A File on....'A City is not a Tree'
Introduction to this File
This article has become one of the classic references in the literature of the built environment and associated fields, having been cited in articles over 40 times since 1980 (25 years after publication) and an unrecorded number of times in books or monographs. [Regrettably online citation indexes do not cover the period before 1980, and generally do not cover monographs and books]. It has an interesting publishing history, having been published in several journals and edited works and has been translated into several other languages.
Biographical Note on Christopher Alexander
Christopher Alexander was born in Vienna, Austria in 1936, raised in England, and holds a Bachelor’s degree in Architecture and a Master's’s Degree in Mathematics from the University of Cambridge, and a PhD in Architecture from Harvard University. He has been Professor in the Graduate School and Emeritus Professor of Architecture at the University of California, Berkeley, USA, since 1963.
He has designed and built more than two hundred buildings on five continents, has been a consultant to city, county and national governments on seven, and has advised corporations, government agencies, architects and planners throughout the world. Much of his work has been based on inventions in technology, including. especially, inventions in concrete, shell design and contracting procedures needed to attain a living architecture
He is the father of the Pattern Language movement in computer science, and author of A Pattern Language*, a seminal work first published in 1977 which was perhaps the first complete book ever written in hyperlink format. He founded the Center for Environmental Structure in 1967, and remains President. In 2000, he founded PATTERNLANGUAGE.COM, (a web site created to allow all people- homeowners, architects, builders, planners, and others, to design their own houses, to design large buildings, streets, neighborhoods, and gardens, in a way that will enhance the earth), and is Chairman of the Board. His latest work The Nature of Order is forthcoming
Professor Alexander was elected fellow of the American Academy of Arts and Sciences in 1996, is a fellow of the Swedish Royal Society, and has been awarded numerous architectural prizes and honors including the American Institute of Architects' gold medal for research (1970).
More biographical and bibliographical information can be found on the web site SOME NOTES ON CHRISTOPHER ALEXANDER.
*Alexander, C., Ishikawa, S., Silverstein, M., Jacobson, M., Fiksdahl-King, I. and Angel, S. (1977). A Pattern Language (New York, Oxford University Press).
A City is not a Tree - publishing history
The article first appeared in two parts in
Architectural Forum in 1965 (Vol 122, No 1, April 1965, pp 58-62 (Part I), and Vol 122, No 2, May 1965, pp 58-62 (Part II), and was subsequently republished in
Design No 206, February 1966, pp 46-55;
Ekistics Vol 23, pp 344 - 348, June 1967;
Hefti Birtingur No 13, 1967, pp 50-72;
Architecture Mouvement Continuite 1, November, 1967, pp 3-11;
Cuadernos Summa-Nueva Vision, No 9, September 1968, pp 20-30;
Stichting Wekgemeenschappen Bergeijk, 2; (1966?), pp.77-108
Approach, Spring 1968, pp 26-27;
It also appeared later in anthologies and other edited works:
Architecture Anthology, (1969), Arizona State University, pp. 580-590;
Tres Aspectod de Matematica y Deseno, (1969), Barcelona, pp. 19-60;
La Estructura de Medio Ambiente, (1971) Barcelona, pp. 17-55;
Human Identity in the Urban Environment, Bell, G & Tyrwhitt, J(eds), Harmondsworth, UK, Penguin Books, 1972;
Design After Modernism: Beyond the Object, Thackara, J. (ed.) (1988), Thames and Hudson, London, pp. 67-84;
Architecture Culture 1943-1968: a Documentary Anthology, Ockman, Joan, ed. (1993), Columbia Books of Architecture and Rizzoli, New York, pp.379-388.
This electronic version appears by permission of the author, and was created by RUDI in 2000. It was taken from Thackara (1988), although the diagrams from the original 1965 articles were used to create our images, and we are grateful to Professor Nikos Salingaros for re-drawing Figures A-D.
Christopher Alexander: A City is not a Tree part 1
RUDI Classics or Files on...
'A City is not a Tree'
by Christopher Alexander
A CITY IS NOT A TREE
CHRISTOPHER ALEXANDER
The tree of my title is not a green tree with leaves. It is the name of an abstract structure. I shall contrast it with another, more complex abstract structure called a semilattice. In order to relate these abstract structures to the nature of the city, I must first make a simple distinction.
I want to call those cities which have arisen more or less spontaneously over many, many years natural cities. And I shall call those cities and parts of cities which have been deliberately created by designers and planners artificial cities. Siena, Liverpool, Kyoto, Manhattan are examples of natural cities. Levittown, Chandigarh and the British New Towns are examples of artificial cities.
It is more and more widely recognized today that there is some essential ingredient missing from artificial cities. When compared with ancient cities that have acquired the patina of life, our modern attempts to create cities artificially are, from a human point of view, entirely unsuccessful.
Both the tree and the semilattice are ways of thinking about how a large collection of many small systems goes to make up a large and complex system. More generally, they are both names for structures of sets.
In order to define such structures, let me first define the concept of a set. A set is a collection of elements which for some reason we think of as belonging together. Since, as designers, we are concerned with the physical living city and its physical backbone, we must naturally restrict ourselves to considering sets which are collections of material elements such as people, blades of grass, cars, molecules, houses, gardens, water pipes, the water molecules in them etc.
When the elements of a set belong together because they co-operate or work together somehow, we call the set of elements a system.
For example, in Berkeley at the corner of Hearst and Euclid, there is a drugstore, and outside the drugstore a traffic light. In the entrance to the drugstore there is a newsrack where the day's papers are displayed. When the light is red, people who are waiting to cross the street stand idly by the light; and since they have nothing to do, they look at the papers displayed on the newsrack which they can see from where they stand. Some of them just read the headlines, others actually buy a paper while they wait.
This effect makes the newsrack and the traffic light interactive; the newsrack, the newspapers on it, the money going from people's pockets to the dime slot, the people who stop at the light and read papers, the traffic light, the electric impulses which make the lights change, and the sidewalk which the people stand on form a system - they all work together.
From the designer's point of view, the physically unchanging part of this system is of special interest. The newsrack, the traffic light and the sidewalk between them, related as they are, form the fixed part of the system. It is the unchanging receptacle in which the changing parts of the system - people, newspapers, money and electrical impulses - can work together. I define this fixed part as a unit of the city. It derives its coherence as a unit both from the forces which hold its own elements together and from the dynamic coherence of the larger living system which includes it as a fixed invariant part.
Of the many, many fixed concrete subsets of the city which are the receptacles for its systems and can therefore be thought of as significant physical units, we usually single out a few for special consideration. In fact, I claim that whatever picture of the city someone has is defined precisely by the subsets he sees as units.
Now, a collection of subsets which goes to make up such a picture is not merely an amorphous collection. Automatically, merely because relationships are established among the subsets once the subsets are chosen, the collection has a definite structure.
To understand this structure, let us think abstractly for a moment, using numbers as symbols. Instead of talking about the real sets of millions of real particles which occur in the city, let us consider a simpler structure made of just half a dozen elements. Label these elements 1,2,3,4,5,6. Not including the full set [1,2,3,4,5,6], the empty set [-], and the one-element sets [1],[2],[3],C4],[5], [6], there are 56 different subsets we can pick from six elements.
Suppose we now pick out certain of these 56 sets (just as we pick out certain sets and call them units when we form our picture of the city). Let us say, for example, that we pick the following subsets: [123], [34], [45], [234], [345], [12345], [3456].
What are the possible relationships among these sets? Some sets will be entirely part of larger sets, as [34] is part of [345] and [3456]. Some of the sets will overlap, like [123] and [234]. Some of the sets will be disjoint - that is, contain no elements in common like [123] and [45].
| diagram A | diagram B |
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| We can see these relationships displayed in two ways. In diagram A each set chosen to be a unit has a line drawn round it. In diagram B the chosen sets are arranged in order of ascending magnitude, so that whenever one set contains another (as [345] contains [34], there is a vertical path leading from one to the other. For the sake of clarity and visual economy, it is usual to draw lines only between sets which have no further sets and lines between them; thus the line between [34] and [345] and the line between [345] and [3456] make it unnecessary to draw a line between [34] and [3456]. Diagrams A & B redrawn by Nikos Salingaros |
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As we see from these two representations, the choice of subsets alone endows the collection of subsets as a whole with an overall structure. This is the structure which we are concerned with here. When the structure meets certain conditions it is called a semilattice. When it meets other more restrictive conditions, it is called a tree.
The semilattice axiom goes like this: A collection of sets forms a semilattice if and only if, when two overlapping sets belong to the collection, the set of elements common to both also belongs to the collection.
The structure illustrated in diagrams A and B is a semilattice. It satisfies the axiom since, for instance, [234] and [345] both belong to the collection and their common part, [34], also belongs to it. (As far as the city is concerned, this axiom states merely that wherever two units overlap, the area of overlap is itself a recognizable entity and hence a unit also. In the case of the drugstore example, one unit consists of newsrack, sidewalk and traffic light. Another unit consists of the drugstore itself, with its entry and the newsrack. The two units overlap in the newsrack. Clearly this area of overlap is itself a recognizable unit and so satisfies the axiom above which defines the characteristics of a semilattice.) The tree axiom states: A collection of sets forms a tree if and only if, for any two sets that belong to the collection either one is wholly contained in the other, or else they are wholly disjoint.
| diagram C | diagram D |
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| The structure illustrated in diagrams C and D is a tree. Since this axiom excludes the possibility of overlapping sets, there is no way in which the semilattice axiom can be violated, so that every tree is a trivially simple semilattice. Diagrams A & B redrawn by Nikos Salingaros |
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However, in this chapter we are not so much concerned with the fact that a tree happens to be a semilattice, but with the difference between trees and those more general semilattices which are not trees because they do contain overlapping units. We are concerned with the difference between structures in which no overlap occurs, and those structures in which overlap does occur.
It is not merely the overlap which makes the distinction between the two important. Still more important is the fact that the semilattice is potentially a much more complex and subtle structure than a tree. We may see just how much more complex a semilattice can be than a tree in the following fact: a tree based on 20 elements can contain at most 19 further subsets of the 20, while a semilattice based on the same 20 elements can contain more than 1,000,000 different subsets.
