Tag Archives: mandelbrot set

The Fundamentals of Urban Complexity

This is part II in an ongoing series of excerpts of an article set to be published this summer in The International Journal of Architectural Research, tentatively titled The Principles of Emergent Urbanism. Click here for part I, The Journey to Emergence.

The qualities of an emergent city

The adoption of mass-production processes, or development, in substitution for spontaneous urban growth in the mid-20th century created for the first time a phenomenon of alienation between the inhabitants and their environment. While the physical features of spontaneous cities could be traced to complex histories of families, businesses, and organizations, the physical features of planned cities owe their origin only to the act of planning and speculation. This has severe consequences towards the sustainability of place as there will not grow any particular attachment by the residents, their presence there being only a temporary economic necessity and not the outcome of their life’s growth. Mass-production of the environment left people as nothing more than consumers of cities where they used to be their creators. A building culture was replaced with a development industry, leaving the landscape culture-less and with no particular sense of identity. This took place despite the evidence that a building which has a unique history and has been fitted to someone’s life, as opposed to speculatively produced, generates market value for that property. (Alexander, 1975) This is why, although the demolition of so-called “slums” to replace them with modern housing projects created a great deal of opposition against urban renewal programs, the demolition of the housing projects later on did not lead to a popular preservationist opposition. They were not the physical expression of any culture.

In additional to cultural patterns, spontaneous settlements also have a peculiar morphology that has not successfully been imitated by modern growth processes. Spontaneous settlement processes give individuals full freedom to determine the boundaries of their properties. Spontaneous settlement is one where total randomness in building configuration is allowed, with no pre-determined property lines acting as artificial boundaries. Buildings and building lots as such acquire general configurations comparable to cell structure in living tissues, unique sizes and boundaries that are purely adapted to the context in which they were defined. In the absence of abstract property boundaries, property rights are bounded by real physical limits such as a neighbor’s wall. (Hakim, 2007)

Very attractive spontaneous cities have a specific pattern of the urban tissue. It consists of similar vernacular buildings that appear very simple when considered individually, but produce a visually fascinating landscape when considered as a whole. This is a form of fractal geometry. In mathematics a fractal is a geometric object of infinite scale that is defined recursively, as an equation or computation that feeds back on itself. For example the Sierpinski triangle is defined by three triangles taking the place of one triangle as in figure 4.

Sierpinski_triangle_evolution

Figure 4. A triangle triggers a feedback function that produces three triangles, which themselves trigger the feedback function to produce nine triangles, and so on. This process can unfold as long as computational resources can be invested to increase the complexity of the object.

The Mandelbrot Set is a much more interesting fractal that is defined as a simple recursive mathematical equation, yet requires a computation to visualize in its full complexity. When computing how many cycles of feedback it takes for the equation to escape to infinity for specific coordinates, figure 5 is the outcome.

Mandelbrotset

Figure 5. The image on the right is a deeper magnification of the image on the left, produced with a narrower range of coordinates as the input of the Mandelbrot set’s feedback function.

In addition to its remarkable similarity to natural phenomena, this form of geometric order informs us of a very important law in geometry: a feedback loop that is fed through the same function will produce an ordered but unpredictable geometric pattern out of any random input.

This tells us why cities of vernacular buildings have such appealing geometric properties at the large scale, despite being often shabby and improvised at the scale of individual buildings. Shanties made of scrap metal and tarp look rough at the scale of the material, but because multiple shanties share the construction process and originate from similar feedback conditions they form an ordered geometric pattern with its specific “texture”. The same process takes place at other scales of feedback, for example the production of a door. Whether the input for one door is larger, taller, wider than another door, if the same production process is employed the two doors will contribute to the overall fractal order of the urban space. This law has been employed not only in traditional and spontaneous cities, but also for modern urban planning initiatives. In the New York City neighborhood of Times Square the structure of billboard advertisements is defined by a building code that determines their configuration in relation to the configuration of the building. The outcome is a unique tissue of advertisement billboards that has become more characteristic of the neighborhood than the buildings themselves, which are not produced by a shared feedback function.

