Geometry

Alexander has had, from the beginning of his career, a particular interest in understanding the organization of space in explicit geometrical terms - this is in part an outgrowth of his early training in mathematics. In his doctoral dissertation, he analyzed form in geometric terms:

Every aspect of form, whether piecelike or patternlike, can be understood as a structure of components. Every object is a hierarchy of components, the large ones specifying the pattern of distribution of the smaller ones, the small ones themselves, though at first sight more clearly piecelike, in fact again patterns specifying the arrangement and distribution of still smaller components.

While at Harvard, Alexander participated in work at the Center for Cognitive Studies, where he was in touch with the developing field of linguistics and concepts related to the structure of languages by Chomsky and others. These influences and the impetus to find a generative process led directly to The Timeless Way of Building and A Pattern Language where his mathematically structured understanding of the design process was melded with an evolving appreciation for the unified and integrated nature of reality, which was taking form around the concept of ‘wholeness’. These books introduce the concepts of ‘patterns’ and ‘pattern languages’.

Alexander’s pattern language theory posits that in the world of the built environment, patterns are generated by certain events and processes which call for a specific spatial configuration to support them. This can be likened to words, which are generated by the need to express. Space and place are created by the interaction of various patterns just as sentences are created by various combinations of words. As a sentence can convey the same or similar meanings with different words and different combinations of words, space and place can, inversely, be made of similar patterns but have distinct differences in actual configuration due to the uniqueness of context, culture and of the personalities of their creator(s). These combinations of patterns (words) into space (sentences, paragraphs, etc.), and its transformation - through human use and perception - into place (story, meaning, knowledge) essentially constitute a language. Patterns can be interconnected with infinite variation to form a rich and complex world. Indeed, it is Alexander hypothesis that traditional cultures unconsciously used what amounted to a pattern language when they created the beautiful buildings and towns of history.

Of course the patterns vary from place to place, from culture to culture, from age to age; they are all man-made, they all depend on culture. But still, in every age and every place the structure of our world is given to it essentially, by some collection of patterns which keeps on repeating over and over and over again...These patterns are not concrete elements, like bricks and doors - they are much deeper and more fluid - and yet they are the solid substance, underneath the surface, out of which a building or a town is always made.

A Pattern Language lists some of the more common patterns that humans have used in recent times. It contains 253 patterns which are organised in a hierarchical arrangement and which cover a large range of scales. Patterns, at their core, are about relationships. While a pattern might describe a particular spatial configuration which can be named - a courtyard for example - A Pattern Language is arranged such that each pattern is made up of smaller patterns and contributes to larger patterns. No pattern need exist on its own as a ‘thing’. In fact, the cross-referencing of patterns in A Pattern Language is an essential component of their use. It was intended that different languages could be built up by users using whichever patterns they felt were appropriate to the task at hand. Further, the authors of A Pattern Language are very clear that users should develop their own, unique patterns and pattern languages based on their particular needs.

An analysis of The West Dean Visitor Centre, based on the patterns of a A Pattern Language, can be found in Appendix 2.

Another theorist whose work has many similarities to Alexander’s is Bill Hillier, who published an influential book, The Social Logic of Space. He has developed a concept of patterns as well:

The aim of The Social Logic of Space is to begin with architecture, and to outline a new theory and method for the investigation of the society:space relation... It establishes a fundamental descriptive theory of pattern types and then a method of analysis...it establishes a descriptive theory of how spatial pattern can, and does, in itself carry social information and content.

Both Hillier and Alexander recognise the fundamental power of seeing the built environment as made of relationships. And they both have developed a theory that provides a valuable analytic tool, one which focuses on the social structures of a place. This is an important alternative to the narrow perspective of most architectural discourse which is centered on formal issues of style and form. Hillier shares Alexander’s view that his narrow perspective has contributed to the many problems that have beset our environments in the last fifty or so years:

...the discourse about architecture that is a necessary concomitant of the practice of architecture is afflicted with a kind of permanent disability: it is so difficult to talk about buildings in terms of what they really are socially, that it is eventually easier to talk about appearances and styles and to try to manufacture a socially relevant discourse out of these surface properties. This cannot be expected to succeed as a social discourse because it is not about the fundamental sociology of buildings.

Despite the basic agreement on the importance of ‘patterns’, the two theories quickly diverge. In fact, in the Social Logic of Space, Hillier attacks Alexander’s theory of patterns:

...the ‘pattern language’ of Christopher Alexander, while appearing at first to be close to our notion of fundamental syntactic generators, is in fact quite remote, in intention as well as in its intrinsic nature. For our purposes, Alexander’s notion of a pattern is too bound to the contingent properties of configurations to be useful for us, while at a more abstract level, his preoccupation with hierarchical forms of spatial arrangement ... would hinder the formation of non-hierarchical abstract notions of spatial relations which, in our view, are essential to giving a proper account of spatial organization.

