fractal-based description of urban form

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Environment and Planning B: Planning and Design, 1987, volume 14, pages 12 3-1 34

Fractal-based description of urban form

M Batty, P A LongleyDepartment of Town Planning, University of Wales Institute of Science and Technology, Cardiff,CF1 3EU, WalesReceived 19 January 1986, in revised form 20 M ay 1986

Abstract. Fractals, shapes with nonintegral or fractional dimension which manifest similardegrees of irregularity over successive scales, are used to produce a consistent measure of thelength of irregular curves such as coastlines and urban bou ndarie s. Th e fractal dim ension ofsuch curves is formally introduced, and two computer methods for approximating curvelength at different scalesthe structured walk and equipaced polygon methodsare outlined.Fractal dimensions can then be calculated by performing log-log regressions of curve lengthon various scales. Th ese ideas are tested on the urba n boun dary of Cardiff, and this revealsthat the fractal dim ension lies betw een 1.23 and 1.29. T he app rop riate nes s of fractalgeometry in describing man-made phenomena such as urban form is discussed in the light ofthese tests, but further research is obviously required into the robustness of the methodsused, the relevance of self-similarity to urban development, and the variation in fractaldimension over time and space.Fractals are irregular shapes whose geometry is scale-d epen den t. At every scale,the degree of irregularity which characterises the geometry appears the same, thisbeing referre d to as self-similarity. A t no scale can the form of a fractal or anypar t thereof be describ ed by a sm ooth function; thus any such function is said tobe nonrec tifiable. Th es e pro pe rties of self-similarity and nonrectifiability lead tothe somewhat startling conclusion that fractals have nonintegral or fractionaldim ension. Typical exam ples includ e rugged bo un dar ies such as coastlines whichhave dimensions betw een 1 and 2, and surfaces such as geom orpho logical lands cape swith dim ension s betwe en 2 and 3. Fractal objects, however, can exist in anynonintegral dimension and are thus not simply confined to easily perceived naturalphenomena (Mandelbrot , 1983).

Coastlines are the most appropriate examples to begin an illustration of theseideas, for the concern of this paper is the application of fractal geometry to urbanform, as described by the bou nda ries of urb an areas. Coastlines are obviously self-similar: as one ap pro ach es, within the ob serv er's view point which fixes the scale,the degree of irregularity always appea rs to be the same. A t finer scales, irregulardetail is a scaled version of that at precedin g scales. T he c ons equ ence of this isthe well-known conundrum that the length of the coastline depends upon thescale. A t finer and finer scales, the coastline beco m es longer and longer, itsultimate length being inde term inate . Th is fact has been know n for many years.For example, Perkal (1958a) refers to measurements of the coast of Istrii made bythe Viennese geographer Penck in 1894, and there have been several attempts atproviding ordered measurement of nonrectifiable curves, such as that initiated bySteinhaus (195 4; 1960 ). Indeed , the conundru m of length is sometimes referred toas Steinhaus's parado x (Nysteun, 1966). How ever, i t was Rich ardson (1961) whowas first to describ e this ph en om en on systematically in formal term s. But he wascontent to leave the matter as an empirical fact, and it fell to Mandelbrot (1967)to develop the formal framework.


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124 M Batty, P A Longley

Indeterminate length is the indicator that such lines have nonintegral or fractionaldime nsion. A straight line has dimen sion 1, a plan e dimension 2, and so on; acoastline fills more of the space than a straight line, but does not fill the entirespace as does a plane, hence i ts dimension would appear to be between 1 and 2.There are curves which continually reenter a region, filling space entirely, such asthe Peano curve, which have a fractal dimension of 2 but a topological dimensionof 1. Th us the m ore rugged the coas tline, the grea ter the fractal dim ension abov eits topo logic al dim ens ion of 1. In theo ry, a perfec t fractal is form ed by a singleprocess which operates repeatedly at all scales, but, in reality, several processes arelikely to be at work in moulding the irregularities characterising phenomena suchas coastlines. Th us it is widely acce pted that fractal dim ension is usually con stantonly over a limited range of scales, and that many phenomena are multifractal.Much present research in applying fractal geometry is therefore concerned withdetermining the ranges over which particular fractal dimensions apply (Kaye et al,1985), and, in several disciplines, two distinct ranges of scale described bydifferent fractal dimensions have been identified (Orford and Whalley, 1983).

