Generating Cities from the Bottom-Up

Using complexity theory for effective design

Michael Batty

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This essay introduces the idea that cities evolve from the bottom up, that patterns emerge as a global order from local decisions. As the future is unknowable, cities must be planned according to this type of complexity. We begin by introducing the idea of hierarchy and modularity as the basis of a generative process that leads to patterns that are self-similar over many scales, patterns that are fractal in their structure. We then present a simple model of growth that generates such patterns based on a trade-off between connecting to the growing structure and seeking as much free space as possible around any location within it. These are the tensions that exist in real cities through the quest by individuals to agglomerate. Once we have sketched the model, we demonstrate how this generative process can be simulated using cellular automata which allows us to think of the logic in terms of rules that represent how existing development takes place. By changing the rules, we can introduce optimality into the process, developing the logic to enable certain goals to be pursued. We conclude with various demonstrations of how idealised plans might be conceived within this complexity which is generated from the bottom up.

1) A new paradigm for city planning
Cities are built from the bottom up. They are the product of millions of individual decisions on many spatial scales and over different time intervals, affecting both the functioning and form of the city with respect to how it is structured and how it evolves. It is impossible to conceive of any organisation that can control such complexity and thus the very question of the extent to which the city might be ‘planned’ is thrown into a new light. Throughout history, plans for cities have been proposed as a top-down response to perceived problems and the realisation of ideals but there are few examples where control has been sufficiently strict to enable their complete implementation. Most development in cities occurs without any central planning yet cities continue to function, often quite effectively without any top-down control. Cities, as part of societies and economies, not only hold together without any top-down control but actually evolve their own coordination from the bottom up, their order emerging from these millions of relatively uncoordinated decisions which express Adam Smith’s characterisation of the economy as being managed by an ‘invisible hand’.

Cities are biult from the bottom up, incrementally, and where they are planned from the top down, the plan is usually a small part of wider organic development

Fifty years ago, cities were first considered to be systems whose functioning was based on many interacting parts and whose form is manifested in a relatively coordinated hierarchy of these parts (or subsystems). Yet systems in these terms were conceived of as being centrally controlled. As the paradigm developed, there was a subtle shift to the notion that the order in many systems and their resulting hierarchies emerged from the way their parts or elements interacted from the bottom up rather than from any blueprint imposed from the top down. The complexity sciences developed to refresh this systems paradigm with the focus changing from an analogy between cities as machines to one based on evolving biologies, whose form was the resultant of subtle and continuous changes in their genetic composition at the level of their most basic component parts. This shift in thinking is wider than cities per se. It is from thinking of the world in terms of its physics to one based on its biology, from top down to bottom up, from centralised to decentralised action, and from planned forms to those that evolve organically.

In this essay, I will argue that a new paradigm for planning cities is required which takes account of how they are built which is largely but not exclusively from the bottom up. It draws on recent developments in complexity theory which in terms of city planning, draws on the traditions first suggested by Patrick Geddes (1915) in his book Cities in Evolution at the beginning of the last century but taken up in earnest in the early 1960s by Christopher Alexander (1964) and Jane Jacobs (1962) amongst others. This paradigm has taken a century in sensitising us to the need to step carefully when intervening in complex systems. Its message is that we plan ‘at our peril’ and that small interventions in a timely and opportune manner which are tuned to the local context are more likely to succeed than the massive top-down plans that were a feature of city planning throughout much of the 20th century. To impress this new style of planning, we will proceed by analogy using metaphors about how cities are formed taken from physics and biology. We will first outline the notion of modularity and hierarchy, of self-similarity and scale in the physical and functional form of cities, and then we will present ways in which basic functions generate patterns that fill space to different degrees. Cities develop by filling the space available to them in different ways, at different densities and using different patterns to deliver the energy in terms of people and materials which enable their constituent parts to function. We will demonstrate a simple diffusion model and then generalise it to grow city forms and structures, in silico. We will allude to city plans in history that demonstrate our need to plan with and along side the mechanisms of organic growth rather than against these processes which has been the dominant style of planning in the past century.

