A Theory of the City as Object

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A theory of the city as object: or, how spatial laws mediatethe social construction of urban space

Bill Hillier*

Space Syntax Laboratory, The Bartlett School of Graduate Studies, University College London, Gower Street,London WC1E 6BT, UK

A series of recent papers (Hillier et al, 1993; Hillier, 1996b, 2000) have outlined a generic process by whichspatial configurations, through their effect on movement, first shape, and then are shaped by, land-usepatterns and densities. The aim of this paper is to make the spatial dimension of this process more precise.The paper begins by examining a large number of axial maps, and finds that although there are strongcultural variations in different regions of the world, there are also powerful invariants. The problem is tounderstand how both cultural variations and invariants can arise from the spatial processes that generatecities. The answer proposed is that socio-cultural factors generate the differences by imposing a certain localgeometry on the local construction of settlement space, while micro-economic factors, coming more andmore into play as the settlement expands, generate the invariants.

URBAN DESIGN International (2002) 7, 153179. doi:10.1057/palgrave.udi.9000082

Keywords: society; city design; axial maps; formal typologies; cultural invariants and variations

Movement: the strong force

The urban grid, in the sense used in this paper, isthe pattern of public space linking the buildingsof a settlement, regardless of its degree ofgeometric regularity. The structure of a grid isthe pattern brought to light by expressing the gridas an axial map1 and analysing it configuration-ally. A series of recent papers have proposed astrong role for urban grids in creating the livingcity. The argument centres around the relation

between the urban grid and movement. InNatural movement (Hillier et al, 1993), it wasshown that the structure of the urban grid hasindependent and systematic effects on movementpatterns, which could be captured by integration

analysis of the axial map2. In Cities as movement

economies (Hillier, 1996b) it was shown thatnatural movement and so ultimately the urbangrid itself impacted on land-use patterns byattracting movement-seeking uses such as retailto locations with high natural movement, andsending non-movement-seeking uses such asresidence to low natural movement locations.The attracted uses then attracted more movementto the high movement locations, and this in turnattracted further uses, creating a spiral of multi-plier effects and resulting in an urban pattern ofdense mixed use areas set against a background ofmore homogeneous, mainly residential develop-ment. In Centrality as a process (Hillier, 2000), itwas then shown that these processes not onlyresponded to well-defined configurational prop-erties of the urban grid, but also initiated changesin it by adapting the local grid conditions in the

*Correspondence: Space Syntax Laboratory, The Bartlett Schoolof Graduate Studies, 1-19 Torrington Place, University CollegeLondon, London WC1E 6BT, UK. Tel: +44 (0) 171 391 1739, Fax:+44 (0) 171 813 4363,E-mail: [email protected] axial map is the least set of longest lines of directmovement that pass through all the public space of asettlement and make all connections.

2The 1993 paper dealt only with global or radius-n analysis,but a series of studies since then have shown that local orradius-3 integration is normally a better predictor of pedes-trian movement.

URBAN DESIGN International (2002) 7, 153179r 2002 Palgrave Macmillan Ltd. 1357-5317/02 $15.00

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mixed movement areas in the direction of greaterlocal intensification and metric integrationthrough smaller scale blocks and more trip-efficient, permeable structures.

Taken together, the three papers describe aspectsof a generic mechanism through which humaneconomic and social activity puts its imprint onthe spatial form of the city. The papers do not dealwith the patterns of activity themselves, but thetheory seems to work because, regardless of thenature of activities, their relation to and impacton the urban grid is largely through the way theyimpact on and are impacted on by movement.Movement emerges as the strong force thatholds the whole urban system together, with thefundamental pattern of movement generated bythe urban grid itself. The urban grid therefore

emerges as a core urban element which, in spite ofits static nature, strongly influences the long-termdynamics of the whole urban system. In the lightof these results, we can reconceptualise the urbangrid as a system of configurational inequalities that is, the differences in integration values in thelines that make up the axial map whichgenerates a system ofattractional inequalities thatis, the different loadings of the lines with builtform densities and land-use mixes and notethat, in the last analysis, configuration generatesattraction.

Space-creating mechanisms

The three papers cited describe a process thatgoes from the spatial configuration of the urbangrid to the living city. But what about the griditself? Is this arbitrary? Would any grid config-uration set off the process? The aim of this paperis to try to answer this question. It will be arguedthat urban grid configurations are far fromarbitrary, but in fact are themselves the outcomesof space-creating mechanisms no less generic thanthe space-to-function mechanisms described inthe three cited papers. The argument runs asfollows. If we examine a large number of axialmaps, we find well-defined invariants as well asobvious differences. What process, we must ask,can produce both. The answer proposed is thatthe invariants arise from a combination of twothings. First, in spite of all their variability, thereare certain invariants in the social forces or moreprecisely in the relations between social forces that drive the process of settlement aggregation.

Second, there are autonomous spatial laws gov-erning the effects on spatial configuration of theplacing of objects such as buildings in space, andthese constitute a framework of laws withinwhich the aggregative processes that create

settlements take place. The social forces workingthrough the spatial laws create both the differ-ences and the invariants in settlement forms. Thelink between the two is again movement, butwhereas the space-to-function mechanism wasdriven by the effect of spatial configuration onmovement, the space creating mechanism isdriven by the influence of movement on space,and so can be considered a function-to-spacemechanism.

The concept of spatial laws is critical to thisargument, so we must explain what this means.

Spatial laws, in the sense the term is used here,does not refer to universal human behaviours ofthe kind claimed, for example, for the theory ofhuman territoriality (as reviewed in Vischer-Skaburskis, 1974), but to ifthen laws that saythat if we place an object here or there within aspatial system then certain predictable conse-quences follow for the ambient spatial configura-tion. Such effects are quite independent of humanwill or intention, but can be used by human

beings to achieve spatial and indeed social effects.Human beings are bound by these laws in thesense that they form a system of possibilities

and limits within which they evolve their spatialstrategies. However, human agents decide inde-pendently what their strategies should be. Likelanguage, the laws are then at once a constrainingframework and a system of possibilities to beexploited by individuals.

In fact, it seems likely that human beings alreadyintuitively know these laws (although theycannot make them explicit), and can exploit themas agents to create social effects through spatial

behaviours at a very young age. Considerthe following true story. A group of people aresitting in armchairs in my daughters flat. Mytwo-year-old grandson Freddie comes into theroom with two balloons attached to weights bytwo pieces of string about two and a half feet long,so that the balloons are at about head height forthe sitting people. Looking mischievous, he placesthe balloons in the centre of the space defined bythe armchairs. After a minute or two, thinkingFreddie has lost interest, one of the adults movesthe balloons from the centre of the space to the

The city as objectB. Hillier

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edge. Freddie, looking even more mischievous,walks over to the balloons and places them backin the centre of the room. Everyone understandsintuitively what is going on, including you. Butwhat is actually happening?

The answer is that by placing an object in thecentre of a space we create more obstruction tolines of sight and potential movement than ifwe place it at the edge. This is the principle ofcentrality set out in the theory of partitioning inChapter 8 of Space is the Machine (Hillier,1996a, b). If we place a partition midway on aline, it creates more and more evenly distributed gain (added distance in summing shortest tripsfrom all points to all others) in the universaldistance (the sum of distances from each point toall others) than if we place it peripherally (in

which case the depth gain is more unevenlydistributed, but is overall less). Because this mustapply to lines in all directions, it follows thatit will also work for objects placed in space. Anobject placed centrally in a space will increaseuniversal distance and interrupt intervisibilitymore than one placed at the edge. Now it is clearthat Freddie not only knows this in the sensethat he can make use of this knowledge in

behaviour, but it is also clear that he can use this surely theoretical knowledge of space toachieve social ends, namely drawing attention tohimself and away from the adults engaged inconversation. It is also of course clear that weknow this about space in the same way asFreddie, but it is also clear as professionals that itis unlikely that we were taught this vital principleof space in architecture school or in maths class.

What is proposed here is that spatial laws, drivenby social forces, account for exactly and only thespatial invariants of cities3. The form of the paperwill be to: examine axial maps and develop anaccount of their invariants as well as theirdifferences; outline and demonstrate the spatial

laws in question; apply these to what willbe called the basic generative process by whichurban-type spatial systems arise; and develop atheory of how the impact of the spatial laws onevolving settlements is driven by two kinds ofsocial forces, which can be broadly termedthe socio-cultural and the micro-economic. It is

proposed that culture is a variable and puts itsimprint mainly on the local texturing of space,generating its characteristic differences, whereasmicro-economics is a constant and puts itsimprint mainly on the emerging global structure

of the settlement in a more or less invariant way.The reason one works locally and the otherglobally is due to the ways in which each usesthe same spatial laws to generate or restrainpotential movement in the system.

