Paper presented at the international seminar: Measuring Urbanity, Technical University of Lisbon, Portugal, 2013.
Lars Marcus
1. Introduction: the need for architectural models of the city
Following Alan Wilson’s outline of the development of urban modelling in his seminal work Complex Spatial Systems (2000), it is clear how urban modelling since its very beginning with von Thünen and its subsequent history has been founded on a geographical understanding of urban space, and, one might add, with a close relation to urban economics. This is not surprising and could be argued to be a natural order of things. In this paper, however, the question will be raised, why we have seen so little contribution to urban modelling from, what might be called, the other major discipline on space, that is, architecture. There are several reasons for raising this question, which will be further elaborated in the paper, but a major issue can be stated at once. Urban modelling aims in most cases not only to describe cities but also to support intervention in urban systems, why it seems that the inherently generative discipline of architecture, as opposed to, the essentially descriptive discipline of geography, would make a natural contributor to the development of urban models.
The geographic foundations of urban modelling are reflected both in how cities in such models normally are conceptually determined and spatially represented. von Thünen (1826), can be said to have set a high standard that has proven deeply influential and robust. With the two concepts distance and cost, combined with a simple spatial representation of cities as points in the centre of concentric distances, he was able, with great conceptual and representational economy, to target, at least in principle, the spatial structure of the economic process of land-use distribution in cities. This basic approach has been further developed through the history of urban geography, not least influentially by Walter Christaller and his Central Place Theory (1933), but has at the same time to a high degree retained its conceptual and representational framework. It is first in recent decades with the explosion of computational power that we can see a true reformulation of these frameworks based in theories on complexity (Batty 2005). Even so, the inherent geographic foundations are still there, more or less taken for granted, why we believe it useful to challenge urban modelling with conceptual and representational understandings of urban space founded in architecture.
According to Wilson (2000), we can see how the early models, by such people as von Thünen and Christaller, developed into more complex models in the 1960:s known as interaction models, but also that there was a particular forerunner to these known as the gravity model (Stewart 1947). For the current aim of attempting to introduce both conceptual and representational ideas from architecture into urban modelling, we will use the basic form of the gravity model as a point of departure. Even though it has often proven both limited and simplistic, it manages to embody in graspable form basic and important entities that still are used and applied in urban modelling.
A more particular background to such a re-evaluation of the gravity model and the interaction models that developed out of it, is that while the gravity model has worked quite well for the needs presented to urban modelling by urban geography and the different directions in the planning profession, primarily traffic planning, it has had great difficulties in answering to the needs in architecture and urban design. A reason seems to exactly be the different understandings of urban space. Where geography and planning, speaking in broad terms, concerns distributions of entities in space, architecture and urban design concerns the distribution of space itself (Koch 2005; Marcus 2010). That is, the built form of cities is in urban geography and planning a distribution in space among many others, while it in architecture and urban design has a privileged position as the medium that structure urban space as an entity in itself, and, of course, in extension different other entities. Since how we model the world inform us on how to act upon the world, it therefore seems to be a need for an architectural model of the city as support for architectural practice.
2. The gravity model: a generic model of the city
The gravity model represents a particular comprehension of urban spatial processes based on a simple correspondence with classical physics and the Newtonian gravity model. Its history is possible to trace back to the 19th century and the first statement is usually attributed to Carey in 1858: “The greater the number collected in a given space, the greater is the attractive force which is there exerted [...] Gravitation is here, as everywhere, in the direct ratio of the mass and the inverse one of distance.” (Wilson 2000). This general concept was first applied to migration (Ravenstein 1885; Young 1924) and later to retail (Reilly 1929). While this model, just as in physics one might say, has proved successful on a pragmatic level and been applied, in different versions, in many tasks and queries in urban modelling, for instance, clarifying the mechanisms behind basic land-use distributions; constructing origin and destination-models for traffic planning and modelling the interaction between urban nodes, it can also be contested, again just as in physics, when it comes to more complex theoretical and empirical issues. Theoretically, as often repeated, there is little to support the transfer of a gravity model from physics to the analysis of the spatial distribution and processes in cities (Wilson 2000, p 151), and methodologically, the geometric and mathematical concepts used to measure its variables often lack the precision necessary to capture the finer grains of urban processes.
