Notes on an Incomplete Architecture



 An On-line Book Draft




First Draft

Presented at the

Congress for the New Urbanism 2000



Michael Mehaffy

900 Cornell Street

Lake Oswego, Oregon 97034

(503) 635-6922



Author’s Note:  This paper was offered as a first draft at the Congress for the New Urbanism, June 2000.   It is a work in progress, with additional material, revisions, footnotes, and illustrations still to come.   Comments and ideas are appreciated.





The beginning of the twenty-first century finds the design professions at a crossroads, with increasingly fractured movements and little overall sense of direction.   Meanwhile, the work of twentieth-century mathematics and related fields have produced startling new insights into the structure of complex processes, with implications for the design professions that have only begun to be understood.  We discuss these insights, and suggest that they offer intriguing possibilities for emerging trends in the architecture of the next century.




The Incomplete Century

The Architecture of Complexity

Trees, and Other Unnatural Structures

Simple Rules, Vast Beauty

Politics, Economics, and the Shape of Culture

The Empire’s New Clothes

Notes Toward a New Science of Aesthetics

Conclusion: The Emerging Shape of the New Century





“I suggest to you that the analogy between aesthetics and logic is one of the undeveloped topics of philosophy.”


-                          -   A.N. Whitehead, 1937


“The final conclusion from (these) discussions… is the importance of a right adjustment of the process of abstraction.  … The higher animals are distinguished from mere life, by their abstractions, and by their use of them.  Mankind is distinguished from animal life by its emphasis on abstractions.  The degeneracy of mankind is distinguished from its uprise by the dominance of chill abstractions...”


-                          -   A.N. Whitehead, 1937


 “When we ourselves become abstractions, we are lost!”


    -   Frank Lloyd Wright, 1930




As I write this, we are experiencing a great deal of hype surrounding the new millennium.  Most of us are well aware that the change of centuries is a mathematically arbitrary occurrence -- based not on natural cycles but on the simple abstractions of the decimal system, and calibrated (slightly in error) to a historical Christian event.  (According to the Chinese calendar, it is currently the year 4625.)  And yet western civilization has always defined and ordered itself according to decades and centuries. Ends of centuries have usually been historically difficult times, bringing together a rush of unfulfilled aspirations, doomsday predictions, and anxieties about the future.  The beginning of each new century has often brought a release of optimism and new possibility, and with it a synthesis of older artistic threads into new fabrics… 


At the beginning of the twenty-first century, it seems that the threads are more like fractured panes of glass, splitting into smaller and smaller shards…  modernism… post-modernism… deconstructivism… neo-modernism… increasingly hostile factions battling and belittling one another, as idealism gives way to cynicism.  The role of artistic culture is challenged as never before, as powerfully complex and corrosive economic processes dominate our lives and increasingly shape our world in ways that we still poorly understand.   We are left disillusioned, isolated, without direction, while the prodigious machine revs up ever higher.


Taken as a whole, it is not easy to defend the quality of the built environment created in the second half of the twentieth century.  An honest student of the nature of things, not blinded by one fashionable artistic dogma or another, perceives a fundamental, qualitative difference – a distinct poverty of beauty and character – that increasingly emerges with time over the span of the twentieth century.   (Tourists betray this perception when they travel to Tuscany or Santorini to admire the complex ramble of vernacular beauty -- not to, say, Houston to admire the crystalline towers.)  Against the vast beauty of nature or the humble richness of vernacular towns, our century’s work compares poorly indeed.  There is now a growing body of mathematical and scientific evidence to support this perception.   This paper is a very brief, perhaps absurdly brief, overview of that growing body of knowledge, and its implications for the design professions.


As we enter the twenty-first century, the well-documented state of the built environment is this:  simple, well-designed vernacular buildings, and the knowledge to make them, are disappearing around the world.  New homes and small commercial buildings the world over are clumsy composites of fakery and schlock.  The fabric of once-integrated neighborhoods has been sliced to shreds by roads and parking lots.  Industrial buildings are remarkably crude and artless.  Shoe-box office and apartment buildings, shoddy imitations of modernist masters, are blighting cityscapes across the globe.


There are people who actually believe that none of this really matters – that the twentieth century’s undeniable advances in living standards and medicine and technology are far more significant than what they regard as the mere external appearance of the machine of society.  Of course it would be nice if our buildings were prettier, but that is really only a matter of cosmetics.  Some of these people, incredibly, are architects.  For them, a few rare pieces of fine art amid the general mess will suffice.  And perhaps they’re right – perhaps it doesn’t matter whether we experience a simple rich beauty in our routine daily lives, whether our buildings complement or desecrate the beauty of the natural environment, whether our civilization is shaped, as, say, Athens was in the time of Pericles, by beautiful and elegantly functioning architecture.


Perhaps the way we shape our built environment has no relation to the way we shape the natural environment.  Perhaps the unsustainability of our buildings has no relation to the unsustainability of our future.


Reductionism is the gift and the curse of our age.  We take the world apart into little pieces and put them back together again in all sorts of interesting ways.  But of course sometimes we have trouble getting all the pieces to fit back together, like the mechanic who discovers a few extra pieces after the car has gone back together.   Then we hope that it doesn’t matter – perhaps the car will run OK after all.   Or perhaps it will run a little too well – perhaps we will find ourselves like the sorcerer’s apprentice, having unleashed something we cannot understand or control.


Of course in most cultures throughout history, our reductionist treatment of the built environment would be regarded with dismay.    Churchill’s famous remark that “we shape our buildings, and thereafter they shape us,” would have found little disagreement before the Second World War.  It is only in our own “modern” era that the beauty and elegance of buildings has been seen as so much sugar-coating over their utilitarian function – their engineered functionalist assemblages.  The architectural arts have had to settle for a game of catch-up, accepting the utilitarian dictatorship but trying to make it the basis of their own art form -- “house as machine,” and other lovely projects of the sort.     


In the last millennium we have made wondrous if imperfect achievements in establishing democracy, open society, human rights, the rule of law – achievements, we should remind ourselves, once widely dismissed as impossibly utopian.  We have seen an explosion of commerce and technological creativity. Yet in an era when it seems that anything is possible, it seems ironically that the only thing that is not possible is a kind of connectedness and wholeness amidst all of this disconnected stuff – an integrity, a genuineness.  Did it ever exist in the past, or are we feeling nostalgia, as some claim, for a past that never was?  By our standards, the ancients were brutish and scientifically ignorant; yet it is easy to feel an admiration bordering on jealousy for the coherent beauty of their art and their built environment.  Through the eyes of an intelligent anthropologist, we can look all around the world through the ages, and we can indeed find that same coherence and beauty.  Science does tell us that we are missing something, that we have lost something in the bargain. The observant student of design history cannot escape the impression that we stand in time and place as an island of disconnectedness and fakery.   The more we try to copy the genuine, the more we destroy it, and the more fraudulent our effort becomes.  Then we quarrel with one another about our mutual forgeries.


