Tag Archives: structured chaos

The genesis of complex geometry

I don’t believe that there is a dichotomy between a supposedly modern and traditional architecture. Instead there exist different geometric processes, and while traditionally builders have employed nesting processes in their work, for perhaps no other reason than it came naturally to them, modern builders have restricted themselves to linear geometric processes due to drawing their inspiration from Cartesian science and engineering.

In attempting to transform architecture into a vessel for artistic expression, modern architects have been trapped by their limited tool set, and the product of their work has often been confusing, silly, or utterly corrupt. There are only so many tricks that one can perform with linear geometry, although computers have extended the reach of those tricks. But the confusion of modern architects becomes even more obvious when they ascribe artistic merits to traditional builders who never aspired to be artists at all. One such instance is the introduction of a recent biography of the 18th century french military engineer Vauban by official starchitect Jean Nouvel, who described Vauban’s fortresses as an early form of land-art and morphing. Jean Nouvel asks, could a man be an artist without being aware of it? Vauban was not an artist at all. Military necessity led him to employ geometric processes that significantly increased the complexity of fortifications, and it is merely incidental that today we find his projects to have artistic merits.

The process through which Vauban’s work became worthy of architectural praise provides the key to the distinction between linear and nesting geometry. Vauban was not himself the inventor of the star fort. Those had been around for more than a century when he began his career for the army of king Louis XIV. The basic star fort was a simple concept: the old masonry walls of the medieval age had shown themselves to be obsolete with the advent of cannons, and they had been replaced with thick banks of earth dug out of trenches whose major flaw was to provide space out of reach of defensive fire at its angles. The angles were thus extended into diamond-shaped turrets in the first pass at a feedback correction, introducing nesting geometry and initiating the first step of the genesis of a fractal.

180px-Neuhäusel1680

A basic, early star fort

While the star fort was successful at resisting attacks, it was not impregnable. A method was devised to capture them by digging trenches in zig-zagging patterns through which troops could assault the walls without being exposed to cannon fire. In fact this is how Vauban built his career, and some of his “plans” for besieging star forts are significant civil engineering projects of their own.

Siege de Turin 1706

The siege of Turin. From an encircling trench, Vauban built successively denser trenches to capture the citadel and take the city, a process that was extremely expensive and time-consuming.

While star forts never truly became obsolete (as medieval fortifications had) until well into the 19th century, military engineers did improve on their effectiveness by correcting their vulnerabilities, which happened to be at the angles they were characterized by. And so, by another layer of feedback, the geometric depth of the star fort concept increased.

Citadelle San Martin

San Martin Citadel, a “second generation” star fort.

Vauban’s great invention was nothing much more than repeating this process of increasing depth one more time, creating what many now consider to be his masterpiece, the Citadel of Lille, a showcase of complex geometry made from the refinement produced by centuries of feedback of the star fort concept.

Citadelle de Lille (2)Nouvelle enceinte de Lille

Citadel of Lille and the system of fortification of the City of Lille, as designed by Vauban

If you only understand Cartesian processes, then the only idea that may come to you to improve on the basic star fort would be to add dozens of diamond-shaped turrets, a change that would most certainly make the concept worse instead of better. The military engineers of the time however were well aware that the diamond turrets were optimal in their shape. What was needed was a shape that extended the diamond, and this was achieved by increasing the depth of the whole object.

Another aspect of the complexity of a geometric process seen in the Lille example is its configuration adaptiveness. The shape of the city and the surrounding landscape is completely random, and the encircling fortifications bend to match this randomness, leading to Nouvel’s claim that it is an early example of morphing. But once again there is no deliberate attempt at morphing going on. Since each component of a star fort is defined as a recursive relational transformation of the basic wall, Vauban only had to design the wall and the other parts aligned themselves as a result of the wall’s configuration. If the outcome has artistic value, it is once again only incidental.

It is important to note that the Vauban extensions to star fortifications did not mean that the simple 3-part star fort became obsolete. In fact many simple star forts were built in the 18th and 19th century in America as the threat was low and the cities to be defended underdeveloped. The difference between a simple fort and Vauban’s complex fort is one of depth and effectiveness, and there is a real cost-benefit choice to make. The star fort only became obsolete when the bunker replaced it, and the early bunkers reset the process of complex geometry genesis by being simple concrete shells in their early incarnations.

When we undertake to create symmetry in an urban environment, we want buildings to be as alike as possible while allowing for adaptation to context. If we understand geometric depth we can build in such a way that poor and expensive buildings have the same basic design in their first levels of geometry, but expensive buildings have many more scales of geometry nested within that basic design. It is not necessary for an entire city to be made of the same materials as materials are one of the last visible scales of geometry, and so we can have a city of mud bricks and marble buildings that nevertheless share 95% of their geometry and beautifully complement each other, while both poor and rich citizens have a home adapted to their situation.