This is why we find in axial maps both differencesin local texture, and invariants in the globalpatterning. The combination of the spatial lawsand the dual processes explains why axial mapsread as a set of similarities and differences. Thepaper concludes with a discussion of the relation

between socio-economic and spatial laws, sug-

gesting that although the creation of the space ofthe city is driven by socioeconomic processes it isnot shaped exclusively by them. Equally funda-mental in shaping city space are autonomousspatial laws that generate more or less equifinaloutcomes from varying processes4.

Differences and invariants in axial maps

First, let us consider some axial maps. By far themost obvious differences between them are

geometrical. On reflection, that is all they couldbe. Axial maps are no more than sets of lines ofdifferent lengths with different angles of intersec-tion and different degrees and kinds of intersec-tion (for example, a line can either pass throughanother or stop on it). Axial maps from differentparts of the world tend to differ in all theseproperties. Figures 14 show four fairly charac-teristic axial maps from different parts ofthe world arranged from the most to the leastgeometric: Atlanta (USA), The Hague (Holland),Manchester (UK) and Hamedan (Iran). It iseasy to see that the impression of more to

less geometric arises because the axial mapsdiffer substantially on the basic propertiesof axial maps. Each has its own distinctive rangeof line lengths and angles of the incidence, andits distinctive intersection characteristics. For

3In this sense, the argument is still within the spirit of thetheoretical framework set out by Martin and March in UrbanSpace and Structures in 1972.

4The question What about planned towns? may of course be raised here. However, in the great majority of cases theplanned element is only the first stage of an urban growthprocess that then will be subject to the same lawful influencesas cities which have grown through a distributed process.


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example, if we use the patterns of intersection, wefind that in Atlanta, the tendency for lines to passthrough each other (rather than to end on other

lines) is very marked at all levels. In The Hague,this is found locally but much less so at the globallevel. In Manchester, this is hardly found globally,and what there is locally is much more broken upthan in The Hague. In Hamedan, it hardly existseither at the global or local levels, except in thecentral public areas of the town.

Differences in the range of line lengths and anglesof incidence seem to follow the intersectiondifferences. Atlanta has a number of very longlines approximating the radius of the system, andlong lines can be found in most parts. At the sametime large areas of the grid maintain a strict right-angle intersection with a northsouth orientation,

Figure 1.

Figure 2.

Figure 3.

Figure 4.


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although with a striking offset grid in the historiccentre. In The Hague, the longest lines tend to beless than the radius of the system, and in generallong lines are peripheral to discrete local groupsof lines. A less strong right-angle discipline ismaintained, there is greater variety in the orienta-tion, and long lines, especially radials, tend to

intersect to others at very obtuse angles. InManchester, the long lines are nearly all radialswell below the radius of the system. The tendencyfor the long radials to have near-straight con-tinuations is even stronger, and the local right-angle discipline is even looser. In Hamedan, thelongest lines are only a fraction of the radius ofthe system and tend to be found towards theperiphery of the system. Even so there is a clearradial structure formed by lines of the secondlength rank, and intersecting with greater angularchange than in the other cases. Locally, we find awhole range of angles of incidence including

near-right-angle connections, but in most casesone line tends to stop on another.

These geometric differences are also consistentlyreflected in syntactic differences.

Table 1 shows the syntactic average for 58 citiestaken from four parts of the world. Each regionalgroup of cities, in spite of differences within thesubsamples, has its own characteristic set ofsyntactic parameters.

What is the reason for these geometrical andsyntactic differences? Why should lines in Iraniancities be, on average, markedly shorter than linesin, say, English cities, or why should Europeancities have a degree of geometric organisationsomewhere between UK and American cities,or Arab cities be less intelligible than Europeancities. On the face of it, the differences seemto be expressions of what we might call spatialculture. For example, in cities in the Arab world,the spectrum between public and private spaces

is often quite different from that in Europeancities. In historic European cities, we find thatlocal areas are for the most part easily permeableto strangers, with public spaces in locally centralareas easily accessible by strong lines fromthe edge of the area. At the same time, frontsof dwellings are strongly developed as facades

and interface directly with the street both in termsof visibility and movement. In many Arabcities, strangers tend to be guided much more tocertain public areas in the town, and accessto local areas is rendered much more forbidding

by the more complex axial structure. At the sametime, dwelling facades are much less developed,and the interface with the street tends to bemuch less direct both for visibility and formovement. The differences in the geometry ofthe axial maps seem to be a natural expression ofthese differences. Even in the case of Americancities, where one of the main factors in creating

the more uniform American grid is thought to bethe need to parcel up land as quickly and easily aspossible to facilitate economic development, wenote that the grid was prior to economic devel-opment and should therefore be seen as a spatialcultural decision to create and use space in acertain way.

However, in spite of these differences, there arealso powerful invariants in axial maps that seemto go across cultures and even across scales ofsettlement. One of the most striking is thestatistical distribution of line lengths. Althoughwe find great variations in the average and rangeof line lengths, we invariably find:

that the axial maps of cities are made up of asmall number of long lines and a large numberof short lines; that this becomes more the case as cities

become larger; and that in general the distribution of line lengths incities approximates a logarithmic distribution.

Table 1

Cities Avg. Lines Conn Loc Int Glob Int Intel

Usa 12 5420 5.835 2.956 1.610 0.559euro 15 5030 4.609 2.254 0.918 0.266uk 13 4440 3.713 2.148 0.720 0.232arab 18 840 2.975 1.619 0.650 0.160

Conn connectivity; Loc Int local integration; Glob Int global integration;Intel intelligibility.


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Figure 5 shows the four cities of Figures 14with the distribution of line lengths on the leftand the logged distribution on the right5.

In practical terms, this means that if, for example,we divide the range of line lengths into 10, wefind that in Atlanta 92.7% of lines are in the decileof shortest lines and only 2% in the eight longest.

In The Hague, the figures are 84.8 and 5%, and inManchester 85.9% and less than 3%. In the muchsmaller case of Hamedan, we find that 90% of thelines are in the four shortest deciles and only 2%in the five longest. Looking more widely, we findthat in London (15,919 lines) 93.3% of lines are inthe shortest decile and less than 1% in the topeight deciles. In Amsterdam (7996 lines) on theface of it a more griddy city the figure is 95.8%in the shortest decile and again less than 1% in thetop eight. In Santiago (29,808 lines), an even moregrid-like structure, the figure is 94.7%, again with1% in the top eight, while in Chicago (30,469

lines), a city we think of as wholly grid-like, thefigure is 97.6% with only 0.6% in the top eight. Itis not always quite so high in large cities. InAthens (23,519 lines), for example, the figure is86%, with 2.3% in the top eight. However, even in

the strange pre-Columbian city of Teotihuacan thefigure is 85%.

If we look at a smaller system, we find the sametendency, although less marked. In Venice (2556lines), for example, the figure is 76.3% with 4% inthe top eight; in Shiraz (1971 lines) in Iran, wherelines are on average shorter than in Western cities,71.7% are in the shortest decile and 8.3% in the

top eight. In the English cities of Nottingham,Bristol and York, the figures for the shortest decileare 78, 63 and 55%, respectively. Even in muchsmaller systems, we can find a strong tendencyin this direction. If we take Old Paranoa, theinformal settlement built by the workers whoconstructed the dam in Brasilia (de Holanda,1977), we find that 32% of the lines are inthe shortest decile and 68% in the shortest two.In the southern French town of Apt, 41% are inthe shortest decile and 59% in the shortest two,while in Serowe (a self-generated settlement in

southern Africa) 32% are in the shortest decile and68% in the shortest two. Even in a small areawithin London we find 24% in the shortest decileand 53% in the shortest two.

As settlements grow, then, the proportion of linesthat are long relative to the mean for thesettlement becomes smaller but the lines

Figure 5.

5In some cases, such as Chicago and Amsterdam, we find aloglognormal distribution. However, the difference between alog and a loglog distribution is much greater than that betweenan unlogged and logged distribution, so these differences arenot pursued here.