Formally the gravity model can be represented as follows:
Where F represents some kind of flow between the locations i and j, for instance people, money or information, M the mass or attraction at the same locations, often understood as their population, and D the distance between them, while G is a constant. We here immediately see the two fundamental features of the basic gravity model. First, a measurement of ‘mass’ or attraction, normally interpreted as a measurement of population and different attributes to population, for instance expenditure, which can be combined with the amount of floor space for shopping at the two locations in a market area analysis for example. Second, a function of ‘distance’, where there is a great variety in modes of measurement, for instance, physical distance, temporal distance or monetary cost. As pointed out by Wilson (2000), we clearly also see the “two obvious ways in which the traditional gravity model can be developed: first, through an attack on the notion of ‘mass’; secondly, by scrutinising the nature of the distance function”. Later on, this is exactly how we will proceed.
First we need to better understand why the gravity model has proved so generic in urban modelling. The reason is that the gravity model, including the wider set of interaction models, immediately answers to one of the two fundamental questions in urban geography and in extension maybe to both, that is, the questions of location and spatial interaction. The first, which is the concern in geographic location studies, either starts with a particular urban activity or process and studies its spatial distribution, or, it starts with a particular area studying whatever activities or processes taking place within it. However, in itself this represents a rather static and not very life-like form of study of cities, since the same people are most likely to be involved in many different activities, just as most urban processes will be related to many locations, that is, in the end we will see a lot of flows between locations. The study of such flows of interaction between locations, depending on their degree of attraction and variety in distance, is exactly the concern of spatial interaction studies where the gravity model in particular has played a significant role. However, it has often been suggested that location studies and interaction studies actually are two sides to the same coin and that a good model therefore should be able to serve both kinds of studies. Such models have also been developed (e.g. Wilson 2000, p 67). Nonetheless, such combined models retain the basic units we found in the classical gravity model, that is, some kind of measure of ‘mass’, often also developed into measures of ‘attraction’, and some kind of measure of ‘distance’, which brings us back to the basic modes whereby one could propose development of the model, that is, new ways of measuring ‘mass/attraction’ and/or ‘distance’.
Apart from this, there is one more fundamental dimension to urban modelling that we need to consider and that is the choice of spatial representation. Wilson, for instance, argues for the supremacy of discreet zones as a means of representation, rather than continuous space descriptions, but there certainly are other options as well, why we in the choice of representation also see a possibility of principal improvement of the model. In the following we will address each of these possibilities in turn, aiming to, through a re-consideration of the gravity model and by way of recent developments in architecture and spatial morphology, transform the generic gravity model into something we call an architectural model of the city.
3. Spatial representation: from discreet zones to topological networks
In any scientific endeavour the first step is to develop useful means to represent whatever it is that one wants to study. We can say that in the end we always develop knowledge on such representations rather than the ‘reality’ they are supposed to represent. This is not saying that we cannot say something meaningful or even ‘true’ about reality, it is simply acknowledging that we never can make full sense of what is ‘out there’. What we can do is to agree on certain frameworks within which we can speak of conditional truths. This is a huge discussion of course and in the current context it only serves the purpose to put emphasis on how critical the choice of representation is in scientific endeavours and how diligent we need to be in our specific case concerning the representation of urban space. The obvious trap here is if the representation is taken for granted, or even mistaken for ‘reality’.
If we return to how cities actually are spatially represented in urban modelling, we can see how Wilson, for instance, posits the choice of representation as a choice between discreet zones and continuous space representations, and convincingly argues for the use of discreet zones, despite what intuitively seem to be the finer resolution of continuous space representations. He mainly prefers discreet zones because of the problems in representing flows conveniently in a continuous space representation. However, this is neither easy to do neatly in a discreet zone representation where the addition of a network of relations between the discreet zones is necessary for this purpose and, on top of that, some kind of dummy links connecting the zones to this network, all adding up to a rather untidy representation.
Given the development in urban modelling in the last decades, we should rather extend the range of representational opportunities quite a bit (Batty 2005). While we could make a long list of new directions in urban modelling, we will here in particular bring in the development of cellular automata and agent-based modelling as a means to create a principal range of representations. Such a principal range makes it more apparent what urban modelling actually is about, and not least what we more precisely mean by the, in this context, ubiquitous notion of space. Urban modelling aims to capture human based activities, which give rise to more general processes in urban space. As such, urban models typically work at the interface between urban structure and urban process, where processes give rise to structures, which in turn structure the processes and so on - the typical metaphor here is a river, where running water carves out a river bed that then directs the water of the river. While both structure and process are inherently intertwined in space-time, and consequently with each other, we need to admit our tendency to separate them and conceptualise structure as primarily space based and process as primarily time based. We can then interpret space as a kind of raw material that process can give a certain structure and time as a raw material that structure can shape into a particular process.