We can look at these examples from Korean traditional architecture for an illustration.

48799484.CIMG0512Tomb_of_King_Tongmyong,_Pyongyang,_North_Korea-2

On the left is a simple house and on the right is the tomb of a great king. Both buildings have the same design, but the building on the right has much greater depth in this design.

Another interesting comparison is between the Golden Gate bridge in San Francisco and the Verrazano Narrows bridge in New York.800px-Golden_Gate_Bridge_from_underneath800px-VerrazanoFromNCLDawn

The bridges are the same in design, but the Golden Gate bridge has more depth within this design, and is for this reason the more famous of the two bridges. That doesn’t mean the Verrazano Narrows bridge isn’t beautiful on its own.

And to make things as simple as they can get, we can compare a Sierpinski triangle with four levels of iteration with one that has six levels.

Geometric depth

The fractal on the right has all the same elements as the one on the left, but also has more.

A lot of the residential buildings we create today would benefit from being more like the Verrazano Narrows bridge. They try to be more than a simple house for a simple family and end up covered in tacky, useless ornament that have obviously been forced into the design. Simplicity, if it is adapted to context, can create as beautiful a landscape as complexity. Postmodernistic nonsense geometry does not. We would be better served going back to the simplicity of 1950’s international style modernism than what is being built by architects today. The best architects would reinvent it with greater depth.

Previous topics

References

Vauban, l’intelligence du territoire

Hommage a Vauban 1969

A modern artist’s homage to Vauban. This artist did not understand complex geometry.

Chaos re-emerges

1960’s psychedelic art?

landscape-chaos

Or the American landscape?

Notice how much negative space is created by the imposition of the grid on a chaotic reality. The simplicity of the cartesian plan is deceptive. It generates complications as the random process of change unfolds.

Decoding paradise – the emergent form of Mediterranean towns

scarano4

Serifos in Greece

Until very recent times, a study entitled Julian of Ascalon’s Treatise of Design and Construction Rules From Sixth-Century Palestine might have been categorized somewhere in-between ancient history and archeology of architecture, if not relegated to the dusty shelves of legal scholarship. Although it deals with one of the most sought-after secrets of architecture, how to build the charming Mediterranean towns of Greece, Spain, North Africa, the Near East and many other places, this is not immediately obvious from the content of the treatise. The reason for this is that the treatise does not so much describe the form of the town as the process for building it, and the process turns out to be emergent. Unless the reader makes the link from process to form, the rules described will make no more sense than the rules for a cellular automaton out of context.

It is tragic that enormous amounts of resources have been spent attempting to recreate the Mediterranean town with no clue as to the underlying source of its complexity. Montreal itself has the world famous Habitat 67, a confusing pastiche of the memories that architect Moshe Safdie brought back from his land of birth, which he had in common with Julian of Ascalon. Habitat 67 was intended to be a low-cost solution to housing, but it never was taken seriously as a model for urban habitat, and its current untrendiness spares it from being labeled fake complexity. That an attempt to emulate the architecture of some of the poorest people of previous centuries would result in an expensive failure testifies to the inadequacy of modern production processes, but also of the wealth inherent in those simple traditional production processes. The beauty resulting from large aggregations of simple buildings has turned many towns into tourist destinations. There is value in process.

The complexity demonstrated by the constructions of pre-modern civilizations may be a direct consequence of their material poverty. Most people will claim that the loss of building quality is a result of culture, and so we must change our own culture through education. That is not a complete answer. Cultures are stored in information technologies and media. The modern era coincides with the invention of printing, making it possible for the first time to reproduce information in large quantities at low costs. As information technologies have progressed and become more affordable, building processes have become increasingly dependent on large amounts of descriptive information, with blueprints describing in every minute detail how to compose a building. And now that CAD software can describe and store nearly limitless information, whole new forms of buildings have become possible.

All of this progress has only enabled builders to become lazier with information. Pre-modern builders, limited to oral communication and their brains to hold information, had to employ very sophisticated means of information compression to communicate and simply remember their cultures. This lead them to rely on simple processes the likes of which are behind the complexity in fractal geometry and cellular automata to build their environments – very short sequences of information that can be utilized to generate fully complex forms. Christopher Alexander even used as an example, in The Nature of Order, the production of a boat that had been coded into a song that the builders recited while creating the boat, adding a mnemotechnical aspect to the storage of cultural information that was essential to pre-modern survival.

