cellular automata and fractal urban form: a cellular ... · urban land-use dynamics with a high...

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Environment and Planning A, 1993, volume 25, pages 1175-1199

Cellular automata and fractal urban form: a cellular modelling approach to the evolution of urban land-use patterns

R White Department of Geography, Memorial University, St John's, Newfoundland, Canada

G Engelen Research Institute for Knowledge Systems, PO Box 463, 6200 AL Maastricht, The Netherlands Received 29 July 1992; in revised form 11 February 1993

Abstract. Cellular automata belong to a family of discrete, connectionist techniques being used to investigate fundamental principles of dynamics, evolution, and self-organization. In this paper, a cellular automaton is developed to model the spatial structure of urban land use over time. For realistic parameter values, the model produces fractal or bifractal land-use structures for the urbanized area and for each individual land-use type. Data for a set of US cities show that they have very similar fractal dimensions. The cellular approach makes it possible to achieve a high level of spatial detail and realism and to link the results directly to general theories of structural evolution.

Introduction Cities, like most geographical phenomena, are complex objects. The intricate mix of urban activities evidently functions with a logic of its own, but a logic nonetheless, as cities are one of the most successful creations of human society. Indeed, as almost all cities exhibit this complexity, it is reasonable to suppose that complexity is in some way an essential quality. This view was developed very persuasively by Jacobs (1961) in her study of New York. Her evidence was anecdotal, but recent work in the theory of dynamical and evolutionary systems provides support at the most fundamental level for the idea that complexity is an inherent, necessary characteristic of cities.

Cities exist to support the social and economic functions of society. But human societies are inherently information-rich systems. The built city must thus in some way reflect the information structures of the society that creates and uses it, and indeed the city is itself an information-rich medium. Thus the complex spatial detail present in a city may be seen not as noise, but as information. Users of geographical information systems (GISs) are perhaps alone among contemporary geographers to realise this, if only implicitly. After all, the primary purpose of a GIS is to store spatial detail—as much of it as possible. But, although city engineers and others know very well how to use particular items of data stored in their GIS, no one has given much thought as to what collectively the data might signify about the nature of cities.

This interpretation is in sharp contrast to the premise that underlies most current geographical analysis. Briefly, the orthodox position is that most complexity is simply noise, obscuring a structure which is essentially simple; with luck and sufficient skill, given the techniques of inferential statistics, the noise can be stripped away to reveal the underlying simplicity. Geographical theory typically predicts relatively simple locational patterns—the regularly spaced retail centres of central place theory, for example, or the concentric land-use zones of Alonso-Muth land-use theory. If these patterns are difficult to find in real data sets, and then only after much statistical processing, this is simply taken as confirmation that the world is unfortunately noisy. A theory such as that used to produce the Alonso-Muth

1176 R White, G Engelen

model can be reformulated as a computable general equilibrium model, and with complicated boundary conditions can then yield complex patterns. But, in this case, the complexity is essentially imposed on the system as an external condition, rather than being generated by the system itself.

Furthermore, most geographical theories—central place theory and Alonso-Muth land-use theory are instances—are static, and rational actors are assumed to interact in a market which remains in a state of stable equilibrium. This approach has produced useful results. The phenomena captured in an Alonso-Muth model, for example, undoubtedly exist in real cities. But, although at the level of the individuals involved in the system the description of their economic behaviour may be quite reasonable, at the aggregate level these models describe a static general equilibrium in which every individual is at a constrained optimum. This is not a fundamentally reasonable characterization of a city, which common sense and experience tell us is rarely if ever in an equilibrium state. Almost all cities are undergoing continual growth, change, decline, and restructuring—usually simultaneously.

By taking the dynamic modelling approach, the major shortcomings of the equilibrium theories can be avoided. The feasibility of this approach has been demonstrated by many workers (for example, Allen, 1983; Allen and Sanglier, 1979; Allen et al, 1984; Clarke and Wilson, 1983; Dendrinos and Sonis, 1990; Engelen, 1988; Haag, 1989; Reggiani, 1990; Weidlich and Haag, 1987; White, 1977; 1978; 1984; Wilson, 1978; 1981; Wilson and Clarke, 1979). Also, in applications to individual cities (for instance, Pumain et al, 1987) it has been shown that it is possible to generate reasonable representations of actual urban forms. In this approach, the focus is on the process, which may or may not lead to a stable equilibrium; but, in any case, the models do not depend on an assumption of equilibrium. The models typically yield results that are relatively complex, both temporally and spatially.

Nevertheless, from the standpoint of our interest in spatial complexity, the approach has one great shortcoming: it is impossible to achieve more than a very crude spatial resolution. As each region in principle interacts with every other region, the computation time grows exponentially with the number of regions. Hence, in general, several dozen regions is the maximum number that can be handled in practice.

In this paper, we use the techniques of cellular automata to develop a model of urban land-use dynamics with a high spatial resolution. A cellular automaton is essentially simple: it can be thought of as an array of cells whose states depend on the states of the neighbouring cells. More specifically, a cellular automaton consists of an n-dimensional rectangular array of cells whose cells can be in any one of several discrete states, a definition of the neighbourhood of a cell, and a set of transition rules which determine the state of each cell as a function of the state of the cells within the neighbourhood. Time is discrete, and all cells have their states updated simultaneously.

Cellular automata are not new. They were developed by the physicist Ulam (1952) in the 1940s and were soon used by von Neumann (1966) to investigate the logical nature of self-reproducing systems. Research on cellular automata has grown rapidly since Wolfram (1983; 1984) showed that these apparently simple systems can generate very complex structures, including fractals, and demonstrated that they therefore provide a useful technique for exploring a wide range of fundamental theoretical issues in dynamics and evolution.

Currently, cellular automata are being used by researchers from a variety of disciplines to investigate questions concerning the origin and evolution of structure (for example, Bak and Chen, 1989; Bak et al, 1989; Barbe, 1990; Kauffman, 1990;

Cellular automata and fractal urban form 1177

Langton, 1986; 1990; Markus and Hess, 1990; Wolfram, 1986a; 1986b). Thus Bak and Chen (1989) show that a few simple models can give insights into a wide variety of phenomena. Their percolation model, for example (but see also Grassberger, 1991), demonstrates the mechanism by which a spatially uniform energy input is dissipated on a consequent fractal structure. The mechanism is relevant to a number of dynamic processes, such as the spread of forest fires and epidemics, the propagation of chemical reactions, and the appearance of turbulence. And it is not coincidental that these are all essentially spatial phenomena, with dynamics that can be characterized as spatial processes. Cellular automata are, by their very nature, spatial models.

