    igur 2 4 Self-avoiding trees with geometric constraints on growth after Nelson andManchester, 1988 .

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    Size and Shape Scale and imension

    the late 1960s. Figure 2.15 a , we show a plastic cast of the lung m deby Keith Horsfield see also est nd Goldberger 1987 nd in 15 b , thegrowth of the lung as a tree structure b sed initially on n ellipse whichexp nds into two with increasing fractal dimension. Finally in Figure

    a

    8

    Of 96

    b 94

    Of 99

    c

    Figure 2 5 Growth of the human lung: a plaster cast by Horsfield ;b idealized growth models; c restricted tree growth both after Nelsonand Manchester, 1988 .

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    82 Fractal Cities

    2.15 c), we show a stylized representation of a severely restricted lung takenfrom the models proposed by Nelson and Manchester 1988 .Our last variants of tree fractals indicate what happens to such structureswhich contract to a p oin t or expand to infinity. If there is no contractionwhatsoever in the branches, then the dimension of the binary tree becomes

    log 2)/log 1/1) which is infinite. In Figure 2.16, we show what happenswhen there is no contraction in a tree structure which has branching anglesof 60. We generate an ever-expanding tessellation of the plane based onregular hexagons, strongly reminiscent of Christaller s 1933, 1966) economic landscapes of central places which we illustrated in Figure 1.23. If weincrease the branching angles to 90 t he n the p la ne is tiled, as with squares Figure 1.22). These forms are highly suggestiveand have important implications for the hierarchy and form generated by central place theory. Wecannot, however, pursue these further here, and t he interested r ea der isreferred to the work of Arlinghaus 1985 .To generate fractals with zero dimension, we set one of the branch anglesto zero. This mean s t ha t the s te m on ly ever generates a single b ra nc h andwhatever the contraction ratio, the dimension is equal to that of a point,zero. We show this in Figure 2.17 where the contraction ratios for the leftand right branches are 0 and 0.8 giving a dimension of log l /log 1/0.8)= I n fact this generates y et a no th er fractal - a spiral; r ead er s who wishto explore the meaning of these forms in greater depth should look at booksby Mandelbrot 1983 and Lauwerier 1991 which both contain many otherexamples. We will return to tree s ha pe s a gain in the last half of the bookwhere we show how such fractals can be grown using geometrically constrained or limited diffusion. But before we conclude our discussion ofdeterministic fractals, we need to introduce one last approach which givesus greater insights into methods for generating fractals, and in particular,shows us how to generate considerably more realistic shapes.

    igur 2 6 Non eontracting trees: infinite tiling t plane

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    Size and Shape Scale and Dimension

    Figure 2 7 Contraction a tree into a spiral2 7 Fractal ttractors Generation Transformation

    So far mos t of the fractals we have generated are strictly self-similar ndnon-overlapping, although the trees of the last section were based on slightrelaxations of these assumptions. Moreover, the w y we constructed thesefractals w s y emphasizing recursion of the generator through m ny levelsof the hierarchy or cascade. There is, however, another w y of generatingfractals which in one sense is little different from the methods we haveu se d so far, ut in another sense exploits the geometric rather th n struct ur al pro pert ies of t he object t hr ou gh its e mp ha sis on the nature of thetransformations involved. This method involves treating fractals as a process of transforming nd contracting a large object into a smaller one, progressively moving towards the ultimate geometric form of the fractal whichis now referred to as a fractal attractor . From wh t we have said so farin this chapter, we have assumed the existence of fractals without dwellingon the ultimate form of the limiting process which we have taken on trust.The great value of the a ppro ach which is based on specifying the n tureof transformations is that there are proofs that the limiting forms for a largeset of fractals exist nd are unique. The mathematical proofs have eendeveloped y Hutchinson 1981 nd Barnsley nd Demko 1985 amongstothers, ut the practical application of this fairly esoteric approach is duealmost entirely to Barnsley 1988a, b).As we have indicated, the essence of the approach is to specify the correctset of geometric transformations which enable the object in question to etiled or formed from copies of the object at successively finer scales. Suchtransformations do in fact exist in the computer programs used to generatethe fractals which we have presented so far in this chapter, although thesetransformations have not be en the par ticu lar subject of our interest. Thebasic idea of Barnsley s 1988a) approach is to approximate the final formof the object which we as sume is t he given s ha pe y a series of transformations which, when applied to ny point on the object, will generate another