Without knowing how traditional cultures were stored, we had no idea how to inspire ourselves from them. Modern and post-modern architects attempted in vain to imitate traditional building using their own, lazy information technologies, and succeeded only in building pastiche of complexity. The breakthroughs in complexity theory of the past decades finally gave us the opportunity to decode the mysteries of historic building cultures by showing us what kind of information to search for. What was right in front our noses suddenly becomes deeply meaningful.

It is to his great credit that Besim S. Hakim went looking specifically for the source of the emergent forms of Mediterranean towns in treatises of building laws. From his study of the treatise of Julian of Ascalon, but also of those of Muslim scholars around the Mediterranean, he was able to identify the underlying process that generates the complex morphology all towns of the region have in common, and that so many have sought to imitate. It is no exaggeration to call this pioneering work in complexity.

The space of Hakim’s search began in the Islamic world, with the treatise of Ibn al-Rami from Tunis in circa 1350. Tracing the origins of the practices described in the treatise, references to treatises written in Egypt, Arabia, Tunisia and Andalusia in previous centuries were researched until the treatise of Julian of Ascalon was uncovered. Written in Palestine to describe the local building customs in order to provide the Byzantine empire with an improved legal system, this particular treatise’s value is its longevity. After propagating throughout Greek civilization as part of a general book of laws (the Hexabiblos), its authority was invoked in decisions dating as recently as the 19th century. Hakim infers the origins of these shared practices, and the shared morphology of regions as far apart culturally, linguistically and geographically, as Andalusia, Greece and Palestine, to customs from ancient Babylonian civilization that had spread to the Eastern Roman Empire.

The goal shared by these treatises is a definition of urbanism as relevant today as it was in Babylon:

The goal is to deal with change in the built environment by ensuring that minimum damage occurs to preexisting structures and their owners, through stipulating fairness in the distribution of rights and responsibilities among various parties, particularly those who are proximate to each other. This ultimately will ensure the equitable equilibrium of the built environment during the process of change and growth. (Hakim, Mediterranean urban and building codes: origins, content, impact, and lessons, p. 24)

Here we see what the underlying error of Habitat 67 was. It was designed as a single static building imitating a process that made a living tissue out of many individual acts of simple building. The codes of the Mediterranean treat the town as a living, whole structure in movement that must be preserved while it achieves equilibrium with a changing environment and society.

Perhaps the most relevant conclusion of this research is the identification of proscriptive and prescriptive rules for building.

Proscription is an imposed restraint synonymous with prohibition as in ‘Thou shalt not’, for example, you are free to design and manipulate your property provided you do not create damage on adjacent properties. Prescription is laying down of authoritative directions as in ‘Thou shalt’, for example, you shall setback from your front boundary by (x) meters, and from your side boundaries by (y) meters regardless of site conditions. Byzantine codes in many instances included specific numeric prescriptions, unlike their Islamic counterparts that tended not to include them. (Hakim, Mediterranean urban and building codes: origins, content, impact, and lessons, p. 26)

A prescription would be a rule that defines in detail what to do in a given situation. A proscription is a template for defining prescriptive rules, a pattern for a rule. Muslim scholars provided mainly proscriptions, but Julian of Ascalon’s treatise was highly prescriptive. Julian was describing in details the local building codes with the idea that they would be used to devise proscriptive rules for the empire. By accident these prescriptive rules became law and remained in force for centuries until their inability to deal with society or physical conditions radically different from sixth century Palestine made them obsolete. Although it means the codes failed to deal with changing circumstances, this gives us the chance to bridge the gap between the physical structure of built towns and the rules that generate them.

The concept of proscriptive rules also helps explain why so many different cultures with specific structural typologies can generate such similar morphology. Hakim uses as an example the problem of views. The Greeks were preoccupied with views of the sea, and their prescriptive rules obliged the preservation of view corridors in new constructions. Muslims, on the other hand, were preoccupied with the preservation of privacy and the prevention of intrusive views from one property to another. This would have very different results structurally, however those two prescriptive rules are based on the same underlying proscription. Local customs and culture could therefore be translated into prescriptive rules using the proscriptions inscribed in building treatises and the emergent morphology of those proscriptions would be symmetric from one culture to the next, while being fully adapted to local conditions.

Another significant fact that strikes out from these treatises is the importance of relationships between neighbors. The Julian of Ascalon treatise describes how to literally embed houses into each other, ultimately making them one continuous, somewhat random building created through iterated steps. But most importantly by proscribing rules as relevant to a neighborhood, Mediterranean urbanism avoids the problem of the absolutist, dare I say “Cartesian” rules of modern planning that are relative to the precisely subdivided lot the building is on. Hakim shows the wastefulness of latter rules in a comparison of the old town of Muharraq in Bahrain with a new subdivision from modern Muharraq.

hakim1a hakim1b

The town on the left was generated using proscriptions based on neighbors, while the subdivision on the right used absolute rules planned with the subdivision. Notice that the configurations on the right waste much of the space in order to achieve a strictly Cartesian, grid-like morphology that no doubt looks orderly to the planners.

