"Compact City Replaces Sprawl", por Nikos A. Salingaros

The compact, geometrically integrated city can and should replace suburban sprawl as the dominant development pattern in the future. This approach to urban planning and design is well established among proponents of the New Urbanist and Smart Growth movements. However, the more radical scenario I propose in this paper is that the compact city should also replace the high-rise, ultra-high-density megacity model. I will present arguments for the compact city from both directions, criticizing both conventional suburbia and the hyper-intensity of the urban core. A radical intervention is required on the part of concerned urbanists. We need to rethink the positioning of individual buildings to form a coherent urban fabric, as well as the role of thoroughfares, parking, and urban spaces. New zoning codes based on the rural-to-urban Transect and the form of the built environment are now available to assure predictable densities and mixed use for the compact city.

1. INTRODUCTION.

Sprawl is a remorseless phenomenon. We see it covering more and more of the earth's surface, whether it is in the form of favelas invading the countryside in the developing world, or as monotonous subdivisions in the United States. Nevertheless, the city of tomorrow (actually, in many parts of the world, the city of today) has a low-rise, compact human scale. If the government does not forbid it (or cannot control it), favelas eventually condense to define compact urban regions, but the same organizing process cannot occur in subdivisions because of anti-urban zoning. A favela can become living urban fabric, whereas its high-priced US analog remains dead. The difference is in the connectivity.

Suburban sprawl has become a self-generating, self-fulfilling machine that produces an enormous amount of mechanical movement, but is not conducive to natural human actions and needs. Sprawl persists because vehicles define a now-familiar self-perpetuating entity: the auto-dependent landscape. Cars enable sprawl, and sprawl needs cars. This suburban machine now circumvents its human creators and feeds in directly to the globalized economy. Yet it wastes untold amounts of time and resources, while trapping those without cars in their homes.

High-rise apartment and office towers are equally unsustainable. The serious threat of high energy costs makes both ultra-high-density environments based on skyscrapers, and low-density suburban sprawl no longer feasible. Ultra-high-density urbanism creates more problems than it solves, in the form of energy reliance that draws on the resources of an enormous surrounding region and shortsightedly depends on an uninterrupted supply of cheap oil. Our only alternative is the smaller-scale, compact city, ideally surrounded by and close to agricultural lands for local food supply. We should produce viable settlements at optimal densities for the human scale, just as body tissue has a compact structure at an optimal density. This can be achieved through thoughtful planning and the appropriate codes.

Urbanism once meant dense city living for humans, but anti-urban forces have (literally) driven people out to the opposite condition: low-density suburban sprawl. The correct solution is not formless sprawl, however, but an intermediate density low-rise compact city that is geometrically integrated. The huge commercial success of postwar suburban growth (a low-density phenomenon) took place because it harnessed genuine and powerful socio-economic forces. It also generated and fed some of those forces by means of clever media manipulation and advertising. Those same forces can be channeled to build a better environment for human beings ”the compact, geometrically integrated city” so as to make an urban environment adaptable as much as possible. Suggestions for achieving that on a theoretical level are offered in (Salingaros, 2005a).

There is nothing wrong with either high density or low density per se, as long as it is well integrated with other densities and is in the right place (not too much of the same thing). People in the past several decades seem to have bought into the false notion of geometrical uniformity, which goes back to the now discredited 1933 Charter of Athens (Salingaros, 2005b). That document introduced notions that turned out to be catastrophic for cities, such as separating functions into single-use zoning, the false economy of scale, and also seductive but toxic images of ultra-high-density skyscrapers, vast open plazas, and uniform housing developments. It gave planners the idea of disintegrating the city into non-interacting components, or at best, ones that interact with each other only at tremendous cost and inconvenience; the opposite of a geometrically integrated city.

2. DUANY AND THE TRANSECT.

Even the best theoretical urbanism is close to useless without changes in our zoning codes, however. The existing codes, more than anything, determine the pattern of urbanism. The planner-architect Andreas Duany and his partner Elizabeth Plater-Zyberk are at the forefront of efforts to reform these codes. They coded and designed the highly successful New Urbanist community of Seaside, Florida in the mid 1980s. The momentum from Seaside propelled traditional town planning again into the mainstream of planning options (Duany & Plater-Zyberk, 2005; Duany et. al., 2000). Duany and his colleagues have built numerous New Urbanist projects around the world, and in each case they work closely with the local government to adopt codes based on urban form instead of the separation of uses. Without a form-based code, one cannot predictably plan a human-scale community. Duany will not work for a community that wants to rebuild itself, but which stubbornly retains its postwar anti-urban codes. He has found out from experience that it leads to time-consuming and irresolvable conflicts.

Using a very pragmatic approach to urban form, Duany classifies different zones according to a Transect (i.e. a cross-section of a continuum) of the built environment, according to intensity and density of urban components. He then proposes that communities ensure their desired urban character by adopting written codes that prescribe it. In Transect planning there are six zones, but the three zones T3 (Sub-Urban), T4 (Urban General), and T5 (Urban Center) (Duany & Plater-Zyberk, 2005) contain the areas that we would identify with a compact, walkable, mixed-use village or city neighborhood. Unfortunately, the single-use zoning of the past sixty years has made such compact patterns illegal. (Note that, as explained below, Sub-Urban is not the same thing as suburban).

I propose that a compact T3/T4/T5 city or town begin to substitute for suburban sprawl everywhere around the world. The compact city is sustainable, whereas both sprawl and the high-rise megacity are not. The Transect codes are ready for immediate use, and should therefore be adopted by government agencies. The β€œlow-density city” we now see erasing farmland is not a city: it feeds off and depletes a vast region that it keeps at a distance, so the functioning city is much larger, has a higher net density than first appears, and is ultimately unsustainable.


3. THE THREE URBAN TRANSECT ZONES OF THE COMPACT CITY.

Transect Zone T3 allows single houses on large lots, with a looser road network than in the higher zones. A Transect-based code limits the density to maintain a relatively rural character. Still, there would be walkable street connectivity to the denser Zones, so that residents are not isolated and forced to use cars for all their daily needs. Thus, T3 is part of the compact city, not estranged from it. (Country houses, on the other hand, would be part of T2, the Rural Zone, which is by definition outside the city). The T3 Zone may be the same density as the dreary suburban tract houses we see in sprawl ”technically referred to as Conventional Suburban Development (CSD)” but other key design elements in the new codes ensure much more housing diversity, walkability, and connectivity.

Transect Zone T4 is the denser Urban General Zone, with houses closer to each other and to the sidewalk. More mixed use is permitted, with corner stores and restaurants within walking distance of most houses. As soon as the density permits, therefore, the mixing of functions is actively encouraged by the Transect-based codes.

Finally, Transect Zone T5 is the Urban Center, thoroughly mixing commercial uses with housing. This is analogous to the neighborhood center or small-town Main Street in early twentieth-century America, as well as the traditional European village. Transect-based zoning supports the compact city from both of the critical standpoints identified earlier, for it also prevents the erection of high-rise buildings and vast parking lots, whose expanse and density destroy the desired human-scaled character of T5. (The height limit in the Duany Plater-Zyberk Transect-based Smart Code is three storeys for T3, four for T4, and six for T5). Other important details, such as sharp curb radii and narrow streets, help to calm traffic.

The urban geometry in these Transect Zones is entirely different from that of sprawl (Conventional Suburban Development): roads and buildings correspond more to the compact small town found at the turn of the last century. Suburban sprawl, on the other hand, is neither a low-density CITY nor true country living; in pretending to be both, it accomplishes neither. The correct Transect codes ensure that the complex urban morphology necessary to support the city for people will not disintegrate into disconnected sprawl.

One crucial point of the Transect is that the three zones T3, T4, and T5 connect to and adjoin each other. Each one is kept by its own code from changing wildly, yet each one needs the other two next to it. Suburbia without an urban center requires constant driving, while a downtown without a healthy mix of uses is dead after business hours (Salingaros, 2005b). The codes prevent the repetition of one single zone over a wide area, thus preventing the monoculture of sprawl.

Theoretical work (Salingaros, 2005a) based upon earlier work by Christopher Alexander (Alexander et. al., 1977) supports Duany and Plater-Zyberk´s practical prescriptions with fundamental arguments about urban form and structure. New Urbanist solutions also draw upon the neo-traditional notions of Leon Krier (Krier, 1998). The same approaches will, of course, also work for the Urban Core (T6), as well as for Natural and Rural Zones (T1 and T2), and the appropriate Transect-based codes apply to those densities as well. Nevertheless, here my topic is the compact city, a human-scaled city to replace both sprawl and the high-rise megacity. The compact city, therefore, involves only the medium-density zones of T3, T4, and T5.


4. SPRAWL IS DRIVEN BY THE CAR.

Sprawl exists only because it is an outgrowth of car activities. In turn, this automobile dependence generates urban geometries that accommodate cars first and pedestrians second. These are the wrong priorities for a healthy life, especially for those who cannot drive: the young, the old, and the poor. The sustainable compact city must be designed for the pedestrian first.

People have been encouraged by the automobile industry and by government agencies promoting the automobile industry to indulge in an impossible and destructive fantasy of inappropriate urban types. In practical terms, sprawl comes about from misunderstanding urban morphology. The needs of the car automatically generate an urban morphology appropriate to the car. Sprawl relies totally on the automobile, and thus follows the dendritic (treelike) geometry of roads. A dendritic geometry is good for the automobile, but is inappropriate for human beings. Sprawl occurs when buildings are erected with no regard or understanding of which connective geometries encourage walking. Suburban sprawl grows uncontrollably, generated by anti-urban zoning codes that achieve the opposite geometry to what human beings need.

Complex urban fabric means condensation, connectivity, and mixing; the opposite of homogeneity (Salingaros, 2005a). And yet, most postwar planning has deliberately spread a homogeneous, amorphous structure over the earth, replacing healthy urban fabric in existing compact cities. Monoculture displaces and stretches its vital connections to complementary nodes, making the functioning city (a much larger entity that encompasses the entire commuting distance) tremendously wasteful of both time and energy.

