Complexity and Urban Coherence

NIKOS A. SALINGAROS Division of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA. Email: salingar@sphere.math.utsa.edu

Journal of Urban Design, vol. 5 (2000), pages 291-316. © Carfax Publishing, Taylor & Francis Ltd.

ABSTRACT. Structural principles developed in biology, computer science, and economics are applied here to urban design. The coherence of urban form can be understood from the theory of complex interacting systems. Complex large-scale wholes are assembled from tightly interacting subunits on many different levels of scale, in a hierarchy going down to the natural structure of materials. A variety of elements and functions on the small scale is necessary for large-scale coherence. Dead urban and suburban regions may be resurrected in part by reconnecting their geometry. If these suggestions are put into practice, new projects could even approach the coherence that characterizes the best-loved urban regions built in the past. The proposed design rules differ radically from ones in use today. In a major revision of contemporary urban practice, it is shown that grid alignment does not connect a city, giving only the misleading impression of doing so. Although these ideas are consistent with the New Urbanism, they come from science and are independent of traditional urbanist arguments.

Introduction

This paper uses scientific principles to understand urban coherence. From small man-made objects -- such as sculptures, pottery, and textiles -- to buildings, the best examples share a particular geometrical quality. Though not usually viewed from this perspective, we will argue that the form of cities and the urban fabric is also governed by the same general rules. Geometrical principles that produce a beautiful sculpture or textile generate a positive emotional response: can we identify similar rules for an urban setting the size of a neighborhood or an entire city? If so, then the resonance with human beings that characterizes great urban environments could be explainable in terms of geometry. The ideas developed here have been encouraged by recent work of Christopher Alexander (Alexander, 2000).

An essential quality shared by all living cities is a high degree of organized complexity (Jacobs, 1961). The geometrical assembly of elements to achieve coherence results in a definite and identifiable urban morphology. It turns out that this morphology closely resembles that of traditional cities and towns: unplanned villages of many different cultures around the world; cities as they were before the middle of the nineteenth century; and to some degree, free squatter settlements. The morphology of a geometrically coherent system resembles planned twentieth-century cities the least of all. Contemporary rules for urban form, which reduce both complexity and connectivity in today's cities, are not capable of generating urban coherence. We analyze why this is so, and offer new rules that do.

Different components of the urban fabric: streets, shops, offices, houses, pedestrian zones, green spaces, plazas, parking lots, etc. connect to generate a successful city, creating an efficient, livable, and psychologically nourishing human environment. The success of the result depends on geometrical coherence. The transportation network defines city form; a city lives and works according to its network of connective paths (Alexander, Neis et al., 1987; Hillier, 1999; Salingaros, 1998). In addition, it will have pedestrian life if its urban spaces accommodate and support pedestrian paths (Alexander, Neis et al., 1987; Gehl, 1987; Hillier, 1997; Salingaros, 1999). A third, purely geometrical factor -- urban coherence -- determines the success of a city, and has its own set of rules. These need to be studied independently of the path structure and the formation of urban spaces.

To achieve geometrical coherence in any system, a tightly-knit and complex whole is generated via general rules. Geometrical coherence is an identifiable quality that ties the city together through form, and is an essential prerequisite for the vitality of the urban fabric. The underlying idea is very simple: a city is a network of paths, which are topologically deformable (Salingaros, 1998). Coherent city form must also be plastic; i.e., able to follow the bending, extension, and compression of paths without tearing. In order to do this, the urban fabric must be strongly connected on the smallest scale, and loosely connected on the largest scale. Connectivity on all scales thus leads to urban coherence.

In living cities, every urban element is formed by the combination of subelements defined on a hierarchy of different scales. Complementary elements of roughly the same size couple strongly to become an element of the next-higher order in size (Salingaros, 1995). Different types of connections tie elements of different sizes together, so that every element is linked to every other element. The strongest connections are local (close-range) ones. Connections between smaller and larger elements, or between internal subelements of distinct modules, are weaker. Repeated similar units do not connect: coupling works either by contrasting qualities, or via an intermediate catalyst. Elements are therefore necessary, not only for their own primary function, but also for their secondary role in linking other elements that cannot couple directly by themselves.

This paper presents theoretical rules for assembling components into coherent wholes, developed outside urbanism. The components that influence urban morphology are then reviewed. We derive and explain geometrical coherence, while at the same time applying the proposed rules to analyze the urban fabric on successive scales. First, we identify the basic interactions between components on the small scale. Fractal interfaces and the auto-catalytic threshold are discussed. We then review modular decomposition, and Alexander's Pattern Language. After this, we examine ordering mechanisms on the large scale. The key role of entropy in alignment forces is explained. Finally, these ideas are applied to cities. We emphasize the need for mixed use, and argue that long-term stability depends upon allowing for emergent connections.

Rules for geometrical coherence

In a general complex system, as for example an organism or a large computer program, certain rules of assembly are followed so that the parts cooperate and the whole functions well. There is little formal difference between such systems and the urban fabric (Lozano, 1990). A few structural rules have evolved in the study of complex systems. Initially stated by Herbert Simon for economics (Simon, 1962; Simon and Ando, 1961), some were re-invented in the context of computer programming (Booch, 1991; Courtois, 1985; Pree, 1995). Others appeared independently in engineering and biology (Mesarovic, Macko et al., 1970; Miller, 1978; Passioura, 1979). Of the many different possible statements of system rules, the following list is critically relevant to urban design.

Rule 1. COUPLINGS : Strongly-coupled elements on the same scale form a module. There should be no unconnected elements inside a module .

: . Rule 2. DIVERSITY : Similar elements do not couple. A critical diversity of different elements is needed because some will catalyze couplings between others .

: . Rule 3. BOUNDARIES : Different modules couple via their boundary elements. Connections form between modules, and not between their internal elements .

: . Rule 4. FORCES : Interactions are naturally strongest on the smallest scale, and weakest on the largest scale. Reversing them generates pathologies.

: Rule 5. ORGANIZATION : Long-range forces create the large scale from well-defined structure on the smaller scales. Alignment does not establish, but can destroy short-range couplings.

: Rule 6. HIERARCHY : A system's components assemble progressively from small to large. This process generates linked units defined on many distinct scales.

: Rule 7. INTERDEPENDENCE : Elements and modules on different scales do not depend on each other in a symmetric manner: a higher scale requires all lower scales, but not vice versa .

: . Rule 8. DECOMPOSITION: A coherent system cannot be completely decomposed into constituent parts. There exist many inequivalent decompositions based on different types of units.

These eight rules are offered as generic principles of urban form. We will be analyzing in some detail where the rules come from, giving original arguments with visual, scientific, and urban examples. The whole point is to convince the reader of their inevitability in assembling a living city. A system's development in time defines an underlying sequence. The smaller scales need to be defined before the larger scales: their elements must couple in a stable manner before the higher-order modules can even begin to form and interact. Elements on the smallest scale, along with their couplings, thus provide the foundations for the entire structure. Requiring a hierarchy of nested scales means that not even one scale can be missing, otherwise the whole system is unstable.