Fundamentals of urban complexity

Christopher Alexander showed in A City is not a Tree (Alexander, 1965) that social and economic networks formed complex semi-lattice patterns, but that people who observed them limited their descriptions to a simple mathematical tree of segregated parts and sub-parts, eliminating connections in the process. (Figure 6 compares the structure of a tree and semi-lattice.) In attempting to plan for urban structure, a single human mind, without a supporting computational process, falls back on tree structures to maintain conceptual control of the plan, thus computing below spontaneous urban complexity, a phenomenon that is consistent with Wolfram’s theory of computational irreducibility of complex systems. (Computational irreducibility states that the only accurate description of a complex system is the system itself and that no abstraction or reduction to a simpler process is possible.) Nikos A. Salingaros later detailed the laws of urban networks in Theory of the Urban Web. (Salingaros, 1998) Network connections form between nodes that are complementary, and therefore the complexity of networks depends on an increasing diversity of nodes. Salingaros describes the urban web as a system that is perpetually moving and growing, and in order to do this the urban tissue has to grow and move with it. Consider for example the smallest social network, the family. Debate over accessory units or “granny flats” has intensified as normal aging has forced the elderly out of their neighborhoods and into retirement complexes, while at the other end of the network young adults entering higher education or the labor market vanish from a subdivision, leaving a large homogeneous group of empty-nesters occupying what was once an area full of children, and often forcing school closures (a clear expression of unsustainability).

treelattice

Figure 6. A comparison of a tree pattern on the left and a semi-lattice pattern on the right. The tree structure is made of groups and sub-groups that can be manipulated separately from others. The semi-lattice pattern is purely random without distinct sub-parts.

These social networks grow more complex with increasing building density, but a forced increased in density does not force social networks to grow more complex. For instance the spontaneous settlements of slums in the developing world show remarkable resilience that authorities have had difficulty acknowledging. Because of squalid living conditions authorities have conducted campaigns to trade property in the slum for modern apartments with adequate sanitary conditions. To the authorities’ befuddlement some of the residents later returned to live in the slum in order to once again enjoy the rich social networks that had not factored in the design of the modern apartments and neighborhoods, demonstrating that the modern neighborhoods were less socially sustainable than the slums.

In commercial networks, space syntax research (Hillier, 1996), using a method for ranking nodes of semi-lattice networks, has shown that shops spontaneously organize around the multiple scales of centrality of the urban grid at its whole, creating not only commercial centers but a hierarchy of commercial centers that starts with sporadic local shops along neighborhood centers and goes all the way to a central business district located in the global center of the spatial network. The distribution of shops is therefore a probabilistic function of centrality in the urban grid. Because the information necessary to know one’s place in the hierarchy of large urban grids exceeds what is available at the design stage, and because any act of extension or transformation of the grid changes the optimal paths between any two random points of the city, it is only possible to create a distribution of use through a feedback process that begins with the grid’s real traffic and unfolds in time.

The built equilibrium

Although they may appear to be random, new buildings and developments do not arise randomly. They are programmed when the individuals who inhabit a particular place determine that the current building set no longer provides an acceptable solution to environmental conditions, some resulting from external events but some being the outcome of the process of urban growth itself. It is these contextual conditions that fluctuate randomly and throw the equilibrium of the building set out of balance. In order to restore this equilibrium there will be movement of the urban tissue by the addition or subtraction of a building or other structure. In this way an urban tissue is a system that fluctuates chaotically, but it does so in response to random events in order to restore its equilibrium.

This explains why spontaneous cities achieve a natural, “organic” morphology that art historians have had so much difficulty to describe. Every step in the movement of a spontaneous city is a local adaptation in space and time that is proportional to the length of the feedback loops and the scale of the disequilibrium. For spontaneous cities in societies that experience little change the feedback loops are short and the scale of disequilibrium small, and so the urban tissue will grow by adding sometimes as little as one room at a time to a building. Societies experiencing rapid change will produce very large additions to the urban tissue. For example, the skyscraper index correlates the construction of very tall buildings with economic boom-times, and their completion with economic busts. The physical presence of a skyscraper is thus the representation of a major disequilibrium that had to be resolved. (Thornton, 2005) The morphology of this change is fractal in a similar way that the movement of a stock market is, a pattern that Mandelbrot has studied. In general we can describe the property of a city to adapt to change as a form of time-complexity, where the problems to be solved by the system at one point in time are different from those to be solved at a later point in time. The shorter the time-span between urban tissue transformations, meaning the shorter the feedback loops of urban growth, the closer to equilibrium the urban tissue will be at any particular point in time.