Although these models yield important insights into properties that are shared by large classes of dynamical systems, by their very generality they are not realistic descriptions of any particular phenomenon. Nevertheless, cellular automata have two characteristics that make them inherently attractive for application to geographical problems. The first is that they are, as noted, intrinsically spatial; and the second is that they can generate very complex forms by means of very simple rules. Thus they offer the possibility of understanding the origin and role of spatial complexity.

Tobler (1979) was perhaps the first to recognize the advantages of cellular models. Couclelis (1985; 1988; 1989) subsequently took up the approach and developed it in two directions: one, a rather philosophical exploration of the links with the theory of complex systems; the other, an examination of possible uses in an urban planning context. Although she implements several simple models, they are not intended to be realistic representations of urban processes. Instead, they are used primarily as aids in thought experiments, and with their help she provides important insights into the nature of geographical processes. Cecchini and Viola (1990; 1992) have also used cellular automata to model the urban growth process. As are Couclelis's, their goals are essentially heuristic, but they attempt, quite successfully, to produce more realistic looking results. Cecchini and Viola view the city as a continuously evolving object whose complex, large-scale structure is the cumulative result of the local application of relatively simple decision rules. Their views are thus in this respect quite close to those that form the basis of this paper.

Several others have also applied cellular automata to geographical problems. Phipps (1989) used constrained automata with stochastic perturbations of the cell states to investigate certain basic principles of spatial structuring. Frankhauser (1991a) showed that a cellular model of tumour growth could also be interpreted to represent the growth of an urban area. And Hillier and Hansen (1984) developed a cellular approach to the generation of spatial structure which they used successfully to model and explain both the built form of French villages and the layout of rooms in houses.

Last, Batty et al (1989), Fotheringham et al (1989), and Batty (1991a; 1991b) have used a closely related technique, diffusion limited aggregation (DLA), to model the growth of built-up areas. These models are also cellular in nature, but only two cell states—vacant and occupied—are possible, and only vacant cells that are in contact with an occupied cell can be converted to an occupied cell. Like cellular automata, DLA models also generate complex forms by means of a simple process, and the forms are often very suggestive of real urban areas. But the DLA process does not seem to correspond very closely to any actual urban growth process. The applications are nevertheless interesting.

In this paper, we will address the issue of complexity in urban structure by developing a cellular automaton specifically to model the evolution of urban land use. The model generates fractal patterns of land use by means of relatively simple yet reasonable rules of spatial behaviour; the process by which order and complexity


evolve simultaneously is thus made explicit. We then cojnpare the urban structures generated by the cellular, model with data from a sample of US cities to show that the model yields a good representation of actual urban form. Last, we use recent results in the theory of cellular automata and other connectionist models to provide additional insight into the reasons the land-use structures evolve as they do, and to demonstrate that a complex, fractal order may be a functional necessity.

A cellular model of urban land use The model developed here is intended to be used to investigate basic .questions of urban form rather than to provide realistic simulations of the development of particular cities. Consequently, it is kept as simple as possible. Nevertheless, in order to be useful, even as a tool for theory development, the model must give reasonable representations of both urban form and the process which generates it; therefore, it must be more specific and hence more complicated than the highly generic models typical of basic research in cellular automata. Similarly, it is much more specific than the models used by Cecchini, by Frankhauser, and by Batty et al to model urban structure, as their models are all essentially simple generic cellular automata interpreted as urban models, rather than models designed specifically to replicate actual urban processes. In this model, each cell state represents a type of land use—for example, vacant, housing, or commercial—and the city grows and its structure evolves as cells are converted from one state to another according to the transformation rules. The model is specified as follows.

(1) The automaton is developed on a 5 0 x 5 0 grid of cells. Each cell must be in one of four states: vacant (V), housing (H), industrial (I), or commercial (C). The grid size imposes some limitations on the treatment of land-use changes. For example, the minimum size for some developments, such as large industrial sites or airports, may be substantially larger than one cell, but the current version of the cellular model would require them to grow incrementally. These limitations do not constitute a serious problem for the present work, which is focused on general theoretical issues.

(2) The net number of cells, Nt (i = H, I, C) to be converted from vacant to each of the urban land uses at each iteration is determined exogenously to the cellular automaton. The number of cells to be converted to each state in the first iteration is supplied with the initial conditions, and the number to be converted in subsequent iterations is determined by applying growth rates for each urban function (supplied as parameters) to those initial numbers. It seems reasonable to treat the growth rate as exogenous, as the growth of a city would normally depend primarily on the role of the city in the regional or national economy, rather than on its internal spatial structure.

(3) The neighbourhood of a cell is defined to be all cells within a radius of 6 cells. In the general case, the neighbourhood thus contains 113 cells, but the more distant cells within this neighbourhood may be given a weight of zero, in which case the effective neighbourhood is smaller. As the array of cells is regular, each cell within the neighbourhood falls within one of 19 discrete distance bands.

(4) In order to reduce the computational burden, a hierarchy of land-use states is defined such that a cell may only be transformed from a lower to a higher state. The order in this model is vacant (the lowest state), housing, industry, and commerce (the highest state). Thus, for example, a vacant cell may be converted to any other use, but an industrial cell may only be converted to commercial use. Because cells that are in an urban state cannot be converted to vacancy, the city can only grow. These restrictions have been relaxed in a more recent version of the model, in which a cell may change from any state to any other state, including vacant.


(5) At each iteration, transition potentials must be calculated for all allowed transitions. These potentials represent the behavioural propensities of the actors who determine urban land use, and they thus form the basis of the transition rules in the model. Transition potentials for a cell are calculated as weighted sums:

Ptj = S 1+ E mkdIhd), (1) \ h,k,d J

where P0 is the transition potential from state / to state j ; mkd is the weighting parameter applied to cells in state k in distance zone d; h is the index of cells within a given distance zone; I,ld equals 1 if the state of cell h = k; otherwise, Ihd equals 0; S is a stochastic disturbance term. The disturbance term is given by

S = l+(-\nR)a, (2)

where R (0 < R < 1) is a uniform random variate, and a is a parameter that allows control of the size of the stochastic perturbation. The stochastic term S has a highly skewed distribution, so that most values are near unity, and much larger values occur only infrequently. Thus most of the potentials Ptj are close to their unperturbed, deterministic values; in other words, the transition parameters such as those shown in table 1 dominate the determination of the transition potentials. As Ptj > 1, every cell in the array has a nonzero chance of transition. In general, cells within the neighbourhood are weighted differently depending on their state and also on their distance from the reference cell, and the mkd may be specified to represent, for example, a standard negative exponential distance-decay relationship. Vacant cells do not receive a weight and thus do not contribute directly to the transition potential.