Rather than the somewhat overly abstract distinction into time based and space based models, one could of course use the more commonly used notions of place based versus people based modelling (Miller 2007). However, whichever we choose we can see how continuous space representations are found at the space edge of the range in that such models seems to specifically origin in space and in a particular understanding of space as a neutral field that can be imprinted by process, while agent-based modelling is found at the time/people edge of the range in that they seem to start in time or people based activity and more particularly in an understanding of time/activity as a kind of code that can be inscribed into agents that in there subsequent actions generate structure. In such a range of representational options, discreet zones can be seen as a kind of continuous space representation that to some degree has been structured by process into particular zones which in turn structure the mapping of other processes, and cellular automata, finally, as a kind of agent based representation of time that to some degree has been processed by structure so that the code is inscribed into static cells, which process the further generation of structure.
This is a highly principal discussion that clearly goes beyond what is possible and necessary to unpack in our current context, but it serves the purpose of clarifying how representations in urban modelling, while appearing to be most diverse, share the aim to capture how urban processes structure urban space and how urban structures process urban time, and that what differs is that certain representations have a primacy in space and others in time, where we must choose depending on our particular needs and resources. Important in our current context is that this range also can serve as a means to characterise other representational forms, for instance, the type of models used in space syntax research, which are of central importance for this paper. These models are typically network based, that is, they focus the relations between activities in urban space rather than the spatial location of such activities per se. In classic terminology we can say that both continuous space representations and discreet zones conceptualise space as absolute, while network models conceptualises space as relative. If we extend things a bit we can also suggest that agent based modelling conceptualise space as relational.
This can be interpreted as if network models are inherently space based; however, we need to remember why relations are deemed so important in network models. The reason is that relations capture potential flows, movements and interactions between locations and their particular activities, that is, relations here come close to representing processes, albeit in the form of structure – we can say that the riverbed is made into a proxy for the running water of the river. Therefore, while all urban models combine structure and process, and in extension space and time, network models seem to do this more deliberately, and, one might add, with great representational economy.
This seems to be the reason for the characteristic property of network models, namely their ability to capture some quality over and beyond the particular activities they map. They do not only focus spatial relations rather than spatial locations – in network models locations are typically defined by their relations rather than against a neutral background – they focus spatial relations between nodes per se rather than relations between the particular activities at these nodes. We here need to make a distinction between networks and systems, where systems is a wider concept that normally denotes a set of relations between an interacting group of components, while network denotes a set of relations in itself, not necessarily concerned with the character of the components it ties together. Such sets of relations, when seriously regarded in themselves, as they are in network modelling, seem to capture some particular procedural quality inscribed in them. As such, network models seem in their very geometry to be able to capture time or activity in the form of a structured process, which make them inherent and powerful space-time models.
We therefore might say that network models are to be found right in the middle of the representational range earlier introduced, and more specifically that they from the perspective of space, on the one hand, can be described as spatial fields that have been structured by processes into a type of one-dimensional discreet zones, that is lines, with the particular character that these zones/lines represent spatial relations between a set of neutral nodes rather than areas characterised by a particular activity. From the perspective of time, on the other hand, they can be described as codes of time or activity that has been processed by structure into a particular form of one-dimensional cells, or once again lines, with the peculiarity of representing sequential steps, that is a form of process, between a set of locations rather than coded cells only influencing adjacent cells.
4. Measuring distance: from physical distance to cognitive distance
Network models concerns relations between nodes and are therefore also inherently concerned with the topic of distance. As touched upon, distance can be measured in several ways, as physical distance, as temporal distance or as some sort of cost. What we choose has tremendous effect on our results of course and is of the highest relevance for what we want to capture. In the context of complexity theory and the growing urge to understand cities bottom-up and as self-organising systems (Batty 2005), recent critique of spatial analysis has focused how one especially has lacked “effective means for representing or dealing with the spatial complexity of a realistic urban environment”, and more specifically, that one has not been able to “incorporate data about a person’s cognitive environment into the analytical framework” (Kwan 2000). It has also been pointed out that: “In geographic space, it is well known that spatial behaviours of humans are directly driven by their spatial cognition, rather than by the physical or geometrical reality. The cognitive distance in spatial cognition is fundamental in intelligent pattern recognition” (Shu et al. 2001). This inability in urban modeling is critical also if we want to generate knowledge that can inform urban design since it blocks the possibility for precise urban modeling at the micro scale, that is, the scale where people in the street experience cities.
