The "toy model" from evolutionary biology
Organized complexity versus empty purity
Element variety is necessary for coupling
The decomposition of coherent complex systems
Stability and emergent connections
Control and the suppression of emergence
Cities evolve their own form
A city becomes alive as it increases its number of connections. Furthermore, the transition from "dead" to "alive" is sudden. This is described by connecting urban nodes using more and more connections until suddenly, they are all interconnected (Section 2). An urban node is some location where a person wants to be -- one's bedroom; one's office at work; a park bench to sit for five minutes when the weather is nice; a particular spot on the sidewalk to stand for ten seconds and enjoy the view. All of these are nodes. Urban connections are created on every scale through roads, isolated bicycle lanes, footpaths, etc. Section 3 proposes the human-mind/urban-web analogy, in which a living city works through its connections to acquire the properties of a "superorganism". It metabolizes and processes information -- one might say it has its own intelligence. For this to happen, however, the number of connections has to be many orders of magnitude greater than is allowed by modernist planning.
Section 4 introduces the law of requisite variety for urban systems. Abstracting all urban elements, whether nodes or connections, allows us to consider how everything comes together regardless of its functional type. A certain threshold is necessary before all elements can connect. This threshold has to do with the different types of urban elements present, and coalescence will never take place with only a limited variety. Section 5 continues the picture of each urban element acting both as a node and as a connector, contrary to the usual monofunctional typology. Like the components of an organism, each urban element is itself highly complex. Alternative decompositions of a city into its components reveal this important duality.
The dynamics of urban form are treated in the last three sections. Section 6 identifies two types of forces that influence urban form -- a genetic "blueprint"; and self-organization due to feedback mechanisms. Planners have to understand how a living city functions in order to provide a correct blueprint (which has not happened since the 1920s). Most important, they have to understand and accommodate positive urban forces, as the city attempts to reorganize itself dynamically. By misunderstanding and trying to suppress those forces, twentieth-century cities have become pathological. The key is allowing for emergence to build the urban fabric, instead of trying to thwart it (Section 7). Planning based on simplistic visual ideas and social control prevents emergence. Finally, even the well-meaning "New Towns" failed because they did not provide for emergent forces, but assumed that everything is static (Section 8). A living urban fabric depends absolutely on dynamic stability in the face of inevitable changes.
2. The "toy model" from evolutionary biology 
The new science of complexity supports traditional proposals for urban design. A result from random graph theory applied to a model in evolutionary biology illustrates what actually happens in creating an organized urban web. It mimics the process of building in history. Let us proceed to connect all the different elements in an urban situation. We try to achieve maximum organization by making adjustments to the components -- moving them around, and modifying them so that nodes and architectural elements connect to each other at a distance. The goal is always to establish connections.
Organization can be studied in terms of pairwise linking. Consider N elements that are initially independent. Pick any pair at random and connect them, repeating this process at every step. Each time, one link is established, and in this way many small chains are created. The length of the largest chain initially will be very small, and grows slowly. At some point two or more chains will link. Up to N/2 steps, elements are linked largely into pairs that are independent of each other. When the number of pairwise connections exceeds N/2 , however, small chains begin to connect into larger chains, and at some point between N/2 and (N/2)lnN steps, many elements will link together to form one giant multiply-linked chain. The larger the system, the more sudden this coalescence. The system has undergone a phase transition from a disorganized into an organized state. Further pairwise coupling will increase the size of this largest chain, but only by small increments, since it may already connect more than 80% of the elements.
This result applies to urban design in the following way. The planning process can either be mimicked with a computer model, or it is carried out gradually in actual building over the years. Connecting nodes incrementally will result in a perceptible improvement in the organization of the overall structure. What is seen is striking and is akin to a phase transition. A certain point is reached when almost everything coalesces -- organization has been achieved. From this point on, every observer will experience the ensemble as being linked together.