(6) At each iteration, sufficient cells are converted to each use so that the net increase in the number of cells in each nonvacant state is equal to the exogenously specified increase, TV, (/ = H, I, C). The cells converted to each state are those with the highest potentials for that state, insofar as the hierarchy of states permits. To begin with commerce, the Nc noncommerce cells with the highest potentials for commerce are identified. If some of these cells are also in the lists of cells with the highest potentials for industry or housing, then a series of stochastic tests is made to determine the function to which the cell will be converted. The probabilities used in these tests are specified as parameters in the input. If as a result of these tests some of the Nc cells in the original list are not converted to commerce, in compensation other cells with lower potentials must be converted in order to arrive at the correct number of conversions. The procedure is then repeated for industry, and, finally, for housing. In the case of commercial functions, however, the gross number of conversions must in general be higher than NT and NH, in order to compensate for the loss of cells that are converted from these uses to uses higher in the hierarchy. In this conversion procedure, only potentials for transformation to the same function are compared. Potentials for transformation to different functions are not meaningful as the potentials for different functions are scaled arbitrarily with respect to each other, depending on the values of the weighting parameters mkd. In a more recent version of the model, transition potentials are compared directly, so each cell is converted to the state with the highest potential.

(7) The simulation is run for as many iterations as possible before sizeable clusters of urban activity impinge on the edge of the array and boundary effects become important. The number of iterations possible is thus effectively controlled by the growth rate specified and the size of the stochastic disturbance, determined by a.


The model was not calibrated directly with any particular city or set of cities, as the object of this research is to investigate general features of urban structure. Nevertheless, sensitivity analysis showed that intuitively plausible transition rules (for example, parameters representing a short-range repulsion effect of industry on housing) generated realistic looking cities, and unreasonable rules did not.

Except as noted, the simulations discussed in this paper were run with the following parameter settings: initial ratio of cells to be converted to commerce,

Table 1. Weighting parameters, mw, for transitions from state / to state j (i -* y). A positive value of mkd indicates that cells of type k in distance zone d increase the likelihood of a transition from i to y; a negative value indicates a reduced likelihood; zero indicates no influence at all. Distance zone refers to distance from the candidate cell, zone 2 being the nearest and zone 19 the furthest.

State, k

Euc.

Vacant -*• C I H

Vacant -* C I H

Vacant -+ C I H

Industry C I H

Industry C I H

Industry C I H

Housing C I H

Housing C I H

Housing C I H

Distance

2 3

1 1.4

commerce 25 25

0 0 4 3.5

industry 0 0 3 3

- 1 - 1

housing - 2 - 1 -10 - 1 0 •

2 2

zone,

4

2

25 • 0 3

0 2 0

2 - 5 •

1.5

-* commerce 25 25 25 • - 2 - 2 •

4 3.5

— industry 0 0 0 0 0 0

-*• housing 0 0 0 0 0 0

- 2 3

0 0 0

0 0 0

-*• commerce 25 25 25

1 1 4 3.5

-* industry 1 1 2 2

- 1 - 1

-*• housing 0 0 0 0 0 0

1 3

0 2 0

0 0 0

, d

5

2.2

- 1 -0 2.5

0 1 0

1 - 3 -

1.5

- 2 -0 2.5

0 0 0

0 0 0

- 2 -0 2.5

0 0 0

0 0 0

6

2.8

-1 -0 2

0 0 0

1 -1

1

-2 • 0 2

0 0 0

0 0 0

- 2 0 2

0 0 0

0 0 0

7

3

- 1 -0 2

0 0.2 0

1 0 1

- 2 • 0 2

0 0 0

0 0 0

- 2 • 0 2

0 0 0

0 0 0

8

3.2

- 1 -0 2

0 0.2 0

0.5 0 1

- 2 -0 2

0 0 0

0 0 0

- 2 -0 2

0 0 0

0 0 0

9

3.6

-1 -0 1.5

0 0.2 0

0.5 0 1

- 2 -0 1.5

0 0 0

0 0 0

- 2 -0 1.5

0 0 0

0 0 0

10

4

-1 -0 1.5

0 0.2 0

0.4 0 0.5

-2 -0 1.5

0 0 0

0 0 0

- 2 -0 1.5

0 0 0

0 0 0

11

4.1

-1 -0 1.5

0 0.2 0

0.3 0 0.5

-2 -0 1.5

0 0 0

0 0 0

- 2 -0 1.5

0 0 0

0 0 0

12

4.2

-1 -0 1.5

0 0.2 0

0.2 0 0.5

- 2 -0 1.5

0 0 0

0 0 0

-2 -0 1.5

0 0 0

0 0 0

13

4.5

-1 -0 1

0 0.2 0

0.1 0 0.5

-2 -0 1

0 0 0

0 0 0

-2 • 0 1

0 0 0

0 0 0

14

5

-1 -0 1

0 0.2 0

0.1 0 0.5

-2 -0 1

0 0 0

0 0 0

- 2 • 0 1

0 0 0

0 0 0

15

5.1

-1 -0 1

0 0.2 0

0.1 0 0.1

- 2 -0 1

0 0 0

0 0 0

- 2 -0 1

0 0 0

0 0 0

16

5.4

-1 -0 1

0 0.2 0

0 0 0.1

-2 -0 1

0 0 0

0 0 0

- 2 • 0 1

0 0 0

0 0 0

17

5.7

-1 -0 1

0 0.2 0

0 0 0.1

- 2 -0 1

0 0 0

0 0 0

- 2 -0 1

0 0 0

0 0 0

18

5.8

-1 -0 1

0 0.2 0

0 0 0.1

- 2 -0 1

0 0 0

0 0 0

- 2 -0 1

0 0 0

0 0 0

19

6

-1 0 1

0 0.2 0

0 0 0.1

- 2 0 1

0 0 0

0 0 0

- 2 0 1

0 0 0

0 0 0

Note: Euc. Euclidean distance; C commercial; I industrial; H housing.


industry, and housing, respectively, 1:4:7; growth rate per iteration, 5%; stochastic perturbation parameter a, 2.5; and transition parameters mkd as shown in table 1.

