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.eduSubmitted for publication in the Journal of Urban Design. (Note that the Figures are not yet available for posting).
ABSTRACT The theory of complex interacting systems, which has been developed in computer and biological sciences, is applied here to urban design. This theory can be used to resurrect dead urban and suburban regions, by re-arranging their geometry to generate connections. New projects may even approach the coherence that characterizes the best-loved urban regions built in the past. These ideas are consistent with, and support the New Urbanism, although they come from science and not from any traditional urbanist thinking. The rules are radically different from the ones in use today. Complex large-scale wholes are built from tightly interacting subunits on many different levels of scale. This provides a scientific method for understanding and controlling the complexity of urban form.
Introduction
A theory of architecture and urban design (Salingaros, 1995; Salingaros, 1998a) has been developed that applies to any scale. This research program uses scientific principles to investigate architecture and design, and is based on recent work of Christopher Alexander (Alexander, 1999). From small man-made objects -- such as sculptures, pottery, and textiles -- to buildings, the best examples share the same mathematical structure. Though not usually viewed from this perspective, cities and the urban fabric are also governed by the same rules. The mathematical principles that produce a beautiful small sculpture or textile generate the same positive quality in an urban setting the size of a neighborhood or an entire city.
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 and livable, psychologically positive human environment. This depends in large part on the geometry (not the large shapes, but the connectivity). Our aim is to generate a tightly-knit geometrical field via mathematical rules. We will identify which techniques generate, and which destroy, the urban fabric. Useful older solutions have been dropped for no good reason, yet anachronistic older solutions are maintained out of inertia, even though they no longer apply.
Every urban element is formed by the combination of subelements. Complementary elements of roughly the same size couple strongly to form an element of the next-higher size. 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 groups, 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 to link other elements that could not couple directly by themselves.
Many theories of urbanism have appeared, and examples of their application abound (covering a broad range of successes and failures). Without getting into a debate over all the competing ideas on planning, traffic flow, social organization, etc., we observe that the urban environment in the twentieth century looks and feels inhuman, regardless of the theory behind its creation. There is something terribly wrong with the geometry of the contemporary built environment for it to differ so dramatically from historical urban environments. It is also clear that there must be an underlying commonality among twentieth-century rules for urban design, which is causing the above effect.
The composition of complex interacting systems
In a general complex system, as for example an organism or a large computer program, certain rules are followed so that the parts cooperate and the whole functions well. There is little formal difference between such systems and the urban fabric. 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 has been developed by the author for urban design.
- A collection of strongly-coupled elements forms a group. There should be no elements without any relationship inside a group.
- Similar elements do not couple. A critical diversity of different elements is needed because some will enable couplings between others.
- Different groups couple via their boundary elements. Communication is primarily between the groups themselves, and not between their internal elements.
- Interactions are strongest on the smallest scale, and weakest on the largest scale. System coherence therefore evolves from small to large.
- The interdependence of elements and groups on different scales is not symmetric: a higher scale requires all lower scales in order to function, but not vice versa.
The coherence of a complex, interacting system may be illustrated as it evolves in the time dimension (although we are going to apply these concepts primarily to space dimensions). During a short time period, strong internal couplings will establish internal equilibrium in each primary group, with little change in the relationship among different groups. In a longer time period, however, the couplings between groups 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, groups of groups will tend towards equilibrium, and so on. The end result is a global equilibrium state for the entire system.
The development of a system in time defines an underlying causality. The smaller scales need to be defined before the larger scales: their elements must couple in a stable manner before the higher-order groups can begin to form and interact. Elements on the smallest scale, and their couplings, are thus 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. Connective rules determine whether a system is coherent or not. In addition, these general rules assess the stability or effectiveness of a complex system independently of what that system is supposed to do.
Components of the urban fabric
Roads, paths, parking, green, residential, commercial, and industrial elements must all be accommodated; even though they are contrasting, they have to coexist harmoniously. Elements of the urban fabric cannot be discussed in isolation from each other. Each urban element can increase in intensity, either by lateral, or by vertical growth. Buildings can increase in number of stories; lawn can progress to bushes, then 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, 1998b). When discussing balance, we have to compensate for the height or intensity of what we are balancing. Urban spaces present special difficulties that are discussed elsewhere (Salingaros, 1999).