With the wrong codes in place almost everywhere today, roads in fact determine the geometry of urban settlements. Let’s examine what happens when the government builds a road to connect two towns. A road in the countryside attracts new buildings along its length, thus linking each building with that particular road and with nothing else. But human beings do not link to a road: they link to work, school, church, medical facilities, etc. Clustering is supposed to occur among linked human activities, and not strictly between houses and a road. It’s the wrong linking, and it destroys the meaning of a city.

The solution is obvious to some of us. Zoning codes should prevent the dendritic growth of buildings along roads, and instead promote an urban geometry that concentrates human connections inward to focus on local urban nodes. Transect-based zoning has the correct zoning codes that do this, replacing anti-urban zoning codes that allow the unrestrained growth of the auto-dependent landscape.


5. LAWS, REGULATIONS, AND THE DEMOCRATIC IDEAL.

I have proposed Transect-based zoning to regulate the development of urban areas of different density. It may appear to a reader that this represents a rather strict set of regulations. The notion of regulations runs counter to our utopian conception of civic freedom, and may cause strong protests if not revolution. In the case of Transect zoning, however, I am simply advocating a REPLACEMENT of very rigid zoning codes that already exist, governing the geometry of buildings and roads. Most people are woefully unaware of how tightly the built environment is now controlled by existing codes on planner's books. They have been sold the false image of suburban freedom. In fact, Transect-based zoning provides MORE choices for development than does current single-use zoning.

Another misconception about Transect zoning and the New Urbanism is that it places severe restrictions on cars. It merely changes the geometry of how they move and where they park. True, in the compact city, the movement of cars is calmed, and parking is no longer dominant and obvious in front of buildings. But cars are not banned, and parking is adequate.

Still, for a variety of reasons, including energy costs and population growth, car use must be curtailed over time. Unfortunately, the immensely powerful car industry has successfully coupled the idea of personal β€œfreedom” with a car purchase, and it has been almost impossible to convince people to reduce car use. They don’t see that giving unlimited β€œfreedom” to the car has to be paid for by the destruction of a city, and of their own human environment. One’s car today represents something almost inviolate β€” a right of ownership and object of fetish all at the same time. It is going to be very difficult to educate people on this point.

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6. THE AUTO-DEPENDENT LANDSCAPE SELF-GENERATES.

The auto-dependent landscape consists of the road surface, parking, and all areas devoted to the care and feeding of vehicles, such as gasoline stations, garages, muffler shops, tire stores, hubcap stores, car dealerships, parts stores, car washes, automotive junkyards, etc. Shopping areas and restaurants take the form of drive-ins or malls set back in a sea of parking. In this way sprawl is a self-generating system with mechanisms for spreading and enlarging itself. In the auto-dependent landscape β€” occupying more than half the urban surface in many regions β€” vehicles no longer serve simply as a means of human transportation, but as ends in themselves.

Since the auto-dependent landscape feeds on and generates much of the world’s economy, it is not feasible to simply eliminate it. Many countries’ industries and economic base depend on producing cars and parts, or petroleum and petroleum products. Global wars are fought over the petroleum supply. At the same time, the auto-dependent landscape is changing the earth and human civilization, so it has to be contained. What is good for General Motors is no longer good for America, to turn around an old American slogan. Car-related activities within a city are still essential for our economies, but they must be kept on the proper geographic scale. The great planning fallacy in our times is trying to mix up (instead of carefully interface) the auto-dependent landscape with the city for people: all that happens is that the former takes over the latter.

Most important, vehicular speed must be calmed. The highways of the auto-dependent landscape are designed to maximize a smooth and fast flow of traffic, without any consideration of human beings outside a car. Those same principles of speed maximization at the expense of pedestrian physical and psychological well-being have been automatically applied to all roads inside the urban fabric, making it anti-urban in the process. My book β€œPrinciples of Urban Structure” (Salingaros, 2005a) offers rules that reestablish the city for people by giving pedestrians priority over cars. Those rules rely on earlier work by Christopher Alexander, published as β€œA Pattern Language” more than twenty-five years ago (Alexander et. al., 1977).

Despite numerous, well-documented presentations of energy/oil depletion issues, people remain blissfully unconcerned about their car-dependent lifestyle. They trust the transnational oil companies to continue providing them with affordable gasoline until the end of time. Gasoline will certainly be available β€” at market price, whatever that may be in the future. I do not add my voice to the doomsayers predicting the end of petroleum, but unsustainable urban and suburban morphologies will simply become too expensive to survive. The compact, small-scale city is sustainable, whereas ultra-high-density skyscrapers and suburban sprawl are not.

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7. SPRAWL IS ALSO DRIVEN BY COMMERCIAL FORCES.

The dream of owning an isolated country villa surrounded by forest draws people out to suburbia, and cheap land draws developers there. At the same time, lower rents and taxes draw business there, following residential growth. But because the form of suburbia is already established by single-use zoning, businesses must locate away from residential areas, and they must locate where there is enough drive-by traffic to sustain them. Since developers and builders have made fortunes out of selling this defective geometry, they simply keep building what they have done for decades. Government perpetuates sprawl by building roads and infrastructure in an anti-urban pattern.

Because business in sprawl depends on attracting the drive-by customer, then, it must announce to all drivers that there is ample free parking everywhere. Thus we have the shopping mall surrounded by a vast parking lot; the office tower in the middle of farmland surrounded by its parking lot; the university campus in the middle of nowhere surrounded by its parking lots, and so on. Urban morphology is determined in most places by highways and parking lots. Again, the priorities are exactly backwards. Thoroughfares and parking lots should conform to a compact urban structure, not the other way around.

The geometry of commercial nodes is generally oriented outwards toward high-speed arterials to attract drivers. Current zoning makes sure that it cannot be oriented toward residential neighborhoods. That must change with new Transect-based zoning. When a community adopts such a zoning code, there will be assigned Transect zones as described above and structured so that stores, schools, churches, and parks are within walking distance of homes. Density increases as T-Zones get higher, but never to the extent of the high-rise megacity that depends precariously upon a vast energy grid. In a Transect-based code, mixed use is allowed in all T-Zones, and the design of streets favors the pedestrian. The first priority is to get rid of the parking lot in front of a store, narrow the streets, and provide a wide sidewalk (Sucher, 2003). On-street parking is fine; as is parking behind, below, and above the store (Sucher, 2003). Parking garages must have liner stores with windows, so that the pedestrian does not walk past blank walls or rows of cars. People are more likely to walk if there are pleasant things to look at on the way.

Sustainable compact cities in place all around the world are now being destroyed by the introduction of anti-urban components. Not only are skyscrapers proliferating as symbols of modernity, but so are more modest typologies that profit one person while slowly degrading the entire city. In Latin America and Europe, for example, a new corner store typology copied from the United States erases the sidewalk and gives it over to parking. If this goes on (along with adopting other similar typologies from the auto-dependent landscape), that will unbalance societies that have depended on a human-scale urban morphology for so long.

Transect-based zoning codes limit the number of storeys in the compact city to three in zone T3, four in zone T4, and six in zone T5. This places a ceiling that protects the urban fabric from the negative consequences of high-rise construction. These problems include: the office tower (which generates traffic congestion for the entire region during rush hour); the residential tower (which generates strongly negative social forces as discussed in (Alexander et. al., 1977; Salingaros 2005a)); and the giant parking lot that comes as part of either of these (and which erases the human environment precisely where it ought to be intensified). High-rise buildings don’t belong in a compact city. Genuine high-density, high-rise city centers do exist, as coded for in Transect Zone T6, the Urban Core. Examples include the downtown Loop in Chicago, Manhattan, Hong Kong, and Sydney. But I do not foresee a future for new T6 Cores, so I have confined the compact city to a T5 maximum density and six-story height limit.

It is a great pity to see cities in the developing world self-destruct as they try to imitate the images of dysfunctional western cities (to them, symbols of power and progress). Cities in southeast Asia and China that had been working fairly well up until recently, such as Bangkok and Shanghai, have in one bold step ruined their traditional connective geometry. Their mistakes include building megatowers, then widening streets and building a maze of expressways to serve the new ultra-high-density nodes. For their entire future, those cities are condemned to be choked by traffic.

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8. LOW SPEED ENCOURAGES URBAN LIFE.

The compact city is a LOW-SPEED city. This feature has to be guaranteed by narrow streets and a special low-speed geometry. Planning has for several decades concentrated upon increasing vehicular traffic flow. This has diminished the livability of cities and urban regions. To rebuild a living environment for people, we need to reverse almost all the traffic-boosting planning measures implemented since the end of World War II β€” that is, rewrite the traffic codes. Roads inside the compact city should not be built to accommodate fast vehicular traffic. Cars should go slowly inside this region. The physical road surface and width will force them to. Transect-based planning calls for thoroughfare design to respond to the context of the T-Zone, not the other way around.

The key is to permit internal access everywhere for large vehicles such as fire trucks, delivery trucks, and ambulances, but in the immediate vicinity of a house cluster around an urban space, all the roads should be woonerven, the Dutch model of very low-speed roads shared with pedestrians (Gehl, 1996). Here we may use narrow roads with occasionally semifinished surfaces. We have forever confused ACCESS with SPEED. Today, fire departments refuse to cooperate with urbanists, insisting on an overwide paved thoroughfare everywhere. The reason is that fire chiefs want to be able to make a U-turn in one of their giant fire engines anywhere along any road.

The compact city mixes shared civic spaces with concentrated arrangements of structures. It defines a highly-organized complex system, in which each component supports and is connected to the whole. A city for people consists of buildings of local character and specific function that contribute to the immersive context of their Transect Zone. This is the opposite of modern β€œgeneric” building types, which are strictly utilitarian and connect only to the parking lot. Fixated on fast speed, governments or developers spend much of their money on paving wide roads and vast parking lots, neglecting the design of urban space. When building a low-speed parking ribbon (described in the following Section), parking costs should be the last priority, thus permitting gravel, and brick/grass surfaces. Such surfaces slow cars down.