In table 1, each block of three rows contains the parameters for calculating one transition potential. In each such block the first row contains the weightings applied to commerce cells in distance zones 2 - 1 9 , the second row, the weights applied to industry cells, and the third the weights for housing cells. For example, commerce is strongly attracted by commerce cells in the immediate vicinity (values of 25 for distance cells 2 - 4 in the first row C) but repelled by more distant commerce (values of - 1 in distance cells 5-19) . It is unaffected by industry (zeros in the first row I), but it is attracted by housing (representing consumers), subject to a distance-decay effect (as shown by the decreasing values in the first row H). As for the other two functions, industry is unaffected by commerce, attracted by other industry in the immediate vicinity, repelled by nearby housing, but weakly attracted to housing located at a greater distance. Housing is repelled by adjacent commerce but attracted to it otherwise (subject to a distance-decay effect), strongly repelled by nearby industry, and attracted to other housing (again with a distance-decay effect).

Simulation results An extensive sensitivity analysis was carried out. The analysis reported in this paper is centred on three sets of four runs each. The four runs in each set differ only in the seed used to generate the stochastic perturbation; they are identical in terms both of parameter values and of initial conditions. The three sets differ only in the initial configuration of land use. They will be referred to in the text and figures as groups 1, 2, and 3. In order to transcend the stochastic variations and arrive at an idea of the typical behaviour of the model, results will frequently be aggregated by group to produce a sort of composite cellular city with generic properties. Analysis is then carried out on these composite cellular cities rather than on the individual stochastic variations produced by each simulation run.

Land-use patterns at iteration 40 for these twelve runs are shown in figures 1-3 . The variety in the individual urban forms generated is striking. It is evident that the specific urban form is extremely sensitive both to random perturbations and to differences in initial configurations. Nevertheless, the constraints imposed by the initial configuration can, to a degree, override the stochastic effect. This is seen most clearly in figure 3, where all four cities of group 3 have been blocked in their northward development by a large industrial zone appearing just to the north of the central business district (CBD). This pattern is the result of the initial configuration in which all industry is located north of the CBD, with no other land use present there. Also, comparing figure 1 with figure 2, one can see that the coherent initial form of the group-1 cities tends to result in a coherent CBD, whereas the scattered initial configuration of group 2 produces, in three of the four cities, a locational pattern for commerce in which there is no strong CBD.

The stochastic effects operate most effectively in the early stages of the simulation and on the periphery of the urban configuration. The reason is that, as the number of occupied cells within a neighbourhood grows, the range of values calculated for the deterministic term in the transition-potential equation [equation (1)] increases, thereby reducing the relative importance of the stochastic term. As a result of this reduced influence, coherent, large-scale zones of a single land use are able to develop. These are evident in all the cities in figures 1-3 . But where they develop is determined largely by the stochastic element, which, operating to maximum effect on the periphery of the city where it is largely unconstrained, scatters seeds around which these zones nucleate. Thus, for example, although all four cities of group 1


(figure 1) show well-developed industrial zones, in each case their number and locations differ because of the stochastic perturbation. The seeding effect can be seen clearly in figure 4, where the industrial cells seeded during the first 10 iterations in the northeast, the south, and the far northwest all grow to large coherent industrial areas by iteration 40.

In spite of the great variety of urban forms generated for a given set of parameters, at a deeper level all the cities are very similar. The ordered complexity that is evident in figures 1 - 3 is characteristic of fractal forms, and, in fact, all the cities shown in these figures exhibit a fractal structure, and their fractal dimensions are all very similar. The fractal nature of the urban land-use patterns is revealed by the scaling relationships that characterize three features: the radial density gradient for each land-use type, the size distribution of clusters of commercial activity, and

• Commerce B Industry • Housing

(a)

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D

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(c) (d) Figure 1. Group-1 cellular cities: land use at iteration 40 for four cities differing only in stochastic perturbation. Initial condition is as shown in the inset, upper left.


the irregularity of the perimeter of the urbanized area. We will examine each of these relationships, as each represents one measure, or class of measures, of the urban form. It is important to have quantitative descriptors of urban form, and in most of the standard measures it is assumed that in some sense the form is simple. The idea that complexity may be an inherent, important feature of form, and not just noise, is relatively recent, and fractal dimensionality is one of the few concepts that is directly relevant to the problem. The scaling relationships just mentioned all yield measures of fractal dimensionality.

Area - radius scaling Cities consist of a more or less dense scattering of urban activities in the space which contains them. As geometrical objects, they may thus be thought of as Cantor dusts in two-dimensional space, or, more appropriately for cellular models,

„*

p

J*% m

y o

• Commerce E Industry El Housing

4 f-â \>

mli

h (b)

1

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1 + 01 sy» »

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Figure 2. Group-2 cellular cities: land use at iteration 40 for four cities differing only in stochastic perturbation. Initial condition is as shown in the inset, upper left.


as Sierpinski carpets (figure 5), with, of course, a stochastic element in the pattern. For such objects it has been shown that

i _ „-iD B> = q (3)

where B is the number of cells occupied by the original object [B = 5 in figure 5), i is the step number, q is the scale reduction factor (in figure 5, for example, at each step the scale of the original figure is reduced by a factor of 1/3), and D is the fractal dimension. (For a more extensive discussion, see Mandelbrot, 1983;

• Commerce • Industry • Housing

u

-D D

'" D

| « V,

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a

r r

: • "

L D ri

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Figure 3. Group-3 cellular cities: land use at iteration 40 for four cities differing only in stochastic perturbation. Initial condition is as shown in the inset, upper left.


Frankhauser and Sadler, 1991). Solving this relationship for D, one obtains

D = ( l g * ) / l g [ - | , (4)

which allows the dimension of the Sierpinski carpet to be calculated directly. The fractal dimension, D (D < 2) reflects the fact, evident in figure 5, that as

the object expands in cell space the number of cells composing it grows less rapidly than the number of cells in the square area necessary to contain it, so that the object becomes more sparse. In particular, the length L of a side of the figure is

H •B

o a

\ )

i

• Commerce m Industry EI Housing

(5)

" "1=3

%

0 a° .

S3

m

3 P W W j i - H B B D ^

Lru ^BF"I

(d) (e)

Figure 4. Five stages in the growth of a cellular city.