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    point on the object but t t he s ame time con tract the first p oi nt into t hesecond. By applying these transformations successively to the point generated so far, the process will ultimately lead to the shape of the attractorbeing filled in . The restrictions on permissible transformations imply thatthey be contractive nd th t t hey r epr es en t th e b es t possible geometricapproximation to the object in question. the transformations are badly chosen, then the object s shape will notbe generated nd the resulting fractal will not be the one that is observedin reality. However, we can first il lustrate this approach for strictly selfsimilar fractals because we can intuitively guess their correct transformations. In fact, we have been doing this throughout this chapter in the computer programs which have been used in their generation. Another featureof the approach is that we must specify the right number of transformations. some are left out, the ultimate object will in some w y be incomplete. The transformations do not have to generate strictly self-similarobjects, they can be self-affine nd they m y tile the objectwith overlappingcopies. In fact the success of the a pp ro ac h is due precisely to this. As wewill restrict most of fractals in this book to those with a dimension between1 nd 2, we can illustrate the typical transformation for any pair of coordinates x nd y in 2-space in matrix terms as

    2.38

    where x nd are the transformed points x nd y based on all three typesof transformation - scaling, rotation nd translation, where a, c nd are the coefficients specifying the scaling nd rotation, nd e nd f are th etranslations associated with x nd y respectively Barnsley, 1988a, b;Barnsley nd Sloan, 1988).The best w y to demonstrate the method is by example. We will showhow the Sierpinski gasket discussed earlier in Section 2.5 nd illustrated inFigure 2.8 can be generated by suitable transformations. From Figure 2.18,we see three transformations of the big into the little equilateral triangles,nd it is clear that successive application of these transformations will yieldthe Sierpinski gasket shown in Figure 2.8. Moreover, it is easy to specifythese transformation because all that is involved are scalings nd translations. ow it is also clear that if we merge the three transformationsshown in Figure 2.18, we obtain the fractal t t he second level of the cascade. Defining the object t level k as k1 the object at the second level isgiven as

    F = ool Fo) U ooiFoU oo3 Fo2.39

    where 11 2 3 are the three transformations nd 0 a combined transformation operator. we use equation 2.39 recursively then we obtain t thenext levelF2 = O F1) = O[O Fo)],

    nd in general for Fk 2.40)

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    Size and Shape Scale and Dimension 8

    igure 2 8 Transforming Sierpinski s gasket2.41

    the transforms are those in Figure 2.18 w hi ch are b as ed on strict selfsimilarity, then the object is generated in the limit as Foo = limk k ndthe Sierpinski gasket is defined as in Figures 2.8 nd 2.18. Barnsley 1988a)summarizes this process of putting the transformations together in the Collage Theorem w hich ensures th t the fractal attractor is generated in thelimit.So far there is little that is radically different from the recursions used inthe generation of the previous fractals in this chapter. In the case of theSierpinski gasket in Figure 2.8 for example, the computer progr m used togenerate this involves a recursion which generates k copies of the initialtriangle on the kt level. I n Figure 2.8 this yields 34 = 81 copies. Even atthis level the ultimate form of the gasket is clear, ut i there h d beenm ny more transformations which were self-affine nd overlapping, thenthe process of generating these y direct recursion could e very lengthy.This is wh er e the second p rt of Barnsley s approach becomes important,nd we will sketch this in the next section.