The coherence of a complex interacting system may be understood as it connects progressively. During a short time period, strong couplings will establish internal equilibrium in each module, with little change in the relationship among different modules. (One analogy is the initial formation of many small isolated crystals in a solution). Over a longer time period, the weaker couplings between modules will take them towards a larger-order equilibrium, while their internal equilibria are of course maintained. The process iterates, so that on even longer time periods, modules of modules tend towards equilibrium, and so on. The end result is a global equilibrium state for the entire system (corresponding to a single complex crystal).

Components of the urban fabric

Many distinct elements are necessary to achieve urban coherence. Roads, paths, parking, together with green, residential, commercial, and industrial elements must all be accommodated; even though they are contrasting, they have to coexist harmoniously. Each urban element can increase in intensity, either by lateral, or by vertical growth. Buildings can increase in number of stories; green can progress from lawn to bushes to trees, which are limited to their natural height. Footpaths are independent of vehicular roads: the former range from a garden path, to a sidewalk, to a pedestrian mall; the latter can increase in intensity from a back alley, to a local road, up to an expressway (Salingaros, 1998).

Some traditional elements of the urban fabric are now suppressed for reasons of style. Among the most important casualties are the connective elements between interior and exterior spaces. From the Hellenistic stoa, to Roman porticoes, to the retractable street canopies of the North African souq, to the canvas awnings of stores and open-air markets, an intermediate space was defined under different conditions and for different occasions. Without this element, the indoor/outdoor transition is too abrupt, and the connection is lost. Front porches are hardly ever used because there is no contrast or coupling to the street. The principle behind the half-covered veranda is to give a feeling of enclosure, while at the same time opening up to the outside world in front of the house.

Another group of urban elements largely missing from contemporary cities includes those defining a pedestrian environment, and its complex interface to other modes of transport. Footpaths, sidewalks, bollards, low walls, arcades, colonnades, covered walks, slightly raised pedestrian crossings, covered bus stops, tree-lined boulevards, small parks, etc. are now considered anachronistic, and are eliminated from today's automobile city. If they re-appear at all, they do so selectively -- as some quaint "quotation" from the past -- and are never integrated into the whole. We are not arguing for the return to a purely pedestrian city; however, these missing components are necessary for any city to achieve geometrical coherence. Simply put, these are the small parts that one needs to assemble a large and complex urban whole.

Despite almost a century of criticism within urban design theory of some of the worst modernist tenets, we are still unable to duplicate the beauty and functionality of neighborhoods built before the second world war. The prophetic analysis by Jane Jacobs (Jacobs, 1961) is ignored by the majority of urban developments today. Empirical rules for generating the urban fabric are given by Alexander and his associates in (Alexander, Ishikawa et al., 1977; Alexander, Neis et al., 1987), later developed into a fundamental theory of order (Alexander, 2000). Solutions in the same spirit are offered by Greenberg (Greenberg, 1995) and by Kunstler (Kunstler, 1996). Urbanists trained in the older, humanistic design idiom will find support here coming from an unexpected, scientific direction.

Coupling urban elements on the smallest scales

The central thesis of this paper is that urban coherence is founded on the small scale, where contemporary urban design is most deficient. We discuss in detail the various processes governing small-scale geometrical complexity that contributes to urban coherence. Geometrical links are related to the theory of fractal interfaces. We will use new results from evolutionary molecular biology to argue that a large variety of connected urban elements is a prerequisite for urban coherence (which finally proves Jane Jacobs's (Jacobs, 1961) insights).

The idea of coupling

"Order on the smallest scale is established by paired contrasting elements, existing in a balanced visual tension" (Salingaros, 1995). What are the smallest urban elements that can be paired in this way? They include everything accessible to a pedestrian at arm's length, and which is used to build up a city. Bricks, paving stones, footpaths, trees, individual parking spaces, walls, doorways, windows, ledges, columns, sidewalks, benches, bollards, etc. must all be created and positioned so as to couple strongly with each other and with a nearby pedestrian (Rule 1). The combination of several pedestrians with pavements, walls, and street furniture defines the smallest modules in the urban fabric.

Already the first examples point to the delicate and dynamic quality of urban modules. Any one of these modules is defined at a single point in space-time. People will move about, whereas the built elements remain fixed. It takes a combination of the two to define a module on that scale, and the module itself evolves with time. Most important, built elements without people do not define a complete urban module. People-people and people-object interactions provide the primary motivation for mankind to erect buildings and cities, a basic fact that is often forgotten (Jacobs, 1961; Whyte, 1980). Coupling between pedestrians and surfaces occurs via the information contained in the built structures (Salingaros, 1999); here we will analyze the coupling between the built elements themselves.

The nature of strong links

Two architectural or urban elements can link strongly in many different ways. Connections depend upon both shape and position. Coupling also connects two points that are linked by function (Salingaros, 1998). A link is established if each element of a pair somehow reinforces the other visually, geometrically, structurally, functionally, or all of these together. Two elements that are simply juxtaposed, but which do not interact in any way, do not couple. They remain unaffected by each other and fail to weave the urban fabric. Just as common is the juxtaposition of elements that weaken each other. Not only are these unrelated, but often a stronger element renders the weaker element ineffective in its present position.

Figures 1 to 5 illustrate visually what we mean by strong coupling, though the process is not limited to what is shown in these examples. Modules form from elements on the same scale (Rule 1), so those parts of elements that couple together are of comparable size, as shown. Notice how in each case the coupled elements have contrasting, complementary qualities. For simplicity, the solutions diagrammed below refer to a plan in two dimensions; it is straightforward to generalize them to three dimensions.

Figure 1. Geometric coupling through contrast in texture.

Figure 2. Geometric coupling through contrast in color.

Figure 3. Geometric coupling through interpenetration.

Figure 4. Geometric coupling through permeability.

Figure 5. Inductive coupling via a common third element.

A useful analogy is to imagine some sort of "friction" between regions A and B in Figures 1 to 4, arising either from contrasting materials, or from the geometry of the interface. If two regions can "slide" against each other, they are not coupled. An isolated element might have properties that give it internal coherence, yet when juxtaposed with its complement, the pair acquires new properties and added strength through mutual support (Figures 1 and 2). The union of two or more elements has to show completeness; not only is an individual element much weaker alone, but a successful grouping is clearly self-contained (Rule 1). A coupling is strong whenever one element needs its complement for greater coherence. Completeness depends on the strength of the overall boundary. The aim is to unify different elements into a higher-level module that acquires its own properties.

Rule 3 states that the boundary elements in a module connect it to another module. Some elements may literally fit together geometrically like pieces of a jigsaw puzzle (Figures 2 and 3). Contrast can work together with interlocking to bind elements closer together (Figures 1 and 2). In other cases, the interface between two elements may preclude joining, so that some "glue" in the form of an intermediate region may be required, which couples to each element's boundary (Figure 5). Inductive coupling -- occurring with the help of an intermediate element -- explains how large complex modules can be formed from many coupled pairs. If A connects to B, and B connects to C, then A connects to C (Figure 5). Pairwise connections usually act in the presence of structural continuities, so that these together with the local bonds define a larger module.