M M M M ULtfflJU uLRxju ftj^ 11 ptj^ 11111111 Fm I IrJi

HnllnrHtillkmnllnn 1111111111] l iAjj M11111 H~H

H I i K rrrn M &ffM MIM M

(b) (c)

Figure 5. Th ree stages in the construction of a Sierpinski carpet: (a) S = 1, (b) S = 2, (c) S = 3 , where S is the step number.

Iteration 10 10000 T

1000 i

10000

10 Radius (units)

100 1

(b)

10 Radius (units)

100 1

(c)

10 Radius (units)

• Point used in regressions to fit inner-zone curve

• Point used in regressions to fit outer-zone curve

o Point not included in either regression.

Figure 6. A r e a - r a d i u s relationships for the composite urbanized area of group-3 cities, for three levels of stochastic disturbance: (a) a = 1.0, (b) a = 2.5, (c) a = 4.0; for iterations 10, 20, 30, and 40 .


and the total number of occupied cells, BT, by

BT = Bl. (6)

Thus the relationship between the size of the object, as measured by the number of cells composing it, and its diameter (fractal dimension), is given by

BT = LD. (7)

Hence, we can write

lgBT = c + D l g r , (8)

where c is a constant and r is the radius of the object. As both r and BT are variables, the fractal dimension appears as the slope of

the relationship between them. Equation (8), therefore, provides a useful way of estimating the fractal dimension of irregular or stochastic spatial distributions. In the case of cities, the total area occupied by urban activities within a given radius of the centre is determined for radii of increasing length. Regression analysis, then, yields the slope and hence the fractal dimension.

This fractal dimension is referred to by Frankhauser and Sadler (1991) as the radial dimension. This measure is location specific in that the result depends on the choice of the origin point for measuring the radii. In principle, the centre of the distribution should be chosen. In this research we use the original centre of the city and ignore the possibility that the city may grow in such a way as to become eccentric to this point. When areas are measured by cell counts on a rectangular grid, as is the case here, rounding errors are introduced when the counts are made within circular zones. These errors can be eliminated by using rectangular zones, and Frankhauser and Sadler claim that this approach gives better results. But, as neither the cellular cities generated by our model nor, for the most part, actual cities operate on a Manhattan metric, we have chosen to retain circular zones, because they correspond more closely to the processes that generate the observed patterns.

Frankhauser and Sadler (1991) discuss two other measures of the fractal dimension of the urbanized area: the grid dimension and the correlation dimension. Neither of these methods involves the use of a single reference point to which all observations are tied. They thus yield measures of dimensionality that are, in a sense, context free. However, the contextual nature of the radial dimension is useful in that the procedure provides additional information on the internal structure of the urban area. In general, the three approaches give different values for the fractal dimension, as Frankhauser and Sadler point out, and comparison of these values for a given city can also yield information on the spatial structure.

The radial dimension was calculated for the urbanized area of the cities shown in figures 1-3 . In order to reduce the problem of stochastic fluctuations inherent in the results of any one simulation, the cell counts for the four cities in each group were summed, and the radial dimensions for the aggregate results were estimated by using two-unit increments of the radius; the last five zones, occupying the corners of the grid, were omitted because cell counts in these zones are dominated by boundary effects. Dimensions were calculated for iterations 10, 20, 30, and 40. The aggregate results for the group-3 simulations are shown in figure 6(b). Graphs (not shown) for the group-1 and group-2 simulations are quite similar; the fractal dimensions for all three sets are given in table 2. In addition, in order to test the sensitivity of the results to the degree of stochasticity, the four simulations in group 3 were rerun with two other values for the stochasticity parameter, a: one lower (a = 1.0) and one higher (a = 4.0). Area-radius relationships for these simulations are shown in figures 6(a) and 6(c).


The cellular cities clearly display a bifractal structure; in each case the area-radius relationship is kinked, with a steep inner segment and a flatter outer part. The inner zone consists of the area within which the urbanization process is essentially complete—although vacant cells remain and the urban structure continues to evolve. The outer zone is the area in which stochastic effects remain important and the urbanization process is still fully active, so that the urban structure has not stabilized. This pattern appears at a very early stage in the simulation; it is already well developed by iteration 10. Thereafter, the bifractal pattern is maintained as the city grows, although the boundary between the two zones shifts outward, and both zones show a slow increase in their dimensionality. The degree of stochasticity does not affect this general conclusion, although the greater the stochastic perturbation in the system, the lower the radial dimension of the inner zone, and the greater the rate at which the city spreads out. The response of the system to the stochastic effect means that the parameter controlling it should be a key one in calibrating the model to real situations.

Frankhauser (1991b) finds the same bifractal structure in a sample of world cities. Furthermore, Frankhauser and Sadler (1991) observe that in the one case where they have historical data, the pattern is relatively stable over a long period. They establish the radial dimension, D, of the Berlin agglomeration for five dates from 1875 to 1980. The rate of increase in D is relatively rapid between 1875

Table 2. Radial dimensions of the urbanized area.

Iteration Inner zone, a Outer zone, a

1.0 2.5 4.0 1.0 2.5 4.0

Group 1 10 20 30 40

Group 2 10 20 30 40

Group 3 10 20 30 40

1.80 1.92 1.97 1.96

1.54 1.73 1.85 1.92

1.52 1.74 1.85 1.93

1.61 1.72 1.76 1.84

1.42 1.48 1.60 1.77

0.93 1.16 1.38

0.17 0.42 0.53 1.09

0.21 0.27 0.61 1.23

0.19 0.49 0.78 1.27

0.76 0.93 1.12

1880 1920 1960 Year

• Berlin • Cell city

2000

Figure 7. Evolution of the inner radial dimension of the urbanized area: Berlin 1875-1980, and group 3, iterations 10-40 (sources: Frankhauser, 1991b; Frankhauser and Sadler, 1991; and simulations).


and 1910; from 1910 until 1980 the rate is slower, and the change is essentially linear. The values for Berlin are plotted in figure 7 along with those for the group-3 iterations graphed in figure 6(b), with each iteration assumed to represent two years, so that the growth rate of the area occupied by the cellular city is approximately 1.025% per year. (In contrast, the growth rate of the built-up area occupied by Berlin as estimated from Frankhauser and Sadler's data is in the neighbourhood of 1.035% over the period 1910-45.) The behaviour of the two series is strikingly similar, but it must be kept in mind that the two growth rates are not equal, and Berlin is hardly an ideal case of unperturbed growth in view of its tumultuous history during the period.