Such a cognitive point of departure in description and analysis of the built environment is exactly what we find in space syntax (Hillier & Hanson 1984; Hillier 1996). In space syntax analysis the means of description have been a primary concern, that is, there has always been a concern for representation and the invention of the axial map has been instrumental in the development of space syntax research. The axial map is a network representation of urban space based on graph theory, constructed from the point of view of a cognitive subject, i.e. an experiencing and acting human being. It is made up of the least amount of straight lines that cover all accessible urban space in the area of analysis, where each straight line (here called axial line) in the map represents an urban space that is possible to visually overlook and physically access. This basic mode of modelling has continuously been developed so that there today is a rich set of analytical measures all keeping the close relation to the cognitive dimension of urban space. Of primary importance is the development of axial analysis into segment analysis (Hillier & Iida, 2005), where axial lines are broken down into street segments on which different cognitive distance measures can be applied, such as, topological (axial line steps) and geometric (least angular deviation). With this as a basis, different forms of interaction can be calculated, such as, centrality and betweenness, which in space syntax research are called integration and choice respectively (Hillier et al, 2012). The formula for integration calculation is:
and for choice:
The ability of such models to capture pedestrian and also vehicular movement has been shown in a great body of studies around the world (see e.g. Hillier et al 1993, Hillier 1996, 1999). Still, at closer scrutiny the axial map turn out to be a rather peculiar entity. As already pointed out, axial maps, including their later development into segment maps, are network representations. Mathematically, networks are normally described as graphs and axial maps are thereby part of a long tradition of graph-theory in spatial analysis and urban modelling. The very field of graph theory famously origins in an urban problem called ‘the Bridges of Königsberg’ (Euler 1735). The normal procedure here is to represent urban elements as nodes and the links between these, usually in the form of streets, as arcs. When specifically modelling a street network the street-junctions are then represented as nodes and the street-segments connecting these as arcs. Peculiar to the axial map is that this is done the other way around - the streets are represented as nodes and the junctions as arcs.[1] This is an unusual representation within spatial analysis and has called for the development of a unifying mathematical framework where “space syntax can be translated into a more familiar analytical frame” (Batty 2004a). Here it is argued that this unusual way of representing urban space in graphs exactly has its origin in the architectural foundation of space syntax, where in architecture space typically is designed from the point of view of an experiencing subject, that is, we do not experience cities as a set of street-junctions where streets are mere links, rather we exactly experience cities as a set of streets where street junctions are experienced as links between these. This conception of urban space that comes natural to architecture is awkward to the more system-inclined conception of space in geography and traditional urban modelling. This small shift in how to represent urban space, which we here exactly want to see as a shift from a geographic model of the city to an architectural model, has deep repercussions on the analytical performance of the model.
But it is not only compared to standard procedure urban modelling in geography that space syntax descriptions are unusual entities. They are also quite different from the point of view of architectural and urban morphology. “The work of urban morphologists has made the greatest contribution in the elemental category. Their studies focus on buildings and their surrounding spaces, lots or parcels, and streets, analysed at different levels of resolution” (Talen 2003). While urban morphology in this way traditionally has focused on descriptions of urban elements, one has within space syntax always stressed the relative or systemic dimension of cities and architecture, that is the description of relations between elements, or as preferred in space syntax, the configuration of architectural and urban elements. In this sense space syntax differs fundamentally from mainstream architectural and urban morphology and rather adhere to approaches more familiar to urban modelling in geography.