The last item of significance, and perhaps the most revolutionary, is how the proscriptions extracted by Hakim are similar in nature to the rules that Stephen Wolfram described to generate emergent complexity with cellular automata. He himself follows a proscription/prescription system, where the proscription is for example the 2 color, one-dimension elementary cellular automaton that made him famous, for which there exist 256 different prescriptive rules of neighborhood, some of which grow in time to make two-dimensional chaotic fractals. Some urban complexity researchers such as Michael Batty have been playing with cellular automata trying to reproduce urban form, but their efforts have taken them on the wrong track. The codes of historic towns behave in the same manner as a cellular automaton. This should be the focus of their research.

Whatever the potential for research, the proscriptions discovered by Besim S. Hakim are still relevant today and can be used to create the prescriptions that we need to implement an emergent urbanism relevant to the problems of today, that is to say the creation of a sustainable city and living urban tissue out of the vast urban fabric of suburban sprawl. Hakim has so far focused his work on the regeneration of historic neighborhoods by restoring the generative codes that produced them, but there is a vast potential to expand his work to non-historic neighborhoods that are in dire need of new life.

Addendum

Four regions, four cultures, one shared process generating a symmetric morphology

sidibousaid

Tunisia

mojacar-from-the-air

Andalusia

scarano1a

Greece

palestine

Palestine

Reference

Besim S. Hakim – Generative processes for revitalising historic towns or heritage districts

Besim S. Hakim – Julian of Ascalon’s Treatise of Construction and Design Rules from Sixth Century Palestine

Besim S. Hakim – Mediterranean urban and building codes: origins, content, impact, and lessons

and don’t forget to look at Besim S. Hakim’s website.

Mr. Besim S. Hakim provided comments for this article

Picture from Alessandra Scarano were also used

Design, configuration and natural form

When did human creations stop being natural? We look at a tower block, a subdivision or a shopping mall parking lot and see the worst of industrial civilization translated into form. We tolerate them as necessary to achieve the material wealth of our civilization. Those human settlements that are still natural we grant special protections through UNESCO and historical preservation laws. We do not have a law that promotes the creation of new historic settlements because we are not quite sure how they are made.

I believe that our mistake is not in the things we make, that there is nothing unnatural about a shopping mall parking lot from a design point of view. What makes the shopping mall parking lots we build so unnatural are errors in configuration of the design elements. To understand this, one must understand the difference between design and configuration.

The form of a tree is an ideal example to illustrate the difference between the two concepts. Any particular species of tree will have a design that is essentially the same from one tree to the next. The design elements in the tree are all the named parts: trunk, branch, leaf, root, bark, and so on. These parts are organized into hierarchical relationships with the whole tree and with each other. We will always find the roots related with the trunk in the same way. This relationship is a design solution that achieves a specific result. However, the position of any of the parts is not fixed. In the DNA of the tree are rules that instruct cells to adapt themselves to the larger context the tree finds itself in. The different design solutions that result from this cellular action will therefore adopt a position that reflects the particulars of time and place, ensuring that the tree’s form is perfectly adapted to its environment. This is why it makes no sense to create a description of the forms of a leaf in order to make another leaf – that form is relevant only to this particular leaf, and another leaf, although it would have the same overall design of parts, will take a completely different configuration.

Adapted to chaos

A chaotic configuration of a standard design

If you’re having trouble seeing this, imagine the following scenario: we take the DNA of a tree and clone it 100 times. Then we lay out a grid 10 trees by 10 trees and watch them grow. What would happen would be that every tree would come out a different way, since the earth around them would be structured differently, the wind patterns would be different, the shade and the moisture would be different. The trees would each have the chaotic, random shape that we know trees to have, yet would all be perfectly symmetrical with one another without being identical. Each clone would adopt a unique configuration of the same design.

When we look at a traditional village, we find that the same house design is repeated time and time again, but configured in such a way that it is differently adapted than the other houses. The reiteration of an often very simple design is all that it takes to create a natural landscape, so long as each house is configured to adapt to its place, and the design elements of the house are themselves configured to adapt to these adaptations.

One design, many configurations

Even today this kind of natural adaptation takes place in modern settlements where planning regulation allows it, or fails at forbidding it.