The same bifractal structure characterizes the locational pattern of each individual land use (figure 8; results for groups 1 and 2 are quite similar; for inner-zone slopes, see table 3 in which US and cellular cities are compared). Furthermore, these results show that the cellular cities have a concentric zonation of activities. As the two slopes for commerce are less than those for industry, which in turn are less than those for housing, the three activities must achieve their maximum relative concentrations in that sequence, moving from the centre to the periphery of the city.(1)

In order to determine whether real cities have similar fractal, or bifractal, land-use patterns, it is necessary to have standardized land-use data tied to a grid system. Passonneau and Wurman (1966) published land-use data on a 5 0 0 x 5 0 0 m grid for twenty US metropolitan areas. Housing data were taken from the 1960 census, and other land uses were established from a variety of maps supplied by local planning agencies. As nearly as possible, they standardized the data from the various sources, and classified the land use as either commercial, industrial, residential, institutional, park, or airport - cemetery; the results were published as maps with a dimension of 80 grid cells on a side. In all cases the urbanized area extends beyond the edge of the map; in the case of the larger cities, such as New York, Los Angeles, and Detroit, only the inner city is shown.

10000 T

13 1000 \

< ioo X

10 i . 1 10 100 1 10 100 1 10 100 , x Radius (units) ,. x Radius (units) . . Radius (units) (a) (b) (c)

• Point used in regressions • Point used in regressions o Point not included to fit inner-zone curve to fit outer-zone curve in either regression.

Figure 8. Area-radius relationships for group 3: (a) commerce, (b) industry, (c) housing.

W The radial dimension for the inner zone of each activity is able to exceed 2 units when few cells in the central area are occupied by the particular activity, so that the density of occupied cells is able to increase with increasing distance from the centre. In other words, whereas the total number of cells in the array must grow with the square of the radius, the number of cells occupied by any particular activity will grow faster than the square of the radius if the proportion of cells occupied by the activity increases with increasing radius. This can only happen, of course, as long as the marginal proportion of cells occupied by the activity is less than unity. Beyond the radius at which all cells are occupied, the radial dimension cannot exceed 2.


For our purposes, in order for the data to be comparable with the simulation results, the entire urbanized area should be available on the map. As that is not possible, only those cities for which a substantial amount of the urbanized area is shown were selected for analysis. For several of these, the quality of the land-use data was suspect. In the end, four cities were retained for analysis: Atlanta, Cincinnati, Houston, and Milwaukee. In order to make the data more nearly comparable with the simulation results, we have placed airports in the industrial category and have combined the institutional, park, and cemetery classifications into an 'other' category. The modified land-use data for the four cities are shown in figure 9 on the original 80 x 80 500 m grid.

For these cities, the area-radius relationship was established both for the urbanized area and for each of the four land-use categories. As areas occupied by water are

• Commerce • Industry E3 Housing E3 Other

(a) (b)

(c) (d)

Figure 9. Land use in four US cities, 1960: (a) Atlanta, (b) Cincinnati, (c) Houston, (d) Milwaukee (source: Passonneau and Wurman, 1966).


not available for urbanization, the cell counts used to establish area-radius relationships were adjusted accordingly. For the urbanized area as a whole, in each case the relationship was highly linear, but no kink was visible. In other words, insofar as these data are concerned, the cities appear to have a simple fractal structure. However, it is more likely that, in fact, their structure is bifractal, but that the outer zone, characterized by the lower dimension, simply lies beyond the area covered by the data. This interpretation is reinforced by Frankhauser's data, which show that a bifractal structure is normal for large cities.

Analysis of each individual land-use type shows that these also are distributed fractally. Graphs for Cincinnati are shown in figure 10. Only one of the sixteen area-radius relationships, that for housing in Houston, is apparently not linear. The others are quite linear. Industry in Cincinnati and Milwaukee and commerce in Atlanta show bifractal distributions. All other relationships are characterized by a single slope; again, it is likely that all of the relationships are actually bifractal, but that the outer zone is missing. Slopes for the inner zones of the three sets of simulation runs and for the four US cities are shown in table 3. For the most part, slopes for the simulated cities lie within the range of values characterizing the actual cities. The notable exception is for industry, where the two sets of slopes are not really comparable, as the simulated cities do not include an 'other' category of land use.

In every case, both for simulated and for actual cities, the slopes are lowest for commerce and highest for housing, with industry having an intermediate value.

1000

100

1000 T

73 IOO {

I 100 10 100 1 10 Radius (units) Radius (units)

(c) (d) Figure 10. Area-radius relationships for Cincinnati: (a) commerce, (b) industry, (c) housing, (d) other land uses.

Table 3. Radial dimensions for individual land uses: cellular and US cities.

Land use

Commerce Industry Housing Other

Cellular group

1

1.09 1.85 2.29

2

1.17 2.41 2.96

3 _

1.03 2.72 2.77

US city

Atlanta

1.00 1.97 2.12 2.52

Cincinnati

1.10 2.11 2.51 3.42

Houston

1.24 1.51 2.76 2.77

Milwaukee

1.27 1.83 2.38 2.17


Thus the simulated cities show the same concentric land-use zones as the US cities. This congruence appears in spite of the fact that the simulation model was not calibrated to data for any actual cities. Were this done, there is no doubt that the fit would be better. It is also worth noting that in the simulated cities, the break between the two zones tends to occur closer to the centre for industry than for the other two activities; and, correspondingly, in the US cities industry accounted for two of the three cases where land use showed a bifractal structure.

Cluster size frequency spectrum The set of cells occupied by a particular land use in a cellular city (see figures 1-3) can be thought of as an object consisting of a number of clusters of various sizes. If it is a fractal object, then it is self-similar. There is no characteristic cluster size: every change of scale maintains the frequency ratio of clusters that differ in size by a given factor. In other words, the relationship between frequency of occurrence and cluster size is hyperbolic, and thus log-linear. Consequently, the log-log plot of frequency of occurrence of each size class against cluster size will be linear if the object is a fractal (Bak and Chen, 1989; Mandelbrot, 1983). Similarly, it has been shown that the Pareto or log-linear rank-size distribution is a characteristic of a fractal distribution (Frankhauser, 1991a; Nicolis et al, 1989; Wong and Fotheringham, 1990).

The cluster size frequency spectrum for commerce was examined for the cellular city groups 1 -3 . (The other functions do not produce a sufficient number of clusters for an analysis to be interesting.) Clusters were defined in the conventional way by using only horizontal and vertical adjacencies, for the sake of consistency with the procedure of other authors. However, this is not necessarily the most appropriate adjacency criterion, especially for cities developed on transportation networks with diagonal routes. The clusters used in the analysis are not necessarily stable. Because the city is growing, individual commercial clusters may also grow. Thus, cluster size frequency spectra are iteration specific.