One explanation for Alexander’s use of a hierarchical organization is his intention that his theory be generative, meaning that it be a useful tool for the creation of good space, and further, that this tool be available to lay people, such that the average person or group of people could begin to influence and control their own environments. This is much more difficult than only trying to describe existing conditions and might explain the necessity of hierarchical organization. For that is what the hierarchy does here, it is strategy which organises the patterns in sequential structure mirroring the process of formation. It does not limit spatial relations to certain fixed patterns, but is instead very flexible. It has nothing to do with creating power imbalances in space, as the above implies. To the contrary, it gives people power to shape their own environments.

The generative intent of the pattern language theory also marks its distinction from another theoretical platform in recent architectural discourse - the concept of typology. This has been investigated by King, who notes that both typology and pattern language concepts recognise a "beauty of culturally determined form." She goes on to point out an inherent inflexibility in typology as a design tool: that is, in holding onto and departing from forms not necessarily relevant to our culture, a designer is limited to a certain range of responses to particular cultural forces:

The concept of pattern language, on the other hand, is directed towards defining a process that will produce the same kind of form as one finds in traditional cultures, but from scratch and on our present cultural premises. At the core of this process is the invention which aims to embody such cultural premises as archetypal configurations, and where an investigation of built form including historical archetypes comes in as part and parcel of making such inventions.

Throughout Alexander’s search for a theory of architecture, he has maintained emotional fulfillment as the primary goal, yet he has attempted to develop his theories with an eye toward intellectual clarity and empirical rigourousness. In his search for ‘wholeness’ or ‘the quality without a name’, Alexander began to see a geometric structure, which, when present, resulted in what people recognise as beautiful.

...there is a specific archetypal structure, which I shall call the "one" for short, that exists in a thousand forms, but that is always at the bottom of all art and all building which lives and breathes. This "one" is a geometric structure, which can be defined in precise mathematical terms. It is an invariant structure, a "presence" which manifests itself in anything which lives, or which is "one."

In The Nature of Order, Alexander has developed what he calls the "Theory of Centers" to illuminate this structure. The "Theory of Centers" posits the existence of entities which Alexander calls ‘centers’. These ‘centers’ are the parts that make up larger wholes and which ultimately make up the "one" or "wholeness" of the universe in its entirety. In turn each ‘center’ is made up of smaller ‘centers.’ Alexander contends that all of reality is made up of these ‘centers’ in constantly evolving interrelation.

In using the word center in this way, I am not referring at all to a point center like a center of gravity. I use the word "center" to identify an organized zone of space -that is to say, a distinct set of points in space, which, because of its organization, because of its internal coherence, and because of its relation to its context, exhibits centeredness - it forms a local zone of relative centeredness with respect to the other parts of space.

In this view it is not only that which we normally recognise as ‘things’ or entities - a book, or a person, for example - which are considered ‘centers.’ A ‘center’ can be a zone of space - like a room or a garden - which does not have clearly defined physical limitations in the sense that a ‘thing’ might.

All the basic things in the world, the elements of which the world is made, are centers in this way. None of them can be exactly bounded. They are all entities which have a fuzzy edge, and whose existence lie mainly in the fact that they exist as centers in the portion of the world they inhabit...When I think of them as centers, I become more aware of their relatedness, I see them as focal points in a larger unbroken whole and I see the world as whole.

‘Centers’ are not necessarily static entities. Two people dancing creates a ‘center’ that moves in space and time. Sunlight streaming in through a window creates a ‘center’ which shifts as the sun moves. In fact, all ‘centers’ are in flux all the time, it’s just that with some the rate of change is slower than we can easily see, so the ‘center’ may seem static. For example, a house may have a fireplace and chimney removed or added, a second story added, windows and doors added, removed, rearranged. Each of these elements are ‘centers’ in themselves, are made up of ‘centers’, and contribute to other ‘centers’ around it.

A critical aspect of this theory is that no ‘center’ exists on its own:

Each center has a certain life or intensity...The life or intensity of one center gets increased or decreased according to the position and intensity of other nearby centers. Above all, centers become most intense when the centers which they are made of help each other.