2) Modularity, hierarchy and self-similarity
There is wonderful story first told by Herbert Simon (1962) which illustrates the importance of hierarchy and modularity in the construction of stable and sustainable systems. Simon tells of two Swiss watchmakers, Hora and Tempus, who both produced excellent but identical watches, each of 1000 parts. The key difference between the watchmakers was in the processes they used to produce each watch. Tempus, for example, built each watch by simply taking one part and adding it in the requisite order to the next until the whole assembly was complete, Hora however built up his watches in subassemblies, first of 10 parts each. Once he had produced 10 sub assemblies, he added these into a large subsystem containing 100 parts. When he has added all his 10 part assemblies into the larger parts of 100, he completed the whole watch by simply adding the 10 larger assemblies together. It took Hora only fraction longer to add the subassemblies. To all intents and purpose the completed watches took the same time and were no different.

street networks are examples of how cities grow from the bottom up: they represent the skeletal structure on which all else in the city hangs

As the fame of the watchmakers grew, they received more and more orders but in this fictional world, the only way they could receive these orders was by telephone. Every time the telephone rang Tempus had to put down the watch and it fell to pieces so he had to start again once he had taken an order. Hora on the other hand simply lost the subassembly he was working on when the telephone rang and he had to put down the watch to answer it. You can see immediately what happened in the long term. Tempus found it more and more difficult to complete a watch as the telephone rang more and more while Hora simply traded off his telephone time for watchmaking for his process was robust. Ultimately Tempus went out of business while Hora prospered and the moral of the story is that sustainable systems which can withstand continued interruptions of this kind are built in parts, from the bottom up as modules to be assembled into a hierarchy of parts.

Modular construction is not simply a functional process of ensuring that component parts of a system are stuck together efficiently and sustainably but a means of actually operating processes that drive the system in an effective way. For example, different functions which relate to how the economy of a city works, depend on a critical mass of population such that the more specialised the function, the wider the population required to sustain it. In short, more specialised functions depend on economies of scale such that the size and spacing of various functions produces a regular patterning at different hierarchical levels. The modules are thus replicated in a way that they change their extent with their scale. We can demonstrate this point using some simple geometry that illustrates how we can scale a physical module, producing a fractal that is similar on all scales. Imagine that we need to increase the space required for planting a barrier along a straight path. If we divide the line of the path into three equal segments, we can take two of these segments and splay them away from the path so that they touch and form an equilateral triangle in the manner that we show in Figure 1(a). This clearly increases the length of the line L (which has an original length of three units) by one unit, so that the new length of the line becomes (4/3)L. We can further increase the length of the line by subdividing each segment into three and displacing the central portion of each of the original segments to form the same equilateral triangle but at a scale down from the original. If this is done for each of the original four segments, then the length of the second line composed of these four segments increases by 4/3. This in turn is 4/3 the length of the original line L and the new line is now (4/3)(4/3)L. We can continue doing this at ever finer scales and the length of the line at scale n thus becomes (4/3)nL.

Figure 1: Constructing a Space-filling Curve: The Koch Snowflake Curve

a) left: successive displacement of the central section of a line at ever fine scales
b) right: application of the displacement rule to the lines defining a triangle shape called a Koch Island

This construction is a recursion of the same rule at different scales and it generates a pattern which is self-similar in that the motif – the triangular displacement occurs at every scale and is in a sense the hallmark of the entire construction. The structure grown from the bottom up produces a shape that is a fractal, a regular geometry composed of irregular parts which are repeated on successive scales which is indicative of the same processes being applied over and over again. The process can be viewed as a hierarchy which is clearly present in the pattern itself but in terms of the recursive process, can be abstracted into the usual tree-like diagram which we show in Figure 2.

Figure 2: The Hierarchy of Composition in Constructing a Fractal

There are several strange consequences to the process that we have just illustrated. If the process of adding more and more detail of the same kind continues indefinitely, the length of the line increases to infinity but it is intuitively obvious that the area enclosed by the resulting shape either in the Koch curve in Figure 1(a) or the Koch island in Figure 1(b) converges to a fixed value. Second, if the line becomes more and more convoluted in filling the plane, then it would appear that the line which has Euclidean dimension of 1 seems to have the dimension of the plane which is 2. This concept of space-filling can be formally demonstrated to be encapsulated in the idea of fractal dimension. The Koch curve in Figure 1(a) has a fractal dimension about 1.26 while a more convoluted line like a fjord coastline has something like 1.7. Rather smooth curves such as the coastline of southern Australia have a fractal dimension of about 1.1. In fact the inventor of the concept of fractals, Benoit Mandelbrot, wrote a famous paper in Science in 1967 which was entitled “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”. To cut a very long story short, objects which are irregular in the way we have shown and which manifest self-similarity are fractals whose dimension lies between the dimension that they are defined by and the dimension of the space that they are trying to fill. In cities, filling the two dimensional pane with particular forms of development from the parcel to the street line and at different densities suggests that their fractal dimension lies between 1 and 2. Thus this dimension becomes the signature of urban morphology which is the outcome of processes that generate fractal shapes (Batty and Longley, 1994).