    2 8 ractals as terated unction ystemsAs the recursion of transformations proceeds according to equation 2.41),the generated points move closer nd closer to the attractor. A t some pointthe ap pro ximati on is g oo d enough for the scale of the fractal generatedbecomes finer th n that of the computer screen on which it is viewed orthe resolution of the printer on which it is printed. When this scale isreached, further application of the transformations will not dd any furtherdetail to the picture. The crucial issue t he n is to get as close as possible tothe best possible approximation to the attractor y generating as few pointsas possible, for once a point is reached on the attractor, all subsequent

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    transformations of that point will also be on the attractor and the picturewill be quickly revealed.Barnsley 1988a) suggests the following procedure: pick any point on thecomputer screen, hopefully and ideally in the vicinity of the fractal in thatthe object is to b e p lo tt ed on that screen. Then pick a transformation fromthe set of transformations at random and generate a new point by usingthis transformation on the o ld point. owi this p ro ce du re is r epe at ed asequence of points will be generated which will only m ov e tow ards theattractor if the transformations are contractive, that is i they scale and dist ort th e object into a smaller copy of t he original initiating object. I n onesense, we are unlikely to pick objects which have no self-similarity becausethere would be as many transformations as points in the original pictureto generate. The art, of course, is to encode the picture in as few a numberof transformations as possible, and for t he object to be tiled by these theymust contract the original shape. owassuming tha t this is the case as inFigure 2.18, a point very near the attractor will be generated with nearcertainty, say by the 10th iteration, i the original point chosen is nearthefractal. Once the point is there, then further applications of the transformations randomly will begin to fill in the form of the attractor and the picturewill emerge like a pointill ist painting composed of tiny dots.

    only on e tran sformati on is ch osen from several, th e pic tu re will beincomplete. som e of the transformations ar e m or e i mp or ta nt to t he picture than others in terms of the amount of the object they generate, thenthese transformations should be picked more frequently. Rather than choosing each transformation to apply with equal probability, we can measurethe importance of the transformation in p ro po rt io n to th e d et er mi na nt o fthe scaling-rotation matrix given in equation 2.38). In short i there are ntransformations, then the probability Pj of applying the t transformationto the po in t in question can be set as

    lajdj j jlPj n 2.42laidi i il

    i=O

    where the coefficients aj, bj j and dj are those associated with the t transformation specified in equation 2.38). The method is thus operated as follows: each transformation should be chosen in accord with the probabilitiescomputed in equation 2.42 and after about 10 iterations, the sequence ofpoints will be on or v er y n ea r to t he attractor and can t hus be plotted onthe screen or p ri nt ed as hard copy. Equation 2.42 is a measure of thearea o f the fractal associated w it h the t transformation. This method ofgenerating fractals is referred to by Barnsley 1988a) as the Iterated FunctionSystem IFS , while the process of r an do ml y generating po in ts bu t in astructured form is called the Chaos Game Barnsley, 1988b). In Figure 2.19,we show four stages in generating the Sierpinski gasket given earlier inFigures 2.8 and 2.18. There are three transformations used and the coeffic ients associated with them are given as al = az = = 0.5, dl = dz = 0.5, ez = 1, e = = 0.5 and PI = pz = P = 0.33 with all other coefficients setto zero, t hu s s ho wi ng t ha t scaling and translation are the only transformations used in constructing this fractal.