This enormously greater variety is an index of the great structural complexity a semilattice can have when compared with the structural simplicity of a tree. It is this lack of structural complexity, characteristic of trees, which is crippling our conceptions of the city.
To demonstrate, let us look at some modern conceptions of the city, each of which I shall show to be essentially a tree.
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Figure 1. Columbia, Maryland, Community Research and Development, Inc.: Neighbourhoods,in clusters of five, form 'villages'. Transportation joins the villages into a new town. The organization is a tree. |
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Figure 3. Greater London plan (1943), Abercrombie and Forshaw: The drawing depicts the structure conceived by Abercrombie for London. It is made of a large number of communities, each sharply separated from all adjacent communities. Abercrombie writes, 'The proposal is to emphasize the identity of the existing communities, to increase their degree of segregation, and where necessary to recognize them as separate and definite entities.' And again, 'The communities themselves consist of a series of sub-units, generally with their own shops and schools, corresponding to the neighbourhood units.' The city is conceived as a tree with two principal levels. The communities are the larger units of the structure; the smaller sub-units are neighbourhoods. There are no overlapping units. The structure is a tree. |
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Figure 4. Mesa City, Paolo Soleri: The organic shapes of Mesa City lead us, at a careless glance, to believe that it is a richer structure than our more obviously rigid examples. But when we look at it in detail we find precisely the same principle of organization. Take, particularly, the university centre. Here we find the centre of the city divided into a university and a residential quarter, which is itself divided into a number of villages (actually apartment towers) for 4000 inhabitants, each again subdivided further and surrounded by groups of still smaller dwelling units. |
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Figure 5. Tokyo plan, Kenzo Tange: This is a beautiful example. The plan consists of a series of loops stretched across Tokyo Bay. There are four major loops, each of which contains three medium loops. In the second major loop, one medium loop is the railway station and another is the port. Otherwise, each medium loop contains three minor loops which are residential neighbourhoods, except in the third major loop where one contains government offices and another industrial offices.
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Figure 6. Chandigarh (1951), Le Corbusier: The whole city is served by a commercial centre in the middle, linked to the administrative centre at the head. Two subsidiary elongated commercial cores are strung out along the maior arterial roads, running north-south. Subsidiary to these are further administrative, community and commercial centres, one for each of the city's 20 sectors.
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Figure 7. Brasilia, Lucio Costa: The entire form pivots about the central axis, and each of the two halves is served by a single main artery. This main artery is in turn fed by subsidiary arteries parallel to it. Finally, these are fed by the roads which surround the superbiocks themselves. The structure is a tree.
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Figure 8. Communitas, Percival and Paul Goodman: Communitas is explicitly organized as a tree: it is first divided into four concentric major zones, the innermost being a commercial centre, the next a university, the third residential and medical, and the fourth open country. Each of these is further subdivided: the commercial centre is represented as a great cylindrical skyscraper, containing five layers: airport, administration, light manufacture, shopping and amusement; and, at the bottom, railroads, buses and mechanical services. The university is divided into eight sectors comprising natural history, zoos and aquariums, planetarium, science laboratories, plastic arts, music and drama. The third concentric ring is divided into neighbourhoods of 4000 people each, not consisting of individual houses, but of apartment blocks, each of these containing individual dwelling units. Finally, the open country is divided into three segments: forest preserves, agriculture and vacation lands. The overall organization is a tree
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Figure 9.The most beautiful example of all I have kept until last, because it symbolizes the problem perfectly. It appears in Hilberseimer's book The Nature of Cities. He describes the fact that certain Roman towns had their origin as military camps, and then shows a picture of a modern military encampment as a kind of archetypal form for the city. It is not possible to have a structure which is a clearer tree. The symbol is apt, for, of course, the organization of the army was designed precisely in order to create discipline and rigidity. The photograph on the [left] is Hilberseimer's own scheme for the commercial area of a city based on the army camp archetype.
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Each of these structures, then, is a tree. Each unit in each tree that I have described, moreover, is the fixed, unchanging residue of some system in the living city (just as a house is the residue of the interactions between the members of a family, their emotions and their belongings; and a freeway is the residue of movement and commercial exchange).
However, in every city there are thousands, even millions, of times as many more systems at work whose physical residue does not appear as a unit in these tree structures. In the worst cases, the units which do appear fail to correspond to any living reality; and the real systems, whose existence actually makes the city live, have been provided with no physical receptacle.
© Christopher Alexander
Christopher Alexander: A City is not a Tree part 2
RUDI Classics or Files on...
'A City is not a Tree'
by Christopher Alexander
Part II
The units of which an artificial city is made up are always organized to form a tree. So that we get a really clear understanding of what this means, and shall better see its implications, let us define a tree once again. Whenever we have a tree structure, it means that within this structure no piece of any unit is ever connected to other units, except through the medium of that unit as a whole.
The enormity of this restriction is difficult to grasp. It is a little as though the members of a family were not free to make friends outside the family, except when the family as a whole made a friendship.
In simplicity of structure the tree is comparable to the compulsive desire for neatness and order that insists the candlesticks on a mantelpiece be perfectly straight and perfectly symmetrical about the centre. The semilattice, by comparison, is the structure of a complex fabric; it is the structure of living things, of great paintings and symphonies.
It must be emphasized, lest the orderly mind shrink in horror from anything that is not clearly articulated and categorized in tree form, that the idea of overlap, ambiguity, multiplicity of aspect and the semilattice are not less orderly than the rigid tree, but more so. They represent a thicker, tougher, more subtle and more complex view of structure.
Let us now look at the ways in which the natural, when unconstrained by artificial conceptions, shows itself to be a semilattice.
A major aspect of the city's social structure which a tree can never mirror properly is illustrated by Ruth Glass's redevelopment plan for Middlesbrough, England, a city of 200,000 which she recommends be broken down into 29 separate neighbourhoods. After picking her 29 neighbourhoods by determining where the sharpest discontinuities of building type, income and job type occur, she asks herself the question: 'If we examine some of the social systems which actually exist for the people in such a neighbourhood, do the physical units defined by these various social systems all define the same spatial neighbourhood?' Her own answer to this question is no. Each of the social systems she examines is a nodal system. It is made of some sort of central node, plus the people who use this centre. Specifically she takes elementary schools, secondary schools, youth clubs, adult clubs, post offices, greengrocers and grocers selling sugar. Each of these centres draws its users from a certain spatial area or spatial unit. This spatial unit is the physical residue of the social system as a whole, and is therefore a unit in the terms of this discussion. The units corresponding to different kinds of centres for the single neighbourhood of Waterloo Road are shown in Figure 11.
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The hard outline is the boundary of the so-called neighbourhood itself. The white circle stands for the youth club, and the small solid rings stand for areas where its members live. The ringed spot is the adult club, and the homes of its members form the unit marked by dashed boundaries. The white square is the post office, and the dotted line marks the unit which contains its users. The secondary school is marked by the spot with a white triangle in it. Together with its pupils, it forms the system marked by the dot-dashed line. As you can see at once, the different units do not coincide. Yet neither are they disjoint. They overlap. |
We cannot get an adequate picture of what Middlesbrough is, or of what it ought to be, in terms of 29 large and conveniently integral Chunks called neighbourhoods. When we describe the city in terms of neighbourhoods, we implicitly assume that the smaller elements within any one of these neighbourhoods belong together so tightly that they only interact with elements in other neighbourhoods through the medium of the neighbourhoods to which they themselves belong. Ruth Glass herself shows clearly that this is not the case.
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Next to Figure 11 are two representations of the Waterloo neighbourhood. For the sake of argument I have broken it into a number of small areas. Figure 12 shows how these pieces stick together in fact, and Figure 13 shows how the redevelopment plan pretends they stick together. |
There is nothing in the nature of the various centres which says that their catchment areas should be the same. Their natures are different. Therefore the units they define are different. The natural city of Middlesbrough was faithful to the semilattice structure of the units. Only in the artificial-tree conception of the city are their natural, proper and necessary overlaps destroyed.
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Consider the separation of pedestrians from moving vehicles, a tree concept proposed by Le Corbusier, Louis Kahn and many others. At a very crude level of thought this is obviously a good idea. Yet the urban taxi can function only because pedestrians and vehicles are not strictly separated. The cruising taxi needs a fast stream of traffic so that it can cover a large area to be sure of finding a passenger. The pedestrian needs to be able to hail the taxi from any point in the pedestrian world, and to be able to get out to any part of the pedestrian world to which he wants to go. The system which contains the taxicabs needs to overlap both the fast vehicular traffic system and the system of pedestrian circulation. In Manhattan pedestrians and vehicles do share certain parts of the city, and the necessary overlap is guaranteed (Figure 14). |
Another·favourite concept of the CIAM theorists and others is the separation of recreation from everything else. This has crystallized in our real cities in the form of playgrounds. The playground, asphalted and fenced in, is nothing but a pictorial acknowledgment of the fact that 'play' exists as an isolated concept in our minds. It has nothing to do with the life of play itself. Few self-respecting children will even play in a playground.
Play itself, the play that children practise, goes on somewhere different every day. One day it may be indoors, another day in a friendly gas station, another day down by the river, another day in a derelict building, another day on a construction site which has been abandoned for the weekend. Each of these play activities, and the objects it requires, forms a system. It is not true that these systems exist in isolation, cut off from the other systems of the city. The different systems overlap one another, and they overlap many other systems besides. The units, the physical places recognized as play places, must do the same.
In a natural city this is what happens. Play takes place in a thousand places it fills the interstices of adult life. As they play, children become full of their surroundings. How can children become filled with their surroundings in a fenced enclosure! They cannot.
A similar kind of mistake occurs in trees like that of Goodman's Communitas or Soleri's Mesa City, which separate the university from the rest of the city. Again, this has actually been realized in the common American form of the isolated campus.
What is the reason for drawing a line in the city so that everything within the boundary is university, and everything outside is nonuniversity? It is conceptually clear. But does it correspond to the realities of university life? Certainly it is not the structure which occurs in nonartificial university cities.