The effort to find a mathematical explanation for what is beautiful and good is of course one that has stretched back thousands of years, spanning the works of Pythagoras, Plato, Vitruvius, and many others.   In the twentieth century, we have come to believe that the question of what is beautiful and good is completely relative and even meaningless.  The real explanation we sought was for what is rational, what is functional, what is logically ordered.  And we felt that in answering it we had arrived at a kind of pinnacle.  We embraced our abstractions, our machines, our crystalline pure geometries.  We felt that we were above time, above history, beyond the messiness of the world.  We had arrived at a wiser, purer age.  We were modern.


At the end of the twentieth century, we have been deeply humbled by our failures, by the revelations of the depth of our ignorance that the new sciences of the twentieth century have brought.  But that itself is a crucial bit of self-knowledge, as Plato himself would have counseled.   And now it is time to set about the humble task of applying the old lessons of history and nature, and the new lessons of mathematics and science, to an exploration of a richer and more satisfying architecture.  After all, it’s a new millennium.




The Incomplete Century


“There are more things in heaven and earth, Horatio, than are dreamt of in our philosophies.”


                                            -     Hamlet, Shakespeare


The beginning of the twentieth century was an almost unparalleled time of contagious optimism in human affairs, and in science and mathematics in particular.   Pioneering work seemed to be closing in on the last secrets of the atom, the workings of physics, the very structure of the universe.  It was almost as though a hidden vault of blueprints for nature itself had at last been found, and we had only to learn to read them. 1


In 1913, Alfred North Whitehead and Bertrand Russell published Principia Mathematica2, a landmark work that sought to lay the foundations of all mathematics.  In particular, Principia Mathematica sought to clear up, once and for all, some contradictions in mathematics that seemed to be preventing a complete and consistent description of nature.


A mathematical equation is, in the end, a nifty little model of some part of the world.  Plug in your starting conditions, crank the machine and voila! You can predict where your artillery shell will land, how thick your beam needs to be, how much current your hodroelectric plant will generate.  As mathematical and scientific descriptions became more and more comprehensive, it began to seem that the entire world could be completely described.  LaPlace’s fantasy of a demon that could know the entire universe from a few simple starting conditions began to seem theoretically and perhaps even actually possible3.


Georg Cantor’s work on the theory of sets in the 1880s led to the discovery of some flaws in this arrangement.4  It was noted by Russell and others that in Cantor’s work there are some very peculiar sets that contain themselves – for example, “the set of all sets,” which is a member of itself.  (How can that be?)  Such self-referential sets posed a real problem.  Mathematics, and set theory in particular, is in the business of breaking the world down into nice neat schemes of smaller and smaller subsets, in a well-organized hierarchical system.  The effectiveness of mathematical proofs depends upon this order.   The tangled “strange loops” that these self-referential elements formed opened the door to inconsistency and even absurdity.  


An example of this absurdity is what is known as Epimenides’ paradox, also familiar as the liar’s paradox.  In English, one can easily make self-referential statements  that logically can neither be true nor false.  Suppose I told you:  “I am telling you a lie now.”  Logically, if I am telling you the truth, then I am not telling you the truth; but if I am not telling you the truth, then I am telling you the truth!   Mathematical descriptions can have similar strange contradictions, with equally bizarre results. 


Principia Mathematica seemed to correct this problem, by constructing the foundation of mathematics out of the rigorous principles of a strict logical scheme.  Self-referential elements were to be excluded through what Whitehead and Russell called a theory of types.  The result, it was hoped, was a consistent and complete mathematical foundation. 


But it was not to be.  In 1931 a young mathematician by the name of Kurt Godel showed ingeniously that in Principia Mathematica – and in any related system – one could not prevent the formation of self-referential elements.5  Godel’s brilliant analysis created a numbering system for Principia Mathematica, and proceeded to twist that numbering system, pretzel-like, back upon itself.  The result was a self-description that failed – one that, in effect, contained its own liar’s paradox.  Godel thereby proved that at least some statements in Principia Mathematica, and related systems, could not be determined to be true or false.  Principia Mathematica, and, by extension, any descriptive system, was proven to be doomed to incompleteness.     


The years that followed brought one proof after another documenting varied forms of incompleteness in other mathematical systems, with broad implications for physics and philosophy.  These so-called “limitative theorems” echoed discoveries of limits in other fields.  Perhaps the best known was in physics, where Heisenberg’s uncertainty principle recognized that at least at the molecular level, an observer cannot escape having an effect upon the results observed – in effect, finding a “strange loop” where the observer is an unavoidable subset of his own observations.  It became clear that the issue was not a technical or trivial one, but a profound limitation of human thought.6  Though it had been recognized in various forms by the Greeks and many others throughout history, it was a limitation that had not been properly integrated into modern scientific thought.


Lewis Carroll, an admirer of Cantor and the other logicians of the nineteenth century, captured the problem beautifully in a very instructive fable.  He told of a country whose mapmakers recognized many inaccuracies and omissions in their maps, and set about creating more and more accurate and complete maps, requiring larger and larger scales.  Finally their map grew to be as large as the entire country. “It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well."  


A map has to be smaller, and simpler, in order to be useful. It as to leave some things out to be a map at all.  For we will be just as lost in a map that is too accurate as in the region it represents.7


The larger lesson is simple but painful for the optimists of the early twentieth century.  It is in the nature of descriptions – mathematical, linguistic, or otherwise – that they are abstractions, and therefore they must be simpler than the regions they represent.   As Whitehead himself said, an abstraction is nothing other than an omission of part of the truth.  Models must leave something out to be models – they must necessarily be simpler than the regions they model.  They will always be incomplete.  


This is a more fundamental limitation than one might realize.  It means that the world will always be more complex than any hierarchical scheme, and fundamentally cannot be reduced to a grand hierarchy.  There are always more things in heaven and earth, as Hamlet tells Horatio, than are dreamt of in our philosophies.  


For those in the planning professions, the lesson is inescapable:  there is a danger – as it happens, a very seductive danger -- of getting stuck in the neat and tidy hierarchical schemes we devise, and losing the vitality of the real world.  Because all planning and modeling systems are inherently incomplete, we must keep our systems open and continuously adaptive.  We must not allow ourselves to get locked in to conceptually attractive schemes that fail to respond to the ever-present and ever-evolving complexities of the real world.8


Whitehead recognized the tendency to confuse our own abstract schemes with reality,  and called it “the fallacy of misplaced concreteness.”  (There are those who believe that there is a great quantity of misplaced concrete in the world today…)  The fallacy is an unavoidable consequence of the nature of modeling and of human thought.  However, although we can never escape from this problem, we can model it, and thereby guard against it.  We can continue to perfect an architecture of openness and adaptation, recognizing that it will never be perfect… but it can be better. 



The Architecture of Complexity


“A man, viewed as a behaving system, is quite simple.  The apparent complexity of his behavior over time is largely a reflection of the complexity of the environment in which he finds himself.”