*Note: Only selected images have been printed along with thetext here.


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themselves get longer. This seems to be invariantacross all cultures in spite of the strong geometricdifferences we have noted. A plot of the log ofthe number of lines against the proportion in theshortest decile for 20 settlements from small to

very large, showing an r

2

of 0.802 (p 0.0001)

5

.This also applies to different-sized chunks of thesame city. If we plot the percentage of linesin the shortest decile against the number of axiallines for four different-sized cutouts from theLondon axial map, showing an r2 of 0.923,p 0.0391. However, even the smallest cutout the City of London with only 565 lines (asopposed to 15919 for the largest system) has70% of lines in the shortest decile, and approx-imates a logarithmic distribution.

Why then are line lengths distributed in this way

and, in particular, what is the role of the smallnumber of long lines? A useful clue comes fromlooking at their spatial distribution. If we take thelines in the longest quintile of the range and makethem the darkest lines in the axial map, we find amarked tendency for the longest lines to be centreto edge lines starting at some distance from theoriginal centre. Figure 6 shows the pattern forLondon, and Figure 7 for Athens. The second rankof lines, however, shows a different pattern ineach case. In London, the second-rank lines forma continuous and relatively dense network pene-trating most parts of the grid. In Athens, thesecond-rank lines pick out discrete grid-like areaswith relatively poor connections between them. Ifwe were to look at, say, Baltimore, the second-rank lines tend to be linked directly to the firstrank of lines, forming a tree-like distribution in

the system as a whole. These patterns suggest thatthe first rank of lines reflects generic properties ofcity growth while the second rank indicatesdifferences in the relation of global to local.

This hint of global invariants and local differencesis reinforced if we look at the syntactic analysis ofthe axial maps. If we take the four cities shownin Figures 14 and analyse them for radius-nintegration (for example, Figure 8), we find ineach case that in spite of the geometric differencesa certain kind of structure is adumbrated: each

city has an integration core the patterns formedby the darker which links a grid-like pattern oflines at the heart of the city almost to the edge inall directions either by way of quasi-radial lines orextended orthogonal lines, in some cases reachingthe edge line but in others falling short. Within theinterstices formed by this overall lighter areas arefound, often with a darkish line as a local focus. Inother words, in spite of the geometric differences,each city has, when seen as a system of config-urational inequalities, a certain similarity ofstructure. This is the pattern we call the de-formed wheel: a hub, spokes in all main direc-tions, sometimes a partial rim of major lines, withless integrated, usually more residential, areas inthe interstice forms by the wheel. This genericpattern was first identified as a deep structurecommon to many small towns, seeming to occurin spite of topographic differences (Hillier, 1990).It was also found as a local area structure inLondon, where named areas such as Soho orBarnsbury typically took an area deformed wheelform with the London supergrid (the mainFigure 6.

Figure 7.


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radials and their lateral links) forming the rim ofthe wheel.

As a global pattern, the deformed wheel holds upremarkably well in larger cities for example, inLondon, Athens and Baltimore. The pattern iseven found in very different kinds of cities. Ifwe look at Venice (Figure 9) without the canals,for example, we find that in spite of its veryidiosyncratic history having grown togetherfrom several islands rather than from a singleorigin we still find a very marked deformedwheel pattern, even though the wheel is muchless easily recognisable than in most cases. Orlooking at Tokyo, which is by far the largestsystem ever analysed, we find a remarkable andeven more complex version of the wheel patternwith several layers of rim which, with the sinuousradials, produce a quasi-grid covers a large part ofthe system. Even the strange pre-columbian cityof Teotihuacan shows, at least a partial realisationof the deformed wheel pattern. Again this nearinvariant of cities is found in spite of thesubstantial differences in syntactic values thatwere shown in Table 1.

In addition to these space invariants, we also findthat if we look at settlements in terms of the sizeand shape of blocks, then we find if not invariantsthen at least a set of pervasive tendencies, once

again set against a background of substantialgeometric differences by region (which we maytherefore expect to have a cultural origin of somekind). These can be seen fairly easily in the axialmaps of Figures 14, and even in the analysedaxial maps, but is perhaps easier to see in blackon white figure ground maps of a Turkish cityanalysed by Sema Kubat shown in Figure 10. Themost obvious near invariant is an underlyingtendency for blocks to be smaller and more convexat or near the centre and larger and less convextowards the edges (Hillier, 2000). However, if werelate block size and shape to the patterns shown

by integration analysis of the axial map, we find asubtler pattern. The lines forming the spokes of thedeformed wheel tend to be lined with larger thanaverage (for the settlement) blocks for most of theirlength, but smaller than average blocks in thecentre (the hub of the wheel).6 In contrast, theareas interstitial to the core tend to have blocksizes between these two extremes. In other words,the distribution of block sizes seems to reflectthe distinction between global and local structure.This is to some extent the case in all the settle-ments shown so far.

Figure 8.

Figure 9.

6This phenomenon is also found in subcentres. In Centrality asa process (Hillier, 2000), it was argued that wherever move-ment is convex and circulatory (ie moves around in a locallytwo-dimensional grid as eg in a shopping centre) rather thanlinear and oriented (as in moving through an urban grid froman origin to a destination), then metric integration was the keyproperty in understanding both the movement pattern and thetype of spatial configuration that tended to emerge under theseconditions.


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Socio-cultural relativities andeconomic universals

We are faced then with a puzzle. The processesthat generate the axial maps and block maps ofcities seem at the same time to produce variants,in the form of systematic differences in settlementgeometry and syntax from one region to another,and also invariants. What kind of process canproduce both? It seems highly unlikely that thesedual patterns are in any sense designed in,although of course they may be in some cases.However, the fact that most settlements evolveover long periods compels us to the view that

the patterns arise from a largely distributed orbottom-up process, that is, from multiple inter-ventions by many agents over time. Even if singleagencies are involved, then even so the fact thatsettlements evolve over such long periods implies

that the process of settlement generation must beregarded as an essentially distributed one. Whatkind of distributed process, then, can producesuch dual emergent phenomena?

Let us first note an important difference betweenvariants and invariants: the variants tend to belocal and the invariants global. Now considera case where a city has grown under the influenceof at least two different cultures: Nicosia.Figure 11 is an analysed axial map of the historicalcore of Nicosia in Cyprus (a city sadly nowdivided). The north-east quarter is a historic

Turkish area, the south-east a historic Greek area.The differences in the texture of the grid aremarked, with the two areas having quite differentgeometries and different emergent topologies: theGreek area has longer lines, more lines passingthrough each other, a different pattern of angle ofincidence, and, as a result, much more local andglobal integration (and a better relation betweenthe two) than the Turkish area. Since thesedifferences reflect typical differences found be-tween systems in Europe and the Islamic world, itis reasonable to regard these as socio-culturaldifferences in the basic geometry of space.

However, when we analyse the area as a wholewe find a typical deformed wheel pattern hassomehow arisen over and above these geome-trical differences, even though the differences bet-ween the Greek and Turkish areas show upstrongly as differences in the degree of integration.

We thus see what appear to be two processesoperating in parallel: one a local process generat-ing differences in local grid patterns and appar-ently reflecting differences in spatial culture insome way; and the other a global processgenerating a single overriding structure thatseems to reflect a more generic or universalprocess of some kind. A clue to this comes fromthe simple fact that the less integrated areasgenerated by the local process are largely resi-dential, and it would be natural to think of theseas the primary distributed loci of socio-culturalidentities, it being through domestic space and itsenvirons (including local religious and cultural

buildings) that culture is most strongly repro-duced through the spatiality of everyday life. A

Figure 10.

Figure 11.


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second clue comes from the fact that when weanalyse settlements syntactically, it is the micro-economic activity of markets, exchange andtrading that is most strongly associated with thentegration core, religious and civic buildings

being much more variably located (Loumi, 1988;Karimi, 1998; Hillier, 2000). In this, of course, theintegration core of public space also reflects thespatiality of everyday life, but in this case it tends

both to the global, because micro-economicactivity in its nature will seek to extend ratherthan confine itself, and also to be culturallynonspecific, in that it is in these activities, andtherefore these spaces, that people mix andcultural differences are backgrounded.