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There are always many systems of activity where university life and city life overlap: pub-crawling, coffee-drinking, the movies, walking from place to place. In some cases whole departments may be actively involved in the life of the city's inhabitants (the hospital-cum-medical school is an example). In Cambridge, a natural city where university and city have grown together gradually, the physical units overlap because they are the physical residues of city systems and university systems which overlap (Figure 15). |
Let us look next at the hierarchy of urban cores realized in Brasilia, Chandigarh, the MARS plan for London and, most recently, in the Manhattan Lincoln Center, where various performing arts serving the population of greater New York have been gathered together to form just one core.
Does a concert hall ask to be next to an opera house? Can the two feed on one another? Will anybody ever visit them both, gluttonously, in a single evening, or even buy tickets from one after going to a performance in the other? In Vienna, London, Paris, each of the performing arts has found its own place, because all are not mixed randomly. Each has created its own familiar section of the city. In Manhattan itself, Carnegie Hall and the Metropolitan Opera House were not built side by side. Each found its own place, and now creates its own atmosphere. The influence of each overlaps the parts of the city which have been made unique to it.
The only reason that these functions have all been brought together in Lincoln Center is that the concept of performing art links them to one another.
But this tree, and the idea of a single hierarchy of urban cores which is its parent, do not illuminate the relations between art and city life. They are merely born of the mania every simple-minded person has for putting things with the same name into the same basket.
The total separation of work from housing, started by Tony Garnier in his industrial city, then incorporated in the 1929 Athens Charter, is now found in every artificial city and accepted everywhere where zoning is enforced. Is this a sound principle? It is easy to see how bad conditions at the beginning of the century prompted planners to try to get the dirty factories out of residential areas. But the separation misses a variety of systems which require, for their sustenance, little parts of both.
Finally, let us examine the subdivision of the city into isolated communities. As we have seen in the Abercrombie plan for London, this is itself a tree structure. The individual community in a greater city has no reality as a functioning unit. In London, as in any great city, almost no one manages to find work which suits him near his home. People in one community work in a factory which is very likely to be in another community.
There are therefore many hundreds of thousands of worker-workplace systems, each consisting of individuals plus the factory they work in, which cut across the boundaries defined by Abercrombie's tree. The existence of these units, and their overlapping nature, indicates that the living systems of London form a semilattice. Only in the planner's mind has it become a tree.
The fact that we have so far failed to give this any physical expression has a vital consequence. As things are, whenever the worker and his workplace belong to separately administered municipalities, the community which contains the workplace collects huge taxes and has relatively little on which to spend the tax revenue. The community where the worker lives, if it is mainly residential, collects only little in the way of taxes and yet has great additional burdens on its purse in the form of schools, hospitals, etc. Clearly, to resolve this inequity, the worker-workplace systems must be anchored in physically recognizable units of the city which can then be taxed.
It might be argued that, even though the individual communities of a great city have no functional significance in the lives of their inhabitants, they are still the most convenient administrative units, and should therefore be left in their present tree organization. However, in the political complexity of a modern city, even this is suspect.
Edward Banfield, in his book Political Influence, gives a detailed account of the patterns of influence and control that have actually led to decisions in Chicago. He shows that, although the lines of administrative and executive control have a formal structure which is a tree, these formal chains of influence and authority are entirely overshadowed by the ad hoc lines of control which arise naturally as each new city problem presents itself. These ad hoc lines depend on who is interested in the matter, who has what at stake, who has what favours to trade with whom.
This second structure, which is informal, working within the framework of the first, is what really controls public action. It varies from week to week, even from hour to hour, as one problem replaces another. Nobody's sphere of influence is entirely under the control of any one superior; each person is under different influences as the problems change. Although the organization chart in the Mayor's office is a tree, the actual control and exercise of authority is semilattice-like.
Now, why is it that so many designers have conceived cities as trees when the natural structure is in every case a semilattice? Have they done so deliberately, in the belief that a tree structure will serve the people of the city better? Or have they done it because they cannot help it, because they are trapped by a mental habit, perhaps even trapped by the way the mind works - because they cannot encompass the complexity of a semilattice in any convenient mental form, because the mind has an overwhelming predisposition to see trees wherever it looks and cannot escape the tree conception?
I shall try to convince you that it is for this second reason that trees are being proposed and built as cities - that is, because designers, limited as they must be by the capacity of the mind to form intuitively accessible structures, cannot achieve the complexity of the semilattice in a single mental act.
Let me begin with an example. Suppose I ask you to remember the following four objects: an orange, a watermelon, a football and a tennis ball. How will you keep them in your mind, in your mind's eye? However you do it, you will do it by grouping them. Some of you will take the two fruits together, the orange and the watermelon, and the two sports balls together, the football and the tennis ball. Those of you who tend to think in terms of physical shape may group them differently, taking the two small spheres together - the orange and the tennis ball and the two large and more egg-shaped objects - the watermelon and the football. Some of you will be aware of both.
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Let us make a diagram of these groupings (Figure 16). Either grouping taken by itself is a tree structure. The two together are a semilattice. Now let us try and visualize these groupings in the mind's eye. I think you will find that you cannot visualize all four sets simultaneously - because they overlap. You can visualize one pair of sets and then the other, and you can alternate between the two pairs extremely fast, so that you may deceive yourself into thinking you can visualize them all together. But in truth, you cannot conceive all four sets at once in a single mental act. You cannot bring the semilattice structure into a visualizable form for a single mental act. In a single mental act you can only visualize a tree. |
This is the problem we face as designers. While we are not, perhaps, necessarily occupied with the problem of total visualization in a single mental act, the principle is still the same. The tree is accessible mentally and easy to deal with. The semilattice is hard to keep before the mind's eye and therefore hard to deal with.
It is known today that grouping and categorization are among the most primitive psychological processes. Modern psychology treats thought as a process of fitting new situations into existing slots and pigeonholes in the mind. Just as you cannot put a physical thing into more than one physical pigeonhole at once, so, by analogy, the processes of thought prevent you from putting a mental construct into more than one mental category at once. Study of the origin of these processes suggests that they stem essentially from the organism's need to reduce the complexity of its environment by establishing barriers between the different events that it encounters.
It is for this reason - because the mind's first function is to reduce the ambiguity and overlap in a confusing situation and because, to this end, it is endowed with a basic intolerance for ambiguity - that structures like the city, which do require overlapping sets within them, are nevertheless persistently conceived as trees.
The same rigidity dogs even perception of physical patterns. In experiments by Huggins and myself at Harvard, we showed people patterns whose internal units overlapped, and found that they almost always invent a way of seeing the patterns as a tree - even when the semilattice view of the patterns would have helped them perform the task of experimentation which was before them.
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The most startling proof that people tend to conceive even physical patterns as trees is found in some experiments of Sir Frederick Bartlett. He showed people a pattern for about a quarter of a second and then asked them to draw what they had seen. Many people, unable to grasp the full complexity of the pattern they had seen, simplified the patterns by cutting out the overlap. In Figure 17, the original is shown on the left, with two fairly typical redrawn versions to the right of it. In the redrawn versions the circles are separated from the rest; the overlap between triangles and circles disappears. |
These experiments suggest strongly that people have an underlying tendency, when faced by a complex organization, to reorganize it mentally in terms of non-overlapping units. The complexity of the semilattice is replaced by the simpler and more easily grasped tree form.
You are no doubt wondering by now what a city looks like which is a semilattice, but not a tree. I must confess that I cannot yet show you plans or sketches. It is not enough merely to make a demonstration of overlap - the overlap must be the right overlap. This is doubly important because it is so tempting to make plans in which overlap occurs for its own sake. This is essentially what the high- density 'life-filled' city plans of recent years do. But overlap alone does not give structure. It can also give chaos. A garbage can is full of overlap. To have structure, you must have the right overlap, and this is for us almost certainly different from the old overlap which we observe in historic cities. As the relationships between functions change, so the systems which need to overlap in order to receive these relationships must also change. The recreation of old kinds of overlap will be inappropriate, and chaotic instead of structured.
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One can perhaps make the physical consequences of overlap more comprehensible by means of an image. The painting illustrated is a work by Simon Nicholson (Figure 18). The fascination of this painting lies in the fact that, although constructed of rather few simple triangular elements, these elements unite in many different ways to form the large units of the painting - in such a way indeed that, if we make a complete inventory of the perceived units in the painting, we find that each triangle enters into four or five completely different kinds of unit, none contained in the others, yet all overlapping in that triangle. |
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Thus, if we number the triangles and pick out the sets of triangles which appear as strong visual units, we get the semilattice shown in Figure 19. Three and 5 form a unit because they work together as a rectangle; 2 and 4 because they form a parallelogram; 5 and 6 because they are both dark and pointing the same way; 6 and 7 because one is the ghost of the other shifted sideways; 4 and 7 because they are symmetrical with one another; 4 and 6 because they form another rectangle; 4 and 5 because they form a sort of Z; 2 and 3 because they form a rather thinner kind of Z; 1 and 7 because they are at opposite corners; 1 and 2 because they are a rectangle; 3 and 4 because they point the same way as 5 and 6, and form a sort of off-centre reflection; 3 and 6 because they enclose 4 and 5; 1 and S because they enclose 2, 3 and 4. I have only listed the units of two triangles. The larger units are even more complex. The white is more complex still and is not even included in the diagram because it is harder to be sure of its elementary pieces. |
The painting is significant, not so much because it has overlap in it (many paintings have overlap in them), but rather because this painting has nothing else in it except overlap. It is only the fact of the overlap, and the resulting multiplicity of aspects which the forms present, that makes the painting fascinating. It seems almost as though the painter had made an explicit attempt, as I have done, to single out overlap as a vital generator of structure.
All the artificial cities I have described have the structure of a tree rather than the semilattice structure of the Nicholson painting. Yet it is the painting, and other images like it, which must be our vehicles for thought. And when we wish to be precise, the semilattice, being part of a large branch of modern mathematics, is a powerfu1 way of exploring the structure of these images. It is the semilattice we must look for, not the tree.
When we think in terms of trees we are trading the humanity and richness of the living city for a conceptual simplicity which benefits only designers, planners, administrators and developers. Every time a piece of a city is torn out, and a tree made to replace the semilattice that was there before, the city takes a further step toward dissociation.