The purpose of this paper is to apply these ideas to the measurement of irregularboundaries defining urban areas, thus extending fractal geometry from nature tosystems which are formed from combinations of man-made and natural processes.Such extensions are not new. Perkal (19 58b ) has speculated on measuring theurban edge of Wroclaw and several researchers have used fractal measures todefine the length of political frontiers, notably Richardson (1961) and latterly Clark(1986 a). Althou gh there are many different processes and constraints which mo uld anurban area, a preliminary investigation into the suitability of fractal geometry seemswo rthwhile. M oreo ver, this is a useful com plem ent to our oth er work, in whichland-use patterns are being synthesised from land-use models, with fractal geometryused as a means for displaying realistic-looking maps (Batty and Longley, 1986).Accordingly, we will first formally define the fractal dimension associated with self-similar curves such as coastlines and urba n bo und aries. Th en we will presen t twom etho ds for me asuring the length of such curves at different scales. Th ese involvesimulating piecewise approximations to the silhouette defined by the curve, akin tothe way one might me asure a curve using dividers. Th ese m ethods are thenapplied to measurement of the urban boundary of Cardiff by means of digitiseddata on the extent of the urban area in 1949 and measurements are generatedfrom which the fractal dim ension can be com pu ted by using regression. Finally, abrief conclusion as to the appropriateness of these ideas in describing city systems,and themes for future research are presented.The definition of fractal dimensionConsider an irregular l ine, X, betw een two fixed poin ts. Define a scale of resolution ,A x0, such that, when this line is app roxim ated by a sequ ence of contiguo us segmentsor chords each of length A x 0 , this yields n0 such cho rds. Now determine a newscale of resolution, Ax ly which is half Ax0, that is, Axx = iA% 0 . Ap plying thisscale, Ax 1 ? to the line yields n1 ch or ds . If the line is fractal, the n it is clear th at"halving the interval always gives more than twice the number of steps, since moreand m ore of the self-similar detail is picked u p" (M ark, 1 98 4, page 293). Formallythis means that

^> 2, and ^ = 2. (1)n0 AxxThe lengths of the approximated curves or perimeters, in each case, are givenas P1 = n1Ax l, and P0 = n0Ax0. From the assumptions implied in equation (1),


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Fractal-based description of urban form 125

it is easy to show that P x > P0, and this provides the formal justification that thelength of the line increases without bound, as the chord size or scale Ax convergestowards zero.The relationship in equation (1) can be formally equated if it is assumed that theratio of the number of chord sizes at any two scales is always in constant relationto the ratio of the lengths of the cho rds. T he nn x | A x 0n0 \ A J C J

where D is defined as the fractal dim ens ion . If halving the scale gives exactlytwice the number of chords, then equation (2) implies that D = 1, and the linewould b e straight. If halving the scale gives four times the num ber of cho rds, theline wou ld enclose the space and the fractal dime nsion wo uld be 2. Eq ua tion (2)can be rearranged asn x = (n0Ax0D)AxfD = XAx^D , (3)

where the term in brackets {n 0Axg) acts as the base constant, X, in predicting thenumber of chords, n l7 from any interval of size Ax{ relative to this base.From equations (2) and (3), a number of methods for determining D emerge.Equation (2) suggests that D can be calculated if only two scales are available(Goo dchild, 19 80). Rearranging equ ation (2) gives D as' f e ) / ' f e ) -However, most analyses not only involve a determination of the value of D butalso of whether or not the phenomenon in question is fractal, and thus more thantwo scales are req uire d. W riting equatio n (3) in m ore general terms as

n = XAx~D , (5)where n is the number of chords associated with any Ax, we can lineariseequation (5) as