An example from physics and chemistry illustrates the coupling process, and how it leads to completeness. A salt molecule is composed of two atoms: an acid and a base. Internal atomic bonds, which determine each atom's internal structure, are far stronger than molecular bonds. It is only the outer electron shell that plays a role in the chemical bond that binds the two atoms together. Molecular coupling occurs when the outer electrons of the acid just fill up the holes in the outer shell of the base. In the bound salt molecule, these outer electrons are shared by both atoms, thus providing interpenetration and a common boundary at the same time. We emphasize that the combination possesses new, emergent properties, since the components of common table salt, an essential part of our diet, are sodium and chlorine, which are individually poisonous.

Jane Jacobs pointed out that diversity in urban uses can become a problem only when elements have a disproportionate size (Jacobs, 1961) (p. 234). Especially on the small scale, units that couple must be of similar size (Rule 1), so any large occupant of street frontage will fail to couple with adjoining buildings because of the size imbalance. The same is true for the megatower set in amongst smaller buildings. Size imbalance among urban units can create desolation by preventing small-scale couplings, although exactly the same kind of contrast in uses at small scale becomes an asset because it enables couplings among the adjoining elements.

Mutual reinforcement

We perceive objects interacting via a geometrical field that is distinct from the other known physical forces (Alexander, 2000). This geometrical field is a function of information, and the interactive force depends on how that information intensifies via combination (Salingaros, 1999). Details of the interaction mechanism depend on a spatial field model that will not be discussed here; nevertheless, a reader can verify these effects intuitively once they are identified. Since the interaction depends on information contained in shape, surface texture, pattern, color, and detail, any approach to design that minimizes such information for stylistic reasons also eliminates the building blocks for urban coherence (Salingaros, 1999).

The idea of mutual reinforcement or harmonization describes this effect. Two elements -- for example, a piece of footpath and a wall -- will couple if they reinforce each other. Each of them in isolation is weaker than they are when juxtaposed. By this we mean their function as well as their aesthetic impact, positive visual impression, or degree of perceived emotional comfort in the user. If they make no difference to each other, then the juxtaposed elements are not mutually reinforcing, and there is no connection. In some instances, removing one will seriously diminish the effectiveness of the other. One may then conclude that they were both contributing to create a greater whole, which is destroyed by the removal of one of its components.

Urban couplings begin on the smallest possible scale, and are needed to bind contrasting or complementary components together into one unit. Possible examples of complementary pairs include: footpath with boundary wall; parking place with a piece of pedestrian canopy; wall with tree; bricks with mortar; paving stones of contrasting colors; entry-way with arcade; column with roof; local street with parking spaces; curb with bollards; etc. Whether such couplings work or not depends on a multitude of factors. The test of the degree to which two elements couple relies on judgements made by the human mind, which, after all, is the most sophisticated known computer. The older, humanistic approach to design looked for such harmonies between components, and gave them priority over streamlining.

Fractal interfaces are an inevitable result of coupling forces

Traditional urban geometry is characterized by fractal interfaces (Batty and Longley, 1994; Bovill, 1996; Frankhauser, 1994). The simplest definition of a fractal is a structure that shows complexity at any magnification. Continuous straight-line or plane boundaries and edges dividing one region from another are an exception rather than the rule in living cities. A successful urban interface resembles either a permeable membrane with holes to allow for interchange, or a folded curtain with an edge that looks like a meandering river on a plan. The first type of interface corresponds to a colander or sieve; a surface so stretched that it is full of holes. The second type of interface represents a crinkly, convoluted surface that fills up volume, in contrast to a flat plane which defines a minimal separation (Batty and Longley, 1994; Kaye, 1994).

Colonnades, arcades, rows of houses and shops with gaps for cross-paths all correspond to fractal surfaces akin to porous membrane filters (Figure 4). Such a permeable interface permits free physical movement across itself for some objects (like pedestrians), while keeping a separation between other objects (like vehicles). Urban coherence depends rigorously on the human scale. Perforations or gaps are therefore useful when they occur on the scales 1 m - 3 m corresponding to the size and physical movements of a pedestrian. If gaps in the urban fabric occur only beyond this scale (i.e., without any substructure on the human scale), they erase the fractal coupling.

Other urban interfaces tend to be convoluted instead (Figure 3). An impermeable building edge couples by interweaving with its adjoining space. Convolution or folding provides a greater contact area that encourages human events to take place there. For millennia, daily commerce depended on filtered pedestrian movement in the marketplace, with human contact and interchange occurring in the folds of a building's edge. Fractal interfaces join built structure to open space, and offer the catalyst for the play among natural urban forces and activities. Folding in the urban fabric is a useful coupling on all scales, from the folds of an architectural element at 1 cm, all the way up to the urban folding that creates a semi-enclosed plaza. Nevertheless, the human connection is established by folding on the human scale.

Using the geometrical couplings from the preceding section explains the fractal morphology of connective boundaries as a consequence of system coherence (Rule 3). We do not propose that urban interfaces have to be fractal just because biological interfaces are, even though there is an obvious analogy. Instead, we offer a scientific explanation: fractal interfaces are a direct result of short-range coupling forces that connect two regions. Couplings over the range of human scales will generate a fractal geometry in the urban fabric, as can be deduced from repeating Figures 3 and 4. Since both biological and urban systems obey universal rules of structure based on connectivity acting on different scales, this explains why the same morphology arises in the two separate disciplines.

As proposed in (Salingaros, 1999), the success of urban space depends on visual and auditory connections between a pedestrian and the surrounding built surfaces. The appropriate boundaries for urban space were derived by considering the geometrical optics of information transmission. Interfaces that maximize signals are either perforated, or convoluted, whereas straight edges are poor transmitters (Salingaros, 1999). This is precisely our conclusion in the present paper, which was reached by considering local couplings. The fractal nature of urban interfaces thus follows independently from three entirely different starting points: (1) maximizing geometrical couplings between urban regions on either side of an interface; (2) providing a setting that will catalyze human interactions; (3) the need for a sensory connection to the user.

The most natural urban interface between buildings and street is a relaxed, segmented curve. This geometry is found in traditional villages and towns. There, walls are aligned in such a way so that the ensemble defines an approximately linear ordering of strongly-coupled units. Each individual façade or section of wall is angled and curved on the small scale, not because of carelessness, but because its shape and alignment are used for local couplings. By contrast, the contemporary practice of strict alignment in urban regions along a straight line, or strict alignment in suburban regions along some arbitrary curve, fails to couple elements on the small scale. Both recent cases are mathematically similar, because they eliminate the fractal quality (i.e., variations on the small scale) of traditional interfaces.

Empty regions do not couple

A minimalist design style for buildings prevents geometrical coherence in an extended urban domain, because the smallest scale influences the largest scale (Rule 7). Regions that contain no information cannot couple among themselves (Rules 2 and 3). Flat, smooth, or shiny surfaces lack internal structure or differentiations. Minimal modules are usually simple and perfectly regular; e.g., square or rectangular. Non-coupling transparent and translucent objects from the "machine aesthetic" of the 1920's have no boundary, so their edge is sharp and abrupt. Figure 6 shows the non-coupling of two juxtaposed empty modules. The reader should not be fooled by the optical illusion of coupling in Figure 6, which the eye creates whenever any two visual designs are aligned with translational symmetry (this point is discussed later).