                                                                - Herbert A. Simon, The Sciences of the Artificial


In the field of mathematics, the decades following World War II saw a new level of innovation and conceptual ferment.  The war had brought teams of Allied mathematicians together in a Herculean effort to break German and Japanese codes.  The first useful electronic computers were developed to rapidly calculate possible code solutions. 


Great mathematical minds like John Von Neumann laid out the essential structure of programmable computers.  Insights they gained from computers seemed to have direct mathematical and logical analogies to other fields.  In one famous example, Von Neumann postulated what would be the minimum number of elements of a self-reproducing machine.  He described a machine that would have to have at least four elements, in two mating pairs.  Later geneticists were shocked to discover that Von Neumann was unwittingly giving a fairly good description of the structure of DNA9.


In the years following, computer development mushroomed.  Content-addressable storage meant that programs and data could interact in entirely new ways, giving computers a breathtaking virtuosity.  Entire new fields emerged, including cybernetics, game theory, artificial intelligence, and systems theory.


Systems theory, a new interdisciplinary field,  grew out of a key organizing feature of the new computers.  Herbert Simon, a pioneer of the new cybernetics, illustrated the feature in the following story.  Imagine two watchmakers, both of whom make beautiful watches, and both of whom receive many phone calls requesting more watches.  Each watch has 1,000 delicate parts.  One watchmaker prospers, while the other goes out of business.  Every time the unsuccessful watchmaker puts a watch down to answer the telephone, the watch falls apart, and he has to start his assembly all over from scratch.  The successful watchmaker builds his watches in sub-assemblies of ten parts, each of which has ten parts, and each of which in turn has ten parts.  When he answers the telephone, only a small part of his work is lost; moreover, he is able to quickly and efficiently build his watches10.


The early computer programs quickly exploited a similar strategy of hierarchical subroutines.  When a program needed to perform a certain task, it executed an appropriate subroutine, a sort of stock “mini-program” within itself.  That subroutine in turn might execute its own subroutines, and so on.  A relatively simple program with a number of stock subroutines could perform a huge variety of chores, by simply recognizing when the appropriate subroutine should be executed.  The programmer could enable the program to make those decisions by a few relatively simple logic tests:  “if this, then do that; if not this, then do something else.”


Simon quickly realized that computers were processing symbols, and that this capability could be used to allow them to solve problems heuristically, in effect by using continuously adapting rules of thumb.   His first artificial intelligence program,  Logic Theorist, was able to actually prove a number of theorems of Whitehead and Russell’s Principia Mathematica – in at least one case, better than the authors. 


Simon’s interest wasn’t so much what computers could do, but what humans could and did do, and how computer simulation could shed light on human cognitive processes.  His economic theory of “bounded rationality” – that managers and markets behave according to heuristics and optimized hunches, not according to rational knowledge -- won him the Nobel Prize in economics in 1984. 


Simon could fairly be described as an over-achiever.  Besides pioneering artificial intelligence, almost single-handedly inventing the new psychological field of cognition, and winning the Nobel Prize in economics, among other endeavors, Simon was one of the pioneers of systems theory.   At its core was the recognition that in many fields of human experience dealing with a large number of elements – biology, physics, economics, and many others – the elements were best understood as a hierarchical system. 


Systems theory had perhaps the most direct impacts in engineering and business management.   Indeed, its impact upon the organization of post-war society has been profound.  Whereas the previous business and engineering models were linear, rational, static, the new models of systems theory were non-linear, heuristic, and dynamic.   Rigid prescriptive methods were discarded and replaced by adaptive, responsive ones.


In The Architecture of Complexity, a landmark paper of 1962, Simon laid out his observations about the tendency toward hierarchical organization in even the most complex systems.   He noted that, as in the example of the watchmakers, there is a natural efficiency in such an organization, one that is exploited by biological and other natural systems.  He also recognized that the structure of hierarchies is far from perfect, and that there is indeed interaction outside of the hierarchical boundaries.  But he also observed a trait he called “near-decomposability,” the tendency to follow hierarchy in the main. 


Foreshadowing the new field of complexity, Simon summarized his insights:


“The fact, then, that many complex systems have a nearly decomposable, hierarchic structure is a major facilitating factor enabling us to understand, to describe, and even to ‘see’ such systems and their parts.  Or perhaps the proposition should be put the other way round.  If there are important systems in the world that are complex without being hierarchic, they may to a considerable extent escape our observation and our understanding.  Analysis of their behavior would involve such detailed knowledge and calculation of the interactions of their elementary parts that it would be beyond our capacities of memory or computation.”11


In the fields of architecture and planning, the “systems approach” became the model of design programming.  A new discipline of “design methods” became established, describing ways of gathering information and establishing the parameters of design.  It seemed we were finally rationalizing what had formerly been a mostly unfathomable artistic process.


Systems theory was a major improvement over the simplistic organizational methods of previous designers.  Adaptation and evolution were explicitly recognized.  The design context was seen as a process, not as a tableau of simple objects.  The Newtonian “billiard ball” model had given way to models of interaction and growth.  But there were still other lessons ahead.



Trees, And Other Unnatural Structures


It has been almost forty years since the mathematician, architect and troublemaker Christopher Alexander shook up the planning profession with a short and highly influential paper titled “A City is Not A Tree.” 12 


Alexander was one of the pioneers of the mathematical analysis and synthesis of design, applying the insights of systems theorists like Herbert Simon.  In 1962, he and Marvin Manheim developed a series of computer program for the hierarchical decomposition of systems for design analysis, called HIDECS.  The tree to which he referred was just another name for a simple hierarchical system.  In his paper, Alexander showed that the designs of urban planners of the post-war period were in fact tree structures, with little overlap between elements.  In a startlingly simple analysis, he showed that as an organizing system for planning a city, the tree lacks real complexity.  The cities formed according to its organization are, in a real sense, unnatural.


In his paper Alexander examined a number of modern “planned” cities and analyzed their structure.  He contrasted that with an analysis of what he called “natural cities” – cities that had evolved over many years through complex behavior and activity patterns.  The natural cities had far richer and more complex relationships than the modern planned cities, with their segregated and lifeless zones of “city center,” “government,” “shopping,” “residential,” “industrial,” etc.   Mathematically speaking, it seems that the older cities had something that the new cities utterly lacked – multiple semi-lattices or networks, overlapping one another in a vastly complex web of relationships that had evolved over many years.   That these relationships appeared to take the form of  “nearly decomposable hierarchies,” as Simon described, was not at all the same thing as saying that they were truly complete hierarchies.  As we have seen, in the totality of nature such a thing is impossible.


The lessons of “A City is Not A Tree” for twentieth century planners were profound:  rather than being more complex and sophisticated than the seemingly “primitive” cities of antiquity, the modern city schemes were in fact cruder and far less complex.  The modern designers, Alexander argued, were bewitched by their own schemes, and failed to notice the subtle complexities of the older places.  As a result their work measured up poorly indeed.