How do we combine all the different components, and in what proportion and patterns, in order to create a successful urban environment? We re-derive from mathematics what was already largely known and forgotten by our technological society, and is also reflected in the New Urbanism. Rules for generating the urban fabric have been given empirically by Alexander and his associates in "The Pattern Language" (Alexander, Ishikawa et al., 1977) and in "A New Theory of Urban Design" (Alexander, Neis et al., 1987), ultimately deriving from his pioneering paper (Alexander, 1965). Solutions in the same spirit are provided by Greenberg (Greenberg, 1995) and by Kunstler (Kunstler, 1996). None of these prescriptions is widely applied today to build new neighborhoods and rehabilitate existing cities.
Coupling elements on the smallest scales
"Order on the smallest scale is established by paired contrasting elements, existing in a balanced visual tension" (Salingaros, 1995). What are the smallest elements that can be paired in this way? They are everything that is accessible to a pedestrian at arm's length. The smallest urban elements are bricks, paving stones, footpaths, trees, individual parking spaces, walls, doorways, windows, ledges, columns, sidewalks, benches, bollards, etc. All of these must be built so as to couple strongly with a nearby pedestrian. The combination of several pedestrians with pavements, walls, and street furniture defines the smallest groups in the urban fabric.
Already the first examples point to the delicate and dynamic quality of urban groups. Any one of these groups 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 group, and therefore the group itself evolves with time. Most important, built elements without people do not define an urban group. People-people and people-object interactions provide the primary motivation for mankind to erect buildings and cities, a basic fact that is often forgotten. Coupling between pedestrians and surfaces occurs via the information contained in the built structures (Salingaros, 1999).
We introduce the idea of reinforcement in the coupling among elements. Two elements -- for example, a piece of footpath and a wall -- will couple if they reinforce each other. Is each of them in isolation just as strong (or even stronger) than they are when juxtaposed? By this we mean their function as well as their aesthetic effect, positive impression, or degree of perceived emotional comfort in the user. If they make no difference to each other, then the elements are not mutually reinforcing, and there is no connection. In some instances, removing one will seriously diminish the effect of the other. In such cases we conclude that one component was contributing to create a greater whole, which is destroyed by its removal.
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. These couplings begin on the smallest possible scale, and bind two contrasting components together into one unit. We are largely dependent on judgements made by the human mind (which, after all, is the most sophisticated known computer). It would be desirable to have a theory by which one may independently compute the degree to which two elements couple. This is the topic of Alexander's recent work (Alexander, 1999); see also a simplified quantitative model of complexity introduced in (Salingaros, 1997).
The nature of strong links
Two elements can link strongly in many different ways. Connections depend on both shape and position. Coupling also connects two points that are linked by function. 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 build the urban fabric. Just as common is the juxtaposition of elements that weaken each other. Not only are they unrelated, but often a stronger element renders the weaker element ineffective in its present position.
A partial listing of visual examples, Figures 1 to 5, illustrates some ideas of what we mean by strong coupling. Groups are formed from elements on the same scale, so elements that couple together are of comparable size, as shown. Notice also how in each case they 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.
An isolated element has properties that give it some 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, i.e., not only is an individual element much weaker alone, but the grouping is clearly self-contained. One should be convinced that each element needs its complement for greater coherence. The whole point is to unify different elements into a higher-level group that acquires its own internal coherence and properties.
An example from physics illustrates the coupling process. A salt molecule is composed of two atoms: an acid and a base. Each atom has a strong internal structure, and it is only the outer electron shell that plays a role in the chemical bond. Internal atomic bonds are far stronger than molecular bonds. 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. We emphasize that the combination possesses new, emergent properties. (The components of common table salt, an essential part of our diet, are sodium and chlorine, which are individually poisonous).
We have to pay close attention to the binding mechanism between any two elements. Which elements in a group play a role in connecting to another group? 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 groups can be formed from 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 define a larger group.