Modern urban plans do not include a dimension of time, and so cannot enable the creation of new networks either internally or externally. They determine an end-state whose objective is to restore a built equilibrium through a large, often highly speculative single effort. They accomplish this by creating a large-scale node on existing networks. In order for such a plan to be attempted the state of disequilibrium in the built environment must have grown large enough to justify the immense expense of the new plan. This is why development will concentrate very large numbers of the same building program in one place, whether it is a cluster of 1000 identical single-family homes or a regional shopping mall, just like the skyscraper concentrates multiple identical floors in one place. Demand for these buildings has become so urgent that they can find a buyer despite the absence of local networks, the standardized building plan, or the monotonous setting. This is not as problematic for large cities for which a single subdivision is only a small share of the total urban fabric, but for smaller towns the same project can double the size of the urban fabric and overshoot the built equilibrium into an opposite and severe disequilibrium.

The mixed-used real estate development has attempted to recreate the sustainable features of the spontaneous city by imitating the morphology of sustainable local economic networks. It has not reintroduced the time dimension in economic network growth. Often this has resulted in a commercial sector that serves not the local neighborhood but the larger region first, consistent with the commercial sector being a product of large-scale economic network disequilibrium. In other developments the commercial sectors have struggled and been kept alive through subsidies from residential development, which is evidence of its unsustainability as part of the system.

References

Alexander, Christopher (1965). ‘A City is not a Tree’, Architectural Forum, vol. 122 no. 2
Alexander, Christopher (1975). The Oregon Experiment, Oxford University Press, USA
Hakim, Besim (2007). ‘Revitalizing Historic Towns and Heritage Districts,’ International Journal of Architectural Research, vol. 1 issue 3
Hillier, Bill (1996). Space is the Machine, Cambridge University Press, UK
Salingaros, Nikos (1998). ‘Theory of the Urban Web’, Journal of Urban Design, vol. 3
Thornton, Mark (2005). ‘Skyscrapers and Business Cycles,’ Quarterly Journal of Austrian Economics, vol. 8 no. 1
Wolfram, Stephen (2002). A New Kind of Science, Wolfram Media, USA

Fake Complexity – CCTV Headquarters

Engineering firm ARUP provided the complexity for this project.

From time to time I happen upon an attempt to “do” complexity that completely misses the point. In this first installment of many “Fake Complexity” topics, the culprit is Rem Koolhaas and his CCTV Headquarters for Beijing.

The choice of Rem Koolhaas is not random. Koolhaas is the only starchitect who understands anything about complexity, making this building that much more tragic. And the sad part is that the CCTV does have some complexity in it – the structural frame of the building is so complex that it had to be generated by computer software, explaining the weird, random shape of the mesh holding it up. The error is that this emergent structure is put to work holding up a horribly corrupt design, and the random pattern of the structure exists to match the similarly random physical stresses the design imposes. The global shape of the building itself is just an aesthetic decision by the architects, it is not the result of a complex process. Complexity only factors in after the architects have done their work. Form here does not follow process.

The fact that the building’s form is not emergent is also the reason why the people of Beijing do not take it seriously as a building. It has already been nicknamed “Big Shorts” by locals, because that’s the form the architects decided for it.

And a comment in passing about the architectural criticism linked above.

You might think that, like a good deal of Koolhaas’s work, the building is as much showmanship as architecture, but it evinces a quiet, monumental grandeur. Some of that is due to the color of the glass, which is a soft gray, almost perfectly echoing the overcast Beijing sky.

What the hell are these people on? Is overcast the new grandeur?

Bonus Fake Complexity Update – Tour Signal by Jean Nouvel

Jean Nouvel was recently announced as the winner of the contest for a second iconic tower in Paris La Défense, which has absolutely nothing to do with him being awarded the Pritzker Prize a few weeks ago. Here is Jean Nouvel’s idea of complexity: a wallpaper of the Mandelbrot Set on the gargantuan atriums of his building. This is an even faker kind of complexity than Koolhaas used, as in that case there was a legitimate emergent process used in the structure of the building. What Jean Nouvel is doing is taking a 2D printout of a complexity-generating computer program and slapping it on the walls of his building. This being kitsch projected to every corner of Paris, it becomes a crime. Thankfully, given the current conditions of global real estate markets, it will meet the same fate as Jean Nouvel’s doomed 90’s project for La Défense, the Tour Sans Fins, and this is why it doesn’t deserve its own Fake Complexity update.

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.