Figure 6. Juxtaposing two empty modules does not couple them.

In cases where empty modules contribute to a larger whole, they are held together by a frame; their boundary plays the connective role (Rules 2 and 3). What we perceive as a built plain unit is in fact the empty region together with its frame. Empty modules can only couple with other elements having internal geometric properties. Coupling is achieved by totally surrounding a void with a structured boundary on the same scale, like putting a substantial frame on a mirror (Figure 7) (Alexander, 2000). Coupling two regions with different textures evolves from Figure 1 to Figure 3 as the texture of one unit diminishes, requiring more of the enclosure mechanism to work; and finally going to total enclosure as the enclosed area becomes empty (Figure 7). Since elements have to be on the same scale to couple strongly (Rule 1), the boundary surrounding a homogenous region should be of comparable size as the region being surrounded (Figure 7).

Figure 7. An empty region is surrounded by a structured border to become a unit.

The coupling shown in Figure 7 works because the internal void contrasts with the complex border, and supports the latter's geometry. The border material could stand alone if configured into a unit without a hole, but a void cannot stand alone as an independent unit. Revising a deep misconception in the twentieth century's architectural and urban design tradition, voids are not units. Using empty modules exclusively -- as in the minimalist design style -- makes it impossible to generate geometrical coherence (Rule 2). If architectural and urban elements cannot couple on the smallest scales, then they can never support the large scale. For this reason, a coherent urban fabric depends just as much on the actual materials, and the shapes of the elementary (smallest) building blocks, as it does on any higher-level connections.

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" (Kauffman, 1995). Such a mixture becomes auto-catalytic. 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 auto-catalytic threshold is probabilistic and sudden (Kauffman, 1995), and proves Rule 2.

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 (Figure 5). 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 (Jacobs, 1961). 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, 1961) (p. 150). 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 (Rule 2). In a separate publication (Salingaros and West, 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.

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 (Courtois, 1985; Simon, 1962; Simon and Ando, 1961). 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 (Rule 8). 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 (Rule 8). A city can be decomposed (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 (Salingaros, 1998); or (C) as external and internal spaces connected by paths and reinforced by buildings (Salingaros, 1999). 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 (Salingaros and West, 1999).

Coupling at a building's edge

A useful alternative decomposition of a city -- and one that illustrates all the points made in this paper -- occurs in terms of basic couplings rather than isolated buildings. We view the geometrical couplings themselves (i.e., the interfaces) as the city's units on a scale of 1m to 10m, while the geometrical objects participating in the couplings are considered secondary. Edges and interfaces are complex, fractal lines that make up a living city: they define spaces and built structures and not the other way around. A city is made up of interactive edges, along which much of the human interaction that makes a city "alive" actually takes place. For example, the space in front and on the sides of a building has to satisfy Rule 1. Do the following couplings work? Pedestrian entrances with street; front door with street or parking; footpath with entrance; footpath with trees or bushes; built elements with existing trees, lawn, or paved plaza; building edge with urban space; building edge with the ground, etc.

Local streets now adjoin but do not connect in any way with house entrances, building fronts, or lawns. Unlike today, footpaths originally connected all the buildings in a neighborhood; the web of pedestrian connections being independent of vehicular traffic (Alexander, Neis et al., 1987; Greenberg, 1995; Salingaros, 1998). In a typical suburban house, the road surface, the sidewalk, the driveway, the front lawn, and the house entrance are all disconnected entities. Proximity does not connect them. Contrast this spatial dissolution to some marvellous road/house couplings from the nineteenth century, when vehicles were horse-drawn. One could pull into an arch, which formed an integral part of the house, and drive through this structure.

Twentieth-century buildings have generally lost their inside/outside connection. Glass walls emphatically do not couple indoors to outdoors; they create informational ambiguity by connecting visually while disconnecting physically and aurally (Salingaros, 1999). Coupling almost always works via an intermediate region: an entrance hall linking the street to the house interior; a roofed corridor as a transition between the inside of a house and a patio or garden; an arcade as a transition between storefronts and a street or plaza; a covered patio as a transition between the inside and the exposed space outside (Figure 5). In contemporary suburbs, people sitting on an open porch are not protected enough from either the road traffic, or from the disturbing feeling of a vast, empty space generated by building set-backs. Without any interface, there is no connection to the open space in front.

That a green area surrounds a building is a very recent notion, and doesn't work because flat lawn provides no boundary (Salingaros, 1999). A lawn helps to isolate the suburban house from its surroundings; it is the opposite of a connective element. The solution offered by the traditional courtyard house makes more sense geometrically. The plainer an element is, the more it needs to be surrounded by a structured boundary (Figure 7). Most successful green areas are surrounded by something: a building, a wall, or a river (Alexander, Ishikawa et al., 1977). Today's flat, uniform suburban lawns couple to nothing. This flawed pattern derives from older palatial estates with vast decorative lawns, which were themselves surrounded by hedges and very high, solid walls (Rule 3). Those walls, although essential for the geometrical coupling, are now outlawed by zoning regulations.

Modular structure of Alexander's Pattern Language

Alexandrine patterns express strong local forces that manifest themselves as either a particular geometry, or as a repeating human action (Salingaros, 2000). By encapsulating the essence of why similar structures arise repeatedly around the world and throughout history, "patterns" represent the most intelligent decomposition of architectural and urban systems that has ever been attempted. Alexander's Pattern Language was misunderstood as being a catalogue of modules, whereas in fact many of the patterns identify interfaces that govern how modules connect to each other (Alexander, Ishikawa et al., 1977). Alexander and his colleagues realized that connective interfaces -- such as boundaries, physical connections, transition regions, and geometrical edges that harbor fundamental human activities -- are essential to creating urban coherence. As in the decomposition of any complex system, architectural and urban interfaces have to be defined with just as much care as the modules themselves.

Alexander looked for patterns of human activity and interaction, and analyzed to what extent the built geometry either encouraged or discouraged them. He thus defined modules of human and social "life" in a way that correlates them with specific geometrical settings. Invariably, these functional modules do not correspond to any self-contained geometrical module, but rather to edges and interfaces in the urban geometry. Here is the alternative decomposition of a living system that follows human activity modules, and which we expect from systems theory. What life a city has occurs as a result of emergence along the interfaces of a decomposition carried out along geometric lines. Emergent properties will not appear directly from the geometrical modules, because those are usually fixed. The exception to this is free, unrestrained building, such as occurs in the favelas of the third world.

In writing the Pattern Language, Alexander wanted above all a method for generating coherence in the built environment. As clearly articulated by Alexander himself, buildings and urban regions designed according to the Pattern Language, although far more accommodating of human movement and interaction than equivalent structures that violate it, have not always added up to a coherent whole (Alexander, 2000). This practical observation is consistent with our interpretation of patterns as modules and interfaces: one can put them together correctly but still not recover (or generate) the emergent properties of a coherent system, such as the essential qualities of great historical buildings or urban regions that have developed over time. Even though a driving criterion for distilling each individual pattern originally was "to what extent does this pattern contribute to generate a unified whole?" achieving system wholeness depends upon the organization of connections outside the Pattern Language.