Alexander’s paper added significant fuel to the mixed-use movement, already championed by Jane Jacobs, the Goodman brothers, Bernard Rudofsky, and others.13  But whereas those observers were making aesthetic comparisons to great vernacular places and pointing out the deficiencies of our own time, Alexander was making a very clear mathematical analysis of the problem – a breakthrough for “hard” sciences, and one that made many people uncomfortable.   This was not the last time Chris Alexander would make people uncomfortable.


The insights of “A City is not a Tree” can in principle be applied to any structure within the human habitat.  For example, consider a common hierarchical house design, dominated by a central space, several hallways leading from it, and each in turn leading to several rooms.  A designer may well develop such a scheme from the “bubble” diagrams common in design.  However, there is no connection between the rooms, other than within the hierarchy.  The number of relationships between spaces is small, and the flow through rooms is limited.


Consider instead a courtyard house, a traditional design common in Mediterranean climates.  Each room is connected to the two rooms on either side of it, and to the courtyard itself.  The number of relationships between rooms is much greater.  Moreover, the flow through rooms is not merely up or down along one axis, but forms a network.  Movement can also occur on short side axes, through the center, and even in a circle.  The presence of such networks and “magic loops” is a key characteristic of great designs throughout history; but instructively, it is remarkably lacking in twentieth century design.


Alexander’s work on the “systems approach” to design led him to become one of the pioneers of early computer use in design.  His book, “Notes on the Synthesis of Form,”14 is a landmark in the field of design method.  Alexander and others attempted to use observations about the behavior of users and their interaction with their environment to map an optimal “fit” between user and environment, and thereby predict and guide optimal design.  


In his work with computers, Alexander made two significant observations.  First, like many observers, he found that the sheer quantity of interactions required to be processed for even a trivial design task can quickly become vast and overwhelming.  Second, over repeated processing he noticed recurring “patterns” of behavior, stable relationships that tended to form over time with similar but varying conditions.   Some of these were so self-evident as to make computer analysis seem pointless and absurd.  


For example, a pattern might be observed in which users frequently stop at the front of a building.  It is learned that they are seeking information about the location of their destination within the building.  They are in fact stopping to read an “entrance directory.”  All of the seemingly unwieldy computer information can be contained relatively simply in this “pattern,” along with a description of the way it would occur under varying conditions.


This insight about “patterns” led Alexander to develop a theory of design archetypes, called a “pattern language.”  Alexander’s decision to bypass the architecture profession and write a popular book on the subject did not endear him to many professionals.  His use of mystical terminology, a result of later explorations of philosophy and metaphysics, did not make matters any better.  Today he remains somewhat estranged from the architecture profession, and at the same time harshly critical of its shortcomings.


Since the publication of A Pattern Language, Alexander’s interest in mathematics has led him to study the geometry of design, and to discover interesting properties of the great designs of the past.  He still believes that there is much for twenty-first century designers to learn from the mathematics of design – particularly in light of new work in the field of mathematical complexity.



Simple Rules, Vast Beauty


z -> z² + c


- Mandelbrot Formula


Most of us at one time or another have marveled at the strangely beautiful Mandelbrot set, the roughly heart-shaped pattern in which one can zoom in on any small feature and find an astonishingly vast variety of pattern and form.  Many do not realize that the formula that generates this vast and often breathtaking beauty is exceedingly simple.  In fact it is the one at the top of this page.


The formula is really not a formula in the usual sense of the term, but an algorithm, a set of instructions for computing a changing value.  The most familiar example of an algorithm is a computer program.  The computer program that generates the Mandelbrot pattern actually looks something like this:


z = 0, c = pixel:

         z = z*z + c

         |z| < 4



The variable Z is a starting point on a two-dimensional array, such as a computer screen.  The algorithm takes the square of z and adds a factor based on the pixel coordinates.  It then checks to see if the new value of Z is beyond a certain range, and based on that test, it then changes Z and starts the whole process over.  As it does so, it uses the value of Z to assign a color to that spot on the screen.  As the process reiterates millions of times, it leaves a color trail across the screen – the familiar and gorgeous Mandelbrot pattern.15


This is an astonishing observation.  How can such a simple process generate such vast complexity?  Such seemingly “natural” beauty?  This is not some mysterious natural process, but a cold, hard, mathematical game.  Something very significant is going on here.


The key to this complexity is iteration – a step-by-step algorithmic process in which the previous results of the process determine the next step, and so on and so on.  As the sheer number of iterations becomes vast, the changing pattern can take on strange characteristics indeed.


The study of such “iterative complexity” exploded with the advent of high-speed computers, and indeed is hard to imagine having much of anything to study without them.  For unless someone devoted thousands of hours of manual calculation time, we simply could not have observed the Mandelbrot pattern and others of the sort, because there are simply too many iterations.  But the interesting thing about iterative patterns is that they are extremely common in nature. 


For many processes, iteration is rather mundane.  For example, in balancing a checkbook, one takes the starting balance, subtracts the first check; then takes that value, subtracts the next check, etc.  But consider a conversation, in which what I say affects what you say, and in turn that affects what I say, and so on and so on, until we have perhaps explored exciting new ideas that neither of us had had before.   In this way, perhaps, over many years we would add new words and ideas, and our language would grow richer and more complex.  It appears that human language and culture has evolved, through a similar process. 


Or consider this iteration: 


Take a number, add it to itself.  Add that number to the first number.  Add that number to the previous number.  Add that number to the previous number… and so on.


The series you get is


1, 1, 2, 3, 5, 8, 13, 21, etc


In mathematics this is called the Fibonacci series, and it can be found throughout the natural world.  Fibonacci patterns can be seen in the formation of leaves on trees, in the beautiful shapes of conch shells,  in the growth of shapes in our own bodies.  The Fibonacci series is actually one of the simpler forms of iteration.


It turns out that the natural world is absolutely crammed with iteration.  As we have begun to understand how iterative processes work to generate such variety, we see iteration almost everywhere we look:  in evolutionary biological processes, in the behavior of gases, in the formation of clouds, in the shape of beautiful rugged coastlines and vast mountain vistas.


In the shape, as we shall see, of beautiful things.


There is an intriguing relationship between the Fibonacci series and the Golden Mean, a long-established proportional method of design used in classical styles.  In the Golden Mean, the proportion of a is to b as the proportion of a and b together are to c.  In other words, take a, enlarge it by a certain percent, take that, enlarge it by the same percent… and so on. 


Iteration tends to create an interesting property found in almost all beautiful structures:  the property of self-similarity at different scales.  Familiar examples of self-similarity are the conch shell, the pattern of leaves and trees, the shape of undulating coast lines, the repeated patterns at different scales of mountain ranges.  Mandelbrot coined a term for these elements that repeat at varying levels: he called them fractals, from the Latin for “broken.”