This suggests a natural explanation for the dualproduction of variants and invariants in urbangrids. On the one hand, a residential processdriven by socio-cultural forces puts its imprinton local space by specifying its geometry andgenerates a distinctive pattern of local differences,

because culture is spatially specific. On the other,a public space process driven by micro-economicactivity generates a globalising pattern of spacethat tends to be everywhere similar becausemicro-economic activity is a spatial universal.This is the critical difference between the twoaspects of the settlement creating process: thesocio-cultural component is idiosyncratic and

local while the micro-economic component isuniversal and global. It is this that creates theunderlying pattern of differences and invariantsthat we find everywhere in settlement forms.

This is the key conjecture of this paper: that theprocesses that generate settlement forms areessentially dual, and through this duality gener-ate the invariant pattern of local differences andglobal similarities that characterises settlementforms. The question then arises: why shouldsocio-cultural life generate one kind of spatialpattern and micro-economic life another? Theanswer, it will be proposed, lies in the fact that therelation between micro-economic activity andspace, like the relation between culture and space,is largely mediated by movement, but micro-economic economics in a universal and globalway, culture in a local and specific way. In whatfollows we will therefore look at spatial andmovement aspects of both socio-cultural andmicro-economic processes and how they affecteach other as a settlement grows.

The basic generative process

We can begin by noting that there is also a set oflow-level invariants, or near-invariants, in urbanspace, which are so commonplace as to be rarely

remarked on, but which are the very foundationof what a settlement is. These are

that most spaces are linear, defined by theentrances of buildings or groups of buildingson both sides; that buildings are clumped together to formdiscrete islands; so that the linear spaces surroundings theislands form intersecting rings and create anoverall system of continuous space (a streetpattern of some kind); and that this is a highly nondendritic configura-tion, that is a pattern that is everywhere ringyrather than tree-like.

The simplest process for generating spatial con-figurations with these properties has been famil-iar since the earliest days of space syntax: therestricted random beady ring process that gen-erates small ring street settlements of a kindfound in many parts of the world (Hillier andHanson, 1984). The process starts with a dyadcomposed of a cell (representing a notional

building) and a piece of open space linked by anentrance so that those inside can come and go into

the outside world. These dyads aggregate ran-domly apart from two restrictions: that each opencell must join full-facewise onto one already in thesystem (joins of closed cells arise only randomly);and no vertex joins for closed cells are allowed(people do not build corner to corner)7.

The pattern on the left of Figure 12 is a typicalproduct of such a process. A beady-ring-typepattern is produced on the way, but this is not ourmain concern here. The overall pattern is that asystem of outward facing islands of built formsvarying in size and creating more or less linearspaces forming intersecting rings has emergedfrom the process. No one designed this. It hasemerged by a process that finds a pathway ofemer-gence by which a global pattern appears

7In the version of the process set out in The Social Logic of Space,the open space of the dyad was the same size as the built cell.In the version shown here, this has been retained, but the builtcells have then been expanded without expanding the openspaces, with the effect that the scaling of open spaces and

buildings approximates real systems more closely.


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from the actions of local agents. A key element ofthe urban system has thus emerged in the form of

a continuous system of open space, permittinginteraccessibility from each part of the settlementto all others.

The pattern thus has enough of the key topolo-gical settlement-like properties (although it lackstheir geometric properties but see below) for usto think of it by Ockhams razor perhaps as thebasic generative process for spatial patterns of agenerically urban kind. But it does not yet look atall like a real settlement. What is missing? Itcannot be just the over- regularity due to the factthat the process has been generated on a regular

grating. The fault seems to lie mainly in thegeometry of both its block structure and its line oraxial structure: blocks are insufficiently compactand lines are insufficiently varied in length. So letus look at two real settlements that seem to havegrown by something like this process and seewhat they have in addition. Figure 13 is the oldself-generated settlement of Paranoa, developedfrom the encampment of the workers who builtthe dam for the lake behind Brasilia (de Holanda,1997). Figure 14 is the settlement of Serowe insouth-west Africa in which the built elements areactually compounds. On the right are radius-nintegration maps of each, and the synergy scatter-gram plotting the correlation between local andglobal integration. On the right of Figure 12 is thesame analysis of the computer-generated pattern.

Two points are of particular interest. First, some-thing like the deformed wheel integration coreexists in both real cases (and in the case ofParanoa cannot be explained in terms of existingroutes in the direction of other settlements, since

there were none except to the south). Second,when we look at the synergy scattergrams, we

find that the r2

-between local (radius-3) andglobal (radius-n) integration is much better thanin the generated case in spite of the fact that itlacks the discipline of an underlying grid. In otherwords, Paranoa and Serowe both display arelation between local and global structure thatneeds to be explained.

Experiments with random lines

We can explore these differences further byexperimentation. We first construct a more or lessrandom rectilinear grid made up of lines that varyin length only a small amount, on average abouthalf the diameter of the overall settlement. Thescattergram gives an r2 between connectivity andintegration of over 0.88,9. We then retain the samemean and range of line length but grow the systemto twice its size. Its diameter is now about threetimes the mean line length. The intelligibility r2

falls to 0.5. We do the same again, increasing thesize of the system until its diameter is about fourtimes the mean line length. The r2 falls to below0.3, as in Figure 15.

It is not difficult to work out what is happening. Ifintegration analysis is carried out on a system

Figure 12. Figure 13.

8Old Paranoa has now been pulled down by the planningauthorities and replaced by a much more regular settlement.9The intelligibility correlation between connectivity andglobal integration is used here rather than the synergycorrelation between local and global because the systems areinitially too small to respond realistically to local integrationanalysis. The argument would however hold up for synergyanalysis.


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with uniform elements much smaller than thesystem itself say a tessellation of square cells then integration will focus on the geometric centre

and fall off towards the edges. As soon as youspecify a system with more or less similardimensions, in this case similar line lengths, thenthe same must happen. As the system becomeslarger, integration will increasingly concentrate inthe centre. The consequences for the intelligibil-ity relation between line connectivity (which isclosely related to length) and integration are thatrelatively longer and therefore better connectedlines will be randomly distributed through thesystem, while integration will be concentrated inthe centre. The more this happens, the less the twowill correlate and the more the local properties

of the system give a poor guide to the globalproperties hence unintelligibility.

If, we then take four lines near the centre andextend them to a length of about 0.75 of thediameter of the system, the effect on both theintegration core and the scattergram is immediateand dramatic. The core, not surprisingly, begins to

go from centre to edge and the scattergramimproves from below 0.3 to above 0.6. However,the scatter is highly non-urban, in that the fournew lines are quite distinct from the rest of thesystem. But in Figure 16, an r2 of .86 is achievedwith a pattern of lines that links laterally atthe edges as well as from centre to edge: thecharacteristic deformed wheel structure. In thisaxial map, 47% of lines are in the shortest decileand a further 29% in the next shortest, almostidentical to Paranoa, where the respective figuresare 52 and 25%.

This suggests that the essential function of thelonger lines against the background of shorterlines is, as we might expect, to give some kind ofglobal structure to the overall pattern, with thelocal structure fitted into its interstices. However,two further points must be added. First, we alsofind that the pattern of long to short lines iscritical not just to the global structure but alsoto the relation between the local and globalstructure. This suggests that the long to shortdistribution is pervasive at all levels of thesettlement and its growth, and therefore needsto be understood as an outcome of a growth

process rather than as one of imposition of aglobal structure. In other words, we need to

Figure 14.

7.0000

Connectivity

Connectivity

Integration

1.3355

Slope = 3.9489Intercept -0.0111R^2 = 0.2979

Mean = 3.6964

Mean = 0.9389Integration

Figure 15.


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understand how the required distribution of linelengths can be produced at every stage of anaggregative process of settlement growth.

Regularities in the configurationaleffects of placing objects

How then can we modify the basic generativeprocess to create these outcomes at every level, sothat the growth process will tend to create notonly a pervasively lognormal distribution at everylevel, but at the same time generate an intelligibleand synergic system with a deformed wheel-typestructure? The answer proposed is that it is herethat spatial laws intervene, driven by the dualsocio-cultural and micro-economic forces impos-ing on space their different requirements forpotential movement.

The laws in question govern the effects on spatialconfiguration of the placing objects (such as

buildings) in space. The laws initially governthe degree of metric integration in the system

measured as the universal distance10

from eachcell in the complex to all others (as opposed to aspecific distance that measures distance from one

cell to one other). The mean universal distance ina complex is thus isomorphic to the mean lengthof the trip by shortest paths within the complex. Itis through their effect on mean trip lengths thatthese laws are activated and govern the evolutionof the urban object.