\nn = I n A D ln A x . (6 )Equation (6) can be used as a basis for regression by using estimates of n and Axfrom several scales. A related form ula involves the length of the curv e or perim eter,P , which is given from equation (5) as

P = nAx = AAx*1"^ . (7)Equation (7) can be linearised by taking logarithms,

l n P = lnX-(l-D)\nAx, (8)where it is clear that the intercepts in equations (6) and (8) are identical and theslopes are related to the fractal dimension, D, in the ma nne r shown. In subsequentwork, we will use equation (8) rather than equation (6), for equation (8) will enableus to check the range of scales used more effectively than equation (6).Several other methods for determining the fractal dimension of curves have beendeveloped, and these involve approximating line lengths by grid intersections andcell-counts (Go odchild, 1 980 ; Dearnley, 19 85 ; M orse et al, 1985), by area definitionof bou nda ries (Flook, 19 78 ; M and elbrot, 1983 ), by area - perime ter relations(Woronow, 1981; Kent and Wong, 1982), and by variograms (Mark and Aronson,1984). We will not pursue these methods here, but they will be considered infuture work.
http://ajcj/http://ajcj/


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Simulating scale approximations: the structured walk and equipaced polygon methodsThe original method used by Richardson (1961) to measure the length of coastl inesand frontiers involved manually walking a pair of dividers along the boundaries atdifferent scales and then determining D from equation (8). To enable the entireperimeter to be traversed, the last chord length which always finishes at the lastcoordinate point is generally a fraction of the step size, and the step sizes used ateach scale usually reflect or de rs of m agnitud e in geometric relations hip; that isA i = a~nAx0, a > 1, which enables each step size to be equally weighted andspaced in the lo g -lo g regression. In Rich ardson's (196 1) research, about sixorders of magnitude or scales were used which is regarded as sufficient todetermine a least-squares regression line.Computer simulations of Richardson's manual method are now well established.Kaye (1978) refers to the method as a 'structured walk' around the perimeter of anobject, and he calls the log-log scatter plot of perimeter lengths versus scaleintervals a 'Rich ardso n plot ' ; this provides a useful visual test of whether o r no tthe ph eno m eno n is fractal. T he struc tured w alk metho d is easy to implem ent on acomputer and here we have used the algorithm developed by Shelby et al (1982)which involves approximating the boundary of an object consisting of line segmentsbetwe en digitised coo rdina tes, with different sized cho rd lengths. T he advantagesof this algorithm consist in its adaptation to the display of the various scaleapproximations on a graphics terminal which enables useful visual assessments to

be ma de of the suitability of the chosen m ap scales. H ere the algorithm isimplemented on a M ICR O VAX II supermini with A utogra ph X 5A display devicedriven by GINO-F graphics software.T he re are two variants involving this m ethod . First, the nu m ber of chord s andperim eter lengths will dep end upo n the starting point along the curve. To red ucethe arbitrariness of this variation, several workers have suggested the structuredwalk be started at several different points, and averages of the results then formed(Kent and Wong , 198 2). Ka ye et al (19 85 ) start the walk at five different po intsalong their curves, but here we have been able to start the walk at each of the

7V(=1558) digitised points which define the boundary of Cardiff, the walkproceeding in both directions towards the endpoints of the boun dary. Th e m ethodis extremely time-consuming, and some runs have taken about two and a half hoursof CPU (central processor unit) t ime on the MICROVAX II, with the machinebeing entirely dedicated to this task.The second variant involves starting the structured walk at different dividerlengths and generating seque nces of pred ictions from the se different lengths. T herange of scales over which the perimeter lengths were computed varied from abouthalf the average chord length associated with the digitised data, to over the maximum

distance between any two coordinate points on the perimeter. T he average chordlength is com pu ted as follows. First, the distances betw een each adjacent pair of(x,y) coordinates, / and z + 1, are co m puted fromd iJ + 1 = [{x-x i + lf + (yi-yi+lY]\ i = 1,..., N-l , (9)

and then the perimeter of the digitised base level curve is computed asp = Yd,i+1. (io)

i = 1

The average chord length, d, of the original curve is therefore given as


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and a lower bound for the chord length used to start the approximation, assuggested by Shelby et al (1982), is taken as ~kd. The maximum distancebetween any pair of coordinates, which in fine-particle science is referred to asFeret's diameter (Kaye, 1978), is given as