Monaco

This is the skyline of Monaco, which by necessity of the small size of the city had to be built piecemeal but yet is still made with an entirely modern building stock. The piecemeal process allowed each building to be configured to its site and thus, despite the fact that the buildings’ design is very basic modern architecture, the whole landscape looks natural. It would be even more natural were the architectural elements also adapted.

favela_rocinha_rio

This the Rocinha favela of Rio de Janeiro. Here the building design is as bare as could be made, the houses being built by poor residents with little capital to invest. But the resulting configurations adapt perfectly to the shape of the hill and the other buildings, and the overall look of the place is that of a human jungle. (If you have the chance to see this summer’s The Incredible Hulk, the movie makes this point by fading from an overhead shot of Rocinha to that of a tropical jungle, subtlety be damned.) The buildings in Rocinha are just as natural as the trees.

How does that translate back into our shopping mall parking lot? It means that although the relationship between the parts, for example the lanes, the spaces and the paint that demarcates them, must be defined, the length of the spaces or the thickness of the demarcations do not have to be identical from one element to another. The chaos of nature requires that they be slightly different from one to the next, and that means that the people who make them must be able to make decisions while they are building. Simply copying an AutoCAD drawing is unnatural. The design must be translated into a language that instructs the builders to make configuration choices while constructing the defined forms. This kind of language is how builders have made traditional towns and how DNA makes organisms.

Separating design from configuration also allows us to make a second attempt at city planning. The plans of modernists all had fixed configurations, and their failure to adapt to their context meant the failure of urban planning. The conflict between design and configuration planning dates back even further, to the 19th century plans for Barcelona and Paris. In Barcelona, Cerda planned a grid of square blocks through which he ran grand diagonal avenues. Those were only two design elements in a very strict configuration that was made possible only by the enormous economic pressure to expand Barcelona. In Paris, Haussmann did not have the luxury of expanding the city with blocks, he had to upgrade a city of blocks that already existed with a new design element, the grand avenue. He deliberately left the configuration of his avenues open until they were completed, and placed them where he met the least resistance. Their effect on Paris is even today essential to life, and they could not have been realized unless their configuration was left adaptive.

What would a natural urban design look like? It must first be a definition of parts that must be related to each other in order to create urbanity. Describe the relationship between the avenue and the streets, the streets and the alleys, describe the relationship between the avenue and the pavement, the pedestrian crosswalks and the shade trees. Describe the relationships with the buildings without delimiting their size and shape. The city builders will then decide in what configuration these elements need to be to fit their context, and the resulting built form of the city plan will be perfectly natural as well as fully planned.

Classicism describes itself as the imitation of nature. Complexity, on the other hand, does not imitate. It is nature, applied to different problems. To create the urban design of our time requires not adopting a certain style or program, but ensuring that any style or program can be adapted to a particular context. It only requires us to use different tools than what we have become accustomed to.

Further reading:

Complex geometry and structured chaos, part I and part II.

Complex geometry and structured chaos part II

Complexity, to employ the definition proposed by Jane Jacobs in the final chapter of Death and Life of Great American Cities, is a juxtaposition of problems. This implies that a complex solution is a juxtaposition of solutions: fractal geometry.

How does the way we build arrive at complex solutions to complex problems without driving the builders to madness? How can we solve problems which exist at every scale in space, but also exist at every scale in time? Let’s take a look at St. Paul’s Cathedral in the City of London.

Let us focus on two different parts of it, the dome and the belltower. At first sight, there is nothing that a dome and belltower have in common. They are two different forms that solve two very different scales of problems. And if they had been built very far apart in two different neighborhoods of the city, one would never even associate them together. Yet in this case they are not only “dome” and “bell tower”, but they are also part of a greater form we call “St. Paul’s Cathedral”. That is to say, their form not only solves the problem of providing a dome and a bell tower, but it also contributes to solving the problem of providing a cathedral. Several scales of solutions are juxtaposed in the same space in order to form a complex solution. How was this result achieved?

Perhaps the architect Sir Robert Wren was a genius, but intuitively we doubt that, since the geometry in St. Paul’s cathedral is very similar to the baroque geometry employed throughout Europe at the time. And when we think back to how the Gothic cathedrals were built, very slowly, sometimes over more than a century, they were necessarily built by more than one architect. If they were all geniuses, then they must have been lucky to find so many geniuses idling about in medieval Europe. That sounds impossible given that medieval cathedrals appear to be even more complex than St. Paul’s cathedral, even though more people worked on their construction over a greater timespan. The sublime Antwerp cathedral, for example, was built from 1351 to 1521, and never completely finished.

There has to be a key to this riddle. How did we lose the skill to make this kind of complexity?