Urban space is supported by the geometry of surrounding buildings (Salingaros, 2005a). Buildings should attach themselves to those spaces, and not to the road. A compact city is defined by internal cohesion achieved via a centripetal (center-supporting) arrangement, versus a centrifugal (directing away from the center) arrangement. Buildings are connected via a network of paths into clusters. A number of buildings should define a cluster perceived by a pedestrian as accessible (a low-speed setting). By contrast, buildings in suburban sprawl are outward-looking and connect to nodes in the far distance, but not to each other (a high-speed setting). There are rarely any local connections in a monofunctional region.

Sidewalks and all pedestrian paths must be protected from unnecessary changes of level, and any other discontinuities (Gehl, 1996). Cars on the other hand, don’t get tired, so their path can easily go around people and pedestrian nodes. Again, that slows them down (anathema to today’s traffic engineers!). Pedestrian paths should be laid out to connect urban nodes, and to reinforce a connected complex of urban spaces (Salingaros, 2005a). A parking ribbon can be designed to snake around buildings and pedestrian urban spaces — not the other way around.

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9. CAR-PEDESTRIAN INTERACTIONS AND THE PARKING RIBBON.

The compact city is a city for people, but it still accommodates cars and trucks. However, surface parking lots interrupt the urban structure and sense of an outdoor β€œroom”; they are dangerous and exhausting for pedestrians, and visually destroy any pleasant walking. They also create runoff from impervious surface, encouraging flooding.

Instead of taking over a vast open area, parking should occur in a ribbon of intentionally constrained road: I am proposing a radically different parking geometry, to be generated by new zoning codes. A parking β€œlot,” then, is just another road, not an open space. These long and narrow parking ribbons will branch into each other, assuming a networked form just like urban streets. A maximum dimension of about two car lengths will be stipulated for the width of any parking ribbon, accommodating only one side of head-in or diagonal parking. Parking ribbons don’t need to be straight, but can be made to fill up otherwise useless narrow spaces.

Furthermore, pedestrians should be given priority when crossing an existing large parking lot. This means building a raised footpath, sometimes covered by a canopy, and also giving it a distinct color coding for visual separation. Giant, uniform parking lots are hostile to human beings and essentially anti-urban. They can be reformatted into parking ribbons by building other structures inside them. Inserting sections of water-permeable surface into giant parking lots will also solve the serious problem of flooding from storm run-off. Such infill solutions can be written into a new code.

On-street curbside parking (either parallel, or diagonal) should be encouraged in the public frontage, but banned from the private frontage, between the sidewalk and building face (Sucher, 2003). On-street parking actually helps pedestrians feel safer on the sidewalk by providing a buffer between them and moving traffic. Sidewalks are not used if there exists a psychological fear from nearby cars and trucks; vehicular traffic parallel to pedestrian flow can be tolerated only if it flows at a certain distance from people. Adjusting the maximum speed of a road (not by speed limit signs, but by its narrowness and road surface) to tolerable limits also achieves this symbiosis. For slightly faster urban traffic, an excellent thoroughfare type to accommodate both car traffic and safe sidewalks is the boulevard, traditionally designed with low-speed β€œslip roads” and parking on the sides.

Parking ribbons already exist in traditional urbanism: as curbside parking on slow-moving roads; and on the sides of a fast-moving boulevard. Most parking garages are indeed wound-up parking ribbons. What I’m suggesting is that ALL parking should conform to the ribbon geometry. A parking lot should never again be confused with an urban space, and cars should never be allowed to take over an urban space.

Another solution is to have orthogonal flow for pedestrians and vehicles (working simultaneously with protected parallel flow). Their intersection must be non-threatening. The two distinct flows cross frequently at places that are protected for pedestrians. In this way, the two flows do not compete except at crossing points. Introducing a row of bollards saves many situations where pedestrians are physically threatened by vehicles. An amalgamation of pedestrian paths defines a usable urban space. This must be strongly protected from vehicular traffic. Any paved space that children might use for play must be absolutely safe from traffic. I discuss all these points at length in (Salingaros, 2005a; 2005b).


10. BEYOND THE TRANSECT WITH CHRISTOPHER ALEXANDER.

Where do the Transect-based codes come from? They are a result of thinking how to create an environment conducive to human life, obtained by comparing present-day with older successful environments the world over. They ultimately depend on traditional solutions, such as those collected in Christopher Alexander’s β€œA Pattern Language” (Alexander et. al., 1977). The Transect’s value lies in structuring a proven form of compact, traditional urbanism in a way that can be used within the existing planning bureaucracy. As AndrΓ©s Duany has so often expressed, he wants to use the system to introduce radical changes without waiting to change the system itself. He calls the Transect-based Smart Code a β€œplug” into the existing power grid used to working in terms of zoning.

There is another approach. Alexander’s new book β€œThe Nature of Order” (Alexander, 2005) is the most important analysis of architecture and urbanism published in the last several decades. Alexander advocates a complete replacement of current planning philosophy, because the existing manner of doing things is so fundamentally antihuman. That may be difficult to implement immediately, but the future of cities does depend upon ultimately applying Alexander’s understanding on how urban form is generated, and how it evolves by adapting to human needs. My own work (Salingaros, 2005a; 2005b) has been profoundly influenced by Alexander’s.

Alexander describes his adaptive design process, giving examples to show urbanists how to tailor it to their own particular project (Alexander, 2005). I will not attempt to summarize his extensive results here, but only wish to point out a key finding. Living urban regions have a certain rough percentage of areas devoted to pedestrians-green-buildings-cars as 17%-29%-27%-27%. Contrast this to a majority of today’s urban regions, which typically have the percentage distribution as 2%-28%-23%-47%. Alexander describes in great detail the succession of geometrical steps that can be taken to convert one type of urban region into another. His approach is to do this one step at a time, and it is eminently practical.

The result is what all of us (Alexander, Duany, Krier, Plater-Zyberk, and myself) want: a human-oriented urban environment. At the same time, Alexander presents a theory of urban evolution, which could be steered either towards a living city, or towards an anti-urban landscape for cars. The point is to recognize the fundamental mechanisms and forces that push towards either goal, and to channel them to what we want. Most important, we should recognize what we really want, since many people (including prominent urbanists) really do want to sacrifice urban life to the auto-dependent landscape, even though they may not openly admit it.

Alexander’s understanding of urban processes probes far deeper than the Transect. Duany and Plater-Zyberk have learned from Alexander, but want to affect immediate improvements. The simplest expedient is a change in zoning codes, such as the Transect-based Smart Code. Today’s urban environment is so fragmented, degraded, and antihuman that such code reform is urgently needed. Once healthy urban fabric begins to grow again, then people can see the advantages of a human-scale built environment. They could apply Alexander’s ideas to generate vital urban regions once again. Anyone who dismisses the New Urbanism as superficial, or as simply a β€œstyle”, needs to read Alexander to really understand urban form.

And yet, I must point out a fundamental difference. Alexander is convinced that genuine urban unfolding β€” the process of sequential adaptation that generates living environments β€” is not possible within current planning practice. He fears that the system is not only misaligned, but is also too rigid to accommodate living processes. The new Transect-based codes, significant as they are in improving an abysmal situation, are not flexible enough, according to Alexander, precisely because they work within the present planning system. Since changing a vast and established bureaucracy is next to impossible, Alexander proposes going around the system. These points raise serious tactical questions.

Defining urban character as inherent in the Transect has begun to reestablish an urban structure that can engender a new urban citizen. The Transect, however, is just a beginning: in addition to these sectional prescriptive codes, urbanists must extend their logic to multiple scales and work through a knowledge of urban adaptive processes (Alexander, 2005; Salingaros, 2005a).

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11. SOME CONTRADICTIONS.

There are several contradictions I feel I need to discuss. First, the limitations of working with a system of permits and construction that is deeply flawed, threaten to neutralize any code-based way of building cities. Alexander (2005) emphasizes that living cities can only come about from an adaptive PROCESS, i.e., building and adjusting urban form step-by-step. This is not easily reconcilable with the present mainstream professional culture. It is, however, the way that traditional building and self-built settlements arose for millennia.

Alexander’s fear is that any system that builds cities without a truly adaptive process will never achieve the intense degree of life seen and felt in cities of the past. That is not the aim of the present code-based system, which instead uses the existing bureaucracy to limit such an evolution of urban form. The gradual evolution of cities, akin to the evolution of individual organisms and ecosystems, is now illegal. What is allowed is a large-scale intervention, regardless if it is catastrophic or nearly so (planners cling to the myth of an β€œeconomy of scale”) (Salingaros, 2005a).

The second contradiction is that a majority of people go along with anti-urban sprawl and high-rise construction without complaining. It is hardly possible to discuss issues of urban form with a contemporary society that has become desensitized through its addiction to technology. Growing up in suburbia with the false notion of unlimited freedom has distanced people from truly human environments. People who enjoy eating junk food in their parked car; who love the ear-damaging loudness of commercial movie theaters and rock concerts; who own a β€œHome Entertainment System” (a monster television/stereo with subwoofer) and another subwoofer in their car, are not going to value the pleasures of a traditional environment β€” it only reminds them of a pre-technological past.

In the present atmosphere, I see Transect-based codes as the best entry-point for bringing a human environment back to our cities. I have discussed these issues with commercial developers, who insist that they are not setting urban typologies: they are only providing what the market wants, working within the existing codes. Clearly, our society has to learn to appreciate good urbanism before Alexander’s work and my own can begin to be applied to cities. The Transect will certainly help to move society in that direction.

Alexander would prefer for codes to be optional and voluntary: accepted by ordinary people on the basis of understanding and sensitivity, and not imposed by law. Duany, on the other hand, is suspicious of media-induced fear and manipulative marketing; those forces push people to reject connectivity and to want to live in monocultures.

The third contradiction is that human-scaled cities must be market-driven and implemented by legislation, but people don’t seem to be ready to do what is required. Any hope for a positive change must come from an educated society that demands good urbanism instead of its β€œjunk food equivalent”. Enough popular support has to build up to pressure elected officials to make the necessary changes in urban codes. Those who need it the most β€” the young, the old, and the poor β€” are either not educated about city form, or have no influence. New Urbanist ideas have been embraced by upper-income groups simply because of their higher level of education. That is not because of any particular attraction between the compact city and any particular socioeconomic class.

