Tag Archives: complex design

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.


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.


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


Vauban, l’intelligence du territoire

Hommage a Vauban 1969

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

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.


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.


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.