Anti-patterns in the contemporary city

In a misguided attempt at forced social engineering, traditional architectural and urban pattern languages were abandoned in the early twentieth century. This act was entirely deliberate, and represented ideas for a new type of city. Le Corbusier formulated his urban proposals into the Athens Charter of 1933, which was subsequently used as a blueprint for post-war urban development. There is no doubt that this reversal of traditional city structure, coupled with the elimination of connective boundaries and interfaces, was based on two false premises: (a) It is desirable to concentrate functions into giant packages; (b) The geometry within each package is homogeneous. Nevertheless, a city contains so many complex human functions that it is impossible to isolate them, let alone concentrate them, so that imposing a simplistic geometry on urban form inhibits the human activities that generate living cities.

One approach to combating the increasing congestion of nineteenth-century cities was to just streamline the geometry. That solution, which helps only in the free flow of vehicular traffic, has generated a stylistic rule responsible for disconnecting urban elements. The visual concept of "streamlining" has been adopted as a universal architectural principle, beyond its narrow application to the expressway. As a consequence, a drastic reduction in the number of different urban interfaces, which ought to exist on many different scales, has made it impossible to generate a coherent urban system. Furthermore, rejecting traditional urban patterns means that people no longer connect to buildings and cities, because human behavioral patterns cannot be contained by architectural anti-patterns (Salingaros, 2000).

Ordering the large-scale urban elements

We introduced mechanisms for connecting the urban fabric on the small scale, arguing that intricacy, connectivity, and organic form are essential pre-requisites of living regions. The search towards such connectivity was pioneered by writers such as Camillo Sitte (Collins and Collins, 1986), Gordon Cullen (Cullen, 1961), Jane Jacobs (Jacobs, 1961), and numerous others (Alexander, Neis et al., 1987; Lozano, 1990; Moughtin, 1992; Moughtin, Oc et al., 1995). That science points towards such connectivity lends unexpected support to the more traditional humanistic approaches. A valid contribution to applications of generic principles of form to urban design is to explain why older techniques work, and why newer techniques -- backed by ideological, philosophical, and technological arguments, and supported by a more recent established tradition -- actually destroy cities.

The remainder of this paper looks at what happens on different scales. Cities are rarely designed as a whole. Aside from artificial examples (which have been severely criticized), urban form is in large part the outcome of economic processes in the land market; juxtapositions of structures regulated by institutions of law and government. In this sense, the ensemble is not designed by anyone, nor is it governed by any aesthetic principle. It is easy, therefore, to doubt that a theory of architecture and urban design can, or should, be applied to any scale. We are going to show how different forms can arise out of geometrical principles as they work on different scales (Salingaros and West, 1999). The results will hopefully demonstrate that the urban fabric can indeed be governed by the same rules as sculpture and buildings.

The strength and range of urban forces

An excursion into physics helps to understand the nature of urban couplings on different scales (Rule 4). Every force f arises from differences in some field U , which represents either a geometrical quality, or a function. It is easy to see where U becomes greater through concentration or intensity. The force f is defined as the negative spatial derivative of the potential energy U of its field, f = -dU/dr . (For readers who don't know calculus, the force can be thought of as a ratio: the difference in potential U divided by the distance dr over which the potential difference is measured). This equation gives a stronger force where the difference in potential is larger. A difference in potential translates into the urban context as a difference in qualities within a short distance; implying a stronger coupling force whenever there is greater contrast in qualities such as texture, color, or curvature of the interface (Rules 1 and 2).

We will now use the above formula to explain two pathologies of twentieth-century urbanism: (1) the lack of coherence inside zoned regions; and (2) the dysfunctional edges of vertically-concentrated functions. The potential U is the same throughout a homogeneous region, so without differentiations, there can be no cohesive forces to hold the region together. This implies that internal contrast -- i.e., complexity -- is needed within any urban region. Zoning into segregated functions and an obsession with a minimalist design style thus rule out urban coherence. A second application of this formula shows why we need connective urban interfaces. Vertical concentration of a function U in monofunctional megatowers creates enormous functional stresses at the building's edge. This occurs because of the tremendous jump in U at an abrupt boundary (representing a very small width dr ).

There exist different types of forces that act on different scales. The above equation also gives a general understanding of their relative strengths and ranges. For comparable potentials, every force is inversely proportional to the spatial dimension, which means that a very strong force acts over short distances, whereas a weak force acts over long distances (Rule 4). This result is verified in nature. Both orbiting satellites and human bodies are held to the earth by gravitation, a relatively weak force. Each body is held together by stronger chemical forces, which depend on the stronger electromagnetic interaction. Finally, the strongest known force holds atomic nuclei together, but has no effect outside the immediate vicinity of a nucleus.

Even in an artificial complex system such as the urban fabric, it is impossible to violate the inverse proportionality between force intensity and range. Juxtaposing large contrasting units generates unnatural forces on the large scale, which overwhelm both the short-range coupling forces, and the weaker long-range alignment forces necessary for urban coherence. Le Corbusier attempted to reverse the intensity and range of urban forces (Rule 4). He conjectured -- incorrectly, and without any scientific evidence -- that this radical re-organization would solve the problems facing nineteenth-century cities in the twentieth century. He never realized that such a reversal was physically impossible, and only succeeded in dissolving the structural interactions between urban units.

Entropy and spatial organization

Coupling established on the small scale via local short-range forces does not necessarily lead to coherence on the large scale. The system needs to create its large scale according to certain ordering principles. Long-range alignment forces differ from the short-range coupling forces discussed in the earlier sections. "Large-scale order occurs when every element relates to every other element at a distance in a way that reduces the entropy" (Salingaros, 1995). Entropy is a concept from physics that measures the degree of disorder. A box of wooden matches scattered randomly on the floor gives a pattern with high visual entropy. The entropy is reduced by carefully aligning the matches into a more regular configuration. It does not have to be a rectangular pattern, but could look like a spider web or a whorl. Mathematical symmetries -- in this case translational, rotational, radial, or spiral -- create large-scale ordering, which lowers the visual entropy.

Another example is to rearrange sticks of different lengths and colors that were initially in a random distribution. Of the infinite possible patterns obtainable, the most unimaginative is one that separates the sticks into neat rows having the same color and same length. By concentrating similar elements together (the foundation for post-war zoning) there can be no short-range couplings on the lowest scale, because there is no contrast. Entropy has been lowered, but by eliminating the smaller scales altogether. This violates Rules 6 and 7. Regardless of any deceptively tidy arrangement, such an ensemble can never achieve coherence because it doesn't have enough complexity.