Ultimately, the most disadvantaged classes of society can least afford the expense of sprawl, yet only those who are better educated see the reality of a human-scale urban environment.

The fourth contradiction is the institutionalization of sprawl. In addition to planning codes, sprawl has been adopted as an unshakeable standard by insurance companies and financial institutions. They are reluctant to finance or insure the compact city, but will automatically help to build sprawl because all their offices and agents have been doing this for decades. That mindset is permanently fixed to the extent that even when natural disasters wipe out vast areas of sprawl, the bureaucracy does not permit them being rebuilt as compact city. An opportunity to finally get rid of anti-urban patterns and to reconfigure our cities is thus missed. All the discussion about wasting time in commuting, and wasting one’s salary on gasoline seems to be for nothing, if it will not influence rebuilding when an opportunity presents itself. This may be interpreted alternately as the bureaucracy doing the β€œsafe” thing; or as criminal willfulness.

Β

12. CONCLUSION.

This essay put forward a radical idea shared by many urbanists today: that the ultra-high-density city is outdated. There are essential differences with other authors, however. Unlike some of my colleagues who abandon any urban principles out of frustration, I condemn suburban sprawl and high-rise buildings as equally unworkable. Supporting AndrΓ©s Duany and Elizabeth Plater-Zyberk, I proposed a β€œnew” ordered urban form: the compact city. This new urban typology looks remarkably like the old geometry of small-town and village living, so it is really a return to traditional urbanism. Where it is radical is that it requires a complete rewriting of the zoning codes. That is essential, since theoretical urbanism is ineffective if the present anti-urban codes remain unchanged.

This essay also contained an implicit condemnation of planners and designers who refuse to distinguish between good and bad urbanism, or to offer any workable solutions. That is the equivalent of doctors refusing to diagnose and cure patients, deciding to give an equal chance to the microbes. Prominent designers talk about the urban condition, labeling the disconnection of our cities (and civilization) as a new, exciting phenomenon: a natural evolution (instead of extinction) of the city. They also accept, without question, the massive destruction of traditional urbanism taking place in China and the developing world as β€œinevitable progress”. Urbanists have a responsibility to intervene; they cannot be neutral observers. From now on, the world can only rely on pragmatic urbanists who are willing to tackle practical issues to create compact cities for humans.

"The Fractal Nature of the Architectural Orders", por Daniele Capo

A discussion of architecture and fractals can lead to ambiguous territory [Ostwald 2001; Balmond 1997; Jencks 1997a; Jencks 1997b; Eisenman 1986]. The aim of the present paper is to test with regards to architectural elements certain concepts that are proper to fractal geometry. The purpose is not to show that the architectural orders are true fractal objects, but rather that how fractal "instruments" can be used to approach certain objects and what kinds of information can be gleaned by such an approach. It is worthwhile mentioning that an architectural element is only approximately fractal, since it cannot have details that are infinitely small; thus, in this regard, I prefer not to speak of "fractal architecture," but rather of architecture "with a fractal nature."

As a guide, we can take the definition of fractal sets F laid out by Flaconer [1990: xx-xxi]:

i. F has a fine structure, that is, details at an arbitrarily small scale;
ii. F is too irregular to be described in a traditional geometric language, both locally and globally;
iii.F often has some form of self-similarity, perhaps approximate or statistical.
iv. Usually the "fractal dimension" of F (defined in one way or another) is greater than its topological dimension;
v. In the most part of the cases, F can be defined in a very simple way, perhaps recursively.

In the field of architecture, Carl Bovill [1996] performed a fractal analysis by measuring, by means of the method of box-counting, the fractal dimension of some works of Wright and Corbusier. In the present paper I would like to make some observations on the implications of an architecture with a fractal nature, and in particular, to demonstrate how a discussion of this kind is well suited to the architectural orders (it appears that thus far the architectural orders have never been subject to a fractal interpretation).

The architectural orders can be taken as meaningful case studies for several reasons. In the first place, the object of study is easily defined; secondly, the analysis can be limited only to vertical successions of elements that are clearly disparate; through focussing on only one dimension, an analysis can be perfomed in a systematic and precise fashion.

In spite of its simplicity, or perhaps because of it, the example of the architectural orders furnishes a very clear image of how fractal analysis can be applied to architecture in general and contributes to the resolution of the ambivalence concerning the meaning of the term "fractal" in the field of architecture.

A DEFINITION OF THE OBJECT UNDER EXAMINATION
The architectural orders taken into consideration are those defined by Palladio in his treatise on architecture (Fig. 1) [Palladio 1992: 30-67]. The analysis will focus on the succession of vertical elements. In essence, one takes the intersection between a straight line and the segments that separate one element from another, and this set of points is considered to be the element in question. This may appear to be an over-simplification, but it is also true that in this way a complete analysis is made possible. Further, the error that might be due to this abstraction can, at the most, result in the de-fractalization of the order. Thus, if we are able to demonstrate a coherence between our abstraction and the fractal hypothesis, then we are justified in saying that the actual order also possesses a fractal nature, perhaps even greater than that which we hypothesized. For now, however, let us accept the idea of investigating the way in which the surface is structured in one dimension.

METHODOLOGY
There are two methods by which the investigation is carried out. The first consists in measuring the box-counting dimension of the set of points defined above. The second consists in an analysis of the relationship between number and dimension of the intervals between the points (that is, of the elements that form the architectural order).

With the first method (Fig. 2), we count the number of small squares that are "occupied" by some point of the set being investigated, and at each successive passage, we divide the side of the square by two. Rather than using the classic method of box-counting, we will use a slightly modified version, "the information dimension" [1], which takes into consideration the number of points that fall in each square. The values obtained are placed on a graph so that the number of squares occupied are laid out on the x-axis, and the logarithm of the inverse of the side of the square are laid out on the y-axis.[2] By means of a statistical analysis the straight line is obtained that best approximates the distribution of points and the coefficient of correlation, which tells how valid it is. The slant of the line represents the fractal dimension of the set of points for the architectural order. In order to obtain the dimension of the architectural order (that is, the dimension of the original drawing, taking into account only the horizontal rows), it is sufficient to add 1 to the result obtained.[3]

The second method (Fig. 3) consists in counting the number of spaces with a length greater than length u, which is varied by halving it. The result is placed on a logarithmic graph in which the number of spaces is laid out on the x-axis, and the value of u on the y-axis. This system derives from the kind of analysis suggested by Nikos Salingaros.[4] In this case as well it is necessary to determine the slant of the line that best approximates the data and the degree of correlation between the two.

It should be observed that the advantage of undertaking an analysis of only one dimension is that the trend of the data can be immediately visualized. Given that we are talking about a set of points and not of a true fractal, it can be expected that, when the investigation is refined, the result tends to zero;[5] what counts is how slowly it does so (that is, how much more it tends towards an authentic fractal). This is easily verified from the point where the curve formed by the data flattens out to the point of becoming horizontal. The fractal coherence can be estimated, then, by means of a simple visual comparison of the data, which is not so automatic if the investigation is extended into the plane. In this case we have the values of the dimensions tending towards one, which is more difficult to perceive with the naked eye.[6]

COMPARISION
The analysis thus conducted leads to a set of results, but it is important that they be compared with a control situation purposely constructed to possess given fractal characteristics. Analyzing our "artificial" order provides us with a kind of litmus paper which allows us to verify the investigation. The limit that we imposed of treating only the succession of vertical elements once again works to our advantage in that we can easily construct a similar paragon. We can imagine reading the succession of elements typical of the architectural orders as a Cantor set. In fact, in the former as in the latter, we find large spaces surrounded by small spaces that are, in their turn, surrounded by still smaller spaces. At this point we must define a model based on a modification of the traditional Cantor set and subject it to analysis (Fig. 4). A first check can be effected by a direct visual comparison, drawing an architectural order and putting it next to a real one. When a certain plausibility of the hypothsis on the basis of the model is observed, we go to the same investigation discussed earlier in the section on methodology, and the results are compared.

RESULTS OF THE INVESTIGATION
The fractal analysis conducted on three orders of Palladio, the Corinthian, the Doric, and the Composite, and on the control model, showed a fundamental coherence with a fractal interpretation.

The "information dimension" method shows that all three of Palladio's orders maintain a certain consistency of the data up to the eighth level, indicating that the value of the dimension is demolished only when the count is based on squares with a small side that is equal to 1/256 of the height of the entire order. If we consider a total height of 10 meters, we can conclude that the fractal coherence is maintained down to a detail of 4 centimeters, which is not surprising considering that, in the architectural orders, that are mouldings that are exceedingly small.

The second method validates the results of the first, showing how the number of elements continues to increase as their height gradually diminishes, a characteristic that is essential to fractal objects. In this case, as above, the most important result is the large interval on which the fractal interpretation has been effected. The trend not being perfectly linear would seem to deny that this is true, even though the coefficient of correlation is still very high, but if the form of the graph is carefully considered, it can be discerned that the most important element is the tendency of the details to grow as their height decreases.

The "control" order, explicitly constructed with a fractal recursion, furnishes results that are very similar to those obtained from the analysis of Palladio's orders, reinforcing still more this interpretation. This helps us to understand that the jumps present in the graph relative to the second method are inevitable.[7]

The fractal dimension measured is not fixed, as can be easily verified in the graphs, but oscillates. The important thing is to notice that it also oscillates for the control order, and is greater than than the theoretical one that is known in this case. The problem therefore reduces down to two facts: the first is that the method itself has limits, as was noted above; the second is that we must not consider the orders as "simple" fractals, but rather as examples of multi-fractals in which diverse dimensions coexist.[8] Even taking into account these limits, it can be brought out in any case that the dimension runs between 0.6 and 0.7.[9] Knowing that these values are approximate by excess, we can in any case affirm that the dimension can be collocated in a position that tends to mediate between 0 and 1, where 0 represents the null set (a total absence of any element of interest) and 1 the completely full set (visual chaos, where every part is filled) (Fig. 5). The geometry of these architectural objects strike a balance between the two extremes, a fact that is held to be extremely important by both Mandelbrot [1981: 45-47] and Eglash [1999: 171].