The laws are essentially clarifications, simplifica-tions and fuller demonstrations of the principlesof partitioning set out in Chapter 8 ofSpace is the

Machine. There it was shown that every time apartition is placed in a system, it has a predictableeffect on universal distance within that system.In that text, four partitioning principles wereproposed for the minimising or maximising ofdepth gain in a system, depth gain being theincrease in universal distance due to the placingof a partition. The principles were: centrality partitioning a line in its centre creates more depthgain than partitioning it eccentrically; extension partitioning a longer line creates more depth gainthan partitioning a shorter line; contiguity making partitions contiguous increases depthgain more than making them discrete; and

linearity arranging contiguous partitions linearlyincreases depth gain more than coiling them up,as, for example, in a room.

In what follows it will be proposed that these fourprinciples11 can be reduced to two laws, onedealing with the relations of spaces and the other

19.0000

Connectivity

Connectivity

Integration

3.4641

Slope = 8.2506Intercept -11.2848R^2 = 0.8755

Mean = 5.7477

Mean = 2.0644Integration

Figure 16.

10For an account of the idea of universal distance see Space isthe Machine, Chapter 3. Universal distance is probably the mostfundamental concept in space syntax. It can be applied eithermetrically or topologically, and allows the redefinition of anelement in a system as no more than a position from whichthe rest of the system can be seen, thus nearly dissolving theelements. (see Space as paradigm in the Proceedings of theBrasilia Space syntax Symposium).

11On reflection, what were noted in Space in the Machine wereempirical regularities, since no theoretical account was offeredas to why they should be so.


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with the relations of objects. Before we introducethese laws however, we will show how thesepartitioning regularities can be interpreted forcellular aggregates. The basic notion we workwith is that of a pair of cells (or boundaries)forming the two ends of a line, and a third cellwhich we wish to place between them. Themethod for calculating the gain in universaldistance is as in Figure 17. Consider a line of

n+1 cells with an object placed somewhere alongit leaving n cells in some distribution on the twosides of the cell with at least one cell on each side.A deviation, d, will be the unit distance aroundthe object that must be added to straight linemovement to go from any cell to any other on theother side of the object. D will be the sum of dsthat are needed to go from all cells to all others, orthe total added to the universal distance on thatline by the placing of an object.

If the object is square and its sides are the samesize as the unit of distance for measuring alongthe line, as in the top case in Figure 17, then d willalways be 2 units of distance. Here we refer tothe 2-unit deviation as a single d. Note that if anobject with, say, shape 3 1 is placed on theline lengthwise, then, as in the middle case inFigure 17, d for negotiating that object will always

be 2 units regardless of the length of the unit,because the trip between the two deviation unitsis parallel to the original line. If, however, the3 1 object is placed orthogonal to the line of

movement (see the bottom case in Figure 41), thena further two units of distance, that is one furtherd, will be added for every parallel line blocked bythe object.

Figure 18 illustrates the principle of centrality: ifwe want to place a cell (the light cell top left)

between two existing cells (dark), does it make adifference where we place it? The answer (mid-

left) is that the more peripherally we place it, theless the increase in universal distance, andthe more centrally we place it, the greater theincrease. It follows (bottom left) that if we placecells evenly along lines, the increase in universaldistance is greater than if we make some gapslarge and others small. It also follows (mid-right)that an object placed in the centre of a space willincrease universal distance more than one placetowards the edge (because the effect on twodimensions will be the sum of linear effects). Theprinciple of extension also follows: if we placea block on a longer line, it increases universaldistance more than if we place it on a shorter line.Figure 19 illustrates the principle of contiguity:cells joined contiguously increase universal dis-tance more than those placed discretely. Finally,Figure 20 illustrates the principle of linearity:contiguous cells arranged linearly increase uni-versal distance more than if they are placedcompactly. It should be noted that these principlesinteract. For example, in a finite space placingone cell contiguously with another will locally

DEFINING DEVIATION, d

line of movement

line of movement

line of movement

For a single unit sized block the total deviation required to go round it is 2 additional unit cells

line of movement

line of movement

line of movement

For a 3*1 block placed lengthwise along the line of movenment, the total deviation is still twoadditional unit cells, since the section of the deviation that is parallel to the line of movementadds no additional distance

For a 3*1 block placed sideways on the line of movement, a further 2 units of distance areadded for each unit of distance the deviation requires from the line of movement

Figure 17.


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increase distance but will also expand the spacebetween the composite object and the boundary,thus reducing universal distance.

The impact of these laws on grids can be explored by constructing experimental grids made up ofmetrically uniform cells (they can be as small aswe like, as long as they are uniform), andcalculating the mean universal distance, or meantrip length, for each. Figure 21 sets out a series of

experiments with grids each with 301 metricallyuniform cells. The cells are circular in orderto avoid the effect of corner joins. Each grid thushas the same number of metric cells and thereforethe same number of distance elements. Differ-ences between grids are therefore purely to dowith the rearrangement of the cells into differentconfigurations. In some cases, the rearrangementhas left one cell that cannot be located in the grid.In each of these cases the cell has been added to

Figure 18.

4

4

4 4

4 4

4 4

4 4 4 4 4 4

4

4

4 4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

2

2

2

2

2 22

2

2

2

2

2

2

4

total depth gain in the immediate

neighbourhood: 20 for 6 cells

total depth gain in the second














This is the principle of CONTIGUITY: the more we make blocks contiguous, the less integrated the surrounding space

Do we minimise depth gain by making blocks contiguous or discrete ? The answer is that other

things being equal, contiguous blocks will always cr eate more depth gain than discrete blocks.

6 6 6 6 6

6

1 2 1 6 1 2

1 2 1 6 1 2

1 2 1 6 1 2

1 2 1 6 1 2

6

6 8

8

6 68

8 8

8 8

8 8

8 8

8 8 810 106 68

2

Figure 19.


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the same position in the grid, namely theintersection of the third column (counting fromthe left) and the third row (counting from the top).Experiments with the sensitivity of the grid tothe addition on one overlaid cell show that anadditional cell overlaid in the centre of theuniform regular grid reduces the mean universaldistance by 0.1% (it will of course slightly increasethe total since there is an additional cell), while

overlaying it on a corner cell increases it by 0.2%.These differences are then one or two orders ofmagnitude less than the effects of configurationalchanges below, and so can be discounted.

Using the regular uniform grid (Grid A) asthe benchmark, we can then vary the configura-tions of grids to illustrate the effect of thefour principles. In Grids B and C for example,

4

4

4 4

4 4

4 4

6 6 4 4

4 4

4 4

4

2

6

6

2

4

4

4 4 6 8 8 6

8 12 12 8

8 12 12 8

16 24 24 1612

12 12

12 12 12

1232

16

16 16

12 1216 16

16 24 24 16

16 24 24 16

16 24 24 16

16

4 4

8 8

8 8

8 8

8 8

8 8

8 8

8 8

8 8

total depth gain in the

immediate neighbourhood: 32






neighbourhood: 128





This is the principle of COMPACTNESS: the more compact we make contiguous blocks, the more we increase the integration of the

surrounding spaces

Given that blocks are contiguous, do we minimise depth gain by making them linear or compact ? The answer is that the more linearly

contiguous blocks are arranged, the more depth gain

Figure 20.

Figure 21.


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we illustrate the centrality principle by placinga block initially in the centre and then in thecorner, while standardising the layout of thecells displaced from the centre. Placing the blockin the centre increases the universal distance

of the grid by 2.6%, while placing it in the cornerreduces it by 0.3%. In Grid D, we take thisfurther by reducing the scale of blocks in thecentre at the cost of increasing them at the edge (acommon form in the centre of towns, as noted inCentrality as a process (Hillier, 2000)). The meanuniversal distance is reduced by 6.3%. If we dothe opposite and make the centre block as large aspossible, and place the small blocks at the edges(the number of small blocks remains the same, asin Grid E), we increase the mean universaldistance by 13.9%, making a total difference

between Grids D and E of just under 20%.

In Grid F, we take Grid E and create a crosslink through the centre. The effect is to increasemean universal distance by 9.1% compared toGrid A, but to reduce it by nearly 5% compared toGrid E.