In any organized object, extreme compartmentalization and the dissociation of internal elements are the first signs of coming destruction. In a society, dissociation is anarchy. In a Person, dissociation is the mark of schizophrenia and impending suicide. An ominous example of city-wide dissociation is the separation of retired people from the rest of urban life, caused by the growth of desert cities for the old like Sun City, Arizona. This separation isonly possible under the influence of treelike thought.
It not only takes from the young the company of those who have lived long, but worse, it causes the same rift inside each individual life. As you pass into Sun City, and into old age, your ties with your own past will be unacknowledged, lost and therefore broken. Your youth will no longer be alive in your old age - the two will be dissociated; your own life will be cut in two.
For the human mind, the tree is the easiest vehicle for complex thoughts. But the city is not, cannot and must not be a tree. The city is a receptacle for life. If the receptacle severs the overlap of the strands of life within it, because it is a tree, it will be like a bowl full of razor blades on edge, ready to cut up whatever is entrusted to it. In such a receptacle life will be cut to pieces. If we make cities which are trees, they will cut our life within to pieces.
Originally published in:
Architectural Forum, Vol 122, No 1, April 1965, pp 58-62 (Part I),
Vol 122, No 2, May 1965, pp 58-62 (Part II)
Also published in :
Design, No 206, February 1966, pp46-55
Ekistics, Vol 23, pp 344 - 348, June 1967
Bell, G & Tyrwhitt, J(eds) Human Identity in the Urban Environment, Harmondsworth, UK, Penguin Books, 1972
This version taken from:
Thackara, J. (ed.) (1988), Design After Modernism: Beyond the Object, Thames and Hudson, London, pp. 67-84.
© Christopher Alexander
John Minett: As the City is not a Tree... it should not be designed as a System
A critique of design theory and method as presently applied to city design
[This article was originally published under the title 'If the City is not a Tree, nor is it a System' in
Planning Outlook New Series, Volume Sixteen, Spring 1975, pp 4 -18]
Most people seem to agree that modern cities are pretty awful. We can be persuaded to enjoy aspects of them, such as thrilling new structures which win design awards, or the feeling of movement on new communication systems, but most of the time they are just plain boring, dull, and tawdry; tinged with a sense of 'welfare' that it's all we can afford. Wherever one goes one sees the same standard answers; stereotyped housing blocks, neat soulless shopping centres, or schools, fire stations or churches (which might be interchangeable), and standard amounts of 'public open space' (ugh!). At the centre of the city there is more standard kit - television mast, faceless tower and slab blocks, and regulation neon signs (when there's electricity) to whoop it up a bit.
Critics, casting around for people to blame latch on to the planner as the villain. He's unimaginative. His thinking is stuck in a rut of neighbourhoods, zoning, densities, and plot ratios. His view of the city is far too simplistic for such a complex machine. We must match the complexity of the city with more complex multi-purpose and intricate development, so we try complex schemes - and what happens? Despite a plethora of ideas and conferences, despite a great deal of endeavour and architectural ingenuity, the plans and sketches which seem to show such high promise turn out to be yet another collection of pedestrian bridges, car park ramps, and plastic paving surrounding anonymous building blocks. We still seem unable to match the life and liveliness of 'real cities'.
Consequently critics have turned their attention to querying the planner's method of operation which requires him to make such assumptions. The problem, they suggest, is not that city planners are inherently simplistic but that the design process they operate forces them to make simplistic assumptions. For example Christopher Alexander (1966) argues that much of people's dissatisfaction with the modern city stems from their artificial organisation into hierarchical groupings of facilities which, he suggests, are based not on the way people use the city, but on the way designers conceive the process of design and apply it to the design of a city. Similarly, Maurice Ash (1969) condemns town planners for what he regards as their attempt to organise people and facilities into idealised patterns based on what he asserts is an unproven theory of settlement hierarchy. Like Ash, Melvin Webber (1968/69) argues that the philosophy of town planning is not founded on social realities. Instead, he suggests, city planners have defined their problems and solutions in idealised terms because they adopted the design style of the parent professions of engineering and architecture, and therefore adopted their design assumptions.
What is design? What are its assumptions?
Herbert Simon (1969) has called design the 'science of the artificial' because it is an artificial creation to catalyse the requirements of man with those of his environment. It can be defined as the ordering of relations between components in order to satisfy predetermined objectives. As a process it can only start when the problem has been defined in terms of objectives to be satisfied; that is when the various aspects or component parts of the problem which together need satisfying have been identified. The solution represents a synthesis which should satisfy the requirements of each component and the problem as a whole. It is this process of synthesis that I would define as Design: a process in which the designer is an innovator, an inventor of solutions.
Investigations into the way designers design show that they go through a series of definite stages. The first is concerned with clarifying the problem by breaking it down into its component parts or functions and identifying their requirements. The second is to gather 'like' components together into 'sets' (i.e. collections of elements which are thought to have something in common). The third stage is to link those components which work together into a system (i.e. a set of components which are linked together in order to perform a function). The aim is to create what Alexander calls 'good fit' between problem and solution.1
For example, imagine designing a house. You break it down into the kind of rooms, or better still activities, which are required, such as sleeping, cooking, sitting etc. [fig. 1]. You analyse those activities which have common or related features (for instance requiring service) and group them as sets [fig. 2]. Then you link those components which require inter-connection (e.g. cooking to eating) into a system [fig. 3]. Structuring the solutions you try to satisfy the known wants and functions involved, by making enough space for the different activities to be accommodated; trying to ensure that those activities which are most linked are put in the closest relationship. But no matter how objective you attempt to be your image or idea of how each part, and the whole, might function, look and feel will also greatly influence any solution you propose. Consequently, although the design process must attempt to be objective, in most cases it will also reflect the designer's own philosophy and values. You can only put forward solutions which you think are satisfactory.2
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| Fig.1 Components | Fig.2 Components grouped to form a 'set' |
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Fig.3 Components linked to form a 'system' |
Examining the design process, the fundamental point emerges that designers create systems. Of all the stages in the design process this is the one where he uses his expertise - this is what the client pays for. Whereas the client should be concerned with helping to define the various parts of the problem, and possibly their grouping into sets, the designer is responsible for ordering the relationships in the system, or between systems and sub-systems, to satisfy functional and aesthetic requirements. The links which make the system are designed to maintain equilibrium between the components, future change being controlled by the links. Consequently, the designer is in a very powerful position. He is creating frameworks relative to the way he thinks they should be. When he creates a system he is not only providing opportunities, but also constraints. His design will stabilise a set of relationships allowing only changes acceptable to the structure. Consequently he must be sure that the relationships he sees are valid, and the links he creates are needed. Identification of a set does not automatically make it a system; it is the links that create the system.
Summarising, the design process would seem to rest on four assumptions:
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That the problem can be defined, in terms of agreeable objectives. |
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That the components can be isolated and their requirements analysed. |
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That there is a 'best-fit' relationship. |
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That the end product can be achieved in reality, because design takes account of the variables in the control of the designer and client. |
Thus in engineering and architecture the aim is to produce an artifact which satisfies a client's known requirements. The operational style assumes that a client's requirements can be identified and arranged into a rational ordered solution to satisfy agreed objectives. Although the design can have built-in 'flexibility' there is an overriding assumption of stability, such that there are ascertainable and agreeable goals based on common values (held by client and designer); that any change will like-wise conform to existing values; that a 'best-fit' arrangement can be found to meet these goals and that controls can be agreed which will allow the plan to be implemented.
The application of design principles in town planning
Although the assumptions underlying design would seem questionable if applied to city planning, their application by city planners is clearly seen in many recent planning reports particularly for New Towns and new Cities. For example, consider the report for the new city in Central Lancashire (for which a Development Corporation was appointed early in 1971).3
In that study an attempt is made to identify all the facilities that the population will require in a 'large' city, and economic levels of provision are ascribed to each facility based on population thresholds [fig. 4]. Facilities are then grouped into sets, which create a hierarchy of 'communities' [fig. 4].
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dance hall cinema variety stores night club museum bowling alley technical college coll, of arts & crafts regional park zoo art gallery law courts theatre botanical gardens specialised shops concert hall reference library county admin. offices sports stadium indust. retraining centre polytechnic |
| Fig. 4 Facilities relative to population thresholds (Central Lancashire: Study for a City) |
3 Township 2 District 1 Neighbourhood |
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Dispersal |
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| Fig. 7 Assumed transport modes to facilities (Central Lancashire: Study for a City) |
Criticism of Planners' design approach
Here then is the nub of the criticism. Because the city planner adopted the operational style of the engineer and architect, he was forced to work on similar assumptions; that he was producing an artifact representing the known requirements of 'the people', that these could be identified and once identified would remain relatively stable, and that there was some rational arrangement of the city which would be satisfactory not only for the present but also for the future. As Webber suggests, the city planner adopted the assumptions of the engineer designing public works including:
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Furthermore the adoption of the architectural / engineering design style assumed the need for a similar approach to implementation, whereby "the sorts of deliberate outcomes accomplished in the centralised decision setting of engineer client relationship, or centrally controlled government enterprise" could be extended to the market place.6 In place of the 'real world' of multiple ownerships and multiple power, an 'ideal world' was required where the planner could play the role of designer for the people, represented by a statutory authority, knowing that his plan could be implemented. As Webber points out, the planners produced a remarkably inventive set of controls - the technical standard (providing minimum levels of quality), the master plan (setting forth overall system design) and the land use regulation (which constrained the locational decisions of individual establishments). But these methods only allow the 'negative' controls of stopping 'unsatisfactory' development.
In Britain the Government went further by providing powers which allowed 'creative' planning - the Comprehensive Development Area (1944) and the New Town (1946), and subsidised local authority housing (1919). In each case the relevant authority could draw up plans and, through its powers of land ownership and finance, could implement its proposals. The trouble is that almost all 'creative' planning has been used for the poorer sections of the community, particularly through the use of subsidised housing for estate development and comprehensive redevelopment of slum areas, and for overspill to New Towns and Town Expansion schemes. The better off, including those who were once worse off but have joined the 'private sector', can afford to move out of the clutches of the 'creative' planner with his concepts of rationally ordered communities. The car-borne mother doesn't have to move where the Gravity Model suggests she should go - the bus-borne mother has no choice!