I do not believe it is an exaggeration to say that iteration, and the related insights of complexity it helped to illuminate, together form one of the greatest scientific discoveries of the twentieth century.  Iteration has become the key insight of the study of complex interactions, where a high number of elements interact.  It has given us access to patterns and structures that were previously inaccessible to mathematical analysis or scientific understanding. Weather phenomena, chemical interactions, evolutionary biology, economic processes and the behavior of markets -- these are just a few of the many diverse subjects that have been illuminated.  Building on the complexity insights of Von Neumann, Simon and others, we have created nothing less than a new science of complex phenomena and their analysis.    


There is an interesting technological advantage to this understanding.  Now that we know where to look for these iterative patterns, we can find them all over in the music and images we wish to record, and develop algorithms to generate copies of these patterns.  In fact this is a key strategy used in much of the information compression technology today.  


We have only begun to explore the relationship between these iterative processes and our aesthetic perceptions. Perhaps in the years ahead we can develop greater insights into the applications of such processes to the development of a truly rich and complex architecture.  We have some fertile material with which to start.



Politics, Economics, and the Shape of Culture


Just now the political debate in the United States is focusing its attention on growth management and “sprawl.”  The Congress for the New Urbanism, an advocate of more traditional land development patterns as an alternative to post-war sprawl, is meeting this year in Portland, Oregon, a city some consider a model of effective growth management policies.   The CNU conference, titled “the Politics of Place,” seeks to examine the Portland “smart growth” experiment, and analyze its successes and failures. 


Although debate within the CNU is certainly lively, it is probably accurate to say that the overwhelming majority of members believe that at least some form of government action to manage growth is appropriate and necessary.  Taking an opposing view, conservative free-enterprise advocates are harshly critical of many of the tenets of new urbanism, characterizing them as unwarranted government restrictions on the freedom of individuals to live where and how they choose.  The enterprisers argue that market processes are the best way to accommodate the desires of individuals choosing to live as they see fit; furthermore, we have sprawl today for the simple reason that Americans have “voted with their feet” and chosen to live on large lots out in the suburbs.  Moreover, these critics point to the inherent inadequacies of imposed top-down solutions and the planning disasters that have resulted.  The last thing we need, they say, is yet more top-down planning as some kind of well-meaning, ill-conceived “cure” for suburban sprawl.  


They have a point.   As we have seen, a well-adapted city does not take a rigid tree-like structure, but must evolve through a complex iterative process, something the market is inherently better-equipped to deliver than any group of planners.  And yet, while conservatives are eager to trash government bureaucracy, they are strangely willing to indulge private business bureaucracy – a bureaucracy that is not responsive to democratic decisionmaking, but responds rather to the choices of individuals with cash.  The more cash, the more response, irrespective of the number of individuals involved.  Conservatives are fond of characterizing consumer choice as a noble thing, the freedom to choose, and so on.  And yet, as a process for shaping our larger world, whatever its complex structural advantages, there is an aspect of it that is fundamentally undemocratic.


Moreover,  the decisions of private business executives are by no means entirely rational, as Herbert Simon’s Nobel Prize-winning work on bounded rationality showed.  (The present author, as a representative of that class, can attest to the truth of this observation.)  Business bureaucracy may perhaps be subject to more effective Darwinian corrections than government bureaucracy, but it too is severely limited by personal biases, political scheming, and the creative abilities and fancies of individual designers.


Then too, it is naive to suppose that government planners had no hand in shaping post-war sprawl patterns.  The influences of government policy in creating sprawl are well-documented, from the selection of interstate freeway routes, to government-subsidized mortgages, to policies that sapped the inner cities of their vitality.  At the very least, we should look to government to reverse the policies that created unsustainable growth in the post-war era.   After all, democratic government is constitutionally charged with promoting the general welfare.  But many conservatives secretly long not for better and more responsive government, but for virtually no government when it comes to economic affairs.  It seems they would prefer a sort of market-controlled anarchy – something that the suburbs at any rate seem to offer in large measure.    


There is an even more fundamental problem with the mechanism of the market, one that may well come to be regarded as the greatest challenge of the twenty-first century.   The market is best understood as a game governed by rules, particularly economic rules assigning valuation in trade.  It follows that all assignments of value must pass through this nexus of the point of sale, the moment in which value must be judged by human beings, according to a body of information and a set of beliefs and assumptions.   Factors of which those human beings are ignorant are not built into the valuation.  These include unanticipated harm to other human beings, future effects upon the environment,  consumption of limited resources,  and other effects in the future.  The market, in a sense, acts quite literally as if there were no tomorrow.  For in fact it does not know about tomorrow.


This is a more profound version of Simon’s bounded rationality.  In fact it is an illustration of the problem of mathematical incompleteness.  For as we saw, no system can take into account the totality of factors with which it is faced;  no economic model can be as large as the region of human values it represents.   Money is an abstraction that represents wealth, which is supposed to be freely exchangeable; yet we know very well that while there is only one spectrum of value, there are varying forms of wealth, including liquid and illiquid, renewable and non-renewable, sustainable and unsustainable.  In translating our human values into monetary form and back again,  it is easy to commit Whitehead’s fallacy of misplaced concreteness:  to confuse forms of wealth, to ignore important values that are not well-represented by wealth, or even to become one of those who know the price of everything and the value of nothing.


The solution to mathematical incompleteness is to keep an open and adaptive system -- or better yet, multiple complementary systems -- that continually self-correct.  In many ways, markets do tend to self-correct.  But their failures, and the resulting need for intervention by public regulatory bodies, is well-known and generally accepted.  The tendency to instability and the formation of speculative bubbles has contributed to crippling economic depressions.  Health and safety issues, environmental problems, and depletion of limited resources are just some of the many human values that are not well-represented by markets. 


The other open and adaptive system is the political institution of democratic government.  In that system, regulations, laws and even the constitutional framework itself can be modified and made more responsive to new conditions.  Within the federal system, smaller and more decentralized systems of states and localities allow voters not only a say but an opportunity to “vote with their feet,” moving to a political climate more suited to their preferences.  The states and localities in turn function as evolutionary laboratories, experimenting with new forms of organization and regulation that, if found worthy, will spread to other jurisdictions. 


In the United States, both of these complementary adaptive systems have functioned remarkably well together, and have delivered an explosion of productivity and innovation in the twentieth century the likes of which the world has never known.  At the same time, there is a widespread unease about what many perceive as an erosion of values, an increasing rat race, a loss of vital aspects of our heritage.


Many observers have commented about the effects of market economics over time on the practices of critical professions such as law and medicine.  Journalism is increasingly replacing solid news with tabloid titillation.  Politics has become a game of advertising, short attention span and sound bites.


We believe that the basis of this unease is indeed a real trend which, if not modified, will result in profound and even disastrous changes in global society.   The twin institutions of market economics and political regulation are simply inadequate to reverse the trend.  A third system must be called upon, a system often ignored in these kinds of discussions, but one that has also made a profound – perhaps the most profound – contribution to human welfare.  That system is the complex of institutions of human culture – science, art, and religion and philosophy.   