While this is a difference that is often pointed out, one can also point to another fundamental difference that maybe is more original. While urban morphology usually describes the city as a collection of urban elements rather than an urban system, it also defines these elements from a rather abstract conceptual point of view, that is in the form of units such as the urban block that are geometrically easy to define for example on a map. One could in Lefebvre-terms say that they concern representations of conceived space rather than perceived space. The morphological descriptions developed within space syntax on the other hand have their rationale from the point of view of human experience and perception. It is here that Seamon (1994, 2004) has pointed to a characteristic in space syntax not so often acknowledged, namely a strong phenomenological strand inherent within its ideas as well as descriptions.[2]
According to Hillier: “From an experiential point of view, cities seem to be about seeing and going. Syntactic analysis confirms this by showing they are structured both to make the physical movement of bodies efficient and to be intelligible to minds.” (2003). He thereby brings in the notion of intelligibility alongside mere physical movement, and that is critical since it translates the moving person from mechanic agent to an experiencing subject that interacts with his or her environment. The argument for the axial line as a metric of distance can then be made: If we make a straight line crooked “we do not add significantly to the energy effort required to move along it, but we do add greatly to the informational effort required” (Hillier 2003). The reason for the success of the axial map in capturing pedestrian movement then, is likely to be its ability to geometrically capture both the energy effort and the informational effort for a moving subject in an urban area.
The three fundamental elements in space syntax morphology, ‘the axial line’, ‘the convex space’ and ‘the isovist’,[3] all have their origin in human faculties of experience; visibility in respect of movement, visibility in respect of co-presence and visibility per se, respectively. In short they are all representations of the phenomenological bottom-line of being in the world. Now, the original thing here is the reduction of these existential notions, that within architectural theory often has lead to concepts difficult to handle analytically,[4] into simple geometry. A less philosophical and more analytical way of conceptualising this experiential quality in space syntax networks is to call them cognitive networks in which the axial line, with or without angular distortion, work as a measure of cognitive distance. This introduction of a cognitive geometry must be the reason why the rather plain appearance of the axial map has proved to have such predictive power. With a straightforward geometry one is able to describe the built environment in a way that simultaneously captures its systemic and experiential dimensions. The axial map can then be said to both embody what Jürgen Habermas call the system and the life-world (Habermas 1984), which is interesting from the point of view of urban modelling which traditionally have been typical system-descriptions.
5. Measuring mass: from discrete density to network density
In the previous section we saw how the network in space syntax could be interpreted as a set of cognitive zones, called axial lines, defined by visibility and accessibility from the point of view of an experiencing subject, and, furthermore, that these zones, or axial lines, in themselves were used as topological units in measuring cognitive distance. What we experience is a sort of descriptive economy, where the same geometric object is made to embody a lot of information in its geometric shape. In the case of the axial line, it at the same time represents a cognitive zone, a link in the network and a distance unit. Taken together, this implies a representational shift into a network model based on cognitive distance. We can conclude that the measurement of distance in this representation has taken the particular form of cognitive distance and that this, furthermore, is embedded in the geometric form of the network itself.
Consequently, when we now turn to the measurement of mass there seems to be a similar possibility to substitute the discreet zones normally used to locate and represent mass with the network connecting these zones. That is, there seems to be a possibility to conflate zones and links into one comprehensive geometric representation in the form of a network in which each link is both link and carrier of the properties of mass found in its adjacent zones. If we, furthermore, construct this network along the lines of the axial map, that is by geometric units that are cognitively defined, as we have seen the axial lines are, we seem to be able to reach an unusual representational economy. Each axial line then, does not only represent a cognitive zone, a link in the network and a distance unit, but can also be made to carry a local property of mass. In short, we seem to have developed a representation that integrates distance and place, and, furthermore, from the point of view of cognition.
Moreover, mass can in such a scheme not only be calculated as a local property represented on each axial line, creating the typical mosaic of such representation, albeit this time in the unusual form of links in a network, but exactly as a property of accessibility. That is, instead of representing some number of density on the different axial lines, we can calculate the accessible mass within a chosen radius, for instance, the accessible mass within a radius of three axial lines. This mimics exactly how accessibility measures such as integration (centrality) and choice (betweenness) are calculated in space syntax, with the difference that here the distance measurer (the individual axial lines) are weighted with a particular value of mass, transforming the calculation from a ‘pure’ accessibility measure to a accessibility to mass measure. This is exactly the kind of measures that have been developed and applied to the software the Space Syntax Tool (Ståhle et al. 2005).
We do not only need to consider on what geometry we want to represent ‘mass’, but more precisely what ‘mass’ it is we want to represent. There obviously are many options here, but in the case we specifically are pursuing here we are aiming for an architectural model of the city, which in this context means a model of the distribution of urban space itself. That is, rather than loading the model with all kinds of data such as the geographic distribution of population, retail or transport nodes, which we of course also can do, we will look for variables of the distribution of urban space itself, that is, geometric or architectural variables as discussed elsewhere.