We then illustrate the principle of extension. InGrid G we displace each vertical segment of cells

between grid intersections one cell to the rightand then to the left on alternate rows. We thusshorten all internal vertical lines with more or lessneutral effects on block sizes. The effect is to

increase mean universal distance by 4.2%. In GridH, we break all horizontal lines close to the centrevertical, creating a pair of lines of fairly equallength at each level. The increase in meanuniversal distance is 1.6%. However, when we

break the horizontal line near the edge vertical inGrid J, thus keeping some lines as long as possibleat the expense of others becoming much shorter,the mean universal distance increases by only0.5%, three times less than with a more central

break in the lines.

The principle of compactness is illustrated in GridK by converting the square central block of Grid Binto a linear block of equal area. The effect is toincrease the universal distance by 6.2% comparedto 2.6% for the square block. We then illustrate theprinciple of contiguity by splitting the linear blockinto two in Grid L. The increase in universaldistance is 1.7% compared with Grid A, butof course it is nearly four times less than for thecontiguous linear block.

These grids are illustrative of course rather thana proper test, because huge combinatorics areinvolved, and in complex situations the fourprinciples will interact. For example, in Grid M,we break many lines, and also make many smaller

blocks in the centre. The result is a decrease inuniversal distance of 1.9% compared with Grid Ain spite of the shortening of lines.

The law of centrality

It is now proposed that these four principles canbe reduced to two formally demonstrable laws: alaw of centrality and a law of compactness. Thelaw of centrality proposes that an object placedcentrally in a space will increase universal

distance more than one placed peripherally.Consider again the line ofn+1 cells with an objectplaced somewhere along it leaving n cells in somedistribution on the two sides of the cell with atleast one cell on each side. Wherever we place theobject, D for one side of the line must be equal toD for the other, since each cell acquires one d foreach cell on the other side of the object. (seeFigure 18 mid-left). For example, if there are xcells on one side of the line and y on the other,then on one side D will be x*y and on the other

y*x. To establish D then, we need only establish itfor one side of the line, since we may then

multiply by 2 to get the total for the whole line.We therefore work by calculating D as the sum ofds for one side of the line.

Suppose then that the object is placed centrally onthe line. It will then have equal numbers of cellson either side. Let m ( (n1)/2) be the numberof cells on each side of the object. Each ofm cellson one side then requires one deviation to go toeach of cells on the other, giving a total of m*m orm2 deviations for each side. The total deviations,D, for the line with a centrally placed object, c, is

then 2(m2

) or m2

for each side:Dc 2m2 1

Now move the object one cell sideways. The totaldeviations for one side will then be (m1)(m+1)and for the other (m+1)(m1) or 2(m1)(m+1)for the whole line. Now m24(m1)(m+1) is anecessary inequality, as for example 3242*4 or4243*5. Similarly, (m1)(m+1)4(m2)(m+2) is anecessary inequality, as 2*441*5 or 3*542*6. Ingeneral: for D(c1c2ycn representing steps away


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19/27

practical purposes a square), then the lessthe increase in universal distance in the surround-ing space. This may be shown by first consideringthe effect, as before, of placing an object on a lineofn+1 cells. We know that the maximum increase

in universal distance for each side is m

2

(m (n1)/2) for the case where the object isplaced centrally. If we then place a discrete objecton a second line with at least one line between thenew and old line, then the gain on the second linewill also be m2, since the objects do not affect eachother (see Chapter 8 of Space is the Machine, for adiscussion of the case where lines are neigh-

bours). In general, the depth gain for singlediscrete objects placed on distinct lines will be2(m2) or n(m2), where n is the number of lines. Therate of increase is therefore linear.

Now suppose that the objects are placed con-tiguously on neighbouring lines. This creates amore complex situation in terms of depth gain,which is illustrated in Figure 20. As we can see,depth gain is least at the edges and greatest in thecentre. With m being the length of the line blockedand n the length of the partition ( the number oflines blocked), the depth gain can be calculated bythe finite series

D n2m2 n 22m2 n 42m2 . . .

n n2m23

which gives a third-order polynomial function

for the increase in universal distance with eitherincreased partition length or line length. Itcan then be compared to the linear rate fordiscrete cells. If blocks are discrete, then universaldistance increases linearly, and if contiguous theincrease is a third-order polynomial function withincreasing contiguity. This demonstrates the oldprinciple of contiguity. However, as we will see

below, we must also unify this with the idea ofcompactness.

Consider the effect of an aggregate of objectsforming an overall shape placed on a regular gridof lines. The shape will increase universaldistance in two directions in the grid, which wecan think of as horizontal and the vertical.Holding m, the length of the line on either sideto the shape, constant, the increase in universaldistance in one direction will be a third-orderpolynomial function of n, the number of contig-uous cells composing that face of the shape.Alternatively, we can hold n constant and vary m,with the same result. These calculations will not

be affected by the number of cells on the adjacentside of the shape, since these will only increaseuniversal distance in the other, orthogonal direc-tion. The overall increase in universal distanceresulting from the imposition of the shape of the

grid will then be the sum of the effects on eachdirection of the lengths of the two different facesof the composite object blocking that directioncalculated by formula (3) applied independentlyto both directions.

Suppose then that the sides are equal, that is theobject is maximally compact, say 2 2:

Holding m constant at, say, 3, the gain in universal

distance will be 2(n2

m2

) 2(22

32

) 72 for eachdirection (made up of the two half-lines), or4(n2m2) 144 for the whole object.

Now alter the shape of the object to a 1 4:

The gain in the vertical direction will now be(4232)+(2232) 180 for each half-line, *2 for thepair of half-lines 360. That in the horizontaldirection will be 2(1232) 18 for the pair of half-lines. The total gain is then 378 compared to 144for the square object. In fact, if we reduce theobject to a linear block of three cells:

then we have 2((3232)+(1232)) 180 for the verticaldirection and 2(1232) 18 for the horizontaldirection, giving 208 which is still greaterthan 144.

The reason for the increase is simple. Since m isconstant, n is the only variable in equation (3).When the block is square then D 2n2. However,if we replace the square object with a rectangularobject, say, (n1) on one side and (n+1) on theadjacent side, then all we need to know is therelation between (n1)2+(n+1)2 for the two un-equal half-lines of the rectangular object and 2n2

for the two equal half-lines of the square object.Since (n1)2+(n+1)2 (n22n+1)+(n2+2n+1) 2n2+2,it follows that (n1)2+(n1)242n2 and that in


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general

n x2n x2on x y2

n x y24

From this it follows that a compact form will

always generate less depth gain than an elongatedform of equal area, and that the differenceincreases rapidly with increased elongation. Aswith the law of centrality, a simple geometricalidea underlies the law of compactness.

Impact of the laws on the basicgenerative process

How then do the spatial laws impact on the basicgenerative process? We have already seen that thesocial forces driving settlement formation are

dual, with a residential component, driven bysocio-cultural forces, and a public space compo-nent, driven by micro-economic forces. Thesecorrespond to a duality in the settlement formitself, with the invariant deformed wheel globalstructure formed by the public space process andthe culturally specific interstitial local backgroundareas formed by the residential process. We alsonote that there is a duality in the spatial laws, inthat the compactness law addresses the physicalcomponent of the settlement, that is the size andshape of aggregate objects (ie blocks), while thecentrality law addresses the spatial component,

that is length of lines, distance of objects fromeach other, and so on. We recall that the output ofthe basic generative process in Figure 12 wasdeficient in both respects: blocks were overlyvaried in their shape and lines were insufficientlyvaried in their length. Our task was to explain thedifferences between the computer-generatedmodel and the real cases by showing how thedual social processes impacted on the basicgenerative process through the intermediary ofthe spatial laws.

Two conjectures can now be proposed. The basicgenerative process guaranteed interaccessibility

but it did not specify its degree or type, that is, itdid not specify a more or less integrated processor a particular local geometry. To control this, onewould need in the first instance to set a parameterfor the compactness law regulating the size andshape of blocks, by specifying, for example, forhow long and where one could continue addingto an existing block and when a new one had to bestarted. Such a parameter would in effect specify

how the compactness law would influence thepattern and degree of universal distance in the

background structure of the system in general.The first conjecture is that it is this localinteraccessibility parameter controlling the gen-

eric block structure and operating through thecompactness law that is set by the residentialprocess and its socio-cultural drivers. It is throughthis that the characteristic local geometry of spaceis created in the first instance in the backgroundresidential areas of the settlement. Where this isset differently by different cultures, we find thekind of differences noted in the different parts ofNicosia (Figure 11). Where it is more homoge-neous, we generate the kinds of generic regionaldifferences in axial geometry that are indexed inthe geometric and syntax values (Table 1) set outearlier.