Like the other two systems, this “cultural system” is open and adaptive.  Science continually seeks to improve itself by finding theories that better explain observations.  Art continually evolves and accretes the contributions of individual artists to the sum of previous work.  Religion and philosophy, though sometimes aiming for rigid certainty,  evolves through pluralism, creating a ferment of competing schools of thought, prophets and thinkers.


In previous generations, the cultural system dominated human affairs, and made key contributions to technical knowledge, aesthetic development, and human values.  But today we live in an age dominated by formal systems.  The workings of game theory point to a process in which the great third system of human affairs is increasingly losing its seat at the table. 


A key problem is that the twin institutions of market economics and political regulation do not work to create anything on their own.  In fact, political regulation is largely an after-the-fact phenomenon, a bridle put on the horse as it is bolting from the barn.  Market economics can be exceedingly prodigious, but it must start with the technology and the creativity of cultural institutions.


The challenge before us, then, will be to restore the truly complementary role of cultural institutions.  While that role was once tacit, in the new age of complex formal processes, it must now be explicit.  I will not comment further on what that challenge will entail, other than to suggest that delegate bodies representing cultural institutions will have to be somehow integrated into the political and economic realms.   This is already beginning to happen on a number of levels;  to cite one modest example, in the United States, the National Academy of Sciences has an advisory board to Congress.


A last observation on this subject is that the paralyzing market fundamentalism of the present day must give way to a realistic understanding of the weaknesses of the market.  The terms of debate must shift from whether government in any form should be withdrawn from economic life, to what kinds of responsive and collaborative roles public institutions should properly play in the future.    



The Empire’s New Clothes


“It does seem to me that the whole thing called ‘modern architecture’ has bogged down with the architects right there on that line… ‘much ado about next to nothing.’”


                - Frank Lloyd Wright, in a letter to Mies van der Rohe, 1947


“When we ourselves become abstractions, we are lost!”


- Frank Lloyd Wright, 1930


After a generation of attempted (and mostly failed) reform, a new form of modernism has yet again become fashionable in architecture at the end of the twentieth century.  Nowadays the architecture schools are full of this neo-modernism, and many of the students and teachers are quite taken by the fashion.  Partly that is because it is a morphology that lends itself to two-dimensional sheets of paper, to museum board and little balsa columns and mullions – perhaps the basis of a new credo, form follows art supply.  But to be fair, the new modernist schemes are often conceptually daring, inventive, witty.  As fine art they can be very intriguing and even compelling.  But in the context of the larger problem of life at the end of the twentieth century, all pretenses and pontifications to the contrary, in the end they are very nearly irrelevant.    Moreover, they are destructive:  there is now mathematical evidence that they, and the modernist principles that underlie them,  have an inherent tendency to destroy complexity and richness in the built environment in which they are inserted, like alien craft.


At the end of the twentieth century, modernism – that noble effort to eliminate bourgeois pretensions in design, that grand historical experiment in purity and simplicity, that honest recognition of the reality of the machine and the need to integrate it with art – modernism has become the tombstone of the twentieth century.  It is time to proclaim the utter failure, and to understand it.  It is time to move on.


Of course the critics of modernism and its destructive influences across the globe are legion; but it would be far more devastating to show that the aesthetic structures and principles that dominate modernism are, in fact, mathematically defective, built on the antiquated foundation of pre-twentieth century mathematics, and hopelessly inadequate to the challenges of the twenty-first century.   


Let us be clear that we are not talking about small-m “modernism,” about the Zen-like simplicity of good minimalist design,  practiced by the Egyptians, Classical Japanese,  and many other great civilizations throughout history.  Rather, we are discussing here what may be called “modernist fundamentalism” – the clinging belief in a modernist gospel, or key elements of it.  That gospel may be summarized as follows:


·        We have arrived at a new and fundamentally different age in human history: the age of technology and the machine

·        The machine (understood as a functional reductionist assemblage) is the central metaphor for the organization of society and the basis of its architecture

·        We must acknowledge the functioning elements of the machine and strip away ornament that does not directly relate to function.

·        We must acknowledge the workers of the machine age and use the productivity of the machine to make a new, more egalitarian society

·        We must celebrate the purer elements and simple geometries that form the machine, through a new minimalist, machine-oriented art: an art of lines, planes, grids, crystals.


And one more, this one probably the most seductive for latter-day adherents:

·        We must deconstruct elements of the machine aesthetic and reassemble them as fragmentary elements in a new art form.


A strong corollary is that that no design elements can in any way copy or incorporate designs of any pre-modern era, and that all design must be radical and new.   For to borrow from the past in any but the most schematic features, is to capitulate to the bourgeois powers of the previous century, and to fail to be bold and inventive:  to fail to be “modern.”


Therefore, modernism eschews vernacular character and ornament in favor of pure geometrical forms.   These forms are bold and dramatic; they look good in magazines, and they give their creators a sense of mastery of form.  Their designers’ excitement is sometimes akin to the apes’ frenzy on seeing the monolith in 2001: A Space Odyssey.  Surely such a strange and unnatural object can only have been made by a very intelligent being.


Or perhaps, as we have seen, such an object is made by a newly powerful being that knows only relatively simple geometries, and is not yet sophisticated enough to incorporate complex networks and iterative processes and the more sophisticated natural forms that result. 


This conception of architecture as a grand order imposed from the mind of a great designer is one that has dominated twentieth century design.  (We note in passing that this conception has its unpleasant echo in the fascist impulses of the century as well.)  Unfortunately, there have not been quite as many great designers as there have been building projects.  In fact, the planet is now crammed chock ablock with bad shoe-box imitations of a relative few rather handsome modern buildings.  There is now a lucrative business for some companies who specialize in blowing up these buildings -- although one may question whether what is being offered in place is any better.


Another belief at the core of modernist doctrine is the belief that history is a ladder at the top of which is a purer, wiser technical age with an architecture to match – our own.   But at the end of the twentieth century, most scientists and historians have abandoned the conception of the ladder of history in favor of something more like a bush, with branches heading out in many directions and sometimes gnarling back upon themselves.


Another central tenet of modernism is that architecture is at its core one of the fine arts.  This reduces the built environment to a kind of macro sculpture gallery, in which a few well-executed pieces reside amid rows of kitsch and shopworn antiques.  But of course architecture is much more than an apprehended piece of art:  it is ultimately the vessel of our daily lives, and therefore it must be shaped in ways that respond to the complexities of our lives.  Failure to do so will have the effect of sterilizing our lives. 


When architecture is seen as a fine art, it abdicates its leadership in the more mundane parts of the built environment.  As a general rule, human beings do not like to live in sculptures.   That partly explains why so much housing consists of cheap set-piece imitations of the architecture of some other time.   Contrast that with the early twentieth century, when architecturally significant arts and crafts and bungalow homes peppered the landscape. 


But the larger problem of the modernist gospel, and the aesthetic that derives from it,  is that it is based upon now discarded mathematical and logical tenets.  They are:

·        The age of the machine is indeed different from other ages, but at the end of the century we have also come to a much greater appreciation of what has not changed and what is structurally universal.