A phase transition in complexity occurs; for cities, as the number of connections between distinct nodes/places passes a certain order, then a very large proportion of all nodes become connected suddenly. This explanation can be used for classifying cities and certainly idealized cities and their geometry, and it may even be linked to the performance of cities. This model of course is like a percolation model used for water flow -- when enough holes appear in a porous medium, suddenly water passes through the material. Similarly, when enough trees are linked to form a forest, then a forest fire will spread throughout the forest. There exist quantitative results on this phase transition between slow and rapid flow.
3. Organized complexity versus empty purity 
Architecture and urban planning can be understood as processes that (ought to) increase the degree of organized complexity. Much has been written about the organization of complexity, especially from the biological point of view. Different processes occurring together generate complexity; and if these are organized coherently, they result in organized complexity. Where very few processes are occurring the situation is not complex to begin with. If, on the other hand, there is complexity but it is unorganized, then we are faced with a chaotic situation. That state is incomprehensible to the human mind, because it escapes our perceptive abilities.
Mankind has always striven to increase the organized complexity of its surroundings, in parallel with a developing intelligence and improved grasp of natural systems. The twentieth century has seen a deliberate reversal of the process. Architects and urban planners became infatuated with visual simplicity and ignored the fundamental process of organization, which is not visually simple. We now have many examples of urban regions where the complexity has been eliminated altogether by suppressing connections. The search for visual purity in the plan has severely curtailed human activities that led to urbanization in the first place.
The principal model for twentieth-century planning, the Ville Radieuse, does not permit the connections that form the urban web. That model allows only pairwise connections between home and workplace, and no others. What we have is a singular bundle of connected node pairs that do not interact. This is equally true between office blocks and high-rise apartment buildings, as it is between factories and suburban tract houses -- the underlying pattern is disconnected. The number of pairwise connections equals N/2 , which is the threshold before the "toy model" previously discussed begins to connect internally. A fully connected graph needs the far greater number of (N - 1)N/2 connections. The necessary linkage for sustaining human life and activities is deliberately avoided in the Ville Radieuse.
Kevin Lynch introduced the mental image of a city as a means for judging its success. Bill Hillier emphasizes the intelligibility of a city as the ease with which one perceives the path structure. Here one may point out the crucial connection between hierarchical organization and simplification. A chaotic process is simplified by organization without necessarily losing any of its intrinsic content. Complex and diverse elements are grouped together so that they cooperate, and as a result they appear streamlined. By contrast, purification is a reducing process that loses much of the information inherent in a system. It is unfortunately very easy to confuse the two, with catastrophic consequences.
We now know much more about the perceptive processes that map the urban web onto the human mind. The two are very much alike, and consist of interacting connective nets on several different levels. An idea, or a path, is established by linking nearby strands of the network. The necessity of having many alternative paths, and comparing them, is the key to reasoned thought. We can be forced into one unique path by a planner, but that is not the way our mind works; it is how a robot functions. The quest for artificial intelligence in machines corresponds precisely in trying to go from mindless simplicity to organized complexity.
The degree of organization of any complex system depends directly on the ratio between the number of connections and the number of nodes. The following comparison is instructive. In conventional digital computers, the number of connections is comparable to the number of nodes (transistors), which is roughly that found in a minimal connected graph. In a brain, however, the number of connections is some four orders of magnitude (i.e., 10,000 times) larger than the number of nodes (nerve cells). Multiply-connected neural computers, which are successful in pattern recognition, are somewhere in-between. The mind-web analogy reveals just how enormous the density of connections must be in a successful urban setting.
4. Element variety is necessary for coupling 
Recent findings in evolutionary biology reveal the need for a variety of connective elements. Consider a mixture of different types of complex organic molecules that were found in an early period of the planet. The likelihood of a chance reaction creating the first life form increases with the number of different molecules in contact with each other. Some molecules will act as catalysts (with a very low probability) for reactions between other molecules, thus facilitating any combination that might take place. Modeling via computer simulations shows a dramatic increase of reaction probability above a certain threshold of molecular variety, known as a "critical diversity". Such a mixture becomes autocatalytic. By contrast, simpler systems containing a sub-critical variety of elements have a vanishingly small probability of reacting.