Understanding the orders, which for centuries have provided the basis for Western architecture, in light of the analysis presented above, brings us to certain considerations. The first is that it allows us to observe, through the analysis of numerical data, how small elements are inserted in a continuous and coherent whole. If we interpret this structure fractally we do not distinguish between the essential and the inessential; everything is essential and so creates in this way a greater (fractal) coherence. It could be said, in this light, that the general form is not what counts the most, but rather, what is really important is the way in which parts hold together. For example, by means of this analysis it could be said that the "abstractions" that reduced the architectural orders to their principal elements (as in certain architectures of totalitarian regimes of the first half of the last century) did not grasp this fact, while an architect like Wright [10], even while not replicating the form of the orders, realized an architecture which, from the point of view of fractals, came very close to them.

The second observation is that the Cantor set constitutes an approximate yet realistic model of the kind of fractal geometry exemplified by the architectural orders. This idea could be carried forward by hypothesizing a simple process of "budding out" which gives rise to structural architectural systems of a similar nature (Fig. 6). Starting with few elements, a pier on a plinth, experiments could be undertaken to see what would happen when a greater number of elements was distinguished, introducing others above and beneath, and continuing this operation a certain number of times. It can be readily observed that this means of proceeding, based on a simple logic, is capable of generating a very high level of differentiation, giving rise in the end to something that comes very close to the actual forms taken by the architectural orders through history. The kind of structuring that we have seen presents us then with a peculiarity from the point of view of perception. In fact, we can affirm that structures with fractal natures are visually very robust. We can observe this by comparing two drawings, one representing Sierpinski's carpet, the other a square subdivided into smaller squares (Fig. 7). By slightly modifying the two drawings it can be seen that the second reflects the effects of modification in an accentuated way, while the first seems effected to a lesser degree. This characteristic of fractal figures could account for the fact that the architectural orders, even while subject to modifications in the parts of which they are constituted, maintain their "order;" if their geometry was not of the fractal nature discussed in this paper, the order would diminish.

NOTES
[1] The "information dimension" consists in keeping a count of the greater or lesser probability that a square will be "filled" by some part of the figure. For the definition, see [Peitgen, et al. 1998]. return to text

[2] For our purposes we consider the height of the entire architectural order, from the ground to the uppermost moulding, to be one unit. return to text

[3] This problem can be taken back to that of the multiplication of two sets with different fractal dimensions, the dimension of which is equal to the sum of the dimensions. Cf. [Falconer 1990]. return to text

[4] For a discussion of Salingaros's position on "fractal" architecture, see [Salingaros and West 1999]. The method that we suggest can also be extracted from the analysis of Mandelbrot [1987] of the Cantor set. The links between the architectural orders and the Cantor set will become evident later in the text. return to text

[5] A fractal object such as the Cantor set will never tend to zero but, in the case of real objects, we cannot ever arrive at this level of abstraction. For this reason we have introduced the definition of architecture as having a "fractal nature," to indicate that architecture which, within certain limits, behaves in a way that is similar to a fractal. return to text

[6] It is possible to propose a different means of representation which would still present the same advantages given above. return to text

[7] In essence, the form of the graph is effected by the fact that the length u has been halved each time. The jumps represent the fact that in those points the height of the elements "jumped" in a more rapid manner. return to text

[8] We are unable to find actual examples of applications of methods of multi-fractal analysis to architecture. Within the limits of this brief paper we can only advance the hypothesis that a similar approach can furnish new information about the geometry of architecture. return to text

[9] 0.6 and 0.7 are the dimensions of the set of points that has been found with our analysis. The dimension of the succession of mouldings varies therefore between 1.6 and 1.7. return to text

[10] For studies of Wright's architecture with regards to fractals, see [Bovill 1996] and [Eaton 1998]. return to text


Daniele Capo, "The Fractal Nature of the Architectural Orders", Nexus Network Journal, vol. 6 no. 1 (Spring 2004), http://www.nexusjournal.com/Capo.html

"Fractal Architecture: Late 20th Century Connections Between Architecture and Fractal Geometry", by Michael J. Ostwald

INTRODUCTION

For more than two decades an intricate and contradictory relationship has existed between architecture and the sciences of complexity. While the nature of this relationship has shifted and changed throughout that time a common point of connection has been fractal geometry. Both architects and mathematicians have each offered definitions of what might, or might not, constitute fractal architecture. Curiously, there are few similarities between architects' and mathematicians' definitions of "fractal architecture". There are also very few signs of recognition that the other side's opinion exists at all. Practising architects have largely ignored the views of mathematicians concerning the built environment and conversely mathematicians have failed to recognise the quite lengthy history of architects appropriating and using fractal geometry in their designs. Even scholars working on concepts derived from both architecture and mathematics seem unaware of the large number of contemporary designs produced in response to fractal geometry or the extensive record of contemporary writings on the topic. The present paper begins to address this lacuna.



This paper focuses primarily on architectural appropriations of fractal geometry to briefly describe more than twenty years of "fractal architecture" and to identify key trends or shifts in the development, acceptance and rejection of this concept. The aim of this paper is to provide an overview for both architects and mathematicians of the rise and fall of fractal architecture in the late twentieth century.

The present paper has three clear limitations or provisions which define its extent and approach. Firstly, it does not question the validity of any specific claims from either architects or mathematicians even though there is evidence to suggest that claims made by both sides are debatable.[1] Secondly, the paper is concerned only with conscious attempts to use fractal geometry to create architecture. A number of prominent examples of historic buildings which exhibit fractal forms have been proposed by both architects and mathematicians. For the purposes of this paper these proposed fractal buildings, including various Medieval castles, Baroque churches, Hindu temples and works of Frank Lloyd Wright or Louis Sullivan, are not considered to be a consciously created fractal designs even if they display an intuitive grasp of fractal geometry. For this reason, the origins of conscious fractal architecture cannot have occurred until after fractal geometry was formalised by Benoit Mandelbrot in the late 1970s even though Georg Cantor, Guiseppe Peano, David Hilbert, Helge von Koch, Waclaw Sierpinski, Gaston Julia and Felix Hausdorff had all studied aberrant or mathematically "monstrous" concepts which are clear precursors to fractal geometry. A final provision for this paper is concerned with the relationship between fractal geometry and the sciences of complexity. While mathematicians and scholars have valued fractal geometry in its own right, architects have generally valued it more for its connection to Chaos Theory and Complexity Science. This is because contemporary architects, like many historic architects, have little interest in geometry or mathematics per se, but value geometry for its ability to provide a symbolic, metaphoric, or tropic connection to something else. Thus, for modern architects fractal geometry provides a connection to nature or the cosmos as well as a recognition of the global paradigm shift away from the views of Newton and Laplace. For this reason, the vast majority of architects mentioned in this paper view fractal geometry as an integral part of, or sign for, Chaos Theory and Complexity Science.

THE RISE OF FRACTAL ARCHITECTURE: 1978-1988

In 1977 the scientist Benoit Mandelbrot's seminal work Fractals: Form, Chance, and Dimension, the first English language edition of his 1975 Les Objects Fractals: Forme, Hasard et Dimension, was published to much critical acclaim. Although Mandelbrot had published some sixty-three papers prior to this date, the formal science of Chaos Theory is widely considered to be defined by this work. However, like the mythopoeic "death of modernism"[2] manifest in the demolition of Yamasaki's Pruitt-Igoe Housing in 1972, this birthdate for Chaos Theory is contentious. What is certain is that within Fractals: Form, Chance, and Dimension, Mandelbrot not only combines his observations of the geometry of nature for the first time but he also makes the first of a number of well documented forays into art and architectural history and critique. Specifically, Mandelbrot concludes his introduction to the book with a discussion of architectural styles in an attempt to differentiate between Euclidean geometry and fractal geometry. In this discussion he states that "in the context of architecture [a] Mies van der Rohe building is a scalebound throwback to Euclid, while a high period Beaux Arts building is rich in fractal aspects."[3] While this is not the first instance of a scientist or mathematician working within the sciences of complexity venturing into architectural territory, it is nevertheless the first clearly recognised example of the attempt to combine or connect architecture with fractal geometry.