·        The reductionist machine metaphor of modernism has long since been replaced in most realms of human activity by the “systems approach” and the complex process metaphor, embodied in the popular phrase “the whole is greater than the sum of its parts.”

·        The role of ornament is often not superfluous, but is an expression of mathematical symmetries in the environment. 

·        Technonogical society no longer creates a clear distinction between a proletariat and a class of capital, and architects have for the most part abandoned any claims to create an architecture of the proletariat.

·        Minimalism, which inserts simple geometries into a complex natural world, tends to cause disconnectedness in the environment

·        Deconstructivism suffers from a “humpty-dumpty” problem:  what may be interesting fragmentary art is not logically promising material for the formation of a well-integrated, meaningful whole environment.


As a corollary, there is a new appreciation for the complex geometries of vernacular architecture and the incredible variety of trans-cultural historical styles.


Why, then, do so many architects still have such a powerful affection for the principles and aesthetics of modernism?  Why do we perceive many rule-breaking modernist buildings as beautiful?  The explanation lies in another revelation of twentieth century cognitive science.   Perception is a process of detecting symmetries, of revealing an order.  The order that architects seek, the symmetries they respond to, are not necessarily those of the real, complex, connected place.  They are often the symmetries of abstractions in the architects’ own minds.  These can also be quite compelling and beautiful – in the abstract realm, and for those who have had the right set of experiences.  That is why so many people still don’t “get” modernism.


I submit that this is a classic example of what Whitehead called “the fallacy of misplaced concreteness.”  Designers manipulate mentally beautiful geometries in their own minds, and become quite taken by them.  When they are executed in reality, the designers still see the beautiful mental forms.   The rest of us, who may not see the architects’ mental gymnastics, tend to have a better view of the natural context that has been imposed upon, often severed, damaged. 


That is one reason modernist buildings do not often “go” with one another – or with anything else.  They hold their power as abstract mental entities, not as elements of a larger composition.  That is an explanation for the common observation that modernist buildings tend to look increasingly ugly with age.  They are not meant to weather, to show fractal patinas, to exist in time.   They are not sufficiently complex.


Notes Toward a New Science of Aesthetics


“And it was then that all these kinds of things thus established received their shapes from the Ordering One, through the action of ideas and numbers.”


- Plato, Timaeus


The work of twentieth century cognitive science has revealed tantalizing clues about the mathematics of beautiful things, and echoed many of the great insights of history.   We suggest that there is still exciting work to be done -- particularly in building on recent work in complexity theory.  Perhaps the age-old dream of a general theory of the properties of beautiful things is within our grasp, and with it new tools for the training of the ordinary designers who do most of the work in civilization.


We have already mentioned the relationship between the golden section and the Fibonacci series and other iterative processes.  This is an example of the generation of self-similarity at different scales, a key component of many beautiful natural forms.  The classical notion of harmony has also been reinforced by recent cognitive science.  In music, it is well known that vibrations in frequencies that are mathematically proportionate sound beautiful to our ears.  A triad chord contains a tonic frequency of x, a dominant frequency of 3x, and a third frequency of 5x. The frequencies of 2x and 4x are heard as higher versions of the same tone. 


The entire system of musical scales is constructed from these tones of mathematically proportionate relationship.   In turn, a great many musical pieces are compositions that alternate between chords built upon these tones, typically returning to the first tone as a final resolution.


The beauty of human faces with proportional relationships is also well known.  It is believed by many modeling agencies that an equal proportion of spacing between the chin and nose, nose and eyebrows, and eyebrows and hairline is one of the most beautiful facial characteristics.


But perhaps the most powerful idea is that of symmetry – not only in the common sense of axial symmetry, but symmetry in the original Greek sense of sym + meter, same measure, or same form, across many different parameters.  The mind perceives a symmetry between the three regions of a face – and also perceives many other facial symmetries, including upswept eyebrows, long eyelashes,  a shape of the mouth that repeats self-similarity in the eyes, and so on.


Similarly, the mind perceives a symmetry in the frequencies of a chord, one that we find quite soothing and beautiful.  We perceive symmetry in the undulating shapes of windswept dunes – not only in the self-similarity of the patterns at different scales, but in the sense of the action of the wind in whipping the sand and gravity in settling it down into a pattern.  (Try turning one of those beautiful sand dune photographs sideways!)


Symmetry explains why some people find certain things beautiful and others do not.  For the symmetries perceived depend upon the previous experiences of the viewer.   Modernism, for example, as we have already discussed, can be quite beautiful under certain conditions, but requires a certain contribution of mental experience on the part of the observer.  Without the experience, there may be no perception of beauty.  


Biologists have found evidence for a genetic basis of the perception of symmetry, particularly in sexual attraction.   This includes not only the apparently innate attraction to the body features of a single sex, but also the attraction to physical characteristics of certain individuals.


Near-decomposability is another property of beautiful things.  Good forms exhibit what psychologist George Miller dubbed “chunks” – a perception of elements in a roughly hierarchical scheme that is perceived as well-ordered.  Again to use the face example, one perceives (face((eyes((mouth(((tooth, tooth, tooth,  and so on.  But while there is near-decomposability, there is also overlap and looping, in proportions, in color relationships, in areas of the face that are continuous and circular.


Yet another property of beautiful things is proportionate boundedness.  A larger scale field is surrounded by a smaller field of proportionate size.  Again to use the face example, the hair tends to form a boundary for the face.  Eyelashes form a boundary for eyes.  Lips form a boundary for the mouth.


In the vast majority of vernacular and traditional architecture, trim forms boundaries at doors and windows.   Window panes are bounded by muntins of a certain proportionate size.  In much of the bad design of the twentieth century, by contrast, windows are extraordinarily poorly proportioned, and have a major negative effect upon the aesthetic strength of the entire structure.   


We previously discussed the complexity of a courtyard house and its network of relationships.  The infinitely circular pattern of possible relationships was noted in particular.  This property of “magic circles” is one that is common to many beautiful designs, from human faces to oriental carpets to the stained glass windows of Gothic cathedrals.  Within the overall boundary (which is not necessarily circular but may be square or irregular), there may be other roughly hierarchic zones, each with its own circular system.  Interrelationships between zones form lattice patterns, even though the entire structure may be strongly hierarchical. 


Interestingly, the Mandelbrot pattern we discussed previously has this same characteristic.


Following is a brief outline of the symmetry characteristics of beautiful things:


Symmetry I:  Likeness at the Same Scale (repetition)

Symmetry II:  Likeness at Different Scales (self-similarity)

Symmetry III:  Likeness within groups at different scales (near-decomposability)

Symmetry IV:  Spatially Proportionate Harmony

Symmetry V:  Likeness to Remembered Structures (symbolic beauty)

Symmetry VI:  Likeness to Genetic Structures (e.g. sexual attractions)


These symmetries can occur in four levels of organization:  Entity/nonentity, multiplicity, hierarchy, and network.