The second conjecture is that with the growth ofthe settlement (and in fact in quite early stages)the public space process, led by micro-economicactivity, sets a global interaccessibility parameterworking through the centrality law. Since micro-economic activity is by nature integrative, this isnot a variable, but a constant. Its effect is alwaysto seek to conserve longer lines and to use theseto minimise universal distance in the larger scalesystem. Since the effects it seeks are spatial, itoperates directly on space and therefore worksthrough the centrality law. The public space

process thus tends to generate the local-to-globaldeformed wheel structure at whatever level of thesettlement it is applied, including, where it isoperative, local area structures. However, this isnot all the micro-economic process does. In its lociof most concentrated activity it will generate not alinear system that minimises universal distance inthe system as a whole, but a locally intensifiedgrid that minimises movement from all originsto all destinations in the local region (see, forexample, the central area of Konya in Figure 10)(Hillier, 1999a, b).

Looking at Konya, we can now see the settlementplan in a new light. We can see how spatial lawsdriven by the dual process have created the keyfeatures of the layouts: a deformed wheel globalstructure, an intensified grid forming the hubof the wheel, and the background of residentialareas. However, there is an important respect inwhich the processes that create these patterns can

be seen as a single process. The operation of thecentrality law is dual, in that it creates both


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integration and segregation. In this it is capable ofreflecting in itself the fundamental duality of thesocio-cultural and micro-economic processes. Thesocio-cultural process, which creates the largerareas of background space in the city, is always a

matter of imposing some restriction on integra-tion and the natural co-presence that follows itthrough movement, while the micro-economicprocess operates of necessity by always maximis-ing integration (minimising universal distance) inorder to maximise natural co-presence in itsspaces. The micro-economic process thereforenaturally occupies that part of the duality of thelaw of centrality which generates the longer linesand the essential structure of the settlement, whilethe socio-cultural process equally naturally occu-pies the obverse side, the production of a largernumber of shorter lines which construct the less

integrated background of mainly residential spacein the interstices of the global structure. Throughthe dual nature of the centrality law, then, thedual process acquires a single expression.

These conjectures require of course a wholeresearch programme to test them, involving bothsimulations of settlement growth and on theanalysis of real cases. However, some usefulpreliminary indications have been gained by

some simple experiments with the impact of thecentrality law on the basic generative process(using at this stage a manual process). Forexample, once we know the law of centrality, wecan use it to maximise universal distance in a

restricted random process by having a rule whichrequires the blocking of the longest line wheneveran opportunity presents itself. Figure 22 is amanually generated outcome from applying thisrule within the basic generative process. Theoutcome pattern is primarily composed of shortlines, and (for the reasons given earlier) has verypoor and unurban local to global synergy of .147,about as low as it can get for a small system). Italso lacks the kind of global structure typicallyfound in settlements, although it does begin toshow signs of an interesting, but overly peaked,log distribution of line lengths (Figure 23). In

short, it shows little sign of the spatial invariantsof settlement we are looking for. In some ways, itis the opposite. Suppose then that we use thecentrality law in the opposite direction, and set upa rule that forbids blocking a line once it hasacquired a length of, say, five cells. This generatesa pattern of many more long lines, as in Figure 24,which does have a good synergy score (0.820), butthe lines do not construct a deformed wheelpattern with interstitial local areas, and as Figure

Figure 22.


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25 shows the approximation of a log-normaldistribution is quite poor.

Suppose then we apply the centrality law in thesimplest and most localised way by setting up arule that says that wherever you are adding a

built form to the aggregate you have to choose alocal location which preserves the locally longerline, but at the cost of continually creating shorterlines. Figure 26 is an outcome of such a process.Its global structure is overly biased towards thecentral horizontal line, but it is centre to edge, andthe local areas are insufficiently structured inrelation to the global core (giving it an urbansynergy score of 0.729), but it does even at thisstage of growth begin to look more like the log-normal distribution of line lengths, as in Figure 27.

This suggests that it may indeed be the duality ofthe centrality law in creating many shorter lines tocompensate for each longer one that is in the lastanalysis responsible for the log-normal distribu-tion of lines lengths in real settlements. However,

although the tendency of the micro-economicprocess to use the longer line output of thecentrality law seems to be invariant, the relations

between these two aspects of the dual processshould perhaps be seen as a variable. Sometimes,for example, the zones of background residentialspace seem to be no more than the by product of

Figure 23.

Figure 24.

Figure 25.

Figure 26.

Figure 27.


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the micro-economic process, while elsewhere Konya would be an example there is a consciousparametrisation of the obverse side of the dualprocesses to create quite substantial regions of theurban grid, sometimes quite distant from the

main settlement structure. In other cases, such asLondon, we find the local areas are much moreclosely related to the global structure, moreaxially integrated into it, and themselves havelocal to global deformed wheel structures.

This kind of variation suggests a rudimentarytypology of settlement forms based on thedifferent balance between the micro-economicand socio-cultural forces. Where the economicprocess is dominant from the beginning, we findlinear or cross-road settlements and these areusually found on major routes between larger

towns, a linear town being global structure only.A deformed grid town is one in which bothprocesses run in parallel. A regular orthogonalgrid town is one in which the local culturalprocess is in the spatial image of the globaleconomic process, as in mediaeval planted townsor early American towns, and where the wholegrid is essentially a micro-economic rather thansocio-cultural creation, as can reasonably be said

both of mediaeval planted towns and earlyAmerican grids14.

We may then be within striking distance of

grasping aspects of the pervasive logic by whichapparently different social forces generate invar-iants in their settlement patterns as well as themore obvious differences. The key issues are theparametrisation of the cultural process whichdefines the local spatial geometry, and the balance

between this and the emerging micro-economicprocess as the settlement grows. In the earlystages of growth, the local socio-cultural processguarantees interaccessibility in the emergingsettlement pattern but little more. It sets aparameter which by deciding the degree or easeof interaccessibility (ie more or less universaldistance) specifies the local geometry of thesettlement, covering both line length, angles ofincidence and block size all factors in inter-accessibility. With growth, the universalistic and

therefore globalising micro-economic process in-creasingly interposes on this process a simpledepth minimising mechanism for each built formplacing decision: conserve long lines, if necessaryat the expense of creating many shorter lines. This

will have the effect of generating a pattern of afew long lines and many short lines, and becausethe choices are regional this will be the case atevery level that is, this process will generate thepervasive log-normal distribution with a few longlines and a large number of short ones at everylevel. Changes in this fundamental pattern ofgrowth will reflect essentially the changing

balance between micro-economic and culturalforces, and this may (as historically in London)alter with the passage of time, with each alterationleaving its mark on the settlement geometry.

However, the core issue is that the inherentduality of the spatial law of centrality is able toreflect the duality of these potentially conflictingsocial forces, and turn what is initiated as a dualprocess into a single process by which the locallyhighly differentiated and globally highly struc-tured pattern of urban space come into being.

A reflection

The deformed wheel structure with its interstitialareas the classic, although not the only, urban

form seems thus to be a product of an essentiallymetric process, optimising metric integration insome aspects, restraining it in others. Some mayhave noticed that this leads to a difficult question.Why should we continue to regard axial maps astopological structures, to be analysed throughtheir graphs, when we have shown that they aregenerated through an essentially metric process?Would we not be likely to arrive at a better pictureof the city if we subjected the axial map to metricanalysis? It has already been suggested that theintensified grids found in centres and subcentresare best understood through metric integrationanalysis (Hillier, 2000). Is it not time to subject theaxial map as a whole to such an analysis, or atleast to a metrically sensitive analysis? In thisway, we could surely counter one of the mainobjections to the axial map as a basis for graphanalysis: that the nodes of the graph representunequal elements.

The problem is that as soon as we introduce ametric dimension to an axial map whether by

14Such economic grids need however to be distinguishedfrom the grids of administrative, garrison or ceremonial townswhich characteristically are not pure grids but interruptedgrids in which many lines, including some major lines, areinterrupted by the facades of major public buildings at rightangles.