Thus critics are suggesting that the stereotyped planned environment is not so much a product of poor imagination on the part of city planners, as a product of the idea of designing cities as if they were a public works artifact. As a consequence of centralised design, instead of catering for the variety of people's aspirations, there has been a tendency to reduce people's needs to formalistic models where the quality of life and environment which comes from individual innovation is squeezed out in order to safeguard 'the public interest'. The city planners' admirable desire to ensure that people have a satisfactory minimum of facilities far too often seems to produce stereotyped results where minima become maxima. All too often it is for one class of society: what Maurice Ash calls 'an alliance with poverty'.7 The idea that the city can be designed as a coherent system made up of a rational arrangement of subsystems is a static concept. The pursuit of city wide equilibrium for whatever high ideals, will result in its stagnation, decline and eventual death.
The design of cities
How then should city planners approach the design of cities? Should they drop the architect/engineering design style, as implied by Webber, and concentrate on social and economic policy? Should they seek to design more complex city structures like the semi-lattic advocated by Alexander?
Whatever the objectives of city planning should be, I do not think we can find another design style (like Bruce Archer, I believe that "the logical act of designing is largely independent of the thing designed"8). Nor do I believe it is possible to 'design in' complexity or deliberately create semi-lattices; for as Alexander has shown (1966) design is inherently concerned with rationalisation and simplification of the parts of the problem into a hierarchical arrangement. So if we can only design in one way, and we cannot design complexity, how can we create complex satisfying cities?
I believe the answer lies in a paradox set by Peter Levin in his paper 'Decision making in Urban Design' (1966). He suggested "perhaps it is a valid aim to try deliberately to achieve a situation in which the functioning of the system is as independent as possible of the design. The design, in other words, should have the minimum effect on human organisation and activities."9 But, as we have seen designers derive their designs from the functioning of the system. Furthermore I don't think you can design a system unrelated to its function. Thus the paradox: but only a paradox so long as the city is regarded as a system.
The city is not a system
Looked at from the viewpoint of its development and operation, the city is not a single system. Rather it is a multiplicity of systems under the control of a multiplicity of agencies. These agencies, both public and private, provide and operate facilities [fig. 9] and both the agencies' attitude to provision and the people's demand for provision is subject to change. Some facilities will grow and others will die without the city planner having much say. Consequently, although he may attempt to design his city plans on the basis of linkages between activity systems, and the use the people make of them, he cannot implement and maintain his plan unless he has autocratic powers to stabilise the world he has attempted to create. Without autocratic powers, which I am sure few people will be prepared to accord, the city planner is wasting his time designing the city as a system. As Margaret Willis has suggested "the narrow and restricted approach of what is primarily a physical plan, and that is devised for the most part independently of the agencies that actually build and administer a town shows the limitation of this type of planning in trying to meet social realities".10 She quotes Professor Peter Wilmot, ". . . If city planning is to respect social criteria, it must increasingly enable the design to work",11 that is work through the agencies which develop and operate the city. As these agencies cannot be welded and subsumed within a single comprehensive system, the city is better regarded as a set of semi-independent systems with each agency responsible for the design and positive direction of its own system [fig. 10].
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| Fig. 9. Examples of Agencies which 'develop' a city |
Fig.10 The city a number of separate system based on agencies |
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It is a more limited area than he attempts to design at present, but it is none the less positive. The city planner can continue to manage and control the spatial organisation distribution, arrangement and visual quality of land use through his development control powers over links between activity systems. He should be concerned with the positive design of public areas and resources as a framework for semi-independent agencies.
Given that each agency will create a rational design solution to its own problem but would no longer be subject to some supra-rationality of how the various activities should best fit together, complexity would be created in the way it occurs in normal cities. Instead of the city being cut up in discreet neighbourhoods around centres with congruent catchment areas for each activity system, each activity system would arrange itself to suit its own demands. Alexander's 'semi-lattice' would result from activity systems solving their problems in their way. Levin suggests that making the function of the city independent of overall design should provide freedom and choice: "A choice of nearby schools for a child, a choice of convenient shops for a housewife, a choice of nearby friends to call on in an emergency, and a choice of routes to work."13 As he suggests this might involve some over-provision of facilities (an anathema to the city planner?), but surely that would be no bad thing for the consumer. We might actually get away from the stereotyped standardised world that seems the only end product available from present city design.
I believe that 'Planning is for People' is in fact a static and paternalistic concept which serves to reduce the quality of life and the environment in which it is lived. City Planning must provide the opportunity for people to enhance their own lives. Andrew Kopkind quoted Catherine Bauer as saying "the worst kind of dictatorship is the kind that gives people what they want, the kind in which you can't tell you're being controlled." As he comments, "if that is one of the possible futures it is too important to be left only to the planners."14
REFERENCES
1 See Christopher Alexander 'Notes on the Synthesis of Form'.
2 In the introduction to Design Methods in Architecture. Anthony Ward queries the 'objective' approach to design which suggests that the nature of design is independent of the thing designed and suggests that perhaps the way a designer goes about designing something does affect the thing that is designed (p. 12). He suggests that "if we are to contribute significantly to the development of design method, we have no alternative but to make explicit the philosophical premises upon which we base our conclusions. Otherwise the logical nature of our task will remain impenetrable." This suggests the beginning of an acceptance by the architectural design theorists that the designer cannot be wholly objective; that there is a place for the subjective as preached by one of the foremost Architectural theorists, Goodhart-Rendall, in the 1930's: "Many modern theorists of architecture, by refusing to let the (design) process disappear from sight, have observed with dismay or neglected to observe that there is then no architecture at the end of it." Goodhart-Rendell (1934). This all echoes the plea of the German philosopher Karl Mannheim (1966) for designers and decision makers to be more self aware and thus gain greater freedom.
3 Robert Matthew, Johnson-Marshall and Partners: Central Lancashire, Study for a City. 1964.
4 Christopher Alexander: A City is not a Tree. 1966.
5 Melvin Webber: Planning in an Environment of Change Part I, Town Planning Review October 1968, p. 192.
6 ibid. Part II January 1969, p. 284.
7 Maurice Ash: Regions of Tomorrow 1964.
8 Bruce Archer: The Structure of the Design Process in 'Design Methods in Architecture' (ed. A. Ward and G. Broadbent, AA Paper No. 4 1969).
9 Peter Levin: Decision Making in Urban Design B.R.S. 1966, p.11.
10 Margaret Willis: Sociological Aspects of Urban Structure Town Planning Review, January 1969, p. 305.
11 Peter Wilmot: Social Research & New Communities Journal of the American Institute of Planners Nov. 1967.
12 See Fumihiko Maki: Linkage in Collective Form in 'Investigations in Collective Form' 1964. Also Jane Jacobs: Death and Life of Great American Cities (1961).
13 Peter Levin: op. cit. p. 11.
14 Andrew Kopkind: 'The Future Planners' in 'America the Mixed Curse' 1969.
Figures 4, 5, 6 and 7 are based on diagrams from 'Central Lancashire, Study for a City' (1967) and are published by kind permission of RMJM (formerly Robert Matthew, Johnson-Marshall and Partners).
Selected Bibliography
ALEXANDER, CHRISTOPHER
A City is not a Tree in Human Identity in the Urban Environment (ed. Bell & Tyrwhitt). Penguin, 1973
Notes on the Synthesis of Form; Harvard University Press, 1964
ASH, MAURICE
Regions of Tomorrow. Evelyn Adams & Mackay, 1969.
GOODHART-RENDELL, L
Fine Art. Clarendon Press, 1934
JACOBS, JANE
Death & Life of Great American Cities. Penguin, 1973.
LEVIN, PETER
Decision Making in Urban Design. Building Research Station Design Series No.49, 1966.
MAKI, FUMIHIKO
Investigations in Collective Form. Washington State University. 1964.
MANNHEIM, KARL
Ideology & Utopia. Routledge, 1966
MINISTRY OF HOUSING AND LOCAL GOVERNMENT
Central Lancashire: Study for a City. HMSO, 1967.
SIMON, HERBERT
The Sciences of the Artificial. M.I.T., 1969.
WARD, A. and BROADBENT, G.
Design Methods in Architecture. Architectural Association Paper No.4, Lund Humphries, 1969.
WEBBER, MELVIN
Planning in an Environment of Change. Town Planning Review, October 1968 and January 1969.
WILLIS, MARGARET
Sociological Aspects of' Urban Structure. Town Planning Review, January 1969.
John Minett presently practices urban design in Phoenix, Arizona, USA. Here he spends much of his time promoting the concept of 'Sociable Cities': places where multi-modal streets support community, traffic is calm, parking is civilized, and places are friendly for pedestrians and bicyclists. It is the antithesis of Auto City which is still the dominant paradigm of the USA. To read more see his new web site www.sociablecity.net (under construction). This article in RUDI was written in the 1970's when he was a lecturer at the School of Planning in Oxford, England. The concept that underlies it continues to underpin his attitude to city planning, and has influenced much of his work and writings since.
John Minett. As the city is not a tree...it should not be designed as a system © John Minett .
Nikos Salingaros : Remarks on a city's composition
Nikos A. Salingaros,
Division of Mathematics
University of Texas at San Antonio
San Antonio, Texas 78249
Dr. Nikos A. Salingaros is Professor of Mathematics at the University of Texas at San Antonio, has a Ph. D. in theoretical physics from the State University of New York at Stony Brook, and has made contributions to mathematical physics, field theory, and thermonuclear fusion. He is the author of more than seventy scientific publications, and has served as associate editor for two journals, and referees for fourteen others.
Since 1983, he has been learning from and working with architectural theorist Christopher Alexander. This interaction has inspired an entirely new direction of research, which uses mathematics to describe aspects of nature that are traditionally regarded as being in the domain of art. These phenomena have so far eluded a scientific explanation. Dr. Salingaros is applying insights gained from complex physical systems to architecture and urbanism. His latest publications combine ideas from complexity theory, fractals, and thermodynamics to develop a mathematical theory of structural form. The same ideas apply to architecture as well as to other complex systems, be they urban regions or biological organisms. From this work, a new picture of urban and architectural form emerges; one that is more consistent with natural structures. As a result, it is now possible to connect mathematically the built environment with the natural environment.