Similarly in the mathematics of complex phenomena four organizing phenomena have been described.  They are called “attractors,” because structures seem to organize around them, much as the rings of Saturn organize around so-called “shepherd moons.” They are the point attractor (entity/nonentity), the circuit attractor (multiplicity), the torus attractor (hierarchy), the strange attractor (complexity).


(To be continued…)


Conclusions:  The Emerging Shape of the New Century


Our condition at the beginning of the twenty-first century is best understood not as a progression of conscious design philosophies, but as an evolution of complex technological culture.  We have gone through phases of industrialization,  imposing regimes of environmental structure,  that in hindsight can be seen as relatively crude.  Our aesthetic systems have progressed through a similar evolution, and are in the process of catching up to recent developments in science and mathematics.   For the aesthetics of the new century, our new lessons are to be found in the complex structures of nature and of history, in combination with the new tools of mathematical analysis and understanding.


Let me summarize those lessons as I have outlined them here:


At this watershed time in history, our reductionist powers have given us enormous technological productivity, but also what may be called a “Humpty Dumpty” problem:  we are finding it very difficult to put things back together.  The problem exists at many levels:  economic, technological, political, and physical – the latter being the primary topic of this paper, the structure of our built environment.


At stake today is the quality of our built environment; at stake ultimately is the sustainability of human civilization.


The first step we must take is to recognize our own shortcomings.  Our intelligence is powerful, but it is fundamentally limited in ways that are not visible.  We are tunnel-visioned without knowing it.  That is part of the nature of intelligence.  The model does not point to the things that it must leave out.  Therefore, we are in danger of getting lost in the model, of imposing our abstractions upon the world and thereby limiting and even sterilizing that world.


The second step we must take is to implement more adaptive and more evolutionary systems, reflecting the insights of twentieth-century mathematics and science.  Some examples:


·        Planning systems must remain open and adaptive.  Top-down planning is likely to fail.  More collaborative, more iterative forms of design and construction should be developed.


·        Planning tools that are as close to the reality of a place should be encouraged.  3-dimernsional models should be favored over two-dimensional drawings.  Mock-ups should be favored over renderings and sketches.


·        The complexities of history should not be dismissed in a rush to create a distinguishing style for our own age.


·        At the same time, the complexities of history can not be parroted – for that is only the simplistic veneer, abstracted from time and context --  but must be understood and incorporated into a genuine synthesis.


·        The complexities of market behavior and its tendencies must be better understood, and mitigated or transformed where necessary by democratic and cultural processes.  The latter will require major institutional transformations which we have only begun to consider.


·        The mathematics of good form and good geometry should be further studied, with the goal of improving the training of architects and designers.  Stylistic pedagogy should be discarded.


·        Tools for better vernacular design should be encouraged in favor of the continued emphasis upon the development of a relatively few “great” professional designers.  These could take the form of popular magazines and websites such as Gustav Stickley’s “The Craftsman” at the turn of the last century; well-crafted computer programs that could respond to local conditions;  and kit or prefab structures, perhaps pre-modified by the user, in a computer-based ordering system. 


·        For architects and other professionals, post-professional certification programs should be developed, verifying the bearer’s study and competence in fields such as urban design, vernacular architecture, and so on. 


·        The task of builders and developers must be seen not as a free-enterprise creation of a disposable product, but a profound change of a non-renewable community resource, with input needed from citizens and from the scientific community.  Developers must see themselves as professionals with a profound responsibility to the well-being of the community, much as (one hopes) medical doctors do.  


·        Manufacturing techniques must incorporate greater vernacular complexity, through technical processes of iteration and variety.  (An interesting example is the manufacture of Cabbage Patch dolls, in which computer programs add subtle unique features to each doll.)


·        New codes and zoning will be needed to allow creativity while assuring the overall beneficial character of the result.  Iterative processes must be exploited. The insights of game theory must be applied to a new generation of codes and zoning.


·        Similarly, public agencies must learn to become collaborators in the improvement of the built environment without imposing a restrictive scheme upon the result.  The public sector must be seen as a catalyst for positive growth, rather than a bridle of negative growth.


In recent complexity theory there is a concept called “emergence.”  It is the tendency of a pattern to form or emerge out of small parts that do not seem to have any relation to that pattern.  In one rather trivial example, the magnified dots of a television screen do not look like a human face, until one draws back and the “big picture” emerges.  Evolutionary processes tend to produce emergence, as individuals without certain traits are selected out, and only those with the traits remain: Harmless Viceroy butterflies over time become identical to poisonous Monarchs as birds tend to eat all butterflies that do not look like Monarchs, and so on. 


Today, many of us are confused and cynical in the face of seemingly disconnected and conflicting events.  The larger pattern is there, if we step back to see it.  Our conscious actions, too, are part of that pattern, and will shape it – for better or worse.    It is time to understand the emerging pattern, and take the next step -- the creation of a new architecture of complexity.




Footnotes (In progress)


1.        An interesting and accessible discussion of this period is found in Douglas Hofstader, Godel, Escher, Bach, Basic Books, 1979.

2.        Alfred North Whitehead and Bertrand Russell, Principia Mathematica, Oxford University Press, 1912

3.        See LaPlace, Pierre-Simon, Essai philosophique sur les probabilités

4.        Douglas Hofstader, Godel, Escher, Bach, Basic Books, 1979.

5.        “On Formally Undecidable Propositions of Principia Mathematica and Related Systems,” Kurt Godel, 1931

6.        See for example, Werner Heisenberg's account in Physics and Philosophy: The Revolution in Modern Science (New York: Harper and Row, 1958).

7.        In Lewis Carroll, Sylvie and Bruno Concluded, The Man in the Moon.  See also Steven Wolfram's marvelous book, A New Kind of Science,

           released after this text was written, and also demonstrating the inherent irreducibility of much of the structure of nature.

8.        There is an instructive analogy in the field of law.  Most of the legal systems of the world operate by the interpretation of fixed statutes.  

           In the common law system used in England and the United States, interpretation follows precedents in the case law.  Thus the law

           continually evolves and adapts to the changing realities of the world.   Another analogy is in democratic constitutional government,

           as opposed to direct monarchy.  In the latter, one person dictates government activity in rigid fashion; in the latter, a continuously

           evolving constitution can be modified by the citizenry as conditions change.  Still another analogy is in the dynamic computer operating

           system Linux.  In contrast to static systems Microsoft Windows or Unix, Linux can be modified and adapted by users, with the result

           that it has become a much more stable and well-adapted system.

9.       See e.g. A. W. Burks, "Von Neumann's self-reproducing automata," in Essays on Cellular Automata, A. W. Burks, Ed. Champaign, IL: Univ. Illinois Press, 1970, pp. 3--74.

10.     Herbert Simon, The Sciences of the Artificial.

11.     Ibid

12.     Christopher Alexander, "A City is Not a Tree",

13.     See for example Bernard Rudofsky, Architecture Without Architects, John Wiley & Sons; 1996.

14.     Christopher Alexander, Notes on the Synthesis of Form, Harvard Paperbacks, 1978.