The point of this result, which has important consequences for urbanism, is that catalytic elements are not explicitly identified as such. There are no catalysts per se, but each molecule (or structural unit) may also act as a catalyst to couple two other units. We start with a random mixture of different units that we know to be components of an eventual organic whole, and which are allowed to interact freely with one another. Every molecule is presumed to play a secondary role as a catalyst, in addition to whatever its principal chemical role may be. It is clear that we need a variety of units, because any single unit might be needed to catalyze a particular connection between two other units. The autocatalytic threshold is probabilistic and sudden.
Urban coherence emerges in an analogous fashion. The formation of a complex interacting whole requires the availability of many different types of urban elements. The reason is that some of those elements need to act as intermediate connectors, to catalyze the coupling between other urban elements. One cannot assemble a living, coherent city by restricting the element variety and mix. The corollary is also obvious -- urban life in the dynamic cities that we know arises almost spontaneously when a critical mixture and density of urban elements has been reached, and disappears when one of those essential elements is removed, isolated, or concentrated. Even if we have the requisite variety of elements, they must be allowed to interact; therefore, segregating urban functions stops the connective process.
This dual, connective role of elements is insufficiently recognized in urban design. After many decades of rigidly stereotyping urban elements according to a single primary function, it is difficult to imagine all their other, secondary functions, and their fundamental role in connecting the urban fabric is ignored. For instance, while it is obvious that we need a road to connect a house with a store, we similarly need stores and houses as geometric connective elements in different situations. Connective elements are eliminated in the drive to "purify" the built environment because their true function is not understood. The mechanism of mutual catalysis is fundamental in complex systems and works in creating living cities the world over, yet it runs counter to what has been taught for decades in architecture schools.
The above result unequivocally supports one of Jane Jacobs's proposals for the generation of life in cities: "The district must mingle buildings that vary in age and condition, including a good proportion of old ones so that they vary in the economic yield they must produce. This mingling must be fairly close-grained". Jacobs outlined cogent economic arguments to support her result; here, our arguments are scientific. Elements of any living environment are not going to be defined by geometrically identical units. In a separate publication co-authored with Bruce J. West ("A Universal Rule for the Distribution of Sizes", 1999), we derive an optimum distribution for project funding in urban construction, which is skewed towards small projects. This formula inevitably precludes most large lump developments, so it guarantees the preservation of old buildings by allowing only a few new buildings into any coherent region.
5. The decomposition of coherent complex systems 
It is surprising, and somewhat alarming, that decomposition theorems for complex systems remain unknown to many authors and planners, who base their work on empirical decomposition schemes, forty years after this work was first published. A functionally integrated urban system is considered to be made up of parts; however, how does one determine those parts? The whole is definitely not reducible to parts and their interaction. Instead, it is called "nearly decomposable", because if it were completely decomposable, each subsystem would behave in a totally independent manner. The whole system would then lose its complexity, and its behavior would reduce to the simple juxtaposition of its constituents. It is the weaker higher-level couplings that provide the essential coherence of a complex hierarchical system.
Even so, decomposition helps in the analysis of a complex system because it reveals its internal structure. Otherwise, the system's complexity will remain a mystery. The choice of what components in a system are the basic ones is arbitrary, however, and depends on the viewpoint of the observer. A city can be decomposed according to the following alternative schemes:
(A) into buildings as basic units (as is usually done) and their interactions via paths; or
(B) as paths that are anchored and guided by buildings; or
(C) as external and internal spaces connected by paths and reinforced by buildings.
Other decompositions are possible, each one of which identifies a different type of basic unit, and builds up the city from an entirely different perspective. All choices may be equally valid, and lead to a partial understanding of the complexity of urban form and function.