Architectural units that do not couple
After describing in some detail the coupling process among different elements, it is necessary to point out an entire class of objects that do not couple among themselves. These elements are characterized by a peculiar geometry. They have flat, smooth, sometimes shiny surfaces, without internal structure or differentiations. Their form is usually perfectly regular; e.g., square, rectangular, or circular. Most important, they have no boundary, so that their edge is sharp and abrupt. This non-coupling class of elements includes transparent or translucent objects with the same smooth surfaces and regular outlines. Such objects define the "machine aesthetic" of the 1920's.
As geometric coupling occurs via mechanisms analogous to the visual contrasts shown in Figures 1 to 5, elements without substructure or boundaries cannot group among themselves. Structureless elements represent emptiness, and have to be treated in the same way as a void. Figure 6 shows the non-coupling of two voids that are simply juxtaposed. 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.
Figure 6. Juxtaposing two voids does not couple them.
Using empty units exclusively makes it impossible to generate a connected urban fabric, whose coherence depends on strong couplings on the smallest scales. If all one has to work with are architectural and urban elements that cannot couple, then the foundations for system coherence are absent. Any couplings on the larger scales are compromised, and will always remain weak and ineffective. 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.
Empty elements can be made to 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, 1999; Salingaros, 1998a). Coupling two regions with texture evolves from Figure 1 to Figure 3 as the texture of one of the units diminishes, thus requiring more of the enclosure mechanism to work; and finally going to total enclosure as the enclosed unit becomes homogeneous (Figure 7). The system rule requiring elements on the same scale (size) to couple strongly means that the size of the boundary surrounding a homogenous region has to be comparable to the region being surrounded (Figure 7).
Figure 7. A void is coupled by surrounding it with a structured border.
The strong coupling shown in Figure 7 works because the void contrasts with the complex border, and supports the latter's geometry. The border material could stand alone as a unit without a hole, but a void cannot stand alone as an independent unit.
Element variety is necessary for auto-catalysis
Computer simulations 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. The result of modeling 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. Simpler systems, consisting of a sub-critical variety of elements, have a vanishingly small probability of reacting.
We draw an immediate analogy with the emergence of urban coherence. The formation of a complex interacting whole has as a pre-requisite 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. It is highly unlikely that one can build up a living, coherent city by restricting the element variety and mix in any way. 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; it also disappears when even one of those elements is removed.
This dual, connective role of elements is not sufficiently recognized in urban design. After many years of rigidly stereotyping urban elements according to a single function, it is difficult to imagine all their other, secondary functions, nor their fundamental role in connecting the urban fabric. 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 connective elements in different situations. The mechanism of catalysis, so fundamental in complex systems, runs counter to what has been taught for decades in architecture schools. And yet, it exists and works in creating living cities the world over.
Entropy and urban organization
"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 on the floor gives a pattern with high visual entropy. One reduces the entropy by carefully aligning them into a more regular pattern. 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, 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, 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. 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 a simple one: alignment forces are long-range, and are therefore weaker than coupling forces, which are short-range. The process of lowering the entropy has meaning only as it relates to human activities. In any living environment, these are not going to be defined by geometrically identical units. Furthermore, the alignment process will not necessarily lead to reflectional or translational symmetry in the plan, which is geometrical order that is not directly perceived on the ground. A plan is seen only by airplane passengers flying over the city. Urban environments that are strongly connected (hence very successful) usually look irregular from the air (Gehl, 1987).
To reduce the entropy (disorder) in an urban setting, an optimum number of connections must be established between all the different groups. There exist distinct levels of scale on which this is achieved. The more long-range connections there are, the more legible the urban fabric is. Different types of connections can be created according to their generative processes, but not through a simplistic visual pattern. Fitting any urban element into an artificial grid could damage it. Alignment must respect each individual group, and not change its internal structure by undoing the couplings between its elements.
The strength and range of urban couplings
An excursion into physics helps to understand the nature of urban couplings on different scales. A physical force f is defined as the negative spatial derivative of the potential energy U of the field, f = -dU/dr. (For readers who don't know calculus, this can be thought of as just the ratio of differences). This equation implies that a force is stronger when the local 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 local texture, color, or curvature of the interface.