Since Leone Battista Alberti heralded modernity (not to be confused with modernism) in architecture, and until the mid-20th century, architects spent their first days in training learning to draw the classical orders. These classical orders supposedly held the finest refinement of western civilization’s building culture, having been in use since Greek antiquity and maybe earlier. It was an architect’s duty to reproduce this culture by learning the orders. Any deviation would certainly cause the doom of civilization. What the orders actually consisted of were fractal nesting rules, settled on more or less accidentally through the ages. Since the abstract concept of fractal nesting would not be discovered until Benoit Mandelbrot’s work in the 1970’s, the orders were simply understood to be unquestionable tradition. Since they were very simple local-form rules, any architect could use them to make his building, and they could be taught to any laborer working on any specific sub-section of a building without his having to know his role in the form of the whole. They could even be used to make simulations of the building, drawings and scale models that would later be used to convince patrons to fund construction. The rules were always the same. Only the problems to be solved changed.

Let’s take a look back at Wren’s cathedral. What does the dome consist of? Nested structures, including columns. What does the bell tower consist of? Nested structures, including the same kind of columns. The two different problems to be solved, dome and bell tower, also happen to share the same nested problems, and when they share a solution to this problem, they become connected into a whole.

Once we are aware of this rule we no longer need a necromancer to reanimate Wren in order to build an addition to the cathedral. We can simply decompile the geometric rules and apply them to solve the new problems we face. Whatever we produce that way will belong to the cathedral as much as the original parts. But we can also extend this to the scale of an entire city. If we apply these geometric rules to build a house or an office tower, it will appear to belong as much to St. Paul’s as the bell tower and the dome do. This enables us to achieve the complexity limit of urbanism. And when we look at all the great cities of the past, Paris, Rome, Venice, Amsterdam, Mediterranean hill towns, what we find is that they look whole because the builders who made them were all using the same rules in order to solve their individual problems. They didn’t realize they were doing it, they were just doing it because that’s how things were done.

If the classical orders were so great, why are they no longer being taught? Up to the 19th century, building technology changed very little, and so simply repeating the tradition was enough to create complexity. When metals and glass became massively affordable in the industrial revolution, architects faced a puzzle. Although the traditions succeeded at creating complex solutions, they were no longer solutions to problems that were relevant to anyone. Some architects experimented with new rules for nested structures using the new materials, more or less compatible with the old rules, and that gave us Art Nouveau and the Eiffel tower, for example. And some more radical architects, such as Louis Sullivan, said that modernity required the invention of a whole new architecture, and this became known as modernism. The modernists were right to declare the classical orders irrelevant, but in their rejection of the very foundations of architecture, the application of simple nesting rules, they also made it impossible for themselves to create complex buildings, and the result is the architectural wreck that unfolded starting in the 1930’s. The worse culprits, no doubt, were those modernists like Le Corbusier and even Albert Speer (bet you wouldn’t think he was a modernist) who favored abstraction and repetition in architecture. Abstraction is only the denial of complexity, the physical nature of our universe. It is the architectural equivalent of playing ostrich.

Post-modernism tried to bring back traditional forms without really giving up modernism, and that was a disaster perhaps worse than modernism was. Since post-modernists did not create nesting rules for their architecture, and on top of that were bringing up forms that were solutions even less relevant now than when they were abandoned, the result was a worldwide goofy architecture that everyone mocks as pastiche.

Some architects have been stumbling upon the right path these last few decades. The most remarkable effort has been the remodeling of the Reichstag in Berlin by Norman Foster.

The old building represented the federalist traditions of Germany, but also had to be adapted to the new philosophy of popular democracy. Foster built a glass dome from which the people can look at their politicians at work while enjoying a wonderful panoramic view of Berlin. Foster nested a new solution to a new a problem within the traditional geometry of the Reichstag, and thus created complexity that is relevant to the problems of today.

Architecture is, ultimately, just the repetitive computation of simple geometric rules to solve complex problems. Necessarily that creates complex solutions, and truly fractal buildings. With the right ruleset, anyone can do architecture, and by extension, great cities. The rules guide your hand.

Cinderella architecture

If you ever find yourself speaking to an architect at a party, most likely the word transparency and the supposed need for it is going to come up over and over. This is a recent concern for the building arts. Modern architecture, traditionally, has been philosophically focused on honesty of materials, or the meaning of forms. Transparency is in a way a renunciation of architecture. Its purpose is to make the form of a building as unnoticeable as possible. Architecture just gets in the way, so making it unnoticeable is the best design choice. Nothing is the new something.

Glass-covered office buildings have been the norm for decades, mostly because corporations are not taken in by architectural fashion and just want regular, boring, functional floor space. The glass curtain wall dominates the office parks and central business districts the world over. The physics of light being what they are, for most of the business day the building offers nothing but a mirrored glass box to the world, or for worse buildings, repeating checkerboard or line patterns (Class II behavior in Stephen Wolfram’s categories). The Houston-style CBDs have been described as soulless and boring, But something strange happens when the sun sets. The soulless, boring cluster of glass boxes becomes a setting worthy of postcards. A glass city at night has a completely different emotional and psychological effect on its visitors, something I’ve witnessed many people express, and have felt myself.