More information is available from Dr. Salingaros' homepage.
© Nikos Saligaros
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers.
Disclaimer
The information provided on the RUDI system is done so in good faith. Neither RUDI Ltd nor any of its respective information providers, licensors, employees or agents accept any liability for the accuracy of any information so provided, and no warranty is given, either express or implied that the information contained therein is accurate or can be relied upon for any particular purpose. The information provided on the RUDI system is not available for re-dissemination.
1. Introduction
What needs to be done to fix inhuman urban form? There is a growing realization that we don't really understand how to build a living environment. I am convinced that the answer lies outside contemporary approaches that derive from architectural modes of thought, in techniques developed for the analysis of complex systems. A large complex system contains an enormous number of internal connections. It is put together from components of various sizes, which connect and interact in particular ways to create a coherent whole. How this occurs in different instances follows from very general rules that were derived in biology and computer science. So far, those results have remained outside mainstream urbanism.
An important exception going the other way is the work of Christopher Alexander. Starting with the classic paper "A City is Not a Tree"(Alexander, 1965), the later book "A Pattern Language" (Alexander, Ishikawa et al., 1977), and his most recent book (Alexander, 2000), his results on architectural and urban form are now applied in computer science and biology. Alexander's work contains many solutions to problems in urban design. His paper originally appeared in 1965, and was hailed as a seminal statement on urban structure; yet despite being reprinted and translated into several languages, it has had little impact on how cities developed since that time. "A Pattern Language" was never adopted by mainstream architects, so the insights offered by Alexander and his coauthors would appear to have been ignored by the profession.
It is time that we appreciated Alexander's mathematical approach for the immensely powerful tools it offers. Such tools provide access to many results in separate scientific disciplines that could be translated into terms relevant to urban structure. Furthermore, the clarity of scientific thought protects human sensibilities against irrational forces in design, which are driven by fashion and the mindless pursuit of novelty. Some of these have become enshrined into our present-day urban design canon, which is now based as much on ideology and ignorance as it is on human needs. Cities ought to be shaped according to some well-tested set of design principles. I would like to derive those rules.
The discussion here will revolve around nodes and their interconnections; how nodes connect to form modules; and how modules connect to form a city. Connections may take various forms: geometrical coupling of structures next to each other (Salingaros, 2000a), visual coupling between a person and the information in a structure (Mikiten, Salingaros et al., 2000; Salingaros, 1999), interaction between human beings, pedestrian coupling of two geometrical or functional nodes via a footpath (Salingaros, 1998), transportation coupling via road or subway between widely separated nodes (Salingaros, 1998), etc. Although I am talking about distinct notions of connectivity, it turns out that they are all related. Geometrical edges, for example, provide both separation across the edge, and a possible conduit for connections along the edge. Urban interfaces act as transverse separators for one type of flow (e.g., cars) while encouraging pedestrian traffic across the interface. For the purposes of this discussion, therefore, I will simply refer to connections as a general, inclusive concept, and not specify exactly which type of connection is implied.
2. Christopher Alexander's "A City is Not a Tree".
The title of Alexander's early paper is catchy if a little misleading. Yes, a living city does not follow a mathematical tree structure, but Alexander's point is that most contemporary cities are trees (Alexander, 1965). Teachers have had the "tree" pattern in mind when teaching city form, thus perpetuating post-war urbanist principles that are based on trees. We now build "tree" cities, and unquestioningly turn older living cities into "trees"; however, whenever we do this, the life of that urban region perceptibly decreases. It is useful to intuitively link urban geometry to "life" -- even though the latter term is not precisely definable -- because one feels its presence immediately. As a result of their geometrical properties (which I will analyze below), modern "tree" cities are not alive in the sense that cities maintaining a more traditional structure are.
Alexander found that a living city is modeled by a mathematical semilattice, in contrast to a dead city, which is modeled by a tree. A semilattice has a vastly larger number of internal connections than a tree of comparable size has. Not only are there many connections in a semilattice, but there is a great variety of them; by contrast, trees have unique connections. I have found it more practical to sidestep the terminology of tree and semilattice, however, and to instead approach the topic from the viewpoint of hierarchical systems. Considering a living city to be a coherent complex system, can we decompose such a system into modules? It turns out that thinking about this problem leads us into a parallel reasoning with Alexander's paper. This is not surprising, since Alexander, along with Jane Jacobs (Jacobs, 1961), first grasped the organized complexity of urban regions. I can try to simplify Alexander's message by re-stating it as follows: "If you can neatly segregate functions or regions on a city's plan, then it represents a tree, and consequently it's not alive".
3. Alternative parcellations of a living city
Decomposition theorems for complex systems were first published around forty years ago (Courtois, 1985; Simon, 1962; Simon and Ando, 1961). I am going to use them to try and understand the complexity of city form. A living city is made up of parts, but how does one determine those parts? The choice of what components in a complex system are the basic ones is actually arbitrary, and depends upon the viewpoint of the observer. That follows because the whole is definitely not reducible to any parts and their interaction. One can define subsystems for convenience, but each subsystem does not behave in a totally independent manner. To help in my analysis, a city may be decomposed in various distinct ways; for example:
- Into buildings as basic units (as is usually done) and their interactions via paths.
- As a collection of paths anchored and guided by buildings (Salingaros, 1998).
- As external and internal spaces connected by paths and reinforced by buildings (Alexander, 2000; Salingaros, 1999).
- As the edges and interfaces that define spaces and built structures (Alexander, 2000; Salingaros, 2000a).
- Into patterns of human activity and interaction occurring at urban edges and interfaces (Alexander, Ishikawa et al., 1977; Salingaros, 2000b).
Other decompositions are possible, where one identifies a different type of basic unit. Any module that can be used as the building block of a complex system will itself have internal complexity and be neither empty nor homogeneous. This allows us to build up the city from several entirely distinct perspectives. Clearly, the shape of the resulting city may look radically different depending on the choice of a basic type of unit used to build it. All choices could be equally valid, and each leads to a partial understanding of the complexity of urban form and function. My point is that a living city is the superposition and balanced compromise between all of these different choices.
Of the five alternative parcellations of a living city offered above, only the first method is recognizable as being part of standard urbanist thinking. The other four, though essential from a mathematical analysis of city form, are still dismissed or are considered irrelevant by most professionals. The only way for students to learn about them is from the writings of Alexander and his colleagues (Alexander, Ishikawa et al., 1977; Alexander, 2000) and Jan Gehl (Gehl, 1987), among others. I don't believe it possible to design or repair urban environments without a thorough understanding of how the space between buildings contributes to -- indeed, provides the foundation of -- urban "life";.
The first approach (1) arranges buildings in some ordering. Unfortunately, this might prevent the generation of useful connections. Geometrical alignment is often substituted for, and in many cases replaces connections between urban nodes. The second approach (2) creates a hierarchy of paths, from protected footpaths, all the way up to expressways. When we build a city starting from footpaths, arranging other urban elements so as not to disturb the path structure, the organization of buildings becomes looser and less symmetric. The resulting geometry is linear and connected; it is neither random, nor chaotic. Historical cities and squatter settlements obey this much freer geometry. Starting with expressways to build a connected web does not work, however, because it reverses the scale priorities (Alexander, Ishikawa et al., 1977; Salingaros, 1998).
4. Urban modules and connective forces
A"module" is any group of nodes (units) with a large number of internal connections (Figure 1). Many of those nodes are also going to be connected to other units outside the module, the purpose of defining a module being to internalize relatively strong connections. Modularization is a process of stabilization, as good modules contain the strongest forces so that the modules (which are larger entities) can interact among themselves more weakly. For an analogy, imagine the thermal motion of particles: the smallest particles vibrate faster, whereas larger clumps vibrate more slowly because they have more inertia. We can then construct modules of modules, etc., according to a hierarchy of forces having decreasing strength.
Figure 1. Six nodes all connected to each other define a module. The nodes' exact position is unimportant.
Coherent systems are defined by strongly-connected units, some of which (though not necessarily all) may be grouped into modules. The elements of a module should not be excessively separated from each other, yet they are not necessarily adjacent. The criterion is not geometrical proximity, but connectivity: connections between internal nodes must be stronger than external connections. Thus, an urban module need not look nice on a plan; and conversely, a geometrically regular grouping of nodes is not automatically an urban module. A group of unconnected nodes next to each other will not form a module (Figure 2).
Figure 2. Nine nodes happen to be geometrically next to each other but are not interconnected. They do not form a module, despite their proximity.
Connectivity could be either geometrical continuity, path connectivity among nodes, or the exchange of persons and information. Buildings couple geometrically by having common walls; or they are connected via an intermediate space (Salingaros, 2000a). This space could contain paths and information that tie together the buildings around it. A pedestrian zone or plaza may or may not be a connective element, depending on whether it is heavily used or not. That, in turn, depends on how nodes are distributed around its periphery so that people need to cross the space. A desolate, empty plaza is not a connective element any more than a parking lot is. Path connections must be designed to encourage the free interchange of users between nodes, and there must be functional reasons for this interchange. One should also not discount informational connectivity in the ground, such as occurs when a floor pattern links visually with surrounding structures (Mikiten, Salingaros et al., 2000).
A busy road separating buildings is a boundary that cuts possible paths between them (Salingaros, 1998). Informational interest in the façades on opposite sides of a narrow street could overcome this separation, unless car traffic inhibits pedestrians from crossing over. This example underlines the mutually supportive roles of informational and path connectivity. Adopting plain surfaces on buildings and floors, and building to setbacks suppresses informational connectivity. The car is a destroyer of pedestrian space by forcing the widening of roads, and by making patterns on building fronts and on the ground irrelevant. On the other hand, the car's positive role is to make urban nodes accessible. Often, a low-traffic road that feeds into a hard-to-reach pedestrian zone enhances instead of hindering the connectivity of urban space (Salingaros, 1999).