Figure 3: Literal hierarchies: transport from a central source

a) left: each link is separate b) right: arranging links into a more efficient structure

There is however a much more literal morphology which is fractal and this is the shape of an object or set of linked objects that form a tree or dendrite. If you want to transport energy from some central source to many distant locations, it is more efficient to develop infrastructure that captures as much capacity for transfer as near to the source as is possible. This is rather easy to demonstrate graphically for if there are 16 points arranged around a circle, then rather than build a link between the source and each of these 16 points, it is more efficient to group the links in such a way that the distance to these different locations is minimised. In Figure 3(a), assuming each single link is of distance 1 unit, then the length of the routes needed in total to service these locations (‘fill the space’) is 16 in comparison with the grouping of these routes into 2, then four, then 8 which is shown in Figure 3(b). The total distance of this arrangement in 3(b) is something between one half and three quarters of the original form in Figure 3(a) depending on the precise configuration although the capacity of the links which take more traffic nearer the source are bigger, and this would incur extra costs of construction. Nevertheless, this demonstrates the important point that where resources are to be conserved (which is in virtually every situation one might imagine), space must be filled efficiently. The tree structures in Figure 3 are fractals with Figure 3(b) illustrating this self-similarity directly while at the same time being a literal hierarchy spread out in space demonstrating quite explicitly the pattern of its construction.

There are many examples of such hierarchical structure in the forms we see in both nature and in made-made systems. Energy in the form of blood, oxygen, and electric signals are delivered to the body through dendritic networks of arteries and veins, lungs, and nerves as we illustrate in the schematic of the central lung system in Figure 4(a). Plants reach up to receive oxygen from the air and down to draw out other nutrients from the soil as in Figure 4(b). Nearer to our concern here and reflecting the discussion of route systems above, Figure 4(c) shows the network of streets in the mid-size English town of Wolverhampton (population circa 300,000 in 2001). It is clear that the traditional street system has grown organically but the ring around the town centre has been planned, imposed from the top down, thus illustrating the notion that what we observe in cities is a mixture of different scales of decision-making. In Figure 4(d), we show one of the Palm islands off the coast of Dubai developed by the construction company Nakheel. This is a wonderful example of how it is necessary to conserve resources when building into hostile media – in this case by reclaiming land from the sea, where transportation and access become the main themes in the way the resort is formed.

Cities develop by filling the space available to them in different ways, at different densities and using different patterns to deliver the energy in terms of people and materials which enable their constituent parts to function

These examples demonstrate that cities are built from the bottom up, incrementally and where they are planned from the top down, the plan is usually a small part of wider organic development. When cities grow at any point in time, we have little idea of what the future holds with respect to new behaviours, values, technologies and social norms and thus it is not surprising that cities grow in an ad hoc manner reflecting the efficiencies and equities that dominate the consensus at the time when development takes place. To illustrate how we can model this process, we can abstract it into two main forces that reflect the desire for space on the part of any individual, developer or consumer which is traded off against the desire to live as close as possible to the ‘city’ composed of other individuals so that economies of scale might be realised. This is a simple model which captures all the ideas we have introduced so far and we will now develop it as a hypothetical simulation.

3) Simulating space-filling growth
Our model is based on two key drivers. First we would all agree that cities exists as machines for enabling us to divide our labour so that we might realise economies of scale, or agglomeration economies as they are increasingly called. Alfred Marshall made the point over one hundred years ago: “Great are the advantages which people following the same skilled trade get from near neighbourhood to one another. The mysteries of the trade become no mystery but are, as it were, in the air.” (quoted in Glaeser, 1996). Our first principle is that individuals must be connected to one another in terms of their proximity to others for the city to exist and this means that new entrants to the city must somehow connect physically to those already there. In contrast, individuals seek as much personal space as possible for themselves and this translates into the notion that they wish to live as a far away from others as possible in the city space. This may translate into living at low densities but as in Manhattan, large apartments in the sky may be another way of realising this quest while there are increasingly innovative ways of meeting this goal by specialising in living in different locations. In our context here, we will embody this second principle as one where people wish to live on the edge of the existing city rather than in the centre, notwithstanding the great variety in these kinds of preference.

