Scientific principles applied to city form help to understand the role of various types of urban connectivity. The degree of "life" in a city or region of a city is tied to the complexity of visual, geometrical, and path connections. There is an optimal distribution of connection lengths in a living city, and violating this distribution removes life from the urban environment. Alternative parcellations of a living city reveal the complex structure that is required to generate human contact, which is the basis for city life. These results are compared to the work of Christopher Alexander.

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:

  1. Into buildings as basic units (as is usually done) and their interactions via paths.
  2. As a collection of paths anchored and guided by buildings (Salingaros, 1998).
  3. As external and internal spaces connected by paths and reinforced by buildings (Alexander, 2000; Salingaros, 1999).
  4. As the edges and interfaces that define spaces and built structures (Alexander, 2000; Salingaros, 2000a).
  5. 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.

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).

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.

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.

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.

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.

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.

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).

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.

Figure 1. Six nodes all connected to each other define a module. The nodes' exact position is unimportant.
Figure 1. Six nodes all connected to each other define a module. The nodes' exact position is unimportant.
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.
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.
Figure 3. Modules internalize connections between their constituent nodes. Three modules connect themselves via organizable forces.
Figure 3. Modules internalize connections between their constituent nodes. Three modules connect themselves via organizable forces.
Figure 4. Six urban nodes ordered symmetrically as seen from the air do not form a good module, because their interconnections are geometrically contorted.
Figure 4. Six urban nodes ordered symmetrically as seen from the air do not form a good module, because their interconnections are geometrically contorted.
Figure 5. Pathological situation consisting of three non-modules without internal connections, so that all connections are among the groups.
Figure 5. Pathological situation consisting of three non-modules without internal connections, so that all connections are among the groups.
Figure 6. Two modules re-organize themselves over time by defining new connections and new boundaries.
Figure 6. Two modules re-organize themselves over time by defining new connections and new boundaries.
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.
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.
Figure 8. Distribution of pathlengths according to 1/x2 law, showing only the three longest paths.
Figure 8. Distribution of pathlengths according to 1/x2 law, showing only the three longest paths.