As soon as we grasp that a living city is not composed of buildings just sitting next to each other, but that the life of a city arises from its ensemble of connections, then the need for the geometry to accommodate those connections becomes paramount. One starts to think of more complex, interweaving geometrical configurations that might support multiple connections, and to look at urban examples from the past that wer e successful in doing so (Salingaros, 2000a). An essential part of this picture is allowing for a multiplicity of alternative connections, either via paths, or via information. Clearly, the attributes of a living city are (i) richness of information, and (ii) the prioritization of pedestrian paths. Those requirements need not in any way impinge upon the web of vehicular connections.
Figure 3. Modules internalize connections between their constituent nodes. Three modules connect themselves via organizable forces.
The join between modules will be successful if it occurs along a region that is weaker than any module's internal connections; i.e., a join should separate the system where there is linkage or transition rather than concentrated structure. Parcellation follows the relative strength of cohesive forces defining a system: strong internal forces hold a module together, whereas weaker forces keep different modules in place within the system (Courtois, 1985). Though oversimplified, the example shown in Figure 3 illustrates the containment of forces within modules: there are 3 inter-modular links in each case, whereas every module contains 6 internal links.
5. Urban modules and geometrical alignment
Having established the notion of urban modules by virtue of their internal connectivity, we need to dispel some misunderstandings in late twentieth-century planning practice. First and foremost is a confusion between connectivity and geometrical alignment. One does not imply the other; significantly, so much effort and cost is spent on geometrical alignment today, and the result damages urban life. Simple alignment in the initial stages of planning is not a contributing factor to urban coherence (Salingaros, 2000a). Alignment comes into play as an organizational mechanism in a functioning urban system, and becomes useful when coherence is emerging from a richly-interacting substructure.
To illustrate what I mean, consider an example as if taken from a city's plan. I will assume a geometry for the nodes and their connections (unlike the symbolic nodes shown in the previous Figures). The six nodes shown in Figure 4 could be buildings of any size in a symmetric grouping as seen from the air, a very common situation nowadays. At the top of Figure 4 we see the geometrical symmetry in the plan, which gives the misleading impression that there exists some form of urban ordering. The bottom of Figure 4 shows where the connections between the six nodes actually lie: that's not what one expects from an ordered group of six urban nodes. Any connective diagram that linked the six nodes with short-range connections would have been preferable.
Figure 4. Six urban nodes ordered symmetrically as seen from the air do not form a good module, because their interconnections are geometrically contorted.
Although Figure 4 illustrates a negative example, it represents a far better situation than exists in many urban regions. After all, the six nodes shown in Figure 4 are mostly connected to each other, even if those connections are not very practical ones. So much of what is built today falls into the category of 'near but disconnected' That corresponds to having the nodes shown at the top of Figure 4 without any connection to each other (see also Figure 2). This urban pathology must be understood as the absence of any need for the separate nodes to communicate. Merely providing potential connections that remain unused cannot build a living city.
6. Homogenization and segregation destroy system structure
Planning rules that concentrate non-interacting nodes prevent urban modules from ever forming. By denying the foundation for an urban system's coherence, it is mathematically impossible to realize a living city. Contemporary cities impose a set of zoning laws that generate a very particular physical structure: vast urban regions with homogeneous sectors and a lot of mechanical movement all over, but with very little life. In the example shown in Figure 5, three non-modules (each consisting of four adjacent but unconnected nodes) link with each other. The pathology of this situation is seen by comparing the links: 3 external links, but 0 internal links in every case.
Figure 5. Pathological situation consisting of three non-modules without internal connections, so that all connections are among the groups.
Concentrating similar functions as in the example shown in Figure 5 violates a system's basic composition: a module's internal connections must be stronger than the connections forming the interface between modules (Courtois, 1985; Simon, 1962; Simon and Ando, 1961). Since similar units do not usually interact with one another, trying to group units of the same type into a module is meaningless. Instead of reducing the forces acting between modules, such a grouping externalizes all its forces. Modular parcellation is effective only when the strongest forces are contained within the modules. Containing forces by coupling strongly-interacting units into modules often results in a geometrically non-obvious partition.
Alexander makes a point in "A City is Not a Tree" that the design of the then new Lincoln Center in Manhattan is fundamentally flawed (Alexander, 1965) . Segregation of performing arts buildings into one region makes no sense because they have no paths. By paths I mean real connections that satisfy a human need to go from one point to another, which has very little to do with where concrete 'footpaths' are actually built (Salingaros, 1998). The buildings are disconnected in the sense of parcellation (2) listed above (see Figures 2 and 5). Will anyone walk over and listen to a symphony in the next building after first going to the Opera? Not likely. There are no paths, and therefore no human connections between the different buildings in the ensemble.
7. Duality between buildings and urban space
Partitioning a system into constituents can be accomplished by means of an appropriate interface. Interfaces between modules comprising a complex system must themselves be complex; they have to couple and connect as well as to separate different units. A system may be partially decomposed into a set of complex semi-autonomous modules and equally complex interfaces that permit joining. That's precisely the way it occurs in biology. Successful system decomposition depends upon the correct distinction between modules and interfaces. Failure to identify the right interfaces prevents the system from functioning after parcellation. The system-level connections in modular decomposition require allowing for enough complexity in an interface.
In "A City is Not a Tree"(Alexander, 1965), Alexander identifies his basic units as geometrical nodes. Each of these could represent a building, or any fixed spot in urban space. Alexander then bases his analysis on estimating the enormous number of connections necessary for those units to define a living city. He concludes that twentieth-century planning practice does not put in -- and its theoretical principles will not even admit the existence of -- the necessary number of connections. Furthermore, permitting alternative connections that enable a system to generate its own complexity contradicts the idea of planning, which supposedly has to completely anticipate all connections. Any two nodes in a mathematical tree are connected by a unique path, so a 'tree'city fits in with this mentality.
My approach here is more general, in that I envision the different types of interfaces as the modules in a living city. For example, a geometrical interface along which people move, and inside of which people interact and perform functions that make cites "alive" forms a module. Its units are combined paths rather than buildings. Such modules are all linked into a network. This object looks organic and fractal; vaguely resembling some strange plant form. The connections determine the buildings' shapes rather than the other way around. Look at a figure/ground reversal on a city's plan: do the exterior spaces contribute to build up the urban fabric, or are they completely taken over by automobiles? Alexander's latest work (Alexander, 2000) is concerned precisely with all those connections.
Older cities were built by designing a continuous urban space throughout the city, as in our third parcellation (3) (Alexander, 2000; Gehl, 1987; Salingaros, 1999). This was an obvious way to do things as long as pedestrian movement was the dominant means of transport in cities: major urban functions occurred in urban space proper. That approach had to be revised to let in cars in increasing numbers, which because of their dominant size and speed displace pedestrians and pedestrian connections. Clearly, however, modernist planners went too far in dissolving urban space entirely, and then cutting expressways through city cores. The importance of urban space was lost in this century when the philosophical emphasis on meaning structures shifted from the space between buildings, to the pure geometry of buildings standing in isolation.
Parcellation (4) builds up a city in terms of basic geometric couplings rather than isolated buildings. Geometrical interfaces are the city's active units, but only if they successfully couple the objects on either side (Salingaros, 2000a). Interfaces are edges representing linear elements, along which a city's "life" is generated. In a typical urban region built today, however, all geometrical components are disconnected, so there is no interactive edge. The truth is that we have forgotten how to create a connective interface. Coupling almost always works via an intermediate region -- the complex, porous, or convoluted edge -- which is eliminated nowadays for stylistic reasons. Unconnected edges serve a purely decorative function.
Alexander and his colleagues realized the importance of parcellations numbered (4) and (5) above, and used them extensively in writing "A Pattern Language" (Alexander, Ishikawa et al., 1977). By studying the most functionally successful and emotionally appealing examples of urban structures in history and from around the world, they discovered that connective edges play a profound role in urban life. Many human activity patterns occur only along geometrical interfaces, the catalyst being the complexity of the interface itself (parcellation number (5)) (Salingaros, 2000b). Modernism deliberately eliminates the interface between urban elements in the pursuit of a "pure" visual style that shows no connections. For this reason, so many Alexandrine patterns seem out of place in today's urban design canon, and being incomprehensible, they are ignored.
8. Control and the suppression of emergence
A complex system that is expected to respond to changing internal conditions -- as for example in diagnosing itself, and correcting internal damage -- needs emergent structures. Self-stabilization, repair, and evolution are properties that do not depend on individual modules, hence they must exist outside of any modular decomposition. Since emergent properties are global, they are also outside the original programmed functions, and cannot be defined at the modular level. In this respect, they are "non-functional" because they do not correspond to the original designed functions. Emergent connections are possible only in a system that is already highly connected and offers a mechanism for additional connections.
It is precisely these evolving properties that generate biological life in an organism; intelligence in the brain; as well as "life" in a building or urban region. To encourage the formation of emergent properties, we cannot apply any single parcellation to the built environment. In all systems, emergence arises from new connections rather than strictly from those contained in the original modules themselves. Whereas the modules are initially fixed, additional connections may arise spontaneously from the interfaces between modules. In the human brain, the multitude of neuronal connections work together to produce consciousness, a property that cannot be understood from the brain's components alone (Edelman and Tononi, 2000).
The comparison between a simplistic aggregate and a system with emergent properties relates to choice: the former is preferred in situations where everything has to be totally controlled; whereas the latter occurs in situations where spontaneous growth is not a threat. In urbanism, the contrast between dead and living regions is stark. Dead cities are rigidly planned so that no spontaneous interaction is allowed between persons; buildings concentrate office or habitation units vertically so that a single entrance may be easily controlled; apartment complexes are usually controlled by having one gate; indoor malls have limited, guarded entrances; etc. Control is further imposed by legislation: no loitering in public; no pedestrians on the street; no sitting on walls; no commerce in residential enclaves; no selling on the sidewalk; etc.