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    Size and Shape Scale and Dimension 8

    igur 2 9 Generating Sierpinski s gasket using IFSThe real power of this method cannot e demonstrated with strictly self-similar objects because these can e generated as quickly not faster in avariety of more direct ways. However where n object is composed ofmuchless obvious self affine copies of itself then the method is truly magical.We will demonstrate this for three objects all tree like shapes which are

    much more realistic th n those shown in the previous sections of this chap-ter. First we show a simple twig which involves three transformations ofthe original object into two which reflect branching nd one which reflectsthe stem. Figure 2.20 illustrates the transformations in terms of the firstlevel of recursion nd the final object after some 10 000 iterations. This twigis d pted from Peitgen Jurgens nd Saupe 1992 who show that t doesnot matter wh t the actual object is which initiates the process becausewhatever object it is it will e scaled down to a point t the resolution ofthe screen before it is plotted; thus it is only the tr ns orm tions of the pointwhich govern the form of the object which is eventually generated. Wemight begin with the Taj Mahal or some equally elaborate object ut aslong as the transformations of the object are those shown in Figure 2.20 atwig will be the ultimate form which we see as n approximation to thefractal attractor.In Figure 2.21 we show n even more realistic tree specified as four

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    Fractal Cities

    igur 2 2 Three transformations defining a twig

    transformations to begin with nd then as five. These too are d pted fromPeitgen, Jurgen nd Saupe s 1990 article nd their 1992 book nd thefigure shows how the object can be improved in realism t hr ou gh theaddition of a single transformation. However, perh ps the best fractal objectcreated by this approach so far is Barnsley s 1988a, b) fern which is shownin Figure 2.22. This is a remarkable demonstration of how a seemingly complex object can be tiled with only four self-affine overlapping but nevertheless contractive transformations. Demko, odge nd Naylor 1985 alsoshow how other conventional fractal objects such as dr gon nd Kochcurves can be generated using IFS. The developments of the IFS approachare only just beginning nd t hey are likely to be manyfold. In particular,there are two w hich are noteworthy. The first relates to showing how verydifferent objects can be transformed into one another. The Koch curve withwhich we beg n this chapter consists of four transformations w hich arestructurally identical to those used to generate the Barnsley fern. the foursets of coefficients { b c, d e nd are com pared for each object, then itis possible to interpolate a sequence of values between those for the Kochcurve nd those for the fern. the objects associated with these interpolatedvalues are plotted in accordance with the order of interpolation, it is poss-

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    Size and Shape Scale and Dimension

    igur 2 2 Realistic trees based on four n five transformationsible to see the Koch curve transform itself into a fern nd vice versa. In factthis m orphogenesis can be animated for the interpolated values are likethose used y animators in the process of in-betweening ; n example isgiven y Peitgen, Jurgens nd Saupe 1992).A related s et of animations ar e e ven mor e p owe rf ul in that they consistof transforming a real object s uch as a tree or fern or Koch curve into nobject with the same num er of transformations ut those with valueswhich specify objects which exist only in mathematical space such as the

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    Fractal Cities

    igur Barnsley s fernMandelbrot nd Julia sets. These are also shown in Peitgen, Jurgens ndSaupe 1992 nd this is t once a demonstration of the fundamental natureof fractal geometry which links real objects with those which only exist inmathematics as well as real attractors to mathematical attractors wh ic hm ight be the strange attractors appearing in chaos theory. Finally, thepower of Barnsley s m ethod is being m ost widely sealized in providing anew approach to image compression. Fractals have been used already fors uch p ur po ses as in those space-filling curves used to store two-dimensional d t in one-dimensional form Goodchild nd Mark, 1987) butBarnsley s approach is more direct. Compressing images by defining IFScodes is immediately pp rent in, for example, the fern which only requiresfour transformations each with six coefficients, making 24 numbers in allto be specified. As Barnsley nd urd 1993 indicate, the real advantagesto such compression are not only t he limited d at a n ee de d but that fractalcompression is, in the last analysis, independent of the ultimate resolutionof the object. This is a fundamental consequence of thinking and articulatingthe world in terms of fractals.