Figure 4: Space-Filling Hierarchies

a) top left: a schematic of the human lung b) top right: a schematic of a tree growing into different media: air above ground and soil below c) bottom left: the road network of a mid-sized English town: Wolverhampton d) bottom right: space-filling in difficult media: Nakheel’s Palm Island in Dubai

Our model can be constructed as follows. Imagine a trader decides to locate his or her base at the intersection of a trading route and a river where the land is fertile and flat. Many cities have grown from such humble origins where comparative natural advantages such as these determine the best location for settlement. Now imagine another individual seeking a permanent location wanders into the vicinity of the lone trader’s base. If that trader happens by chance to get within the neighbourhood of the existing trader, that trader decides to locate there although there may be many traders in the wider hinterland who do not enter into the neighbourhood and never find the emergent settlement. However a certain proportion will find the settlement with a certain probability and given enough time and traders, the settlement will grow. From these simple principles, we can demonstrate the form of the growing city. In Figure 5(a), we show a schematic of the location process. The individuals are arranged around a circle well outside the location of the settlement which is fixed at the centre of the circle with the red solid dot. This is where the original trader locates. Each individual is a sold blue dot and they begin the movement in search of the location using a random walk. They decide at each step to move up or down or left or right randomly and in this way walk across the locational plane. If they move to a cell adjacent to the fixed solid red dot, they settle; they stop any further walking and turn red which shows they are now fixed, stable and no longer in motion. The first one to do so is shown by the shaded red dot adjacent to the initial red dot. That is all there is to it. You can see the final form, so you know what will result but if you had not seen this result, then many would guess that the result would not be a tree-like structure but a compact growing mass.

a) c)

Figure 5: Generating clustered city growth using diffusion-limited aggregation
We show the progression of this in Figure 5(b) and to talk this through, what happens is that as soon as a trader settles next door to the existing red dot, the chances of another trader finding that new trader settlement as opposed to any other one increase just a fraction. As time goes by, the linear edge pattern that is characteristic of the growing tip of the cluster begins to emphasise itself and increasingly a trader finds it impossible to penetrate into the fissures in the growing cluster. Traders then are more likely to find the growing cluster at its edge and in this way, the cluster begins to span the space. If this were to produce a growing compact mass then it would have dimension nearer to 2 – the Euclidean dimension but in fact it has a fractal dimension between 1 and 2, about 1.7 so empirical work in many fields has determined. This like any dendritic structure, is a fractal, and it is easy to see the self-similarity that is contained in its form. Break off any branch and you can see the entire structure in the branch just as you can usually see the entire structure of a tree or a plant in its leaf. As we increase the resolution of the grid or lattice on which this walk takes place, then we get finer and finer tree-like structures where the fractal structure is readily apparent as we show in Figure 5(c).

Plans for ideal cities are not grown by a generative logic, because plans are conceived of all-in-one-piece. The notion of an uncertain future is never in the frame

This form is generated by a process which is called diffusion-limited aggregation (DLA) which has been found and used extensively in physics to grow crystal-like structures and to examine ways in which one media penetrates another, such as oil diffusing into water. In fact you can see similar patterns if you pour concentrated liquid soap into ordinary bath water and this is also reminiscent of the way the Dubai palm island resort has been ‘forced’ into the sea. It is a general principle that a substance with a higher density creates such patterns when infused into a substance of a lower density. A model of course is only as good as its assumptions but it is possible to tune this DLA model to produce many different shapes, some of which bear an uncanny resemblance to those that we find in real cities. For example, we can ‘tune’ the DLA to produce sparser structures if we relax the criterion that the individual who settles must exactly touch the already settled structure. We could, for example, set a distance threshold for this, or we could insist that more than one individual must already have settled. In this way, we can change the density, growing structures that are very heavily controlled in their dependence to what has gone before, generating linear structures all the way to compact ones where the degree of control over where traders are allowed to settle is very weak. Some of these extensions are illustrated in my book Cities and Complexity (Batty, 2005).