Less than twelve months after the English language publication of Fractals: Form, Chance, and Dimension the architect Peter Eisenman exhibited his House 11a for the first time. A few weeks later, in July of 1978, House 11a became a central thematic motif in Eisenman's housing design produced during the Cannaregio design seminar in Venice.[4] Although this project was not publicly exhibited until April, 1980 it nevertheless marks the first widely published appropriation by an architect of a concept from complexity theory.[5] Specifically, Eisenman appropriated the concept of fractal scaling - a process that he describes philosophically as entailing "three destabilizing concepts: discontinuity, which confronts the metaphysics of presence; recursivity, which confronts origin; and self-similarity, which confronts representation and the aesthetic object."[6]
House 11a, a composition of Eisenman's then signature "L"s combines these forms in complex rotational and vertical symmetries. The "L" is actually a square which has been divided into four quarters and then had one quarter square removed. Eisenman viewed this resulting "L" shape as symbolising an "unstable" or "in-between" state; neither a rectangle nor a square. The three dimensional variation is a cubic octant removed from a cubic whole, rendering the "L" in three dimensions. Each "L" according to Eisenman represents an inherently unstable geometry; a form which oscillates between more stable, or whole, geometric figures. The eroded holes of two primal "L"s collide in House 11a to produce a deliberately scale-less object which could be generated at whatever size was desired. This is exactly what Eisenman attempted for a competition for housing in Venice. His Cannaregio scheme ignored the existing fabric of Venice and sought rather to affirm the presence, or absence, of Le Corbusier's unbuilt hospital plan for the site. Through the creation of a fictional past, a false archaeology, the proposal voids the grid of Corbusier's hospital leaving absence in place of fictional presence. "These voids act as metaphors for the subject's displacement from its position as the centered instrument of measure. In this project architecture becomes the measure of itself."[7] Then Eisenman placed a series of identical objects at various scales throughout the Cannaregio Town Square. Each of these objects is a scaling of House 11a, the smallest object being man height but obviously not a house, the largest object plainly too large to be a house, and the house sized object paradoxically filled with an infinite series of scaled versions of itself rendering it unusable for a house. The presence of the object within the object memorialises the original form and thus its place transcends the role of a model and becomes a component and moreover a self-similar and self-referential architectonic component.[8] House 11a is effectively scaled into itself an infinite number of times forming a kind of fractal architecture.[9]
In the twenty years that followed Eisenman's publication of House 11a more than two hundred architectural designs or works of architectural theory have been published which have laid claim, in some way, to aspects of fractal geometry or some related area of the sciences of complexity. While Eisenman has produced more than a dozen projects that have relied upon fractal geometry and its characteristics a large number of international architects, including Asymptote, Charles Correa, Coop Himmelblau, Carlos Ferrater, Arata Isozaki, Charles Jencks, Christoph Langhof, Daniel B. H. Liebermann, Fumihiko Maki, Morphosis, Eric Owen Moss, Jean Nouvell, Philippe Samyn, Kazuo Shinohara, Aldo and Hannie van Eyck, Ben van Berkel and Caroline Bos, Peter Kulka and Ulrich Königs and Eisaku Ushida and Kathryn Findlay, have followed his lead. Two of Eisenman's projects provide useful points of reference for fractal architecture during this period.
Eisenman's 1985 project Moving Arrows, Eros and other Errors (or the Romeo and Juliet project) is a turning point in the development of concepts appropriated from Complexity Science into architecture. At the core of the generative methodology underlying this project is the process of scaling.[10] For Eisenman fractal scaling confronts "presence, origin, and the aesthetic object" [11] in the context of the site, the building program, and its means of representation. While scaling is already present in various ways in Eisenman's earlier projects it is in Moving Arrows, Eros and other Errors that it takes on a greater importance. Betsky records that by

[u]sing a formula developed by the scientist Benoit Mandelbrot, which determines the 'self-sameness' or autonomous replication inherent in certain figures, [Eisenman] mapped plans of vast territories over each other. This technique questioned architecture's relation to a 'normal scale' and 'problematized' the concept of human perspective.[12]

But why appropriate scaling? The feedback mechanisms and fractal forms associated with order in seemingly chaotic systems are, for Eisenman, a means of destroying the stability of architecture and undermining the anthropomorphic orthodoxy that has sustained architectural theory since Vitruvius. Eisenman argues that,

[f]or five centuries the human body's proportions have been a datum for architecture. But due to developments and changes in modern technology, philosophy, and psychoanalysis, the grand abstraction of man as the measure of all things, as an originary presence, can no longer be sustained, even as it persists in the architecture of today. In order to effect a response in architecture to these cultural changes, this project employs an other discourse, founded in a process called scaling.[13]

Moving Arrows, Eros and other Errors is the result of a dual appropriation of fractal scaling and the narrative structure of Romeo and Juliet; as drawn from three different versions of the story by Da Porto, Bandello and Shakespeare. The literary narrative is used by Eisenman to dramatise the meeting of the "the 'fictional' and the 'real'"[14] . In doing so Eisenman attempts to deny the possibility of the origin of a concept meeting reality and thereby destabilise a conventional paradigm in architecture. In the same way Eisenman appropriates fractal geometry to undermine the scale specificity of conventional anthropomorphic architecture; another long unchallenged paradigm in architecture. Anthony Vidler suggests that both of these attempts are successful.

In the complex process by which the Romeo and Juliet landscape is generated, there is no sense of an aesthetic or even a natural 'origin' that gives it meaning. Rather, the forms are produced in a seemingly implacable autogeneration of grids, surfaces, and their punctuation that stems from an equally autonomous procedure called by the author 'scaling.' Referring to the random and fractal geometries of Mandelbrot, this method applies a notion of continuum to all scales and all intervals between scales that represent objects in nature, and produces new objects by virtue of their superpositioning … The result is nothing stable, nor anything preconceived; it exists as a complex artifact marked by the traces of the procedures that generated it.[15]

Perhaps the culmination of Eisenman's fascination with fractal architecture is the project Choral Works, which Eisenman designed with the assistance of the philosopher Jacques Derrida. In Choral Works Derrida's seminal text on Plato's Timeaus combines with the semiotic play upon Chora, Choral, etc., to create a twin textual and formal (or geometric) example of a fractal en abime. In this project, actually not a building but a small garden, both time, in the form of precedents, and space, as a dislocation of Le Corbusier's rediscovered Venetian hospital, are self-referential and are present in a variety of controlled iterations.[16] Eisenman claims that

At each scaling [of the design] aspects of the changes in time, changes in rivers, borders, etc. are introduced. Thus reverberations occur not only in scale but in time, resulting in self similar, but not self same analogies. It is as if there were infinite reflections in an imperfect mirror.[17]

Scaling, self-similarity and self-referentiality are all present in Choral Works although now these operations have taken on a more philosophical and less geometric presence. Choral Works is less obviously derived from geometric iterations than House 11a or Moving Arrows, Eros and other Errors. It must be remembered that in the late eighties many philosophers including Gilles Deleuze and Felix Guattari (whose works were becoming widely influential at that time [18]) had appropriated fractal geometry to explain complex and often unrelated concepts.[19] However, while architects enthusiastically embraced fractal geometry in the early to mid eighties, this situation was to turn around dramatically in the early nineties, although signs of a change had started to appear much earlier.

THE FALL OF FRACTAL ARCHITECTURE: 1989-1999
As early as 1988 some architectural writers were deriding their colleagues' obsessions with Chaos Theory, nonlinear dynamics and fractal geometry. At this time, Michael Sorkin, then architectural critic for the Village Voice, opens his critique of the work of Coop Himmelblau with an apologetic warning that he intends to resort to a discussion of Complexity Science and fractals. Not only does his manner suggest some latent embarrassment about the topic but he even takes the unusual step of attempting to justify his actions with the claim that they are relevant to the profession - an argument that seems out of context given Sorkin's otherwise aggressive approach.[20] In Post Rock Propter Rock: A Short History of Coop Himmelblau Sorkin declares that "[c]haos may be a little overfamiliar nowadays, especially in its studied inscription in architecture. However, the idea behind this latest upheaval in physics does have real implications for us." [21]
Barely two years later, in 1990, Aaron Betsky described Eisenman's Biocentre at the J. W. Goethe University of Frankfurt in terms of a conventional geometric system that is corrupted by fractal geometry. "To safeguard [the] architecture from disappearing completely … Eisenman then meshed fractal geometry with" Euclidean geometry, "'infecting' one geometry … with an equally available one." [22] Here fractal geometry is metaphorically described as a form of virus or parasite inflicting architecture - conventional Euclidean geometry is the antidote. Fractal geometry, the source of the outbreak, is not necessarily critical to the design; rather it is merely the most "available" of a number of possible sources of "infection". The tide had started to turn and the relationship between architecture and the sciences of complexity was now increasingly viewed with cynicism and suspicion.
By 1993 a few architects were even starting to categorically deny any connection between their design philosophy, Complexity Science and fractal geometry. For example, the Iranian born graduates of Cornell University, Gisue Hariri and Mojgan Hariri, open their 1993 manifesto for architecture with the statement that "[w]e do not believe in Chaos, we do not follow Trends, and we despise Kitsch." [23] By highlighting these three terms in italics Hariri and Hariri not only emphasise these concepts at the expense of their argument, they also infer that Chaos Theory and fractal geometry are merely a trend that, for them, is equated with kitsch. Once they have carefully distanced themselves from this perceived taint they feel that they can state the theoretical position that governs their design work.

It is the intension [sic] of our work to bring together in an equilibrium the Mind that disintegrates and categorizes and the Soul that is in constant search for universal unity of all things and events … Examples of this concept The Unification of the Opposites can be found in modern physics at the sub-atomic level where particles are both destructible and indestructible; where matter is both continuous and discontinuous, and force and matter are different aspects of the same phenomenon. Life in general and Architecture in particular are like force and matter intertwind [sic]. It is the events and the smallest experiences in life that form Visions of architecture.[24]

It is ironic indeed that the remainder of their philosophical position is derived from a loose understanding of quantum physics, sub-atomic particle theory and natural systems theory.[25] Nevertheless, they are not alone in their attempts to deny any connection between their architecture and Complexity Science. Perhaps one of the reasons for this dramatic disavowal might be found in the growing number of satirical descriptions of the relationship between architecture and fractal geometry. Paul Shepheard suggests that in 1994 the constant quest for the new resulted in "a furor of nonconsensus" [26] in architectural theory. In order to illustrate the confusion of the time he provides five derogatory descriptions of un-named architectural role models. The first description, which appears to be a synthesis of Peter Eisenman, Daniel Libeskind and Morphosis, commences with a veiled insult.