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    Size and Shape Scale and Dimension

    9 Idealized Models of Urban rowth and ormThis has been a long but necessary chapter, long because it is essential tohave at least a rudimentary knowledge of all the key developments in fractal geometry and fractal forms before we begin our applications to urbanform, necessary because it puts us into a posit ion to begin speculating onthe geometrical properties of city shapes and how we might see them asfractals. In the rest of this bookwe will be dealing with two basic propertiesof urban form which involve, on the one hand boundaries to urban development, and on the other hand the growth of cities, their size, shape anddensity as we might perceive them in terms of fractal clusters. Wearehowever, already in a position to say something about how strictly selfsimilar fractal forms compare with idealized city shapes which have beensuggested down the ages and which we briefly reviewed in Chapter 1. Asa conclusion to this chapter, we will present some of these speculations.The Koch curve was used by Mandelbrot 1967 as an idealized modelof a coastline because its fractal dimension was close to that estimated forthe west coast of Britain D 1.262 , while its geometric properties nicelycaptured the way a coastline might repeat its form at different scales. Infact, as we will show in Chapter 5, urban boundaries are somewhat likecoastlines in terms of the extent to which they fill space, and thus the Kochcurve might also be a good model for a city boundary. There is, however,an obvious but perhaps serendipitous comparison we can draw here. Mandelbrot 1983 noted that the slightly distorted H tree such as that shownin Figure 2.13 b reminded him of the 17th century fortress works of theFrench Engineer Vauban, and in the same way the Koch is land might belikened to the regular fortifications suggested by Renaissance scholars asencompassing ideal cities and actually implemented in many new townssuch as Naarden and Palma Nuova Morris, 1979; Rosenau, 1983 . In Figure2.23, we show a selection of ideal town plans produced during the Renaissance in Europe and echoing the classical ideals of Greek architecture andurban design as portrayed for example by Vitruvius.The regular sorts of fortification which are such a feature of Europeantowns during these centuries were in fact based on the notion of maximizing the amount of cross-fire which could be directed from the town at anapproaching army. tthe same time, this increased the amount of wallwhich had to be defended, and inevitably what was built was somecompromise between these conflicting ideas. The town plans shown in Figure 2.23 in fact show fortifications which are built, not on the triangularKoch island which was used to illustrate the meaning of fractals in theearly sections of this chapter, but upon regular pentagons, hexagons andoctagons. Although these Koch islands only show detailed perturbation ofthe initiator down one or two levels, it is clear that their fractal dimensionsover a couple of orders of scale are close to that of the Koch curve. Onelast point relating to these ideal towns involves the fact that most organically growing towns prior to the post-industrial age are radially concentricin that the town develops around a seed site, usually a palace, temple ormarket complex which is linked to other towns and to the surrounding

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    9 Fractal Cities

    igur 3 Idealized plans: fortified Renaissance towns and Koch islands from Morris1979 .

    suburbs by a regular patterns of radial roads. These features are even clearfor the ideal towns s ho wn in Figure 2.23.ur second example also relates to the Koch curve. In Chapter 1, we dealtwith the Radburn layout of residential housing in which it was possible tovisit any place alongside the branches of roads servicing the area withoutcrossing any of these roads. Such layouts were shown i n Figure 1.19 ndwe discussed their properties in terms of the H tree in n earlier section ofthis chapter. Arlinghaus nd Nysteun 1990 have suggested that fractalssuch as the Koch forest Figure 2.3 nd the Cesaro-Peano sweep curve Figure 2.7 might be used as designs which maximize the amount of linear space for mooring boats i n a marina. They illustrate t he i dea usi ng theCesaro-Peano design which we show in Figure 2.24 where they speculatethat the elaborate nature of the mooring is more likely to coincide with thedistribution of preferences i n the related popul at ion t han a design basedon routi ne mooring along a waterfront. Moreover, as t he amount of boatmooring is also increased dramatically by such space-filling designs, thedensity of development would be greater nd costs per unit of developmentwould likely be lower for each participant in the scheme.The last example we will introduce here involves n idealized version ofthe fractal growth model which we will i ntr oduce from Chapter 7 on. Wehave already presented several ways of viewing Sierpinski s gasket nd nelaboration on this w ou ld be to tile a square initiator with five copies ofitself in the manner shown in Figure 2.25. Mandelbrot 1983 refers to

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    Size and Shape Scale and Dimension 93

    Condominium po dn lltlllip

    igur 4 Maximizing linear frontage using the Koch forest fromArlinghaus and Nysteun, 1990 .