Segregation and concentration of functions, zoning, and uniformization all reflect a simplistic view of a city that negates its basic complexity. The identification of similarly-sized buildings as the fundamental units of a city already destroys its coherence by denying all of its other possible decompositions. Furthermore, the simple alignment of buildings that do not interact in any way decomposes a complex system completely, thus reducing it to a simplistic aggregate. Urban practice has unfortunately done that, and continues to do so, without realizing the damage it is doing to the urban fabric. Just as in a living organism, one cannot undo the whole without killing it. Despite a superficial orderly appearance, most contemporary cities are simply a collection of disconnected parts defined on just two or three scales.
6. Stability and emergent connections 
I am not proposing an anarchic view of architecture; quite the opposite. Systems that develop purely randomly are rarely driven to any form of ordering, either simple, or complex. As in biological organisms, profound structural and functional complexity is carefully governed by both genetic blueprints, and delicate regulatory mechanisms based on feedback and balance. Indeed, the breakdown of these governing agencies leads to pathologies such as cancer, or the unsuccessful repair of the system after an external pathogenic invasion. This is where a living city differs from a favela: the former has additional ordering that fixes the latter's problems while not killing the positive degree of life present. The point is to harness forces so that they cooperate.
Connective forces act on urban geometry, driving it towards a unique morphology in each particular instance. Architects wishing to impose their own imagined order ignore the very forces that are trying to shape the environment. Actions include forbidding people from creating diagonal paths and forcing them instead to an inconvenient pavement. Chasing away street food vendors instead of building kiosks ignores a clue that there exists a need for prepared food at that spot. Contemporary urban design aspires to maintain its appearance against urban forces. That is an ultimately futile quest, because it attempts to block the natural processes of self-organization. Those forces will forever work against any imposed forms, and an enormous amount of energy is going to be expended to maintain the original design, preventing the emergence of connections.
The basic notion of stability in physical systems underlines that states are long-lived only if they do not have to be propped up -- if their energy is such that all inevitable small changes reinforce that state instead of disturbing it drastically. A dynamically stable urban state is one that has an enormous number of geometrical and functional connections on many different scales. Some are going to be cut as new ones arise. These time-dependent processes are self-sustaining on the average. In the same way, traditional buildings that connect well into the urban fabric stabilize that region as a result of their design. Contemporary buildings as a rule don't connect at all -- they fail to create human environments because their architects misunderstood (or vainly hoped to reverse) the direction in which urban forms evolve naturally.
7. 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 "nonfunctional" 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.
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 "nonfunctional" 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.
8. Cities evolve their own form 
Zoning non-interacting units together creates pathological non-systems, such as functionally concentrated commercial downtowns and homogeneous residential suburbs. 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.
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. Christopher Alexander already pointed out that this structure is a tree, and is therefore not alive ("A City is Not a Tree", 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.
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.
This paper addresses the question of how urban form develops in time to achieve a living city. This is probably the most misunderstood topic in architecture and urbanism today, because the twentieth century concentrated on often irrelevant aspects of form and style -- which deal with static structures. And yet, we all know that a city is a dynamic entity, and thus it cannot be understood from static processes alone. It is biologists who have the best tools to help us in this endeavor. There are two related mechanisms that apply directly to understanding development in complex forms. First, the development of biological structure -- as in the developing embryo -- depends on both genetic coding (a predetermined plan) and chemical adjustments during growth (feedback in real time). Second is the complementary idea of evolutionary development, whereby selection changes the type of the organism -- though not the individual organism -- over several generations. Starting with a complex, organic, dynamic picture of a living city, these mechanisms can help to clarify how natural and human actions generate the urban fabric.
In this essay, I have re-used sections dealing with the evolution of urban complexity from three of my previously-published papers (listed below). Although many urbanists already know those papers, I felt that separating the above sections from their original context, and bringing them together for the first time gives a better overall perspective of how important time-development is for generating healthy urban structure.
. "Theory of the Urban Web", Journal of Urban Design 3 (1998), pages 53-71.
. "Complexity and Urban Coherence", Journal of Urban Design 5 (2000), pages 291-316.
. "Remarks on a City's Composition", RUDI -- Resource for Urban Design Information (2001), approximately 14 pages.