Here is a man who scatters chaos on paper and talks about randomness and fractional theory. He calls the scatter the plan of a building. Anything will do--twigs purloined from a pigeon's nest, notes transcribed from the Song of Songs--a scribble he did with his eyes shut, like a shaman in a trance drawing in the dust of the Nevada desert. His building is built. It appears like a mirage in the wasteland of the city, a histrionic essay of joints and materials. He claims the building is ambiguous-he says it is like the chaos of modern life-he tells us all that it is profound.[27]

Although Shepheard's description is strongly reminiscent of Sorkin's 1991 critique of the "daffy postfunctionalist methodology (form follows … anything!)" [28] -- a design process that culminates in tracing the "outline of last night's schnitzel" [29] -- it is the way in which the use of fractal geometry in architecture starts to be associated with caricature that is consequential.
By the time Alberto Pérez-Gómez presented Architecture as Science: Analogy or Disjunction at the 1994 Anyplace conference in Canada, he had to make a deliberate effort to discuss Chaos Theory and fractal geometry as a side-line or accessory to the rest of his presentation on the differences between phenomenological hermeneutics and theories of science. In this way he effectively distances his argument from the taint of nonlinearity while judiciously relying on it to support his position. Pérez-Gómez, realising that what he is about to do is "unfashionable" [30], commences his comments on fractal geometry with the informal statement that before progressing to the main theme of the paper he "would like to explore the potentially fascinating consequences of Chaos Theory for architecture. This [being] a popular topic these days." [31] Pérez-Gómez's outline of the paradigm shift associated with Chaos Theory and fractal geometry is an exemplary model of accuracy and scholarship; he has even read the key scientific texts. Yet, throughout the paper, his description is laced with a delicate tracery of sarcasm and wit. Chaos theory embodies "a formidable and exciting realization" he states. "We have at last 'discovered' that the ancient analogical assumptions that drove traditional architecture and science were not merely foolish dreams." [32] Architects are described as playing with these ideas, using them as a form of authority to legitimise their actions and augment their philosophies. "I cherish", he says, such "stories about a living world and the life of minerals, about the body without organs, about nature as a machine without parts." [33] The tone of Pérez-Gómez's paper is difficult to dissect. He clearly believes that fractal geometry and Complexity Science have much to offer yet his manner is cynical, or at best, wistful.
In the same year, 1994, Christoph Langhof published Imagination is more Important than Knowledge where he too apologises for lowering the tone of a journal to discuss fractal geometry. "Our world" he says, "- if you would excuse the trendy word - is becoming more and more fractal." [34] Why would people like Pérez-Gómez and Langhof feel obliged to apologise for discussing geometry? Perhaps the reason may be traced to the rapid growth of interest in complexity. As Paul-Alan Johnson records, Chaos Theory may have only been "formulated in the 1970s" but within a decade it had become "a booming business" [35] world-wide. Yet within architecture, it had shifted smoothly from being the favoured theoretical influence of the early eighties, to being the conceptual bête noire of the early nineties.
When in 1995 Charles Jencks belatedly published a polemical call for architecture to model a new, cosmogenic or fractalesque aesthetic, (a position that he had developed from his study of the sciences of complexity) the critics were sufficiently forewarned that they were able to respond with a flurry of damning reviews.[36] Perhaps this reaction was complicated by the cult of personality surrounding Jencks, or maybe it was justified. But the fact remains that his call for a fractal architecture of complexity was not only savaged by the critics, it appears to have been largely ignored by an architectural profession that now considered fractal geometry dated.
From the first recorded reference to fractal geometry in architecture barely fifteen years had passed before these once cherished concepts had become anathema. However the cycle from enthusiastic acceptance to almost complete rejection is not complete and signs have begun to appear which suggest that a cautious re-acceptance of complexity is occurring. In 1996 when Carl Bovill published his impressively researched book Fractal Geometry in Architecture and Design, a new stage in the ongoing curious and contradictory relationship between architecture and complexity theory was reached. Bovill, more than any other writer in architecture, immerses himself in the mathematics of complexity. He argues that fractal geometry is a powerful tool for architects, but a tool that has to be used wisely. To date this understated work has been well received, perhaps because its modest aims are well supported in the text. Whether or not Bovill's research signifies a genuine resurgence of interest is unclear at this time. Similarly not all architects stopped designing fractal buildings.
Throughout the nineties the architectural firm Ushida Findlay produced a series of highly inventive projects using Golden Sections and fractal geometry (often in combination) to generate powerful spatial forms. Their S Project, an urban master plan, presents fractal geometry in a particularly compelling manner. The S Project is a major transport interchange for Tokyo located at the intersection of a number of arterial roads and a rail line. The design explores the notion of "city as house"; an idea given renewed currency by the realisation that natural systems posses similar patterns at multiple scales. It is this same realisation, that fractal geometry operates at many scales, that is lacking in so many architectural works that claim a fractal heritage. In many ways, because large scale landscape features are amongst the most recognisable fractal forms, the master plan is an obvious subject for the use of fractal geometry. In the S Project, Ushida Findlay are able to propose a fractalesque network that incorporates systems of "flow and clustering" operating simultaneously at many scales. Regardless of whether the design caters for road traffic or pedestrians it provides a system that "can accommodate the innumerable encounters of freely moving persons who drift throughout the city." [37] Ushida Findlay describes the S Project as a vessel designed to accommodate the "Brownian movement" of people, cars, trains and information. The result is "a new terrain - a new kind of topography" [38] that possesses dynamic similarities at many scales.[39]

CONCLUSION
For almost twenty years there has existed an intricate, constantly shifting relationship between architecture and fractal geometry. At times this dependence is diffuse, and modes of theoretical transference are subtle, symbolic or semiological. At other times wholesale appropriations of geometry take place and large fragments of theory are pirated away from their originating discipline and used opportunistically. As Peter Downton evocatively suggests, on

… dark nights knowledge is sometimes smuggled over the difficult terrain at disciplinary borders by radical thinkers. It is urgently introduced in clandestine meetings and infiltrated by stealth into the mainstream of the discipline without the blessing of the powerful upholders of conventional orthodoxy, the high priests of the dominant paradigm.[40]

At other times analogies are drawn, both by mathematicians and by architects, that call upon the opposing body of theory to submit to an array of duties, ranging from menial, pedagogical roles to heroic, evidential ones.
Throughout the period of this interdisciplinary relationship few from one side have commented on the other side's position. That is, few architects have discussed the way in which architecture is used by scientists and mathematicians working in the sciences of complexity and conversely, even fewer scientists or mathematicians have noted the way in which architects borrow scientific or geometric theories from complexity. A small number of architectural writers, including Peter Fuller, Charles Jencks, John Kavannagh, Paul-Alan Johnson and Norman Crowe [41] are clearly aware that another side of the relationship exists, that mathematicians have made incursions into architecture.[42] But only Pérez-Gómez has even obliquely considered this relationship in a critical sense, concluding deftly that "Mandelbrot's view [of architecture] is hardly different from Prince Charles's opinion" and that "the relationship between geometry and architecture imagined by Mandelbrot and some of his architectural fans is thoroughly classical, simply mimetic in the traditional sense." [43] Examples of the obverse case, that is mathematicians realising that architecture has appropriated from fractal geometry, are even more uncommon. Only the scientist Peter Coveney and the journalist Roger Highfield seem to be aware of, or willing to remark on, the fact that architects are developing their own interpretations of Complexity Science and fractal geometry. In a brief survey in their 1996 book Frontiers of Complexity, the Search for Order in a Chaotic World, Coveney and Highfield comment on developments in the non-scientific fields that have arisen from a study of complexity. They state, with some consternation, that "[c]omplexity has offered a 'cosmogenic' cocktail - the motifs of fractals, catastrophic theory, and chaos - that has caught the imagination of architects."[44] Their promising footnote leads only to Jencks's The Architecture of the Jumping Universe; a minimal recognition but nevertheless better than any other.[45]
These fragments of history are pieced together here to give a brief overview of the often tortuous alliance, the sporadic shifting from amour to intrigue, that has characterised the relationship between architecture and fractal geometry for more than twenty years. The simple reconstruction offered here, while representative of the major shifts in the relationship, is necessarily superficial. Not all architects turned away from fractals in the early nineties and, in the last five years, the signs of renewed enthusiasm for complexity are chimerical at best. In time it might be possible to tell which way the relationship will shift. Whether or not it will mature and stabilise (the state wherein cross-appropriation is mutually recognised) remains largely unclear. Similarly, while this general history, which is woven from fragments, records a reasonable overview of the changes that have occurred it can not and will not suffice to explain all of the roles that fractal geometry has been forced to play in architecture, or architecture in fractal geometry.

NOTES
[1] See: Michael J. Ostwald and R. John Moore, "Charting the Occurrence of Non-Linear Dynamical Systems into Architecture." In Simon Hayman ed. Architectural Science: Past, Present and Future. (Sydney: Department of Architectural and Design Science, University of Sydney, 1993), 223-235; Michael J. Ostwald and R. John Moore, "Fractalesque Architecture: An Analysis of the Grounds for Excluding Mies van der Rohe from the Oeuvre." In A. Kelly, K. Bieda, J. F. Zhu, and W. Dewanto, eds. Traditions and Modernity (Jakarta: Mercu Buana University, 1996), 437-453; Michael J. Ostwald and R. John Moore, "Icons of Nonlinearity in Architecture: Correa - Eisenman - Van Eyck." In Vikramaditya Prakash ed. Theatres of Decolonization: (Architecture) Agency (Urbanism). Vol. 2 (Seattle: University of Washington, 1997), 401-422; Michael J. Ostwald and R. John Moore, "Spreading Chaos: Hayles' Theory and an Architecture of Complexity." Transition, No. 52/53 (1996): 36-53. return to text

[2] Charles Jencks, The Language of Post-Modern Architecture (London: Academy Editions, 1987), 9-10. return to text

[3] Benoit B. Mandelbrot, The Fractal Geometry of Nature (New York: W. H. Freeman and Company, 1982), 23-24. return to text

[4] Jean-François Bédard, ed. Cities of Artificial Excavation: The Work of Peter Eisenman, 1978-1988 (Montreal: Canadian Centre for Architecture, 1994), 54. return to text

[5] In Violated Perfection Betsky incorrectly refers to Eisenman's Romeo and Juliet project, Moving Arrows, Eros and other Errors, as being produced in 1976. If this date were correct it would make the "Romeo and Juliet" project the first instance of an architectural appropriation from chaos theory, some twelve months before the English publication of Fractals: Form, Chance, and Dimension. The real date of the exhibition of Moving Arrows, Eros and other Errors is 1986 thereby confirming Eisenman's House 11a (or the contemporaneous design for Cannaregio housing) as the first widely published instance of an architectural appropriation from chaos theory. Cf. Aaron Betsky, Violated Perfection: Architecture and the Fragmentation of the Modern (New York: Rizzoli, 1990), p. 146; cf. Peter Eisenman, Moving Arrows, Eros and other Errors (London: Architectural Association, 1986). return to text