The patterns that the interior of an office building takes are controlled by its inhabitants, and, so far as the architect who designed the thing is concerned, purely random. And what happens at night to a glass tower is that the architecture of the building, the geometry controlled by the architect, blacks out and becomes indistinguishable from the background, while the lit floors are exposed for everyone to see. This nocturnal geometric transformation we can call the phenomenon of Cinderella architecture.

That a soulless glass city becomes beautiful at night cannot be accounted for only by lighting. A parking lot is lit up at night as well, and no one feels any better about it. Instead, I believe that the phenomenon of Cinderella architecture is an example of structured chaos. The office buildings, while no longer being themselves visible, create common frames within which random events are pictured. These random events can be as simple as lights being switched on or off, and can go into deeper details such as the placement of furniture. The result, seen from far away, is a geometric pattern that is overwhelming in complexity, especially when occurring in thousands of windows over dozens of buildings.

Looking for evidence of this phenomenon, I went to Paris’ version of the Houston CBD, the La Défense office city in the first ring outside Paris, in the sunset hours of a working day. La Défense holds a reputation for having the most boring kind of corporate architecture, but even this small district benefits from the Cinderella effect. Here are some pictures that I hope will illustrate my point.

The regular day-time modern business district.

The same spot at night.

The lower floors of the building on the left are not occupied and the visual complexity of the building suffers from it.

This building was one of the first built on the site, and is one of the most obsolete, but it lights up as much as any other at night, if not more.

How can the most routine kind of modern corporate architecture be creating such complexity? The answer is that it is not the buildings that create the complexity, it is the people living in them, adapting their environment to suit their activities. What makes it beautiful is the common frame that this adaptation takes place in. The frame is the architecture, the complexity is the adaptations that people make with it.

Complex geometry and structured chaos

Fractal geometry has infiltrated popular culture since it was formalized in the early 80’s from the works of Benoit Mandelbrot. While it has been used to study the form of cities by researchers such as Pierre Frankhauser and Michael Batty, the insights to be drawn from this field of mathematics have not yet penetrated the field of urbanism, defined as the construction of cities. Connecting the fractal city by Nikos Salingaros approaches the topic by asking what type of city is fractal, without going into depth as to how a fractal is made. Christopher Alexander, in his second tome of The Nature of Order, The Process of Creating Life, begins to develop profound ideas on the topic, which he had hinted to in The Oregon Experiment and A New Theory of Urban Design.

The basic quality of fractal geometry is that it is recursively-defined geometry; it must be described in terms of itself. A triangle, in basic euclidean geometry, is defined by the connection of three vectors at their extremities. Euclidean geometry is built up by combining basic elements into different shapes. A point becomes a line, which becomes a triangle, which becomes several different kinds of polygons, and so on. (A famous introductory architecture textbook, Architecture: Form, Space and Order by Francis D. K. Ching uses this method.) Fractal geometry does not take this approach of combination. Instead of using a triangle to make a square, in fractal geometry we use a triangle to make another triangle, such as this Sierpinski triangle:

A Sierpinski Triangle

At each step we use the results of the previous step and repeat some procedure, in this case either adding two copies of the previous object below the current one (composition) or replacing the three large triangles each by a copy of the object (decomposition). Both approaches will generate the Sierpinski triangle over an infinite number of repetitions.

The words generate and infinite are very important. It is these two words that make fractal geometry so completely different from euclidean geometry, which can be drawn instantaneously. Because fractal geometry is recursive, it is in theory infinitely complex, and the only way to see what a fractal object will look like is to run the computation that generates it until we grow tired of watching the process unfold. It is, by its own nature, surprising, unpredictable, and thus emergent.

The idea of objects substituting themselves for copies of themselves is nothing that revolutionary. It is the basic process that underlies all living things. In a living system a starting point, the embryo, contains a program, DNA, that will be multiplied into trillions of cells. The cells follow the transformations described by their DNA codes by taking certain actions depending on their environmental factors and previous states. (Alexander uses the example of a bone, whose shape evenly distributes structural stress across its surface, by claiming that the form of a bone emerges from a program telling cells to add bone mass where the stress is most intense.) Because living systems are the result of recursive transformations, it should not be a shock that they exhibit the properties of fractal geometry. The inward-out, decentralized growth of living things makes possible complexity in nature. Benoit Mandelbrot made this obvious when he wrote The Fractal Geometry of Nature, a book that pretty much started the fractal revolution by providing a mathematical framework for understanding real physical space.