4) Real world cities and patterns of complexity
There are many examples at different scales of the way cities are structured along dendritic lines mirroring the lines of energy that serve their distant parts. We glimpsed an idea of this in Figure 4(c) where we abstracted the street network of Wolverhampton but cities are not pure dendrites. Different networks are superimposed on one another for different kinds of transport ranging from different modes requiring different networks through to social and electronic networks which underpin the way people trade and communicate. In Figure 6, we show a map of inner London where the streets are coloured according the energy they transport, in fact using the proxy of road traffic volumes which give some index of both capacity and congestion or saturation. This is also highly correlated with patterns of accessibility which mirror the proximity of places to each other.

Figure 6: The Organically Evolving Network of Surface Streets in Greater London Classified by Traffic Volume

Street networks are excellent examples of how cities grow from the bottom up for they represent the skeletal structure on which all else in the city hangs. As we can see from the way cities grow, transport and land use are intimately related. Indeed, in the 1930s as urban sprawl first became significant in Britain, ‘ribbon’ development became the pattern of transport linked to land use that was the subject of fierce control in the quest to contain urban growth. It is possible to see this connectivity in many patterns of urban growth. The picture of the way the eastern US city of Baltimore in Maryland has grown over the last two hundred years that we show in Figure 7 is a clear illustration of the way development proceeds along radial routes from the traditional centre, particularly at the turn of the last century when streetcars, bus and railways systems dominated. Although this pattern is breaking down as cities become more polycentric and specialised in their parts and as new kinds of central business district such as ‘edge cities’ become established, it is still significant.

Such planning through incremental evolution has not been the history of most city plans hitherto but our science is evolving to meet this challenge

These patterns recur at different scales although the notion of them being faithfully reproduced at every spatial scale needs to be tempered with the obvious fact that individuals are diverse in their tastes and values and thus heterogeneous in their actions. Moreover the sort of similarity that occurs in cities is statistical self-similarity rather than the rather strict self-similarity that we saw for example in the construction of the Koch snowflake curve in Figure 1. In fact, although the pattern of transport routes in cities is generally radial, focussing on significant hubs, and organised according to a hierarchy of importance which mirrors different transport technologies, capacities and speed of transmission, street systems illustrate the space-filling principle quite clearly. At the local level, there is more conscious planning and design of street systems particularly in developments which are self-contained for purposes of the actual construction themselves as well as their financing and sales. For example, residential areas are often formed as small single entry streets into houses arranged around cul-de-sacs for purposes of containment, traffic management as well as security. We will return to these ideas in the next section when we speculate on how such structures can be formed in a more conscious sense through explicit design and planning but it is important to note that these patterns do recur across different scales as can be seen in their statistical distributions as well as their physical self-similarity.

Figure 7: The Two Hundred Years of Urban Growth in Baltimore


The obvious question is how far can we get with the diffusion-limited aggregation model of the last section in generating simulations of real structures that we see in the transport development of London in Figure 6 and urban development of Baltimore in Figure 7? This kind of model is of course a demonstration of how two principles or forces interact to produce a structure that resembles certain features of the modern city. It is not intended as anything other than a graphic way of impressing the notion that bottom up uncoordinated change leads to highly ordered structures – fractals – which emerge from this comparatively simple process. One can begin to illustrate how one might make this more realistic but it is a far cry from the kinds of operational models that are used routinely for strategic planning by government and other agencies. The model is made more realistic simply by planting it into a space or terrain that has real features. In Figure 8, we show four different simulations of development in the town of Cardiff, Wales which takes the coastline and rivers which define that area. We set two seeds, one at the historic centre and one at the dockside and let the DLA model operate in the manner we have shown in Figure 5. From this, we realise quite quickly that the river cutting the town in two makes a difference to the rate of growth in parts of the town while the fact that Cardiff has two centres shows how difficult it is to generate a pattern that gives the right historical balance to each. This is not surprising because none of the factors that affected this competition between the two centres is contained in the model. A more detailed discussion of the simulation is presented in our book Fractal Cities (Batty and Longley, 1994). The model however can only simulate patterns that are a consequence of its assumptions. Yet these kinds of simulation also provide a means of demonstrating and testing various future hypotheses about urban form. By way of providing some sense of closure to this essay, we will show how such models can be used to help to generate effective designs.