[6]Peter Eisenman, "Eisenmanesie." Architecture + Urbanism, Extra ed. (August 1988): 70. return to text

[7] Ibid., 14. return to text

[8] Charles Jencks, "Deconstruction: The Pleasures Of Absence". in Andreas Papadakis, Catherine Cooke, and Andrew Benjamin eds., Deconstruction: Omnibus Volume. (London: Academy Editions, 1989), 119-131. return to text

[9] When examined in detail, from a scientific perspective, the concept of "fractal architecture" is problematic. See: Michael J. Ostwald and R. John Moore, "Fractal Architecture: A Critical Evaluation Of Proposed Architectural And Scientific Definitions." in Kan, W. T. ed., Architectural Science, Informatics and Design (Shan-Ti: Chinese University in Hong Kong, 1996), 137-148. return to text

[10] Peter Eisenman, Moving Arrows, unpaginated. return to text

[11] Peter Eisenman, "Eisenmanesie", 70. return to text

[12] Aaron Betsky, Violated Perfection …, 146. return to text

[13] Peter Eisenman, "Eisenmanesie", 70. return to text

[14] Ibid., 71. return to text

[15] Anthony Vidler, .The Architectural Uncanny: Essays in the Modern Unhomely (Cambridge, Massachusetts: MIT Press, 1992), 130-131. return to text

[16] Jacques Derrida and Peter Eisenman, Chora L Works, - Jeffrey Kipnis and Thomas Leeser eds. - (New York: The Monacelli Press, 1997). It should be noted that Eisenman variously calls the project "Choral Works" or "Chora l works" (the latter being a Greek pun). Choral works is usually the correct title for the project. Kipnis uses the pun instead as a title for the book (not the project) because the book looks at Greek philosophy as well as Eisenman's project. return to text

[17] Peter Eisenman, "Eisenmanesie", 137. return to text

[18] See: Gilles Deleuze and Felix Guattari, A Thousand Plateaus: Capitalism and Schizophrenia. Trans. Brian Massumi. (Minneapolis: University of Minnesota Press, 1987); Gilles Deleuze and Felix Guattari, What is Philosophy? Trans. Hugh Tomlinson and Graham Burchill. (New York: Verso, 1994). return to text

[19] Cf. Alan Sokal and Jean Bricmont, Intellectual Impostures: Postmodern Philosophers' Abuse of Science, (London: Profile, 1998). return to text

[20] Ironically, as Jencks notes in a 1996 interview with the author of this paper, Sorkin, who has expressed his reluctance to affix the label of chaos theory on any work of architecture for fear that it might be read as over fashionable, should himself by 1993 be producing designs that are in part inspired by his readings in complexity. Jencks states that "it is completely and utterly rich that someone like Michael Sorkin, who is now seven years later designing chaos cities, is claiming that it is out of date. He should have had a little more insight into himself, than to have denigrated the idea in other peoples work and then done it. Come on - Mea Culpa. Often the people who damn fashion are those who are about to be victims of it …" Cf. Michael J. Ostwald, Peter Zellner and Charles Jencks, [An interview with Charles Jencks.] "An Architecture of Complexity: Interviewing Charles Jencks." Transition, No. 52-53 (1996): 28-35, quote on p. 29. return to text

[21] Michael Sorkin, Exquisite Corpse: Writings on Buildings (New York: Verso, 1991), 346-7. return to text

[22] Aaron Betsky, Violated Perfection … , 148. return to text

[23] Gisue Hariri and Mojgan Hariri, "Architects' Philosophy." Architecture + Urbanism, No. 274 Is. 7 (July 1993): 81. return to text

[24] Ibid. return to text

[25] Cf. Gisue Hariri and Mojgan Hariri, "Villa, The Hague: The Netherlands, 1992." Architecture + Urbanism, No. 274 Is. 7 (July 1993): 118-121; cf. Kenneth Frampton, "On the Work of Hariri and Hariri." Architecture + Urbanism, No. 274 Is. 7 (July 1993): 82-83. return to text

[26] Paul Shepheard, What is Architecture: An Essay on Landscapes, Buildings, and Machines (Cambridge, Massachusetts: MIT Press, 1994), 15. return to text

[27] Ibid., [my italics]. return to text

[28] Michael Sorkin, "Nineteen Millennial Mantras." In Peter Noever ed., Architecture in Transition: Between Deconstruction and New Modernism (Munich: Prestel, 1991), 111. return to text

[29] Ibid. return to text

[30] Alberto Pérez-Gómez, "Architecture as Science: Analogy or Disjunction." In Cynthia C. Davidson ed. Anyplace, (Cambridge, Massachusetts: MIT Press, 1995), 67. return to text

[31] Ibid., 70. return to text

[32] Ibid. return to text

[33] Ibid. return to text

[34] Christoph Langhof, "Imagination is More Important than Knowledge." Curtin University Architecture Document (1994): 41. [my italics] return to text

[35] Paul-Alan Johnson, The Theory of Architecture: Concepts, Themes and Practices (New York: Van Nostrand Reinhold, 1994), 242. return to text

[36] See for example the critiques of Peter Davey, Christian Norberg-Schulz, Giles Worsley and Richard Weston; Peter Davey, "The Architecture of the Jumping Universe." [Review.] GSD News: Harvard University, Graduate School of Design (Fall 1995): 40-41; Peter Davey, "The Scientific American." Architectural Review, Vol. 198 No. 1183 (September 1995): 84-85; Christian Norberg-Schulz, "The Jumping Jencks." Byggekunst: The Norwegian Review of Architecture, Vol. 77 No. 7 (1995): 399; Giles Worsley, "The Architecture of the Jumping Universe." Perspectives on Architecture, Vol. 2 No. 15 (July 1995): 18; Richard Weston, "A New Architectural Style is Born-Again," Architects' Journal, Vol. 201 No. 21 (May 25, 1995): 52. return to text

[37] Ushida, Eisaku. Findlay, Kathryn. S Project Program. Gallery MA Books. Tokyo. 1996. unpag. return to text

[38] Ibid. return to text

[39] See: Ostwald, Michael J., "Fractal Traces: Geometry and the Architecture of Ushida Findlay." In Leon van Schaik ed., Ushida Findlay, (Barcelona: 2G, 1998). 136-143. return to text

[40] Peter Downton, "The Migration Metaphor in Architectural Epistemology." In Stephen Cairns and Philip Goad eds., Building Dwelling Drifting: Migrancy and the Limits of Architecture. (Melbourne: Melbourne University, 1997), 82. return to text

[41] The architectural historian Crowe discusses Mandelbrot's views on architecture in some detail as a means of explaining a different way of appreciating patterns at multiple scales. Crowe mostly reiterates Mandelbrot's assertions for architecture without comment although he finally concludes that for Mandelbrot "the presence of a natural sense of visual detail that relates to scale may well explain why such buildings as prismatic glass skyscrapers soon become boring to many people. This insight might also be considered for our negative reaction to a building or interior that has too much ornament and so appears to us as chaotic." Cf. Norman Crowe, Nature and the Idea of a Man Made World: An Investigation into the Evolutionary Roots of Form and Order in the Built Environment (Cambridge, Massachusetts: MIT Press, 1995), 119. return to text

[42] Two papers by the author with R. John Moore published in 1995 and 1997 are, to date, the most detailed works on the topic. See: Michael J. Ostwald and R. John Moore, "Mathematical Misreadings in Non Linearity: Architecture as Accessory/Theory," in Mike Linzey ed. Accessory/Architecture. Volume 1. (Auckland: University of Auckland, 1995), 69-80; Michael J. Ostwald and R. John Moore, "Unravelling the Weave: An Analysis of Architectural Metaphors in Nonlinear Dynamics," Interstices, Vol. 4 (1997): CD ROM. return to text

[43] Alberto Pérez-Gómez, "Architecture as Science …", 72. return to text

[44] Peter Coveney and Roger Highfield, Frontiers of Complexity: The Search for Order in a Chaotic World (London: Faber and Faber, 1996), 339. return to text

[45] Stewart and Golubitsky in Fearful Geometry also comment on appropriations from mathematics by architects but they are talking about Euclidean geometry not fractal geometry. See: Ian Stewart and Martin Golubitsky, Fearful Symmetry: Is God a Geometer? (London: Penguin, 1993). return to text

FOR FURTHER READING. The following works cited in this article can be ordered from Amazon.com by clicking on the title


Benoit B. Mandelbrot, The Fractal Geometry of Nature (New York: W. H. Freeman and Company, 1982)
Jean-François Bédard, ed. Cities of Artificial Excavation: The Work of Peter Eisenman, 1978-1988 (Montreal: Canadian Centre for Architecture, 1994)
Aaron Betsky, Violated Perfection: Architecture and the Fragmentation of the Modern (New York: Rizzoli, 1990)
Anthony Vidler, The Architectural Uncanny: Essays in the Modern Unhomely (Cambridge, Massachusetts: MIT Press, 1992)
Jacques Derrida and Peter Eisenman, Chora L Works, Jeffrey Kipnis and Thomas Leeser eds. (New York: The Monacelli Press, 1997)
Paul Shepheard, What is Architecture: An Essay on Landscapes, Buildings, and Machines (Cambridge, Massachusetts: MIT Press, 1994)
Paul-Alan Johnson, The Theory of Architecture: Concepts, Themes and Practices (New York: Van Nostrand Reinhold, 1994)
Norman Crowe, Nature and the Idea of a Man Made World: An Investigation into the Evolutionary Roots of Form and Order in the Built Environment (Cambridge, Massachusetts: MIT Press, 1995)
Peter Coveney and Roger Highfield, Frontiers of Complexity: The Search for Order in a Chaotic World (London: Faber and Faber, 1996)
Ian Stewart and Martin Golubitsky, Fearful Symmetry: Is God a Geometer? (London: Penguin, 1993)
Carl Bovill, Fractal Geometry in Architecture and Design
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University of Maryland Newsletter: Carl Bovill, Fractal Geometry in Architecture
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