The principle behind urban organization is that alignment forces are long-range, and are weaker than coupling forces, which are short-range (Rule 4). Alignment must therefore respect each individual module, and not change its internal structure by undoing the couplings between elements. To reduce the entropy (disorder) in an urban setting, an optimum number of long-range connections must be established between all the different modules (Rule 5). There exist distinct levels of scale on which this is achieved (Rule 6). Different types of connections are created according to their generative processes, but not through a simplistic visual pattern. Human activities do not depend on visual symmetry in the plan, which is geometrical order that is not directly perceived on the ground. Urban environments that are strongly connected (hence very successful) usually look irregular from the air (Gehl, 1987).

The geometry of built form as revealed by the natural uncontrolled growth of cities is fractal and not random (Makse, Havlin et al., 1995). This makes a world of difference: urban evolution is a connective process on all scales; the opposite of what a random process would be. Ordering imposed by planning tries to counteract this mistaken notion of "random" growth (Batty and Longley, 1994). Most successful urban regions tend to be more "nearly ordered" rather than "nearly disordered" (Hillier, 1999). Approximate linearization is a consequence of human movement, and leads to one form of clear urban ordering. This does not imply perfectly straight or parallel lines, but rather a relaxed linearization of urban form induced by the path structure. Regardless of the approximately linear ordering forces due to the transportation network, a city can never be aligned completely without losing its geometrical coherence.

Reducing entropy does not generate local connections

Rectangular grid alignment is a useful entropy-lowering technique for urban complexity created by uneven topography, as for example in hill-towns such as Priene and San Francisco; i.e., too much variation in the ground level (the z-axis) can be countered by a straight x-y grid. This is confused with, and has replaced older techniques for generating strong couplings on the small scale. We are now obsessed with lining up objects, even though a straight-line interface prevents most geometric couplings. (Using the friction analogy introduced earlier, smooth, perfectly straight interfaces don't couple with each other). Figures 8, 9, and 10 illustrate three distinct cases of ordering. In Figure 8, non-interacting elements are aligned, just as in a contemporary city. The opposite case, where interacting elements show no overall alignment, has a decidedly organic form (Figure 9). Human dynamics linearize a city so that its plan is much more aligned than Figure 9 (Hillier, 1999).

Figure 8. Elements aligned but not coupled.

Figure 9. Elements coupled but not aligned.

Figure 10. Coupled elements aligned.

Figure 10 has both coupling and alignment (with more symmetry than is required for a city plan). It is reminiscent of the designs on oriental carpets and ancient Chinese bronze vessels, where bilateral symmetry is used because those patterns are seen frontally. A city is ordered in an approximately linear fashion by its transportation network, so its plan doesn't need such reflectional symmetries. Nevertheless, many urban planners judge a design by how it represents visual abstraction from the air -- as in the disconnected twentieth-century model shown in Figure 8 -- and don't allow for urban functions on the ground to generate their own coherent form. Large-scale ordering may be imposed, but it has to be done delicately, and with an understanding of the relative strength of all the underlying forces.

Architects today use grid alignment instead of pairwise connections as a general design technique. The premise behind this idea is false: no short-range connectivity comes from aligning edges along a rectangular or any other grid (Rule 5). This basic misunderstanding has become so pervasive, however, as to assume an unshakable authority. People now imagine an absolute three-dimensional grid permeating all of space, to which one aligns urban elements; not only buildings, walls, and paths, but also bricks, windows, doors, steps, ledges, manicured bushes, strips of lawn, and even rectangular planters. Aligned elements are believed to connect to an invisible rigid frame, hence to each other. Since there is no such grid, the imagined connections are non-existent.

Two analogies are the making of a quilt; and playing with LEGO blocks. In the first case, sew patches of material together, paying attention to the local connections but not to any overall pattern, so that the quilt is flat but its seams are not aligned. Contrast this with merely laying the same patches in an exact orthogonal pattern on the floor, but not sewing them together. In the second case, use LEGO blocks to build a connected, three-dimensional toy. Contrast this with laying LEGO blocks down on a table in a perfectly rectangular pattern, but without joining them. In both latter cases, picking up any patch or block will not pick up the rest; they are aligned but are not attached. In the same way, historical cities are complex and connected, whereas contemporary cities are aligned but disconnected.

An apparent order that depends exclusively on grid alignment is deeply misleading, because it gives a false impression of connectivity where none exists. A simple test of connectivity is the following: is the urban fabric stable under deformations of the plan? That is, if we shift elements around just enough to break the rectangular grid, is the city still connected? Most often it isn't; it falls apart after the linear ordering is lost, because its elements were never connected in the first place. Suburbs are disconnected by choice. On the other hand, vertical couplings (i.e., apartments on top of shops, or offices on top of apartments) are stable since they are not affected by horizontal perturbations. Unfortunately, those traditional couplings are shunned by planners obsessed with segregating functions.

How the large scale influences the small scale

A connective field in three dimensions permeates the coherent urban fabric. Its properties are entirely distinct from the imaginary grid alignment discussed previously. Short-range couplings tie units together on the small scale, and these are reinforced by weaker intermediate and longer range connections (Rules 4, 5, and 6). A city's overall ordering and shape is influenced by the requirements of its functions, topography, and circulation. In a coherent system, all components are interconnected, so that each element affects every other element in some way. The elements together generate a morphological field that interacts with every individual element, and this interaction can be either positive or negative.

In a coherent structure, a single element of any module will be influenced by all the local forces generated by the other elements of that module, and indirectly by elements outside the module. Neighboring elements will interact with each other, unless prevented from doing so by deliberate isolation. The positioning and even the shape of any particular element will thus be influenced by all the other elements in the whole (Alexander, 2000). Of course, an urban element could be given almost any shape, and placed in any position whatsoever, but that will generate unresolved forces. When the shaping and positioning of an element are perfect -- that is, the interaction forces exerted by all the surrounding elements are accommodated -- the result is seldom symmetric, nor perfectly straight. This plasticity gives life to traditional cities.

The evolution of a complex system in time was discussed in an earlier section, and the sequence leading to coherence was identified as small to large (Rule 6). Most hierarchical thinking in urbanism today is framed in terms of the opposite sequence: large to small (see (Friedman, 1997) for a summary of hierarchical theories of urban planning). Even under the guidance of an overall organizing principle, assembling the urban fabric in principle proceeds from small to large (Alexander, 2000; Alexander, Neis et al., 1987). A flawed urban plan is immediately obvious by its high visual symmetry, which usually means that small-scale structure has been sacrificed to accommodate the largest elements first. Any strict top-down order is imposed order, which violates Rule 7.

The causality expressed in Rule 7, which states that the large scale depends on the smaller scales, should not be misinterpreted. Once properly established, the large scale is much more difficult to change because it includes so much substructure. All of the included subunits must be moved along with the large-scale module. By contrast, it is relatively easy to change the smaller scales, which do not depend on the larger scales (Habraken, 1998). Rooms can be re-arranged without changing the rest of the house; houses can be moved without changing the road grid; a neighborhood can be entirely rebuilt without affecting the rest of the city. This opposite one-way dependence of change in a complex system is sometimes expressed as: "the large scale dominates the smaller scales" (Habraken, 1998).