In a first analysis, cluster size frequency counts were aggregated for all twelve of the cellular cities (that is, groups 1-3 were combined), for four different iterations

° Iteration 40 • Iteration 30 ° Iteration 20 x Iteration 10

(a) (b) 1000 ,

>» In

g ioo -T 0< 1 >>v

& 10 J > .

1 10 100 1 10 100 Cluster size (cells) Cluster size (cells)

(c) (d)

Figure 11. Cluster size frequency spectra for commerce in cellular cities: (a) composite of groups 1, 2, and 3 for iterations 10, 20 , 30, and 40; and iteration 40 for (b) group 1, (c) group 2, (d) group 3.

1UUU T

100 f

10 1


[figure 11(a)]. The cluster size frequency spectrum is linear for all four periods; and, except for iteration 20, the slopes, giving the fractal dimensions, are all very similar. In terms of the commercial cluster size distribution, then, these twelve cities collectively seem to have been organized very quickly into a particular fractal structure and have then maintained that structure, except for a temporary reorganization or 'crisis' around iteration 20.

For each of groups 1-3 , the aggregate frequencies at iteration 40 are plotted in figures 11(b)-11(d). In each case the relationship is essentially linear, with slopes ranging from 1.33 to 1.71. By way of comparison, the cluster frequency spectra for the four US cities are shown in figure 12. The relationship for Atlanta, which has very few larger clusters, is clearly not linear; but for the other three cities it is, with slopes falling between 1.01 and 1.75. With respect to cluster size frequencies, then, the simulated cities tend to be quite similar to the US cities, at least as they were in 1960, and the cellular and the real cities each have a fractal structure.

100

100 T

10^

(b)

to

1 \ D

\ p

]

• ^ L , -~V B ,

1 10 100 1 10 100 Cluster size (cells) Cluster size (cells)

(c) (d)

Figure 12. Cluster size frequency spectra for commerce in four US cities, 1960 . (a) Atlanta, (b) Cincinnati, (c) Hous ton , (d) Milwaukee.

Perimeter-length scaling We have established that cellular cities are characterized by bifractal densities and fractal cluster frequencies. Their boundaries are also fractal. A fractal line has the property that its measured length increases as the length of the unit with which it is measured decreases. The relationship can be established by plotting the number of steps required to make one trip around the boundary against the length of the step used, for a number of different step lengths. Once again, the relationship is log-linear, and the slope gives the fractal dimension of the boundary. There are several other methods of estimating the fractal dimension of a line, and all give somewhat different results. Batty and Longley (1987) provide a thorough discussion of fractal urban boundaries and their measurement, so the problem will not be dealt with here.

The urbanized area of a typical cellular city consists of a large primary cluster and a number of small outlying clusters, many of them consisting of a single occupied cell. For the purpose of measuring the perimeter of the city, we ignore the outlying clusters and measure the perimeter of the main cluster with five different


units of measurement: lengths of 1, 2, 4, 8, and 16 grid-cell units. The procedure was carried out for several cities; the result for the city shown in figure 2(c) is presented in figure 13. The perimeter was measured at iterations 20, 30, and 40. The slopes, giving the fractal dimension of the boundary, are almost identical at the three time periods; all are in the range 1.41-1.45. This is additional evidence that the cellular cities are quickly organized into a structure which is maintained as they continue to grow and develop. An empirical control is provided by Batty and Longley (1987), who determined the fractal dimension of the boundary of Cardiff to be in the range of 1.2-1.3, with the range of values being the result of differences in measurement technique.

• Iteration 20 • Iteration 30 o Iteration 40

~ 1 10 100 Step size

Figure 13. Perimeter-length scaling for a cellular city of group 2 (created from seed 3) at iterations 20, 30, and 40 [see figure 2(c) for a depiction of the city].

Discussion We have demonstrated that the cellular simulations and real cities have fractal structures; but what is the significance of this fact? Recent work in the areas of emergent computation, artificial intelligence, and artificial life, where cellular automata and related techniques have been used, suggests that cities, like many other phenomena that characterize the living world, must have fractal properties.

Langton (1986; 1990) uses cellular automata to investigate the fundamental properties of self-organizing systems. For a given automaton, one cell state can be characterized as the quiescent state. The set of transition rules can then be divided into those that set a cell to the quiescent state and those that do not. Langton finds that, in general, the behaviour of the automaton depends, in a crucial way, on the proportion, A, of rules that lead to the quiescent state. Specifically, for low values of A, the automaton tends to evolve to a simple steady state or limit cycle. For higher values, the behaviour typically becomes essentially random, or chaotic, so, statistically, the behaviour is again simple. But, for a very limited intermediate range of values, the automaton develops transient structures of indefinite size and complexity. In other words, in this range the simple rules of the system produce highly organized, complex, evolving structures—structures which are often fractals. Langton refers to this regime as a phase transition between the simple order of the low-A automata and the simple stochastic noise of the high-A automata. Langton (1990) calculates the Shannon entropy, H, of the patterns generated by the automata, and shows that the complex structures correspond to only a very limited intermediate range of values of H. In one sense, it is not surprising that a cellular automaton can produce highly organized complex structures, as it has been demonstrated that a universal Turing machine is formally equivalent to a cellular automaton, and so some cellular automata must, in principle, be capable of performing any computation. Nevertheless, it is striking that the complex structures appear only as a transition between order and chaos, and thus require a finely tuned system. But why would such a system be fine tuned?

1UUU

Z 100

6 3


Kauffman (1990), using random Boolean networks, has reached conclusions that suggest an answer to this question. In his Boolean networks, the state of each node depends on the state of the nodes to which it is connected, according to a set of Boolean transition rules. Some rules permit nodes or groups of nodes to become locked in a particular state; these network elements are thus 'frozen'. If the proportion of rules which can lock nodes is relatively low, the behaviour of the network is essentially chaotic, with large attractors in small basins of attraction. Conversely, with a high proportion of rules capable of locking node states, the system quickly settles into a relatively short limit cycle or a stationary state; the system now has a limited number of large basins of attraction containing relatively small attractors. The transition to the second type of behaviour occurs when the proportion of rules permitting frozen states is just sufficient to permit the locked nodes to percolate—that is, to create a reticulation of mutually locked nodes that spans the network. The critical value of the proportion depends on the size of the system and its connectivity. The percolation of locked nodes stabilizes the system sufficiently for ordered behaviour to appear.

