Tag Archives: symmetry

Chaos re-emerges

1960’s psychedelic art?


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


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.


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










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

Fitness is about symmetry

On City Comforts they make the case that Daniel Libeskind’s abominable extension to the Victoria and Albert museum could be made to fit with the museum and South Kensington in general. To support this they link to this Good City post on traditional neighborhoods and modern architecture, which argues that the only thing necessary for a modern building to fit in a traditional neighborhood is that the site plan be well integrated into the public space. As evidence for this they show a picture of a modern house in Lincoln Park, Illinois. The house in that picture actually has qualities that the Libeskind addition doesn’t have, mainly that it has several scales of geometry symmetrical with its context. The walls of the modern building are symmetrical with the walls of the historic buildings, and all buildings are thus linked together into one fabric. The windows are different, but that is the outcome of adaptation to changing needs, which requires new scales of geometry.

Integrated site plans are a necessary but insufficient condition of good urbanism. The idea that you can make a whole out of an anything goes architectural approach was rejected even by nihilist-leaning Rem Koolhaas. In complex systems symmetry is found because it is the most physically efficient process. Remembering that the definition of symmetry is a preserved structure after undergoing a transformation, we can more clearly define what kind of modern architecture is good for a historic neighborhood. A new building fits into a historic neighborhood if its geometry can be derived from an old building with the fewest possible transformations. That is to say, all reusable scales are reused, and the modern building has all the useful new geometry that we require in our time. What makes a neighborhood historic are those reusable scales and not the age of the buildings. The building in Lincoln Park reuses the wall scales of its neighborhood. Now try picturing it with bright rainbow walls and neon. Does it fit anymore? No more than does the Libeskind extension.

Rue d\'Aubervilliers, Paris 19

Here is a similar building in Paris’ 19th borough. Its site plan is as urban as could be made, but it is definitely not Parisian. It has no scales in common with its neighborhood. This happened despite the fact that the majority of real estate in Paris is modern. Modern Paris goes unnoticed because it is symmetric enough with its neighborhood that it contributes, or at least doesn’t take away from, the city as a whole.

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.