Coming back to our preferred subject matter, cities and their construction, there is something very profound going on in the construction of cities if Mr. Frankhauser and Batty can calculate an index of “fractality” that is higher for some cities than for others. It means that in the process of building cities humans have unconsciously created complexity by adopting certain processes at certain times, and forgotten or abolished them at other times. That is a fascinating topic of discussion, a debate which has been at the heart of the profession going back perhaps further than the vastly different paths taken by Haussmann and Cerda in the 19th century, and one that must be at the heart of the profession of urbanism today more than ever. What is the alternative to planning and deliberate design of cities, which have nothing but a history of failure to show for themselves?

I’ve explained in a previous article how the very purpose of building cities is to create networks of buildings that handle chaos, the everyday uncertainty of future needs against the permanence of individual buildings. The very act of growing a city is an act of differentiation, creating something different from what currently exists as part of the city’s network of buildings in order to fulfill a need that the existing building stock cannot fulfill. (In other words, adaptation to changing circumstances.) These differentiations can be the creation of new public spaces, such as the boulevard construction initiated by Haussmann that provided large-scale connectivity to the city of Paris, as well as the construction of new, unforeseen buildings. But this is where the architectural design approach to urbanism runs into a major problem: how can beauty and order be created out of something that must necessarily be different from everything else?

The answer to that question is hidden in Benoit Mandelbrot’s greatest discovery, the Mandelbrot Set.

The algorithm that generates the Mandelbrot Set is, like those behind all the beautiful complex structures, extremely simple. It is a chaotic algorithm that “spins” within the boundaries of 2 and -2. For given coordinates in the plane made up of the normal and complex numbers, each coordinate will either spin forever in the orbit of radius 2, or escape after a determined number of iterations. The coordinates which never escape are defined as being part of the set.

A black-on-white picture of the set is by itself very intriguing, but the true beauty of it is not revealed until we apply a system of transformation to the coordinates that were thrown out of it. If, each time we throw out a pair of coordinates, we assign to it a number equivalent to the number of iterations it took to figure out it didn’t belong in the set, we will form groups of chaotic equivalence. And once we apply a single, shared transformation (a “DNA code” for the chaotic equation) to these sets, in this case defining a color for each iteration that threw out some coordinates, applying this color to these coordinates while drawing the Mandelbrot Set, we will generate this kind of geometry:

This is what I refer to as structured chaos. By applying a shared system of transformation to chaotic events, we obtain complex geometry. Shared transformations are the source of the new symmetric property of fractals known as self-similarity, and they are also the source of the wholeness and beauty of those chaotic systems called life, including cities.

Reflecting on the way cities have been built throughout history, the most beautiful places have been those that have shared transformations while creating differentiations. The city of Venice, which continues to inspire architects despite their inability to live up to its beauty, is a perfect example of shared transformations creating wholeness out of chaos. But to understand how to create symmetry by self-similarity, one has to be able to decompose buildings into their different scales and chaotic fields (differentiated elements).

The tradition of teaching the classical orders in architecture was once an imperfect approach to granting architects this skill. The classical orders are one form of transformation system, where large-scale elements, the column, the entablature, are decomposed into smaller-scale elements, the capital, the shaft, which form the large scale elements. And so when many architects, trained to share this transformation system as part of their skill set, worked on completely different buildings, their work could easily form a larger whole; whenever they hit similar problems, they would employ the similar solution they were trained to employ. While two buildings may have completely different sizes or roofs, or one could have a bell tower while the other didn’t, if both buildings had windows and columns, the windows and columns would be made the same way, and thus symmetrical to each other. This is how every building in a city was tied together in a web of geometric relationships, and it is the density of these relationships that gave cities their quality of wholeness and beauty. This property goes beyond the scale of the classical orders. It is also true of ancient Asian cities, or the mythic New York of the 1940’s.

Mythical Manhattan

Sadly the unending race for pure originality and the abandonment of hierarchical geometry by the architectural profession has made the creation of such cityscapes impossible. This has made the modern architectural profession largely parasitic of the city, and their professional ruin easily explained. Given what we now understand about generating geometric wholeness out of chaos, is there anything that justifies, other than a desire for euclidean perfection, the creation of rigid euclidean plans for cities? I have not been able to find any. The work of urbanism must be about two fundamental aspects: defining a system of transformations that will apply to all unforeseeable acts of construction in the city, at all scales, and creating the connective public space that will bind the different buildings together.

Returning to Nikos Salingaros’ question, the kind of a city that is a fractal is the kind that is made by applying shared transformations to chaotic events. This is the holy grail of urbanism in the 21st century. With this knowledge we can finally surpass the classical city, bury the demons of Le Corbusier and the C.I.A.M. while embracing all that technology has to offer to urbanism.