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using an analysis based on metrically uniformelements, or by weighting, say, line segments forlength configurational analysis produces not anenhanced version of the kind of picture given

by the line graph analysis, but a very different

picture: one that essentially picks out geometriccentrality in the system, as we saw when we usedmore or less uniform line elements in the pseudo-system shown in Figure 15. If we applied this to acity like London, it would have the effect that ashort alley off Oxford Street would seem to bemore integrated than, say, the Holloway Road. Inone sense it is of course, since it is closer to thegeometric centre of London. However, in a moreimportant sense we would seem to be losing oneof the most important aspects of the integrationanalysis of the urban system: the substitution of apicture of geometric centrality by a picture of

centrality in the line topology, one that identifiesgeometric centrality but then draws it out towardsthe edges of the system in all key directions, andeven including parts of the edges of the system.

The question is then: which is the true picture. Isthe one brought to light by the radius n analysis ofthe line graph in some sense identifying proper-ties that are truly of the nature of the urbansystem and essential to its functioning? One thingis clear. Metric analysis of a large-scale system isvery much poorer in its capacity to postdict themovement structure. In experiments carried out

in 1986 (Hillier et al, 1986) on axial maps whosesegments were weighted for length and used asthe units of analysis, this very propensity toassign too high a movement prediction to linesadjacent to strong lines and too low a predictionto syntactically stronger but more remote lines,destroyed the normal approximate agreement

between integration and movement. This suggeststhat the axial map, analysed as a line graph, mightafter all be capturing something that is of theessential nature of the urban system.

What can this be? There are two aspects to apossible answer: one substantive and to do withurban reality, the other cognitive and to do withhow we interact with urban reality. Substantively,the empirical effect of the line inequalities in theurban system is to create a disjunction betweengeometric centrality in the system and topologicalcentrality in the line map. In effect, centralityis topologically stretched from the geometricalcentre to form links with the edge in all directions.In doing so it also structures the object by creating

a relation between the local and the globalorganisation. The benefits of these are obviousenough: strangers are provided with easy-to-readroutes from edge to centre and out again, and thesystem acquires local to global intelligibility and

synergy. In contrast, it is easy to see that a systemwithout the line inequalities in the right place andof the right type will degenerate into a labyrinth.

In fact, in terms of the micro-economic processesthat create the deformed wheel structure, we findan even stronger argument when we consider thesettlement not in isolation but as part of the widersystem of settlements. Figure 22 illustrates whatin Chapter 9 ofSpace is the Machine was called theparadox of centrality. On the left are threenotional settlements, each with its own internalintegration core. But when (on the right) we join

them into a single system and analyse theintegration pattern for the system of settlements,we see that integration shifts to the edges of thesettlements. Clearly, if we consider each settle-ment on its own, then the internal pattern ofintegration will approximate the internal move-ment structure, while if we consider them as asystem of settlements the edge pattern will reflectmovement in the overall system.

This is of course exactly what happens in realsettlements. Movement patterns invariably have alocal aspect and a global aspect, the former

reflecting circulation within the system, the lattermovement in and out of the system. Insofar asmovement is driven by the micro-economicprocess, it generates both the intensified localgrids of the centres and subcentres by reflectingthe need to minimise distance from all points toall other points within the zone, and the linearlinks from the local to the global scale of thesettlement, reflecting the need to minimise dis-tance from certain points to certain others at thelarger scale (Hillier, 1999a, b), including into andout of the system. Over time, this tension betweenthe internal and external movement economies ofthe settlement is the fundamental reason whycentrality tends to shift towards the edges of thesettlement, unless strenuous efforts are made toinhibit it.

The deformed wheel structure is the key mechan-ism for this inhibition. In Figure 22, we draw fournational settlements, each with a deformedwheel case linking edge to centre (top left). Topright, we connect the edge lines of the four. The


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core goes to the edge of each. Bottom left, welink centre to edge lines to each other. The corepenetrates into, but not across, the settlements.Bottom right, we extend the centre to edge linesinto neighbouring settlements. The internal cores

of each are nearly restored.

Cities as discrete geometries

The second reason why we might suspect that theaxial map captures essential properties of theurban system is cognitive. The analysed axial mapseems to approximate the intuitive picture wehave of an urban system to an unexpected degree.A simple reason for this would be that human

beings are excellent judges of simple lineardistances when, for example, throwing a stone

or a spear, or a ball of paper into a waste paper basket. However, this comparatively secure jud-gement of distance quickly breaks down when thesystem becomes nonlinear and involves changesof direction. This would make simple sense inevolutionary terms. Distance is a comparativelysophisticated and recent concept, and there is noobvious reason why we would expect to judge itas well in the highly nonlinear situations created

by human settlement as when we are dealing withdistance as a simple extension of bodily reach.

Complex spatial systems seem then to be dealt

with cognitively through something more ele-mentary. What might this be? The obviouscandidate is discrete geometry: that we cognisecomplex spatial systems like cities as assemblagesof interrelated geometrical elements rather than ascomplex patterns of metric distance (Goodmanand ORourke, 1994). Discrete geometry is theapplication of the techniques of discrete mathe-matics such as graph theory to systems of discretegeometric elements, such as lines, convex spacesand visual fields. Space syntax, we now can saywith hindsight (there was not much discretegeometry about when we started), is the applica-tion of discrete geometry to architectural andurban systems considering these first and fore-most as systems of space.

If our cognitive representations of complex spaceare indeed discrete geometrical, then the strongestcandidate as the element in the discrete geometrywould be the line. Lines have the two keyproperties of being both very simple and veryglobal. All we need to know is how far we can see

from a point. Put more theoretically in terms ofthe city as a total visibility field we can followPenn (and adapting Peponiss beautiful conceptof informational stability as those regions in aspatial system that do not change topologically

with movement (Peponis et al, 1997)) in arguingthat a linear clique (a set of points which can allsee each other) preserves informational stabilityfor longest for moving individuals and thus offersthe most economical although not the mostcomplete picture of an overall system (Penn,2001). Other discrete geometrical representations,such as visibility graphs (Turner et al, 2001) forexample, give a much more complete account ofthe complexities of urban space, but it is notobvious that they would form the basis of acognitive representation of the city as a whole.There is too much local information for the global

picture to be clear. An axial map maximises localsimplicity as a means to picturing global complex-ity. With visibility graphs, it is the other wayround. Analysis of how we give directions incomplex spatial systems (Hillier, 1999a, b) sug-gests that the axial maps may not be too far fromthe way we represent them to ourselves, that is asa matrix of lines where changes of linear directionare the key items of information that becomeorganised into the whole picture.

If we intuit the spatial structure of the city as a

discrete geometry, then it is reasonable that weshould analyse it by treating the discrete elementsas the nodes of a graph. We are tempted to addto this: that we represent the urban system toourselves not simply as a discrete geometry, butas a simplified discrete geometry, in the sense thata series of near straight lines of the kind that arecommonly found in cities (Hillier, 1999a, b) areinternally represented as a line, so that the wholesystem comes to resemble an approximate grid. Ifthis is the case, then it would be no more than acase of the imposition of a Euclidean frameworkon non-Euclidean inputs as argued by Petepan

and her collegues commenting on (Okeefe andNadel, 1978)

ythe hippocampus appears to impose aEuclidean framework on non-Euclidean inputsOKeefe and Nadel (1978), who see in thisprocess an instantiation of a Kantian a priorinotion of absolute spaceywe propose that indistorting the sensory inputs, theses spatialmaps impose an order and a structure that


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our spatial conceptual representations requirey(Peterson et al, 1996)

Since it is also the line topologies that seem tocorrelate with movement in the different parts ofthe system, it seems hard to avoid the conclusion

that the line representation of the city is not just aconvenient simplification but something thattouches the essential nature of the city. This doesnot mean that it cannot be improved or brokendown more than it is now. However, it does seemlikely that any future configurational analysis ofthe large-scale structure of cities will need toinclude some representation of its linear dimen-sion as currently expressed, although perhapscrudely, in the axial map.

With or without the axial map, this account ofhow urban space is generated has unavoidableimplications for how we model the city. Models inthe past have used the fundamental concept ofmass and the Newtonian mathematics of gravita-tional attraction as the guiding theoretical entities.The integration equations play the same role inconfigurational models as the Newton equationsdo in attraction-based modelling. But they do soon the basis of a discrete geometrical representa-tion of the spatial structure itself, one t

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A Theory of the City as Object

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