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. This result is verified in nature. 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 electromagnetic interaction. Finally, the strongest known force holds atomic nuclei together, but has no effect outside the immediate vicinity of a nucleus.
In an artificial complex system such as the urban fabric, it is possible to violate the fundamental relation 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. 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 interactions between urban units.
Reducing entropy does not generate local connections
Rectangular grid alignment -- a useful entropy-lowering technique -- has been confused with, and has replaced older techniques for generating strong couplings on the small scale. A straight-line interface, however, prevents most of the geometric couplings described earlier. 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). A reader might be reminded of the Nolli plan of Rome; nevertheless, Baroque Rome was more aligned than Figure 9 because complex human dynamics inevitably generate large-scale ordering in a city.
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 (more 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 works in hundreds of different ways on the ground, but not visually from the air, so its plan doesn't need such exact reflectional symmetries. This is a point of great contention among urban planners. The urban fabric has to avoid the disconnected twentieth-century model shown in Figure 8, and to allow urban functions 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 the underlying forces.
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. Unfortunately, most hierarchical thinking in urbanism today is erroneously 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, putting together the urban fabric must necessarily proceed from small to large (Alexander, 1999; Alexander, Neis et al., 1987). A flawed urban plan is immediately obvious by its high visual symmetry, which usually means that all small-scale structure has been sacrificed to accommodate the largest elements first. Any strict top-down order is imposed order, which is geometrically inflexible.
The fallacy behind grid alignment
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. This basic misunderstanding has become so pervasive, however, as to assume an unshakable authority. People now imagine a three-dimensional absolute grid that permeates all of space, to which one aligns urban elements; not only walls and paths, but also bricks, windows, doors, steps, ledges, manicured bushes, strips of lawn, and 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, consider sewing 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 connected, whereas contemporary cities are disconnected.
The disconnected random city
The preceding arguments lead to an inescapable but frightening conclusion: that the contemporary city is simply a collection of disconnected parts on all scales. Despite a superficial orderly appearance, it is mathematically random. For several decades, every act of construction and urban intervention has as its intended or unintended result the dissolution of the urban fabric. Paradoxically, our civilization is now trying to connect cities electronically, after having taken them apart geometrically. There is a dark irony in this. Many experts predict that electronic connection will solve urban problems, but never address the tearing of the urban fabric that has led to the present state of alienation.
We now have cities (including suburbs) that are as deliberately random as our technological knowledge can make them. Every piece, on whatever scale, is disconnected from every other piece. There is no measurable degree of 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 population at that point loses any humanity left to them. 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.
Recognizing the coherent city
Every point of a coherent urban fabric connects to every other point. How does one recognize this attribute? We are able to see only a small cluster of buildings and spaces at any one time, and should be able to determine if these connect to each other or not. A coherent city is much more than that: a local node connects to every point throughout the city's extent. The same node in a small town will feel and act differently from an identical node situated in the middle of a great city. There is a difference between a row of stores on a village main street, and a similar row of stores in a European capital, because in both cases, this region derives its energy from the rest of the city (Hillier, 1997). In the first case, the energy might come from 2,000 people, whereas in the latter case, it could be 2 million people.
These considerations explain the formerly crucial positioning of important nodes in the geographical center of cities. Nowadays, however, the 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).
Some suggestions for connecting a city sequentially
A general rule for creating urban coherence follows loosely from hierarchical scaling. "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; Salingaros, 1998a). The ratio between scales in the urban fabric should correspond very approximately to powers of this number, which equals the logarithmic constant. Many ideas discussed below have already been suggested by other authors, and supported by convincing arguments. Nevertheless, there are always counter-arguments that prevent their adoption. Up until now, it was sufficient to condemn a particular design solution by labelling it as old-fashioned, romantic, not innovative, or not modern. Our proposals are based on the mathematical structure of complex forms, which cannot be dismissed so easily. One may argue against our results only on scientific, and not on stylistic grounds.