Figure 8: Simulating Growth Using DLA in the Spatial Landscape
Centred on the City of Cardiff

The control over the development is tuned to decrease systematically through the simulations from the top left to bottom left in clockwise order

5) Generating idealised cities
We need a greater degree of control over our simulation process than that provided by the diffusion-limited aggregation model or its variants. In fact the way we generated the previous clusters is using a generative algebra that lies at the basis of many pattern-making procedures called automata. An automata is usually defined rather generally as a finite state machine driven by inputs which switch the states of the machine – the outputs – to different values. The outputs from the machine may then be used as inputs to drive the process of state transition through time and this generative process can be tuned to replicate the sorts of patterns that we have been discussing in this essay. For example, the input to the DLA model is an individual who moves in a cell space and if certain conditions in the space occur, the individual changes the state of the cell from undeveloped to developed. This is of course is done in parallel for many individuals. The idea that the space might be characterised as a set of cells simply gives some geometric structure to the problem and although we have taken for granted the fact that cities are represented in this way in these simulations, for automata in general, and spatial automata in particular, they can be any shape and in any dimension.

Figure 9: How Cells are Developed

a) left: an 8 cell neighbourhood around a central cell in question is applied to b) middle: each cell in a lattice. If one or more cells in the lattice is of a particular state, in this case developed (black), the state cell in question in the neighbourhood (black hatch) changes to developed. If this rule based on one of more cells in the neighbourhood is applied to every cell in the lattice, the result is c) right: the set of cells around the central cell (black) become developed (black hatch).

The automata we use here to generate physical development are called cellular automata (CA) where we assume a regular lattice of (square) cells in which development takes place by changing the state of each cell from undeveloped to developed as long as certain rules apply. The elements of CA are thus: a set of cells which can take on one of several states, in this case developed or undeveloped, extendable to different kinds of development; a neighbourhood of 8 cells in the N-S-E-W-NE-SE-SW-NW positions around each cell in question; and a set of transition rules that define how any cell should change its state dependent upon the configuration and state and possibly attributes of cells that exist within the neighbourhood of the cell in question. Now if we apply this model starting with the initial condition of one cell in the centre of the lattice being switched on – developed – and apply the rule that if there is exists one or more cells in the neighbourhood of any cell, this will generate a diffusion around the initial cell which mirrors the process of successive spreading of the phenomena, just as a physical substance with some motion might diffuse. The diffusion is square because the underlying lattice is square but we can easily develop versions where the diffusion is circular if we so configure the lattice. We show this diffusion and the rules of engagement in Figure 9. If we then modify the rules by noting that the number of cells in the neighbourhood of an undeveloped cell must be only one, then we generate the diffusion in Figure 10(a) and if there are one or two, then the simulation generates Figure 10(b). There are literally millions of possibilities and the trick is of course to define the correct or appropriate set of rules. Wolfram (2002), in his book A New Kind of Science, argues that such automata represent the fundamental units on which our universe is constructed. Although we have more modest ambitions here, this kind of automata can be tuned to replicate many different generative phenomena which characterise many different forms of city.

Figure 10: Regular Diffusion Using CA: Patterns Reminiscent of Idealised Renaissance City Plans

a) left: with only one cell in the neighbourhood b) right: with one or two cells developed

To generate ideal cities using such automata, it is necessary to begin with a set of realistic rules for transition. Ideal cities are often designed to meet some overriding objective function, to minimise density as in Frank Lloyd Wright’s BroadAcre city, to maximise density as in Le Corbusier’s City Radieuse, to generate formal vistas and garden squares as in Regency London, to generate medium density new towns with segregated land uses as in the first generation of British New Towns, and so on. A rather good example which can be generated using cellular automata principles is the plan for Georgian colony of Savannah in the New World. Developed in 1733 by General James Oglethorpe, we show the plan in Figure 11; the CA rules might be imagined in analogy to the way we generate development in Figures 9 and 10.

Figure 11: The Colonial Plan for Savannah Georgia

a) left: the original neighbourhood plan b) right: the plan in 1770

Usually plans for ideal cities are not grown using a generative logic because plans are conceived of all-in-one-piece so-to-speak and the notion of an uncertain future is never in the frame. However CA allows us to generate plans that evolve through time and we can continually change the rules so that the idealisation is a shifting vision. In a sense, the plans which are grown in Figures 9 and 10 have stable rules which may or may not be considered as ideal objectives to be attained. To conclude our demonstration of this kind of logic and the intrinsic complexity of cities in that their ideal form is never certain, we will return to the DLA model and tweak the rules a little so that a system-wide objective might be met. Imagine that the agents in our model move randomly in the manner we described earlier, that is to all points of the compass; this can be simulated using CA by assuming that where the state of the cell is an agent, then the cell changes state according to the movement. If an agent is at cell i, j, and it moves to cell i+1, j in the next time period, then the cell state switches accordingly, from the cell where the agent is located to the cell where it is newly located. Our first rule then is simply cell state switching from the place where the agent was located to its new location. But we also have a rule that says that if the agent is located at cell i, j and there is another agent fixed at a cell in the neighbourhood of i, j, then the agent remains fixed and the cell on which it sites changes to the stable state. Note in this version of the CA, the cells contain mobile or fixed (stable) agents or have no agents within them at all. The cells have three possible states which are appropriately coded but this is still a CA with two sets of rules.