F=max{rf. .+ 1 } . (12)i, i + 1Kaye (1978), amongst others, suggests that an appropriate upper bound for chord-length approximation is ~ i F .The problem with the structured walk method is that, if there are deep fissuresin the curve, this can cause perturbations in the Richardson plot through the chordlength suddenly falling to the lengths where such detail is picked u p. Th is can thendistort the dimension, so an alternative method known as the equipaced polygonm ethod h as been suggested by Kaye and Clark (19 85). In fact, this m ethod wasfirst implemented by Schwarz and Exner (1980) as an artifact of the particularimage analyser used in measuring particle boundaries, and has subsequently beendeveloped by Orford and W halley (1983). Th e method involves using the chordsassociated with the original set of digitised points defining the curve, not newpoints on the curve as used in the structured walk m ethod . T hu s, in the equip acedpolygon m etho d, in general the cho rds are not equal in length. Perim eter lengthsare computed by taking the sequence of chord lengths between adjacent coordinates,then between coordinates which are spaced at more than one pair of coordinatesapart, and so on. If these sequence s are cons tructed geom etrically to ensu re mo reequal weighting in the regressions, the sequences taken would be based on adjacentcoordinates, every second, every fourth, every eighth coordinate pair, and so on.For each sequence, an average chord length is computed for use in the log-logregression.

As in the case of the structured walk method, averages of perimeter lengthsbased on different starting points along the curve reduce the degree of arbitrarinessposed by the prob lem. A part from the obvious advantage of reducing distortionsin the plots and resultant fractal dimensions, the method is easy to compute, for itavoids the time-consuming trigonometry of the structured walk, by using Pythagoras'stheorem to compute chord lengths between given points as in equation (9) above.In fact, the method uses about one tenth of the computer time used in each run ofthe structured walk.The geometry of urban form: the urban edge of Cardiff, 1949The boundary marking the extent of the urban area of Cardiff was defined fromthe 1:25000 Ordnance Survey map published in 1949, as part of a larger projectconcerned with measuring changes in the urban boundary of Cardiff from 1880 to thepresen t day. T he usual problem s of definition were encountered in determining theedge of the urb an are a, and several rules of thum b were invoke d. Typically,allotments and other urban fringe land-uses were excluded, villages linked to theurban area by ribbon development were included, man-made alterations to riversand coast were included, but large landed estates which subsequently become partof the urb an fabric were only included if developm ent had surrou nded them. T heentire definition process emphasised the obvious problems that urban processesand constraints operate at different scales, thus throwing some doubt on the fractalcon cept of self-similarity in this contex t; bu t perh ap s no mo re dou bt than exists inother areas of the physical sciences where fractal concepts have been shown toapply only over restricted scales. Nev ertheless, the complex concaten ation of


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processes and constraints which form the urban area does lead to visual formswhich, superficially at least, appear fractal.On ce the bou nd ary had b een defined, it was digitised to within 1 mm resolution ;the coastl ine contained some 900 points, whereas the urban boundary was basedon 155 8 poin ts. Figu re 1 shows the digitised ou tlines as well as a coarseapproximation to the boundary produced by the structured walk method, which isabo ut thirty times the scale of the original data. N ote the way the appro xim atingpolygon touches the original boundary, and note the equal chord sizes, apart fromthe last chord . Th e coordinates were init ially processe d with the M ICR OP LO Tsoftware (Bracken, 1985) in which data files are first created on a BBC Micro andthen subsequently cleaned up and transferred to the M ICR O VAX II where all thesimulation was done.