While the "Theory of Centers" provides a general framework for understanding how wholeness is created in space by a specific arrangement of ‘centers’, it does not provide explicit explanation for how it actually creates "a geometric structure, which can be defined in precise mathematical terms." It does not provide an explicit explanation for how ‘centers’ help each other. This Alexander does with his formulation of "Fifteen Fundamental Properties." These are geometric properties which Alexander has observed in those entities having the quality of wholeness:

What I did was straightforward and empirical. I simply looked at...examples of things, comparing those which had more life, with those that had less life. Whenever I looked at two examples, I could determine which one had greater "life" or greater wholeness, by asking which one of them generated a greater wholeness in me. ..assuming with as much confidence as I felt to be real and reliable, that what I measured here, would also be shared with others...I asked myself this question: Can we find any structural features which tend to be present in the ones which have more life, and tend to be missing in the ones which have less life? In other words, can we find any recurrent geometrical structural features whose presence in things correlates with their degree of life?

The Fifteen Fundamental Properties are:

Levels of scale

Strong Centers

Boundaries

Alternating Repetition

Positive Space

Good Shape

Local Symmetries

Deep Interlock and Ambiguity

Contrast

Gradients

Roughness

Echoes

The Void

Simplicity and Inner Calm

Not Separateness

Alexander notes that this is in some ways a rough sketch of the elements of the field of ‘centers’ and that there could be greater or fewer properties. He also makes the point that not all artefacts and processes have all the properties at once, but that it is the number and strength of these properties which influence the intensity of life in a thing. In other words, beauty lies on a continuum of sorts - things with more of the properties in stronger configuration are more beautiful than those with less properties in weaker configuration. Also, it is possible that one thing or place which has many properties might be less beautiful than another which has fewer properties but whose properties are more strongly defined.

In The Nature of Order, as well as in other writings, Alexander gives examples of these properties from both the man-made and the natural worlds - that is, from the material world in general. This thesis will illustrate the Fifteen Properties in Part II, which presents a series of projects as examples.

It is Alexander’s contention that these properties are not limited to man-made artefacts and environments, but are a feature of all physical reality, including the natural world:

If we are to use the theory of wholeness - and the concept of life - as the basis of all architecture, it would be nice to know that this wholeness together with the properties which bring center to life, is a necessary feature of material reality, not merely a psychological aspect of things which arises during perception of works of art...the structure of centers I call wholeness, goes deeper than mere cognition, is linked to the functional and practical behavior of the natural world, not only the architectural world, and is as much at the foundation of physics and biology, as it is of architecture.

In the last half century there have been significant new developments in science that seem to support Alexander’s contention that his theory of wholeness fits with new scientific understandings of geometry. The area of science often referred to has ‘complexity theory’, ‘systems theory’ or, more provocatively ‘chaos theory’, holds congruence with Alexander’s notions that we need to rethink our understanding of order, so that we might attain a deeper understanding of how order is created in the world. In his survey of the new science of chaos, Gleick lists a series of particular geometries that have been discovered in the recent decades:

...phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them. Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction.

The area of scientific enquiry known as ‘fractals’ is considered by some a harbinger and contributor to the study of dynamical systems. Gleick cites Benoit Mandelbrot’s work on fractals as one of the first steps in developing a new ‘geometry of nature’ and the idea of fractals closely mirrors Alexander’s theory of ‘centers.’

Often scientists drawn to fractal geometry felt emotional parallels between their new mathematical aesthetic and changes in the arts in the second half of the century. They felt that they were drawing some inner enthusiasm from the culture at large. To Mandelbrot the epitome of the Euclidean sensibility outside of mathematics was the architecture of the Bauhaus. It might just as well have been the style of painting best exemplified by the color squares of Josef Albers: spare, orderly, linear, reductionist, geometrical. Geometrical - the word means what it has meant for thousands of years. Buildings that are called geometrical are composed of simple shapes, straight lines and circles, describable with just a few numbers. The vogue for geometrical architecture and painting came and went. Architects no longer care to build blockish skyscrapers like the Seagram Building in New York, once so hailed and copied. To Mandelbrot and his followers the reason is clear. Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world.

Perhaps the most striking parallel between Alexander’s work and the work of those studying dynamic systems theory is the focus on process as a critical ingredient. Classical scientific methods had a stagnant abstraction inherent in them, which prevented a deep understanding of complex phenomena, and - due to a lack of empirical understanding - these phenomena could only be labelled as ‘chaos’. By focusing on process, certain scientists in recent decades have begun to ravel secrets of complexity and are gaining an understanding that there are dynamic and complex systems at work in nature:

Those studying chaotic systems discovered that the disorderly behavior of simple systems acted as a creative process. It generated complexity: richly organized patterns, sometimes stable and sometimes unstable, sometimes finite and sometimes infinite, but always with the fascination of living things...The tradition of looking at systems locally - isolating the mechanisms and then adding them together - was beginning to break down.

This leads us to the next section of this essay, where an understanding of Alexander’s theories of process are explicated.