Living cities on the other hand are more messy geometrically, and contain multiple paths offering alternative routes both to pedestrians and to cars. Buildings tend to be intertwined and not too spread out, with mixed uses and a reasonably small number of stories. Building complexes are composed of connected smaller buildings with multiple entrances rather than being concentrated vertically into a giant single building. One also finds here a proliferation of "non-functional" urban elements such as small parks, low walls, benches, street vendors, sidewalk cafés, kiosks, etc. This vital interweaving of commerce with daily life, passing time with strangers, and socializing in public provides the dynamic foundations of life in a city. The ancient marketplace or agora was not only a center of commerce, but was at the same time a center for socialization and political and intellectual interchange.
9. Cities evolve their own form
Zoning non-interacting units together creates pathological non-systems, such as functionally concentrated commercial downtowns and homogeneous residential suburbs (Salingaros, 2000a). As it is necessary to link these two groups strongly for communication and transportation, long-range connections generate enormous external forces that eventually lead to the functional choking of cities. The new situation in turn generates new configurations in the urban structure, which planning can guide in either a positive or negative direction. Left to themselves, people will attempt to relocate their business or residence in response to urban forces.
The connections responsible for emergent phenomena arise from having many alternative choices connecting one subsystem with another. Being able to choose depends on both urban geometry, and legislation. Choice is not present when all the nodes connect via a unique path. Emergence, and thus evolution, are impossible in a totally planned city that offers no choice between possible alternatives. System evolution generates connections that cross both modular boundaries and distinct scales to connect one subsystem with a much larger or much smaller structure: such connections are extra-modular. Other system connections are going to be rearranged or cut. To understand the evolution of urban morphology, we need to examine how a city changes its connections over time.
Figure 6. Two modules re-organize themselves over time by defining new connections and new boundaries.
Any parcellation of a city into modules -- even if those modules make the most sense structurally as well as functionally -- will have to rely on the state of the city at that particular time. Yet we know that the functions and nodes in a city are always changing. Systems have a roughly hierarchical ordering, in which smaller interacting components are associated into larger components (but don't necessarily fit neatly into them). The smaller components are continually altered or are being replaced by other components, and this alters the internal composition of the modules. Interfaces that are responsible for system connections are modified by these changes. New connections representing emergent phenomena will have to be accommodated; how that is done cannot be decided beforehand.
The opposite approach from segregated planning was tried in the not-so-recent New Towns, which are made up of a collection of artificial villages. This parcellation doesn't work very well either. Such ideal cities appear more human on paper, because their modules are based on working older prototypes. They also follow system laws by being decomposed into self-contained modules, each module consisting of strongly-coupled units such as houses, shops, schools, parks, etc. Alexander already pointed out that this structure is a tree, and is therefore not alive (Alexander, 1965). Why this is so is more subtle than in the case of the functionally segregated modernist city, and has to do with emerging forces between modules.
Figure 7. Utopian city built from non-interacting modules generates a living form by forging inter-modular connections. This process destroys the originally neat parcellation.
An ideal city built from non-interacting village modules would immediately start to unravel. People will find employment in a different module; others will move to another module but keep their friends, relatives, and shopping at their former module; shops will change so that people go outside their own module; a deteriorating neighboring school or simply the desire for higher quality forces a family to send its children to school in another module; etc. Social and commercial forces cut internal connections and generate new strong connections between and outside the modules. The carefully-planned system decomposition undoes itself, making the original large-scale partition into modules inapplicable. The system becomes degraded because it is not designed to accommodate emergent connections.
10. The distribution of connective lengths
It is extremely difficult, if not impossible to plan a living city all at once. We are left with no choice but to shift our thinking from rigid planning imposed on urban structure, to a time-dependent process that guides the natural evolution of a city. Alexander's latest work (Alexander, 2000) analyzes how the geometry of a living city evolves over time. In this paper, I have tried to indicate the two opposite endpoints away from which a city tries to evolve: (A) the segregated zoned non-system with only long-range connections; (B) the utopian cluster of artificial villages having only short-range connections. A living city lies somewhere in-between these two rigidly planned extremes, though much closer to (B) than to (A). Moreover, a city's viability depends on the freedom to rearrange its connections over time.
These two extreme connective models for a city are characterized by their mutually exclusive connection lengths. What is the optimal distribution of connective lengths in a living city? A mathematical result on the distribution of sizes (Salingaros and West, 1999) answers this question. Systems depend on components of different magnitudes, and the distribution of those magnitudes is optimal when they obey an inverse-power scaling rule. This scaling rule says that the number of connections of each length is inversely proportional to their length raised to a power between 1 and 2. Short connections are thus much more common than long connections, and the longer the connection is, the less frequently it should occur (Figure 8).
Figure 8. Distribution of pathlengths according to 1/x2 law, showing only the three longest paths.
A functioning urban fabric -- living neighborhoods connecting in a mutually beneficial manner to each other, as well as to dissimilar urban regions -- contains connective lengths that obey an inverse-power distribution. Going back to the duality between nodes and connections discussed in an earlier section, the inverse-power rule applies also to the distribution of urban spaces. Urban spaces have to be provided for groups of people in increasing numbers: very many appropriate for small groups of people, and only a few that can accommodate many people. The objective is to encourage personal interactions according to the same distribution: many intimate or brief daily contacts of small groups of people in urban space, with provisions made for the less frequent congregation of larger groups.
Support for this conclusion comes from an incredible variety of complex systems that obey the above scaling rule, from DNA structure, to power-laws from economics, to all fractal forms (Salingaros and West, 1999). Inverse-power scaling is ubiquitous in nature, and is found in a wide range of both natural and man-made phenomena. The distribution of links on the World-Wide Web follows this rule (Albert, Jeong et al., 1999). Perhaps the most relevant example has to do with the distribution of neuron lengths in simple invertebrate animals (Watts and Strogatz, 1998). Nature has already solved the problem of how to connect the nodes of a complex organism in an optimal manner. A close relation exists between inverse-power scaling and 'small-world' networks, whose details I will now describe.
The distribution of connection lengths plays a key role in how a fully-connected network functions. Networks that appear in both natural and artificial systems lie in-between two extremes: (A) Random networks characterized by random links; and (B) Regular networks consisting of only nearest-neighbor links (Watts and Strogatz, 1998). In the former, the pathlengths cluster around some distribution mean, therefore most links are much longer than nearest-neighbor links. Reconnecting a system of type (A) by disconnecting many long links, and replacing them with near-length connections; or lengthening a few of the initially short connections in (B) to generate medium and longer connections leads to a 'small-world' network, which has vastly improved connectivity properties over either random or regular networks (Watts and Strogatz, 1998).
Inverse-power distributions characterize systems that have no fixed scale; i.e., that function equally well on all scales (Salingaros and West, 1999). In practice, inverse-power distributions have a lower cut-off at some smallest allowed length, which is the nearest-neighbor link, and their average length is some multiple (between 3/2 and 2) of the smallest length. This favors the smallest connection lengths. By contrast, the characteristic or average length of a random distribution is some fraction (roughly 1/3) of the size of the whole system, representing the maximum possible length. Because the modernist city and suburb lack small-length connections, monofunctional zoning pushes the characteristic length of urban connections past the random average, and closer to the maximum distance.
11. Ecosystems and geometry
Cities can learn from the theoretical modeling of ecosystems. Biological ecosystems are complex overlapping systems composed of modules of organisms of different sizes. It makes as much sense to define a rectangular habitat for some animal as it does for a "housing sector". Isolating plants and animals into their own segregated sectors destroys an ecosystem. A fine-grained geometry that allows mixing is a prerequisite for life. We can create an artificial reef by dumping old cars and refrigerators on the sea floor; within a few years it is teeming with marine life. A crystal clear mountain lake (which is high on our list according to aesthetic value) is essentially dead, whereas an opaque green pond full of decomposing logs and branches is usually rich in life forms.
Other than geometry, neglected urban qualities include dynamic evolution and stability. Ecosystems are dynamic in the sense that their internal composition and boundaries are changing continuously. No-one plans an ecosystem, but the wrong kind of intervention (either by humans, or by catastrophic natural events) can destroy it forever. Stability in ecosystems is founded upon the existence of different sizes of modules: each reacts differently to perturbations. A simple model shows that large ecological modules react more slowly to perturbations, whereas small modules react faster. This built-in diversity guarantees some basic stability to different types of perturbations.
A city needs the same sort of resilience to changing conditions that a healthy ecosystem has. I don't know how to design this, but it's clear that the solution must come from a set of urban laws -- yet to be derived -- that allow a city to evolve its own life, and to maintain it over time. Not only must the conditions for urban "life" be legislated into a set of guidelines that help the urban fabric to cohere in the first place, but the laws must then guide the evolution of life in a positive rather than a negative direction. We require a set of evolutionary laws, which are the opposite of rigid design laws such as monofunctional zoning. Furthermore, those laws have to allow the reconnection of urban units so as to maintain or increase the degree of life in the environment.
Our civilization is intelligent enough to accomplish what it wants. The problem is that a major segment of today's population actually wants dead urban regions. People seek the very things -- such as a simplistic monumental geometry, monofunctional zoning, priority for automobile traffic, fenced-off commercial and residential blocks, and forcing all poor people into huge apartment blocks -- that destroy the life of a city. The poor have picked up the same images, so after moving up in society they inevitably join with other middle-class citizens in killing their city. Urban legislation creates the type of city we have today; a radically different legislation might re-create a living city once again, if people can be convinced that their lives and their children's lives would become better.
12. Conclusion
My purpose in this paper was to present new theoretical results on urban structure that follow from the parcellation of coherent complex systems. These results drastically alter our conception of a city as simply a juxtaposition of buildings, neatly lined up. A city becomes alive only if its geometry permits an enormous number of changing connections, which allows it to evolve much as an organism does. The connections responsible for a city's "life" themselves define alternative decompositions of city form. A clear picture emerges, of a city whose complexity is based on many more short-range connections than long-range connections. Cities need to re-establish a vast number of nearest-neighbor couplings, as well as a sizable number at various intermediate lengths. A living city's central characteristic, moreover, is that it is constantly readjusting all of its links. Any planning effort must therefore help rather than hinder this natural process of reconnection.
13. Acknowledgment
I am grateful to the Alfred P. Sloan Foundation for supporting this research. Many thanks to L. Andrew Coward and Jorma Mänty for useful suggestions.
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