The perimeter of the digitised boundary determined from equations (9) and (10) gaveP = 3104.456 units, with the average chord length d = 1.993, from equation (11),and the Feret diameter F = 43 2.9 35 , from equ ation (12). Th ese measures areuseful to keep in mind when we discuss the applications of the structured walk andequipac ed po lygon me tho ds. We will deal first with the structured walk meth od.For a given chord length used to start the sequence of predictions of perimeterlengths, a complete series of ten chord lengths are used in the approximations,starting from the finest level of scale now given by A x0 and moving to coarser scales,A i . The sequence of chord lengths is computed from Axn = 2n{d}(n = 0 , 1 , . . . , 9)where d is the start length which is always a function of d, the average chordlength. Thus, for example, where d = id , which is the lower bound recommendedby Shelby et al (1982), the sequence of chord lengths used are in the followingra tios { i , 1 ,2 , 4 , 8 , 1 6 ,3 2 ,6 4 , 1 2 8 , 2 5 6 } . In this case, A x 0 ~ l , a nd A x 9 ~ 5 1 0which is much larger than Kaye's (1978) upper bound of iF . To provide some feelfor this range of approximations, we have plotted the approximated boundaries ofCardiff for Axn(n = 0, 1,..., 8) in figure 2. W ith Ax9, the boundary is approximatedby a single cho rd, which is clearly inap pro pria te. Inde ed, even with A x 7 andA x 8 , the ap prox ima tions are too coarse to be of mu ch use. Th is is clear from

Figure 1. The digitised urban edge of Cardiff in 1949 and a typical scale approximation.


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figure 2 which shows that this kind of visual test is essential in selecting anappropriate range of measurements for use in the subsequent regressions.Our experience in using the structured walk method involved selecting tendifferent starting values of the chord length d, and generating ten sets ofme asure m ents for each of these starting values. T he values of d chosen were6 = OA d, 0.5d, 0.6d, O.Sd, d, 1.5d, 2d, 3d, Ad, and 5d. From the sequencesgenerated, it is clear that several of the chosen measurements are the samebetween series, but each of the regressions developed below involves different setsof m easu res. Before w e turn to these results, a visual com pariso n of each of theten sequences generated is contained in the Richardson plots in figure 3 whichshow the ten measures of In P versus In A x for each of the ten starting values of(5. These plots are all on the same scale for comparative purposes and also showthe values of id, d, ?F, and F.At this point, it is appropriate to describe how the equipaced polygon methodhas been applied. Th e Richard son plot associated with this meth od is also shown

Figure 2. A sequence of scale approximations to the urban edge of Cardiff.


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as the last plot in figure 3. W her eas the ch ord leng ths in the struc tured walk arein geometric relation, it is the chord paces between the given digitised pointsdefining the bound ary which are geometrically order ed. Th e sequence of ten chordlengths is as follows. T h e initial cho rd length, Ax0, is computed as the average ofall single-step segments; the second length, Ax l9 the average of all two-stepsegments; the third length, A i 2 , the average of all four-step segm ents; and so on,with the rest of the sequence being provided by 8, 16, 32, 64, 128, 256, and 512step segments. With 1558 co ordina tes, the last set based on 5 12-step segmentsgives just over three cho rds which from figure 2 is not a good app roxim ation. Infact the process is a little more complicated than this: for all steps over the singlestep length, averages are computed from alternative starting points to remove thearbitrariness of always starting with the first set of coordinates.

Before we consider the results of the regressions, we now need to consider howwe can systematically narrow the range of results we are able to generate, and to50003000

u< D

.6 1000u 500

200

hdd \F F

0.610 100 1000

20001

6 = 0.4 d 6 = 0.5 d

|500040003000chord length6 = 0.6 d 6 = 0.8 rf 6 = l.Od

6 = \.5d 6 = 2.0d 6 = 3.0d

6 = 4.0 d 5 = 5.0 d J equipaced polygon method

Figure 3. R i c h a r d s o n p l o t s b a s e d o n t e n s t r u c t u r e d w a l k s a n d a n e q u i p a c e d p o l y g o na p p r o x i m a t i o n .