    L

    K o K K 2 K 3igur 5 Sierpinski s tree: an idealized model of fractal growth.

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    2.18

    2.43

    Fractal ities

    similar designs as Sierpinski carpets although in this case, the design is adendritic form rather than that of an object into which holes are punched.We will refer to this the n as a Sierpinski tree in the spirit of Mandelbrot.t is easy to see that this tree is formed by tiling the initiator with five copiesof itself, each scaled by one third the original size of the initiator. Thususing equation 2.19 the dimension is D log 5)/log 3 1.465. In Figure2.25 we show three ways of presenting this idealized tree. In 2.25 a), weshow the tree as a strictly self-similar fractal in which we form the attractorby tiling an aggregate unit, a square, by five smaller squares at each levelof hierarchy. This is the same way we introduced Sierpinski s gasket; itrepresents the way we might measure the dimension of this fractal i wewere approximating its form at different levels by changing the scale onwhich we were viewing and measuring it.However, in Figure 2.25 b), we show the dendrite as a spanning tree ateach level of the hierarchy. t is from this figure that we might measurethe perimeter of the fractal. Using equation 2.16 with N k =k and = r ka ss um in g the l en gt h of eac h diagonal of th e u ni t s qu ar e at k = 0 to beand t hat there are four diagonal s pa ns to the perimeter of each square atthe appropriate level, then the perimeter of the entire tree at level k wouldbe given as

    Lk =krk =4 12 ~ rThe third way to examine t he Sierpinski tree is as a g ro wi ng fractal andthis is shown in Figure 2.25 c). In this case, we now h ave a fixed scale rand a varying yardstick or radius Rk at scale k Now the mass of the fractalcan be measured as its number of parts Nk where this number is growingas the scale k o f the tree gets larger. Then

    Nk= ~ r =frEquation 2.43 is of t he s am e form as e qu ati on 2.31 which is the relationship between mass and linear scale for a g ro wi ng fractal. I n short, this isthe relation that we are seeking and which we will exploit from Chapter 7on when we develop real versions of the Sierpinski tree as the skeleton onwhich most cities develop.Noting that the fixed scale r of the growing fractal is 1/27 3-3, then theperimeter can be calculated for each scale k as 4 1/2/27)5 k where we againu se t he four diagonal s pa ns as t he pe ri me ter of each basic u ni t w hos e si deis 1/27 of the unit square. This is a reasonable model of a growing dendritealthough its construction is somewhat different from those trees presentedin an earlier section. t is easy to use IFS to generate the tree for like theSierpinski gasket, its only transformations are scaling and translation. Thereare five transformations and the coefficients are given as aj =dj = 1/3 forall i; bj = Cj 0 for all i; el =z = e = 0; ez = = e=4 = 2/3 ; and es= 1/3 The tree is shown in Figure 2.26 for four stages in its generation.In the next part of the book, we will begin to apply these ideas to morerealistic looking cities than those with which we have concluded here. Inthis we will link our ideas back to som e of those p re se nt ed in our survey

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    Size and Shape Scale and imension 95

    igur 6 Generating Sierpinski s tree using IFSof urb n form in Chapter n particular in the next chapter we willdevelop fractal models of cities in the m nner in which realistic fractalswere first modeled as statistical simulations of self similar lines nd sur-faces. We will look once again at the simulation of coastlines nd thendevelop more idealized models based on simple location theory nd onprinciples of hierarchical dependence. To do this we will develop yetanother model of self similarity this time of a stochastic not deterministicvariety but we will still restrict our models to two dimensional m ps ndour fractal dimensions will still range between 1 nd 2