Coupling at a building's edge
In designing the space in front of a building, we have to ask the following questions: Do the pedestrian entrances connect to the street? Does anyone go from the front door to the street, or is the actual entry elsewhere? Is the footpath leading to the entrance aligned arbitrarily with the road grid, or is it equally arbitrarily given an "artistic" curve? What unit couples with this path to enhance it? Are trees and bushes used to couple with and reinforce other elements? Conversely, do the built elements respect and support existing trees on the lot? If there is a lawn or paved plaza, do these relatively empty elements couple to anything? Are the components of a buildings's edge used for coupling to each other, to a pedestrian, and to the ground?
We now intentionally de-couple cars from pedestrians. Local streets adjoin but do not connect in any way with house fronts and lawns. 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. There are some marvellous road/house couplings from the nineteenth century, when vehicles were horse-drawn. One could pull into an arch and drive through this structure, which formed an integral part of the house. Unlike today, footpaths really connected all the houses in a neighborhood, the web of pedestrian connections being independent of vehicular traffic (Salingaros, 1998b).
Front porches are hardly ever used because there is no contrast or coupling to the front lawn. 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. 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. Such connections are found in neighborhoods built before the second world war. On a low-traffic local road or cul-de-sac, the road surface could be amalgamated into the surrounding front lawns and trees to define a green/road playing area. To slow down the traffic, stones, bricks, and gravel can be used for paving instead of asphalt (Alexander, Ishikawa et al., 1977; Gehl, 1987), a solution applied very effectively in Holland.
The resurrection of arcades
Connecting internal to external space is one of the crucial tasks of urban design. Coupling most 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. All of these inductive couplings, however, are now suppressed for reasons of style, with the dubious goal of "streamlining" forms. Twentieth-century buildings have generally lost the inside/outside connection. Glass walls emphatically do not couple indoors to outdoors.
Arcades work via a combination of inductive and geometrical couplings to tie exterior with interior space as seamlessly as possible. Similar solutions have developed independently in the traditional architectures of cultures around the world. 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 indoor/outdoor space is defined under different conditions and occasions. Without these, the transition is too abrupt, and the connection is lost. There exists an infinite variety of intermediate urban spaces between totally open and totally enclosed that one may utilize.
Uncoupled houses and yards
How do houses couple to each other? House-house couplings are absent from modern suburbia. Each house is connected, via a road, to a place of work, school, and stores; but there is absolutely no reason for each house to be geometrically next to its neighbor. The absence of inter-house links means that, mathematically, modern suburbia is random, despite the misleading visual appearance of nearly identical houses lined up in neat rows. This is a geometrically incoherent situation, which cannot form a stable system. Houses can only link to each other indirectly, and then only via complementary elements such as neighborhood stores, surrounding roads, and common front or back yards.
The fenced suburban back yard suffers from a fundamental lack of coupling and as a consequence, it is unused much of the time. New houses orient themselves towards the street. The back yard does not connect directly to the street, having only a connection through the house. Children prefer to play in the front yard, which can be inadequate and too close to car traffic. One solution is for the house to geometrically enclose the back yard, as in a traditional courtyard. Another is to bring the back yard to the side, with a real door directly to the street, at the same level as the front door to the house. This option couples the back yard with the street more than with the house. Having a prominent, direct connection to the street gives the back yard an identity.
That a green area surrounds a building is a very recent notion, and doesn't work because flat lawn provides no boundary (Salingaros, 1999). The solution offered by the traditional courtyard house makes more sense mathematically. The plainer an element is, the more it needs to be surrounded by a structured boundary (Figure 7). The most successful green areas are surrounded by something: a building, a wall, or a river. 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. Those walls, although essential for the geometrical coupling, are now missing, having been outlawed by zoning regulations.
Coupling houses through intermediate elements
A classic European pairing couples stores to housing in three or four-storey buildings, with apartments on top of each store. The commercial space successfully contrasts and couples with the residential space. 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. American cities employed this example of vertical coupling widely in the last century 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 apartment buildings.
A coupling popular in England is to have a five to ten-house cluster share a giant back yard, so that houses couple via the green area. Children have a 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. A group 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. Nowadays, however, such house/park clusters have been destroyed by cutting the green off by a surrounding road, which undoes the coupling.