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this end , we have used five criteria. Firs t, we have used the range 0.4d < AJC < 0 .5Fto select those observations which are app rop riate. Second, we have used theR icha rds on plots to identify o utliers for exclusion. In particu lar, whe n AJC > F ,then the algorithm always gives the same perimeter length because it always closesthe single chord on the last coo rdina te point. Such points show up h orizontallyon the plots and must be excluded. Th ird, the scale approximation m ust beaccep table visually. A n exam ination of figure 2 suggests that appr oxim ation s withten chords or less are not satisfactory in representing the overall shape, and thusmust be excluded. Fou rth, the r2 measure of fit should always be better than 0.95,and, fifth, the standardised variation in average perimeter length for each A* shouldnot be greater than 10% of the mea n value. Th is also enables poor approxim ationsto be excluded.For each of the ten starting values of 6 in the structured walk method, and forthe equipaced polygon method, we have performed regressions on all ten pointsshown in the Richardson plots in figure 3, on the first 9, the first 8, 7, 6, then 5,below which it is no t appr op riat e to carry out such least-sq uares fitting. T heabsolute values of the slopes of the regression lines |/?| = (1 D) are shown intable 1 along with the r2 values, but, as Shelby et al (1982) indicate, such r2 valuesshould n ot be used to assess good ness of fit in the strict statistical sense. In table 1,the figures which are in bold type involve regressions in which the observationsmeet all the five criteria mentioned above, and this narrows the range considerably.Note that the fractal dimension is given by adding 1 to the absolute slopes intable 1, that is, D = l + | j8 | .Table 1. Slope and r2 values from regressions of the data shown in the Richardson plots infigure 3.Star t ing va lues (d )a

OAd0.5 d0.6 d0.8 d

d1.5 d2.0 d3.0 dA.Od5.0 dE q u i p a c e dp o l y g o n m e t h o d

N u m b e r o f ' o b s e r v a t i o n s ' 5100.269(0.953)0.278(0.969)0.279(0.975)0.292(0.976)0 . 2 9 1(0.982)0.293(0.975)0.282(0.969)0.274(0.945)0.254(0.924)0.245(0.915)0.285(0.993)

90.244(0.959)0.258(0.975)0.263(0.975)0 . 2 9 1(0.966)0.297(0.977)0.309(0.980)0.304(0.984)0.303(0.972)0.284(0.958)0.276(0.953)0.294(0.994)

80 . 2 3 1(0.947)0.255(0.963)0;254(0.964)0.266(0.973)0.276(0.983)0.308(0.971)0 . 3 1 5(0.982)0.327(0.984)0 . 3 1 3(0.981)0.308(0.984)0.285(0.994)

70.207(0.944)0.236(0.956)0.236(0.963)0.254(0.962)0.278(0.975)0.280(0.980)0.293(0.989)0 . 3 3 1(0.977)0 . 3 3 1(0.983)0 . 3 3 1(0.996)0.275(0.995)

60 . 1 7 7(0.961)0 . 2 1 1(0.956)0 . 2 1 6(0.953)0 . 2 3 1(0.957)0 . 2 6 1(0.967)0 . 2 6 1(0.980)0.303(0.987)0 . 3 0 1(0.986)0.306(0.989)0 . 3 2 1(0.997)0.265(0.996)

a S t a r t i n g v a l u e s i n e a c h s e q u e n c e o f t h e s t r u c t u r e d w a l k s .b N u m b e r o f ' o b s e r v a t i o n s ' o f p e r i m e t e r - c h o r d l e n g t h s u s e d i n r e g r e s s k

50 . 1 5 5(0.969)0 . 1 8 0(0.975)0 . 1 8 5(0.966)0 . 1 9 8(0.974)0.234(0.963)0.254(0.963)0.289(0.979)0.282(0.985)0.328(0.996)0.329(0.996)0.253(0.998)

>ns. The f i r s t va lue ineach row-column is slope | /? |; the second value in parentheses is r2.