Figure 12: Optimising Growth by DLA: New Development on Leeward Side of Existing Development

a) left: if cells in black on the windward side are developed, the central neighbourhood cell is developed b) right: a typical outcome of the generative process

Now imagine that there is a strong wind blowing from south west to north east and therefore any agent which is on the windward side of an occupied cell will not fix themselves there. So whenever an agent comes within contact of an agent already fixed on its leeward side, it continues to be mobile. Thus the development moves continually away from the point where the first agent locates and what happens is that a line of cells is established across the space. In fact it is quite hard to guess what happens and it is necessary to run the simulation to see what the ultimate form the model might be. We show this in Figure 12 which is the picture of the city formed when the two principles of contact to the existing agglomeration and the need for as much space as possible are linked to the general objective of locating on the leeward side of already existing development. CA shows how this objective might be met.

Therefore any goals that we might have for the future city are contingent on the present, hence continually subject to revision and compromise

5) Next Steps
There is much still to say about how cities are formed and evolve, how we might best understand and then simulate them, and most importantly, how we should design plans which enable them to function in more efficient and equitable ways.

This essay has broached the idea that cities evolve into an unknowable future that is always uncertain. Therefore any goals that we might have for the future city are contingent on the present, hence continually subject to revision and compromise. In the past, cities have been designed in a timeless future where sets of objectives have been defined to be achievable as if the city were cast in timeless web, and it is of little surprise that few cities have ever achieved the aspirations set out in their plans. Complexity theory broaches the problem of the unknowable future and the way cities evolve from the bottom up, incrementally as the products of decisions that might be optimal at any one time but always subject to changing circumstances. This would appear to be a far more fruitful and realistic way of generating cities that meet certain goals with the goals continually under review as the city emerges from the product of decisions which might be optimal in the small but whose global effects are unknowable in the large, until they emerge.

There are ways in which the processes that we have introduced here might be steered in more centralised ways and it is the challenge of thinking in these terms that the complexity sciences are attempting to grasp: how control and management, planning and design which traditionally have been configured and treated from the top down might best be meshed with systems that grow and evolve from the bottom up. The answers probably lie in notions about hierarchy and the extent to which we might intervene and manage processes that generate hierarchies organically from the bottom up (Batty, 2006). As we learn more about how cities evolve in these ways, it is my contention that we will learn to plan less as we identify points of pressure and leverage. There, effective intervention and design in small, incremental ways might lead to large and effective changes that go with the flow, and do not fight against the grain. Such planning through incremental evolution has not been the history of most city plans hitherto but our science is evolving to meet this challenge.


7 References

Alexander, C. (1964) Notes on the Synthesis of Form, Harvard University Press, Cambridge, MA.

Batty, M. (2005) Cities and Complexity: Understanding Cities with Cellular Automata, Agent-Based Models, and Fractals, MIT Press, Cambridge, MA.

Batty, M. (2006) Hierarchy in Cities and City Systems, in D. Pumain (Editor) Hierarchy in the Natural and Social Sciences, Springer, Dordrecht, Netherlands, 143-168.

Batty, M. and Longley, P. A. (1994) Fractal Cities: A Geometry of Form and Function, Academic Press, San Diego, CA.

Geddes, P. (1915) Cities in Evolution, Williams and Norgate, London.

Glaeser, E. L. (1996) Why Economists Still Like Cities, City Journal, 6 (2), online at

Jacobs, J. (1962) The Death and Life of Great American Cities, Random House, New York.

Mandelbrot, B. B. (1967) How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, Science, 156, No. 3775, 636-638.

Simon, H. A. (1962) The Architecture of Complexity, Proceedings of the American Philosophical Society, 106, 467-482.

Wolfram, S. (2002) A New Kind of Science, Wolfram Media, Champaign, Illinois.