Clusters of houses
According to the scaling rule stated above, a coherent group of houses should theoretically be more than 3 lots in size. We already mentioned how to couple several houses via a common back or front yard. A less effective way to define a multi-house cluster is via paths. Footpaths orthogonal to the local roads can leave spaces between some houses, and these cross-paths establish multi-house clusters as a suburban unit, by surrounding them. Lots today adjoin each other on three sides; however, older homes have a very useful back alley, and cross-paths between every few houses. A neighborhood can be built up from such multi-house clusters.
An urban group 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 is defined by a surrounding road, and the house clusters and four larger buildings surround the park. In the development of Savannah, this group was repeated several times, before the need was felt for other types of elements, exactly as required by the system rules. Unfortunately, this important lesson is misinterpreted, with the focus directed on the early city's rigid rectangular grid. One reads of this example's modularity, with no mention of its connections and couplings.
Lessons from the third world
We can learn a lot 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, the growth is owner-controlled, and tends to follow the mathematical laws of a developing complex system very closely. Of course, the actual 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 almost perfect 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.
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 Athens Charter. This attitude unfortunately turns architects against science, by dismissing natural urbanism and trying to substitute something artificial in its place. It also ignores all the tremendous progress made during the last few decades in understanding the rules of organic growth, which apply to the urban fabric. On the one hand, humanity advances through a deepening scientific understanding of biological systems; on the other, we continue to shape our cities according to simplistic ideas that have been scientifically discredited.
The necessity for commercial elements
We cannot just repeat groups that combine houses, paths, roads, and green areas indefinitely, but must contrast them with something else so as to define groups on a larger scale. The only choice is commercial space; the mathematics makes this a necessity. The next higher unit in scale could be a group of house clusters tied to neighborhood stores, some parking, day-care centers, etc. When there are sufficient residential units to support neighborhood stores, we must allow them to appear as an integral part of the fabric. 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.
Our rules underscore that stores must be spatially integrated with housing clusters; connected as much as possible to the residential part. Even though many residents wish to segregate residential from commercial areas, that destroys the coherence of their neighborhood. Unfortunately, this way of thinking predominates, so that new commercial elements remain unintegrated into the urban fabric, being accessible only by car. One of the greatest obstacles to the integration of commercial space is the homogeneous modern 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.
Different roads (according to their different speeds of flow) should be 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 it must be isolated by boundaries to protect the adjoining regions.
Connecting groups on the largest scale
The explicit stylistic dictate for homogeneous large-scale forms tears the urban fabric. Successful large urban elements have a rich internal complexity, and an enormous number of links to adjoining urban elements. Whereas contrast is essential on the small scale, it can be destructive on the large scale. One cannot juxtapose large areas with contrasting functions; there must be substructure giving rise to connective boundaries, and transition regions. Much of what is built today juxtaposes two or three different high-density functions: the giant office building next to an expressway, stores next to an enormous parking lot, a busy highway next to private houses, a high-rise apartment house next to a vast lawn. Concentrated similar units with an abrupt interface do not couple; instead, one damages the other.
Suppose that we connect contrasting urban units -- say, shops, offices, apartments, streets, footpaths, sidewalks, and trees -- into a group. If this group 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 industry. Even then, we should not just repeat this new, larger whole, but instead look to define an even larger complementary group that might contain some of the same ingredients. The point is not to repeat any unit monotonously, 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.
A green area will work only if it is internally differentiated as well. Successful parks are not uniform, but strongly couple paved footpaths, gravel trails, grass, cultivated bushes, trees, and wild growth. Undeveloped forest left in corridors, however small, 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. It is necessary to couple it via a connecting border consisting of commercial or 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 their center.
Conclusion
Several suggestions were made that could dramatically improve the coherence of the urban fabric. The proposals were based on a set of mathematical rules. These results are valuable because they support urban solutions that instinctively work, and at the same time they invalidate 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 is the basis for an obsession with the segregation of functions. Retail areas have been torn out of residential neighborhoods , leaving suburban tracts that consist entirely of 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. As a result, the modern city is internally disconnected: in mathematical terms, it is random. 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 into the scientific laws of architecture is supported in part by a grant from the Alfred P. Sloan foundation. I am grateful to Christopher Alexander, whose ideas have influenced this paper to an enormous extent.
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