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For the structured walk method, there is still a large variation in fractaldim ension from 1.155 < D < 1.289 and from table 1, it is quite clear that as finerand finer scales come to dominate the regression, the lower the value of D. Thisimplies that there is greater irregularity at coarser scales, but it may also indicatethat where the scale is below the level of resolution of the digitised boundary, thatis , where Ax < d, then no further detail is picked up and the boundary must becon side red E uc lide an . Th is is the case for the first four starting values (first fourlines) in table 1, and, if these are excluded from consideration, the range of D isfrom 1.234 to 1.289. In fact, the rule of thumb suggested by Shelby et al (1982)that d should begin at about id should be reevaluated in future work, so that thevariat ion around d can be considered.

The equipaced polygon method gives much more acceptable results in theseterms, for the basic level of detail sampled is no lower than the digitised baselevel. In fact, the fractal dim ensions pro du ced are much mo re stable than th osegenerated by the structured w alk m etho d. Th ese range over 1.253 < D < 1.285for regressions involving the first eight or fewer values. M oreov er, the Rich ards onplot in figure 3 is the best of those shown there in terms of r2 performance andlinearity. We have also run the equipac ed po lygon m ethod for up to to five hu nd redchord sizes, where the increase in step length between coordinates is based on anarithme tical incre ase, that is 1,2 , 3, 4, 5, ... step lengths. In this case, the range offractal dimensions from the first ten to the whole five hundred chord sizes variedfrom 1.249 to 1.327 which is a much narrower range than that shown in table 1for the structured walk applications.ConclusionsThe fractal dimensions of a wide range of physical phenomena have been studied(Burrough, 1984), but rarely has the irregularity of artificial boundaries beeninvestigated. Richa rdson's (1961) results on the Germ an and Sp anis h-P ortug ues efrontiers, which produced values of D of 1.15 and 1.14, respectively, are theclosest exam ples to the ideas suggested here. T he rang e of variation in fractaldimension for coastlines, however, is quite wide, Richardson's (1961) result ofD = 1.25 for the west coast of Britain, corrected by Shelby et al (1982) to 1.267,which inspired Mandelbrot (1983) to suggest that the Koch curve with D~ 1.262as a good model for a coastline, seems to be towards the upper limit.

T he range of variation record ed in table 1, from 1.15 < D < 1.33 which can benarrowed to 1.23 < D < 1.29, seems to be largely an artifact of the methods used,and it remains to be seen whether or not the values calculated here are characteristicof other cities.This range of variation in D associated with different methods is not unusual

(Burrough, 1984), but it does hasten the search for methods such as the equipacedpolygon m ethod which are m ore robust than the structured walk. A metho d whichcombines the advantages of both has been proposed by Clark (1986b), and histests appear promising. Cell-counting method s of the sort used by G oodch ild(1982) are also worth exploring, although preliminary tests on this data set revealthat these are cons iderably less accu rate than the walking or pacing meth od s. It isalso clear that appropriate selection of the sequence of scale approximations, andthe reduction of any arbitrary variation through averaging are essential in producingappropr ia te D values.Finally, the processes which structure urban form and urban edges should beinvestigated further with respect to the types of irregularity characterising differentcities in time and spac e. Th is might be acco mp lished by having regard to w haturban theory suggests about the concatenation of processes, but also by recognising