In such an architectural model we see two fundamental modes by which mass, or attraction, in an urban network can be increased. First, through spatial densification, that is, increasing the potential intensity of human activity in a particular location by adding floor-space on top of each other. Second, through spatial differentiation, that is, increasing the potential diversity of human activity in a particular location by dividing space, or land, into different spaces, or parcels. The variable density clearly has been intensely discussed within urban modelling and is, as a matter of fact, an inherent part of any gravity model, although it more often is calculated as population density than building density. The variable of diversity, on the other hand, is far less developed and applied in spatial modelling, while it is far from unknown in this context. It also constitute a measurement of another kind than density, since it is a relative measure and as such demands further attention. The diversity variable will be dealt with separately in the following section.
Building density in cities have established measures such as FSI (Floor Space Index), that is, the amount of built floor area per land area, where the latter most often is limited to the area of the parcel built upon. This measure could be extended by being combined with variables of open space such as GSI (Ground Space Index), which measures the amount of land built upon, once again often limited to the area of the parcel built upon, and often referred to as the built footprint. Such combined measures of density have been investigated, developed and tested in the Compact Sprawl-concept (Ståhle 2007) as well as the Spacematrix-concept (Berghauser Pont & Haupt 2009). The fundamental problem with density measures of this kind has to do with the problem how to define the area units they are calculated upon. There is certainly no problem in calculating the amount of floor area per se, the problem begins when we are to define the area we should divide this amount with to reach a density measure. This problem is part of one of the most fundamental and pervasive problems found in geography, referred to as the MAUP-problem (the Modified Area Unit Problem). Whatever density we look for, how we define the area we calculate it on will have a strong impact on the final density value. So, should we divide by plot area, as suggested above, or should we also include part of the street and what about shared amenities, such as parks and open space in general. These questions have always presented a fundamental and irresolvable problem for spatial analysis in geography.
However, this problem seems to be possible to overcome by changing our mode of not only measuring but conceptualising density discussed above, that is, by measuring mass, or density, as accessible mass rather than mass, or density, per area unit. Not only does this transform the measure from, what we have called, a system measure to a life world measure by setting the data in an experiential or cognitive perspective, but it also gets rid of the MAUP-problem, since we no longer need to define an area. We simply calculate how much built floor space is accessible within a certain radius from each and every axial line in the network and then represent these values by, for instance colouring each and every axial line according to some chosen scale. We then have created a model in the form of a network where each part, in the form of an axial line, does not only represent a cognitive zone, a link in the network and a distance unit, but also a carrier of a local property of mass, in this case represented by the building density, furthermore, originally calculated as accessible density rather than density per area unit.
6. Measuring attraction: adding network difference to network density
Moving on to diversity as a second variable of mass, it clearly seems possible to proceed in a similar manner. At the same time and as earlier touched upon, to measure diversity is a trickier business than measuring density, since it is a relational value. While density can be described in absolute terms, diversity always concerns values in comparison to something. This is probably a reason why we find far less such measures as well as far less discussion on urban diversity in quantifiable terms. This does not mean that it is not a common topic in the general debate in urbanism or other urban discourses, far from it. It is an increasingly important topic in many directions of urban theory for all kinds of reasons. In urban economics, the globalised economy and the international competition between cities and regions that arise from that, point out the issue of diversity as maybe the major contributor in buildings attraction and generating economic growth. In urban sociology, it is clear how diversity is a given and major characteristic of our contemporary multi-cultural societies and how it creates both many of its conflicts and attractions. In urban modelling, the understanding of cities as a prime case of adaptable complex systems, makes it clear how diversity is one of the drivers in such systems and therefore one of the most critical variables necessary to understand.
The origin of much of such debate on diversity in cities is the writings of Jane Jacobs, for whom diversity can be seen as the major theme throughout all of her books. More specifically it is spelled out in ‘the Death and Life of Great American Cities’, in her four criteria for generating diversity in cities: more than one primary function, short blocks, buildings of varying age and dense concentration of people. Of these we find, as we shall see, the condition of buildings of varying age to be, maybe a bit surprising, most promising from our current perspective of developing a measure of diversity for our urban model. While the condition of more than one primary function certainly has a strong influence on the degree of diversity in cities, it seems to deal with pure programming of diversity rather than having any obvious geometric property of the kind that we are looking for in our architectural model. The notion of short blocks, on the other hand, seems rather to deal with accessibility, or quite exactly the configuration of cognitive distances dealt with in space syntax theory and discussed above. Finally the notion of dense concentration of people clearly concerns the idea of density in general and as such has already been incorporated into the model. The condition of buildings of varying age then seems principally different from the others and at first sight maybe the one of both least theoretical interest and least practical applicability – how do we plan cities with buildings of varying age or, for that matter, build a theory around such a thing.