Re-connecting the random city

The complexity of a coherent system is proportional to its size. The same node in a small town is less intense than an identical node situated in the middle of a great city, because in both cases, this region derives its energy from the rest of the city (Hillier, 1997). A row of stores on a village main street might connect to 2,000 people, whereas for a similar row of stores in a European capital it could be 2 million people. This effect is of course present only if the urban fabric is connected. In a living city every node connects to every other node, so that a component node is influenced by the size of the entire system, i.e., the city's extent. Paradoxically, our civilization is now trying to connect cities electronically, after having taken them apart geometrically. Many experts predict that electronic connection will solve urban problems, but never address the need for reconnecting the urban fabric.

Important nodes were formerly positioned in the geographical center of cities. Nowadays, connections have been cut, so downtown locations are not necessarily the most connected ones. Even so, many businesses feel that a central location still offers connective advantages. Despite the overall electronic connectivity of modern cities, only certain geographical regions in the world show a high level of creative activity. The reason is that they have the right coherence for fostering commercial performance and creativity. Complexity -- consisting of a mixture of research and educational institutions, together with culture and communications (including a major airport), all connected properly -- provides the right matrix for knowledge-based activities (Garnsworthy and O'Connor, 1997).

In large regions of the world, there is at present no measurable degree of geometrical coherence, except in pockets of older cities preserved for tourism, or neglected by the disconnecting process because they became slums. Eventually, however, urban renewal goes in and destroys even those regions, cutting their geometrical connections with a surgical precision. The disadvantaged resident population may at that point lose any humanity left to them, strictly as a result of the changed urban geometry. Today's disconnected cities fail as an environment for a large portion of the healthy population: children, teenagers, mothers with babies, and older people; as well as handicapped persons of all ages. The solution is to reconnect every piece of a city, on whatever scale, to every other piece.

Integrating commercial elements into suburbia

Rule 1 underscores that stores must be spatially integrated with housing clusters; connected as much as possible to the residential part (Jacobs, 1961; Lozano, 1990). Many residents wish to segregate residential from commercial areas, but don't see how that destroys the coherence of their neighborhood. Unfortunately, this way of thinking predominates, so that new commercial elements remain unintegrated into the suburban fabric, being accessible only by car. One of the greatest obstacles to the integration of commercial space is a homogeneous parking lot that destroys green spaces and paths. The rules outlined here can be applied to create internally-coherent, partially-paved parking spaces coupled with green, which will bear no resemblance to the no-man's-land of vast asphalt surfaces now covering our cities.

When there are sufficient residential units to support neighborhood stores, they will appear as an integral part of the fabric unless forbidden by zoning. Although suburban sprawl creates serious transportation problems because of low density, the false "economy of scale" argument used against small stores clearly doesn't hold . Planners are baffled by the re-appearance of the small grocery in suburbia -- as a corner gas station plus convenience store -- because, according to the twentieth-century design canon, it is not supposed to exist. Modules combining house clusters, paths, roads, and green areas cannot just repeat indefinitely, but must eventually contrast with something else to define a module on a larger scale. The only choice is commercial nodes such as neighborhood stores, day-care centers, etc.

Coupling also helps us to understand paths as interfaces. Different roads (according to their different speeds of flow) are defined by their complement; whether it be green spaces, parking spaces, or stores. Green and commercial strips contain the footpath or sidewalk, and combine to form the broad, tree-lined European boulevard (Greenberg, 1995). Commercial elements arise most naturally (and are most successful) when they are coupled to both pedestrian paths, and local roads. The coupling for a road with very low traffic changes discontinuously as the vehicle flow increases, however. When the traffic exceeds a certain threshold, it becomes necessary to isolate the pedestrians from the road. After a second threshold, a road cannot couple to any active urban element, so an isolating boundary must protect the adjoining regions from vehicles.

Some suggestions for assembling a city hierarchically

To create a new, coherent built environment, a significant portion of existing structures will have to be modified, or replaced entirely. That goes for suburban housing tracts as well as for downtown megatowers, whose shape and use are going to be drastically altered. Even more so than the modification of buildings, the geometry of public space, parking lots, urban plazas, parks, sidewalks, and roads must be re-done. Many authors have suggested rules for assembling a city in a more connected manner, supported by sensible arguments (Alexander, Ishikawa et al., 1977; Alexander, Neis et al., 1987; Gehl, 1987; Greenberg, 1995; Kunstler, 1996; Lozano, 1990). Spurious counter-arguments condemning those ideas as old-fashioned, romantic, not innovative, or not modern enough have unfortunately prevented their adoption until now.

Hierarchical composition of urban elements

A city is assembled from coupled units on different scales. To help in getting a rough idea of the relationship between successive scales, we recall an earlier proposal: "The small scale is connected to the large scale through a linked hierarchy of intermediate scales with scaling factor approximately equal to 2.7" (Salingaros, 1995). The ratio between scales in the urban fabric should correspond very approximately to powers of this number, which equals the logarithmic constant. We do not look for an exact scaling relationship; the point is to allow urban elements to form in a hierarchy of different scales (Rule 6) (Alexander, 2000; Lozano, 1990). Boundaries of boundaries create smaller and smaller scales, and this process continues into architectural forms. All components must be coupled as tightly as possible, with boundaries serving as an intermediate connector between similar-size elements (Rule 3).

Functional and geometrical forces generate an approximate scaling hierarchy if the connective framework is allowed to develop freely as much as possible. Fractal interfaces, connective boundaries, geometrical couplings, and groupings all rely on smaller and smaller pieces that define the joins and intermediate elements (Rules 4 and 5). Geometrical coherence can only be achieved if one applies rules that treat different design scales on an equal footing, making sure not to neglect any particular design scale. The coupling rules work to achieve qualitatively different results on different scales. There exist fundamental mathematical reasons why architecture and urban design, if allowed maximum freedom, will develop by following scale-independent processes (Salingaros and West, 1999).

Our present-day obsession with straightness is really a method for suppressing hierarchical structure. A straight line is bounded by its endpoints, so any perfectly straight design element has a single fixed scale, which corresponds to its length. A complex or relaxed curve, on the other hand, has subscales that are defined by its inflections, and is a far richer object mathematically because of its hierarchical substructure. Straightening out a line removes all of its intermediate scales, and therefore any possibility of geometrical interaction and coupling on those smaller scales. For this reason, traditional and indigenous architecture is only approximately straight; it is extremely well-connected, which is made possible by defining smaller scales by means of small departures from straightness.

Coupling houses through intermediate elements

In modern cities, each house is connected via a road to a place of work, school, and stores; but there is absolutely no geometrical reason for each house to be next to its neighbor (thus violating Rules 1 and 2). Houses can only link to each other indirectly, via complementary elements such as neighborhood stores, surrounding roads, and common front or back yards (Rule 2). The intermediate connective elements exist on a different scale, yet all scales up to the size of the subdivision itself are removed nowadays (Salingaros and West, 1999). Even if a house is coupled to its own yard -- which is certainly not the case for the majority of houses built during the twentieth century -- we are left with house/yard pairs uncoupled from each other. Many people want that, however, having confused the need for privacy with geometrical isolation.