Furthermore, it appears that successful structural evolution of the system is possible only when the frozen elements percolate. In order for the system to engage successfully in 'hill climbing' on a fitness surface by means of small random structural changes, the fitness surface must be sufficiently correlated that the fitness level at any one point carries some information about the fitness of nearby points. This is not the case with the nonpercolating systems characterized by large chaotic attractors: a small change in the structure of the system typically produces a very large arbitrary change in the attractor, and hence in fitness. In other words, systems with very similar structures tend to have very different fitness levels, so a given structure carries essentially no information about the fitness of even very similar structures. Furthermore, systems which have too high a proportion of frozen elements, so that they are significantly beyond the point at which the frozen elements just percolate, also have difficulty with evolutionary hill climbing, apparently because they lack flexibility.

For systems imbedded in variable environments—and this is the case for essentially all biological and social systems—the ability to evolve is essential: without it, they cannot survive. Langton (1986) notes the correspondence between the critical value of X in his cellular automata and Kauffman's percolation threshold. It thus seems likely (Langton, 1989) that Kauffman's results are general, and that evolvability requires a system to be just at the transition point between order and chaos. This implies that the appearance of complex structures by self-organization depends on a balance between order and disorder. In fact, evolved structure may be characterized as an elaborate combination of order and disorder.

It is clear that the results of Kauffman and Langton are relevant to the cellular cities, in spite of some important differences in the models used. The stochastic perturbation effect permits the urban model, despite the relatively simple deterministic component of its dynamics, to emulate the entire range of behaviour observed by Langton as he varied A. Thus cellular cities with a low level of perturbation (low a) have a relatively simple geometric form, and would be characterized by a low Shannon entropy; and those generated with a high value of a have a largely random structure, with no large-scale features, corresponding to a high entropy. But the cellular cities discussed in this paper, and real cities as well, have very complex forms, with structure apparent at various scales, as shown by the fractal dimensions. And fractal dimensions, or power-law spectra of features, are characteristic of systems just on the threshold between order and chaos (Bak and Chen, 1989; Bak et al, 1989; Kauffman, 1990). Furthermore, as we have seen, the systematic


differences in the fractal dimensions of the different land uses imply a concentric zonation of land uses at the scale of the entire city. There is thus a global correlation of cell states in spite of the local nature of the rules generating the patterns. This also, according to Langton (1990), is a characteristic of the phase transition between order and chaos.

Last, if Kauffman's results apply, it may well be that the reason cities seem to be fine tuned to the phase transition point at which fractal structure appears is that they must be so if they are to be able to evolve. And as cities are evolving structures, evolution maintains them at the critical point where continued evolution is possible.

Concluding remarks The cellular urban models discussed in this paper appear to capture some of the key features of urban structure in a relatively realistic way. The urban land-use patterns generated by the model are intuitively reasonable, and are also in general accord with traditional models which are used to predict concentric zones. The essential feature of the patterns is their spatial complexity, with structure present over the full range of scales, from the scale of the individual cell to that of the entire city. Quantitatively, this complexity is manifested as a set of fractal dimensions, and these measures permit a comparison of the cellular cities with actual urban areas, which are found to be characterized by similar fractal dimensions.

Because the model is a cellular automaton, it is possible to interpret its behaviour in terms of the general principles of self-organization and structural evolution that are emerging from basic research on cellular automata, Boolean networks, and related connectionist models. This promises real advances in the theoretical understanding of the nature of cities insofar as they are complex, self-organized, evolving systems. Already, as we have seen, theoretical investigations suggest that complexity is a necessary feature of cities, and that cities that are too simple in their structure, either because they are overplanned or because they are not structured enough, will probably not evolve successfully and will therefore eventually cease to function effectively. Such results could be valuable in helping to establish general guidelines for planning policy.

At the other end of the scale, because the cellular technique permits dynamic modelling to be combined with a high level of spatial detail, it also holds promise as a modelling tool for a number of practical applications. The model discussed in this paper embodies the traditional homogeneous plain. This is entirely appropriate for a theoretical investigation such as the present one, where the aim is to uncover general principles by exploring the generic behaviour of the model. But for applications to specific cities, the homogeneous-plain assumption must be dropped, because local conditions must be taken into account.

This is easily accomplished. In another version of the model (Engelen et al, 1993; White and Engelen, 1992), the evolution of the urban form depends not only on its own previous state, but on exogenous conditions as well. These are of two types. First, there are fixed features, such as rivers and parks, that occupy part of the urban area. These appear as arguments in the transition functions and can thus affect the land-use pattern, but they are not themselves changed. Second, the cells that are subject to urban development are characterized by suitability values for each type of land use. These suitabilities may depend on such factors as slope, soil type, or access to the transportation network. Together, the fixed features and the suitabilities represent the particular local site characteristics that affect the development of the city. It should thus be possible to model particular cities realistically and with high spatial resolution.


For this sort of detailed modelling, it would be advantageous to integrate the cellular model with a GIS (White and Engelen, 1992). In many cases, land-use data that could be used to form the initial configuration in a simulation are already available in a GIS. So also are data on fixed features and the various factors that determine suitabilities for various activities; indeed, GISs are routinely used to calculate suitabilities. A n d simulation results could be returned to the GIS for further processing or for comparison with other data sets.

T h e power and versatility of the cellular modelling technique combined with the detailed spatial data held in a GIS would not only permit realistic modelling of specific situations but also might permit a more complete understanding of the phenomena being modelled. Currently, geographic theory is somewhat divorced from practical applications because the theoretical abstractions depend on simplifying assumptions that are often both unrealistic and difficult to relax. But in the new theoretical approach, exemplified by cellular automata modelling, the theory arises from and is inseparable from the detail. Complexity is seen not as noise, but as information, and so detail is not a distraction: it is essential for understanding the system. Models based on cellular automata and related techniques may thus ultimately make it possible to bring powerful theory to bear in a realistic way on very local and specific problems—problems of the sort that planners and other practi t ioners must deal with on a daily basis.

Acknowedgements. We thank Serge Wargnies and Inge Uljee in Maastricht, and Ken Warren in St John's for their help with this research. This work was supported by the Research Institute for Knowledge Systems, Maastricht, the Social Sciences and Humanities Research Council of Canada, and the Memorial University of Newfoundland.

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cellular automata and fractal urban form: a cellular ... · urban land-use dynamics with a high...

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