The trap here is if we focus the notion of buildings too strong, rather than the spaces in cities in which buildings evolve, that is, the by different property rights defined domains we call parcels, plots, lots or properties. That is of course a space of quite a different nature than the one’s earlier discussed, legally defined rather than physically defined, but it clearly represents a critical and inherently urban space. And if we start to think about it, we can see how fundamental such division of space into several separate spaces is in any architectural endeavour, also of a more direct physical kind - what is architecture if not the art of dividing space by walls – where the primary rationale behind this exactly is the aim to generate diversity, that is, generate discrete spaces for separate and different uses. This clearly is the major rationale behind the division of building space into separate rooms, but we can see this to an equal degree also in the agricultural landscape, where a major concern is the division of land into discrete parcels not only for the sake of separating ownership but equally to create order between different agricultural land-uses, where it becomes obvious how an increased land-division clearly creates an increased agricultural diversity. On a deeper level it has also been shown how such increased land-division has a direct impact on an increased bio-diversity.
It therefore seem to be well-founded to see the vertical division of land as a most fundamental spatial technique whereby one can increase diversity, in a similar manner to how the horizontal addition of floor-space is a fundamental technique in increasing density. If we more specifically introduce this concept to modelling the architecture of the city we can make the following argument. The particular domain of the plot or the property represents the presence of an actor in urban space, in the form of the proprietor or the like, and, furthermore, a very precise location of the influence of that actor in urban space. Within this domain the actor is free to act, obviously keeping within the frames set by different institutions, for instance, the local planning regulations or the particular property rights of the property, which both, most characteristically, have very definite limits to their range of influence – the boundary of the municipality in the first case, and the particular piece of land represented by the property in the latter.
Such actors, furthermore, normally will develop particular strategies for their maintenance and development of their plots of land. An area with comparatively many plots then seem to have the potential to carry more such actors and thereby more strategies for maintenance and development and, moreover, more diverse such strategies. In the end, such an area seems to have the potential to more easily develop a diverse content than an area with comparatively few plots and hence few actors and strategies. Consequently, it seems to be exactly this division of land and creation of potential diversity in actors and strategies that over time can generate the notion of buildings of varying age, propagated by Jacobs, and it is this hypothesis that makes us believe that we here have a geometric variable that can influence urban diversity.
If this represents the argument for what particular spaces that can be of interest for the variable of urban diversity we now need to turn to the issue of how to measure such a variable. As we have already touched upon, diversity is more difficult to measure than, for instance, density. While there are less such measures, and especially far less empirical studies, that concern diversity than density, this does not mean that there are none. Among various available indexes to measure diversity used in the literature, there are two that are more often used, these are the Simpson Diversity Index (SDI) and the Neighborhood Diversity Index (NDI) (Maly and Leachman, 1998). The first one has been used since the 1940:s, mainly in the biological and environmental fields, to quantify the biodiversity of a habitat in an area. The second one is mostly used in studies on the ethnic diversity in cities.
Both of these indices are possible to use for several types of diversity measurements, like housing types, tenure forms, age groups, retail and so on. SDI concerns the calculation of diversity within an area or group but does not compare the results with a larger sample, which may cause both confusing results but above all results that are difficult to compare. NDI, on the other hand, is preferable in the current context since it calculates diversity as a comparison within a particular area in comparison to a larger group. The equation for NDI can be formalized as follows:
D = 1/2(|Ca-Ta|+|Cb-Tb|+|Cc-Tc|+|Cd-Td|)
Where C is the group percentage (categories a, b, c, d, for instance: categories of retail) for the whole city district, and T is the group percentage for the plot. An area reflecting the composition of different functions will have a low D, while an area consisting predominantly of one group will have a high D. The final values are put as percentages, where a low value indicates a high diversity and a high value a low diversity. In order for easier comparisons, d can be equated to 100-D, where d then will be a value between 0 and 100 percent (0<d<100).