A classic European pairing couples stores to housing vertically in three or four-storey buildings, with apartments on top of each store. The commercial space successfully contrasts and couples with the residential space (Rule 1). The more intense use of the commercial space weights it as several times the residential space, so a mix of 1:2 or 1:3 is appropriate, validating the proposed scaling (Rule 6). American cities employed this example of vertical coupling widely until the 1940's. The point is that the residential units couple indirectly to each other via the commercial units, which doesn't occur in today's strictly residential four-storey apartment buildings (Rule 2). The converse -- zoning all high-rise office towers so that their bottom four storeys are residential -- is the quickest way to bring downtowns to life.

A coupling popular in England is to have a five to ten-house cluster share a giant back yard or private square, so that houses couple via the green (which covers about three times the area of a house). Children then have a car-free minipark to themselves instead of separate, smaller back yards. The United States had the corollary of this solution in the 1920's: a cluster of houses sharing a giant front yard, which was maintained by the City. Groups of houses all facing a minipark existed in some form or other. One version has houses built facing a square or cul-de-sac road. Among the most successful spots for this pattern is around a lake. Today, however, such house/park clusters have been destroyed by cutting off the green by a surrounding road, which undoes the essential coupling.

Footpaths orthogonal to local roads leave spaces between some houses (Salingaros, 1998). These cross-paths establish a multi-house cluster as a suburban unit, by surrounding it. Lots today adjoin each other on three sides; however, older homes have a very useful back alley, and cross-paths exist between every few houses. An urban module consisting of eight five-house clusters, together with a neighborhood park and four commercial or civic elements, became the basis for initial planning and growth in Savannah, Georgia (Bacon, 1974). Each five-house cluster was defined by a surrounding road, and the house clusters and four larger buildings surrounded the park. This module was repeated several times before the need was felt for other types of elements, exactly as required by the system rules.

Lessons from the third world

We can learn a great deal by studying the natural growth of the urban fabric as it occurs in the favelas and squatter settlements of the third world. Unrestricted by official zoning ordinances or a rectangular road grid, the growth is owner-controlled, and tends to follow the structural rules of a developing complex system very closely (Lozano, 1990). Of course, the living conditions are deplorable, with a near total lack of sanitation, water, utilities, etc. Nevertheless, beneath the squalor and misery lies a real-world illustration of urban coherence. Another important point is that the growth of shanty or indigenous towns respects and follows the natural topography as no other urban form does (Ribeiro, 1997). Ideally, we would wish to add some (but not too much) alignment to the favela model.

An interesting development illustrative of natural urban forces has occurred with the influx of squatters into nineteenth-century colonial cities. In parts of Cairo, people have taken over the flat roofs of commercial buildings, so that today, there is a separate two-dimensional squatter city built on top of imposing office buildings. Here is an officially unacknowledged vertical coupling between residential and office space. In the southern states of the USA, homeless persons inhabit the space underneath some elevated highway interchanges: a vertical coupling between residential and transportation space. As the forces behind these phenomena are not understood, they are treated as a nuisance, and remain uncoordinated. The population pressure, however, guarantees their continued existence.

Most texts on urbanism condemn favelas for their "formless sprawl". Their authors little understand complexity in either form or function, and are following the declaration of war on cumulative urban forms and organic, continuous structures, as codified by Le Corbusier in the 1933 Athens Charter. Note the causality of scales expressed in the typical favela: the smaller scales -- such as individual buildings -- often precede the large scale that is defined by a path and road network. This causality is reversed in planning, where the large-scale infrastructure is laid down first, to be followed only much later by houses and other buildings. One sees in hybrid systems of slums, where a government lays down a rectangular grid of wide roads, while leaving the building of houses up to the residents, a notable lack of organic coherence such as is found in totally free systems.

Stability and emergent connections

We are 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 (Gehl, 1987; Whyte, 1980). 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.

Connecting modules on the largest scale

Successful large urban elements possess a rich internal complexity and an enormous number of links to adjoining urban elements (Jacobs, 1961). Whereas contrast is essential on the small scale, it can be destructive on the large scale (Rule 4). As discussed previously, one cannot juxtapose large areas, each concentrating similar functions, along a sharp interface. Substructure has to appear, giving rise to connective boundaries and transition regions (Rules 2 and 3), otherwise one region damages the other. Much of what is built today abruptly juxtaposes two or three homogeneous large-scale forms having different high-density functions: the giant office building next to an expressway, a cluster of stores next to an enormous parking lot, a busy highway next to private houses, a high-rise apartment building next to a vast lawn. These archetypes of contemporary architecture violate Rules 5 and 6.

Suppose that we assemble complementary urban units -- say, shops, offices, apartments, streets, footpaths, sidewalks, and trees -- into a module (Rule 1). If this module forms a working unit, it should be coupled with something else, of roughly the same scale, to form an even larger unit. The possibilities might be a civic or government building, company center, sports complex, large hotel, or a small non-polluting industry. Even then, we should not just repeat this new, larger whole, but instead look to define an even larger complementary module that might contain some of the same ingredients. The point is not to repeat any unit monotonously (Rule 2), but to achieve coupling on all scales. There is nothing wrong with repeating subunits in a larger whole, but the repetition itself does not create the connections: it is the common boundary elements that do (Rule 3).

A green area will work only if it is internally differentiated as well (Jacobs, 1961). Successful parks are never uniform, but assemble paved footpaths, gravel trails, grass, cultivated bushes, trees, and wild growth. Undeveloped forest left in corridors, even narrow ones, helps to achieve the appropriate variety needed for internal coupling. Environmentalists argue that strips of wild green provide a minimal urban habitat for some wildlife (Van der Ryn and Cowan, 1996). A large urban park, however, is safe only when it is heavily visited (Jacobs, 1961; Whyte, 1980). It is necessary to couple it via a connecting border consisting of commercial and residential elements, preferably not cut off by a road. A continuously populated rim guarantees a safer green area during much of the day. The city can connect to larger parks by injecting urban elements and their paths, and by establishing populated fingers cutting towards the center.

Conclusion

Several suggestions were made that, if applied, could dramatically improve the coherence of the urban fabric. The proposals were based on rules for geometrical coherence derived from complex systems theory. These results are valuable because they support urban solutions that instinctively work, while invalidating popular but destructive methods that are in wide use today. Since the 1940's urban planners have followed rules whose effect is to sever short-range connections. A fundamental misunderstanding about urban geometry leads to the segregation of functions, which has now become an obsession. As a result, the modern city is intentionally disconnected: in mathematical terms, it is random. Retail areas have been torn out of residential neighborhoods , leaving suburban tracts that consist entirely of isolated houses and ornamental lawns. At the same time, residential units have been torn out of commercial centers, leaving an empty shell at night. It was thought that alignment and repetition of identical units would connect them, but it doesn't. Implementing the rules given here can solve many urban design problems, or at least lead to a clearer understanding of their causes.

Acknowledgments

The author's research is supported in part by a grant from the Alfred P. Sloan foundation. Christopher Alexander's ideas have influenced this paper to an enormous extent, and I am grateful to Debora Tejada for her suggestions.

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