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different types of irregularity at different scales and over different sections of thesame pheno m eno n. Histo rical variations in fractal dim ension are also likely, for thedevelopment of cities has been influenced by processes whose form and scale haschanged o ver tim e. In the quest to synthesise overall land -use patte rns which fullycharacterise urban form and structure, research needs to be started on measuringthe degree of irregularity in pa ttern s whose parts are disjoint. Th ese are all them esfor future research.Acknowledgements. Beryl Collins typed the original manuscript, Tracy Dinnick drew thediagrams, and Ian Brack en advised on the digitisation of map bou ndarie s. Th is pap er wasfirst presented at a Special Session on Quantitative Residential, Retail and Migration Models,at the Institute of British Geographers Annual Conference, University of Reading, Berks,7 January 1986 .ReferencesBatty M, Longley P A, 1986, "The fractal simulation of urban structure" Environment andPlanning A 1 8 1 1 4 3 - 1 1 7 9Bracken I, 1985, "Computer-aided cartography with microcomputers: a practical guide toMICROPLOT" WP90, Department of Town Planning, University of Wales Institute ofScience and Technology, CardiffBurrough P A, 1984, "The application of fractal ideas to geophysical phenomena" Journal ofthe Institute of Mathematics and its Applications 2 0 3 6 - 4 2Clark N N, 1986a, "Fractal harmonics and rugged materials" Nature 319 625Clark N N, 1986b, "Three techniques for implementing digital fractal analysis of particleshape" Powder Technology 4 6 4 5 - 5 2Dearnley R, 1985, "Effects of resolution on the measurement of grain 'size'" MineralogicalMagazine 49 5 3 9 - 5 4 6Flook A G, 1978, "The use of dilation logic on the quantimet to achieve fractal dimensioncharacterisation of textured and structured profiles" Powder Technology 2 1 2 9 5 - 2 9 8Goodchild M F, 1980, "Fractals and the accuracy of geographical measures" MathematicalGeology 1 2 8 5 - 9 8Goodchild M F, 1982, "The fractional Brownian process as a terrain simulation model"Modeling and Simulation 1 3 1 1 3 3 - 1 1 3 7Kaye B H, 1978, "Specification of the ruggedness and/or texture of a fineparticle profile by

its fractal dimension" Powder Technology 21 1 -16Kaye B H, Clark G G, 1985, "Fractal description of extra-terrestial fineparticles" Departmentof Physics, Laurentian University, Sudbury, OntarioKaye B H, Leblanc J E, Abbot P, 1985, "Fractal dimension of the structure of fresh anderoded aluminium shot fineparticles" Particle Characterization 2 5 6 - 6 1Kent C, Wong J, 1982, "An index of littoral zone complexity and its measurement" CanadianJournal of Fisheries and Aquatic Sciences 3 9 8 4 7 - 8 5 3M and elbr ot B B, 196 7, "How long is the coast of Britain ? Statistical self-similarity andfractional dimension" Science 156 6 3 6 - 6 3 8Mandelbrot B B, 1983 The F ractal Geom etry of Nature (W H Freeman, New York)Mark D M, 1984, "Fractal dimension of a coral reef at ecological scales: a discussion" MarineEcology Progress Series 1 4 2 9 3 - 2 9 4Mark D M, Aronson P B, 1984, "Scale-dependent fractal dimensions of topographic surfaces:an empirical investigation, with applications in geomorphology and computer mapping"Mathematical Geology 1 6 6 7 1 - 6 8 3Morse D R, Lawton J H, Dodson M M, Williamson M H, 1985, "Fractal dimension ofvegetation and the distribution of arthropod body lengths" Nature 314 7 3 1 - 7 3 3Nysteun J D, 196 6, "Effects of bou nda ry shape and the concep t of local convexity", in, DP 10,Michigan Inter-University Community of Mathematical Geographers, Ann Arbor, MIOrford J D, Whalley W B, 1983, "The use of fractal dimension to quantify the morphologyof irregular-shaped particles" Sedimentology 3 0 6 5 5 - 6 6 8Perkal J, 1958a, "O dlugosci krzywych empirycznych" Zastosowania Matematyki 3 2 5 7 - 2 8 6 ;translated 1966 by R Jackowski, as "On the length of empirical curves", in, DP10,Michigan Inter-University Community of Mathematical Geographers, Ann Arbor, MI


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