Just as in the case of our density measure, we can choose to calculate this diversity measure, that is, how many plots that are accessible within a chosen radius compared to a larger group, from each and every axial line in the network. We then have both a measure of density and diversity that are measured, so to speak, through the network, using cognitive distance and calculated from each and every axial line or segment, whichever we choose, and therefore can be represented on the network itself. What one would like is of course a combined measure of mass, constituted exactly by the two variables of diversity and density, calculated in the manner extensively discussed above. The exact mathematical form of such a combined value must be developed empirically, but we certainly already at this stage can see the principal strength of such a conception of mass in urban modelling.
6. Conclusion: a network based cognitive entropy model
We then seem to have travelled to the end of the path set out in the beginning of this paper, a re-consideration of the traditional gravity model in urban modelling, retaining its principal components, distance and mass, but substituting both the conceptual understanding of these variables as well as the way they are measured with new concepts and measures, and, on top of this, proposing the network as representational mode for such a re-constructed model. We also set out, through such a procedure, to not construct a model of cities in general as much as an architectural model of the city, implying that typical variables such as populations and economic transactions, by which such models normally are replete, are to be left out, aiming instead to take hold of the geometric dimension of urban space, here understood as the architecture of the city. Hence, the whole endeavour can be seen as a project aiming to take a classic urban model, which typically is based on a geographic understanding urban space, and translating it to an architectural model of the city - architecture being, so to speak, the other discipline of space. Thus, the typically analytical models developed in geography are introduced into the typically generative discipline of architecture.
As such, the model is still constituted by the components distance and mass, but now translated into spatial variables particular to architecture, where distance now is conceptualised and measured as cognitive distance rather than physical distance, reflecting the typical point of departure in architecture of space as perceived space. Mass, furthermore, is translated into the two principal variables of density and diversity, correlating to, on the one hand, the fundamental architectural process of adding floor space as a means to generate density, and, on the other hand, dividing space into separate parcels as a means to support diversity. Both of these are, moreover, measured, not as entities distributed in space but as entities distributed through space, that is, they are not limited to measures on particular local geographic units, such as a particular area, but as measures of accessibilities from a location within a chosen radius through the network, once again translating them into cognitive or experiential measures of these variables. Finally, the network is chosen as form of spatial representation in such a model. What we in the end seems to have reached is an outline of something we can call a network based cognitive entropy model, which we propose to be a powerful model of the architecture of the city. Its further mathematical formalisation and empirical testing is yet to be fully developed, but there is a wide range of studies in support of the individual measures one by one why this final endeavour at least looks promising.
References
Hillier, B., Space is the machine, Cambridge University Press, 1996.
Marcus, L, Steen, J. 1999, “Physical planning for economic growth — a study of urban areas”, Proceedings Second International Space Syntax Symposium, Universade de Brasilia, Brasilia.
Marcus, L. 2000. Architectural Knowledge and Urban Form – The functional performance of Architectural Urbanity. TRITA-ARK Academic dissertation 2000:2. KTH.
Marcus, L. 2001, “The impact of land-division on long-term occupation – the possibility of such a thing as natural occupation”, Proceedings Third International Space Syntax Symposium, Georgia Tech, Atlanta.
Marcus, L. 2007, “Spatial capital – an outline of an analytical theory of urban form”, Proceedings Sixth International Space Syntax Symposium, Istanbul.
Shu, H., Edwards, G., Qi, C., 2001, “Cognitive distance”, in Jun S., Sharatchandra P., Runsheng W., (Eds), Proc. SPIE, Object Detection, Classification, and Tracking Technologies, Vol. 4554, p. 290-296.
Ståhle, A., Marcus, L. & Karlström, A. (2006) “Place Syntax – Accessibility with axial lines”. Environment and Planning B: Planning and Design, submitted.
Wilson, A.G., (2000) Complex urban systems.
Notes
[1] More precisely one can say that within the axial map each axial line is both node and arc.
[2] In recent papers by Bill Hillier this is brought to the forefront and discussed in depth (Hillier 2003a and b). In the future we therefore might expect as many references to Merleau-Ponty as to Levi-Strauss in space syntax literature.
[3] The isovist was introduced by Michael Benedict (1984).
[4] For example the concept of Genius Loci (Norberg-Schultz, 1980).