Structure-Preserving Transformations
2002 Alexander 4 pp.
The Nature of Order, Book Two: The Process of Creating Life

Structure-Preserving Transformations

A decorative flourish or ornament centered at the top of the page.
A decorative flourish or ornament centered at the top of the page.

1. 2 / STRUCTURE-PRESERVING TRANSFORMATIONS FURTHER DISCUSSION

Let's start again. On the right, there is a sketch of a square drawn on a sheet of paper. Below that, I show various ways you might modify the square, add something to it, transform it.

If I ask you to modify it in a way which preserves or continues or extends the structure which exists in the square, you will probably draw something like one of the (A) sketches in the first row below.

A simple sketch of a square.
A simple sketch of a square.

The original square

Five sketches labeled A showing transformations of a square that preserve its structure. From left to right: 1. A square with an 'X' inside. 2. A square with a small dot in the center. 3. A square with a small dot on the right side. 4. A square with a dashed outline around it. 5. A square with a smaller square inside it.
Five sketches labeled A showing transformations of a square that preserve its structure. From left to right: 1. A square with an 'X' inside. 2. A square with a small dot in the center. 3. A square with a small dot on the right side. 4. A square with a dashed outline around it. 5. A square with a smaller square inside it.

A. Transformations of a square which preserve its structure

Three sketches labeled B showing transformations of a square that destroy its structure. From left to right: 1. A square with a wavy line passing through it. 2. A square with a small dot in the center. 3. A square with a dashed outline around it.
Three sketches labeled B showing transformations of a square that destroy its structure. From left to right: 1. A square with a wavy line passing through it. 2. A square with a small dot in the center. 3. A square with a dashed outline around it.

B. Transformations of a square which destroy its structure

If, on the contrary, I ask you to modify the square in a way which destroys or damages or contradicts the structure which exists in the square, you will probably draw something like one of the (B) sketches in the second row.

In both cases, your intuition tells you roughly what to do. Intuitively, we understand the concept of preserving or destroying structure. This means, of course, that in some form we must have an intuitive idea of the structure which exists. That concept is not new: the structure which exists is, of course, the wholeness as I defined it in Book 1. It is the field of centers. But

we must also have an intuitive idea of a transformation which preserves or extends a structure, and an intuitive idea of a transformation which destroys or contradicts a structure. This is new. Except in chapter 1 of this book, I have not previously (in Book 1) suggested that the wholeness which exists contains a seed or direction that points the way toward those transformations which are kind to it and away from those transformations which are unkind to it. But the demonstration I have just given shows that there is indeed some way in which a transformation of a structure which exists can be kind or notkind — structure-preserving or structure-de-destroying, more consistent or less consistent with the structure that exists.

A simple square with a single dot in the center.
A simple square with a single dot in the center.

Square with a dot

A square with a central dot and a cluster of small dots to its right.
A square with a central dot and a cluster of small dots to its right.
A square with a central dot and a cluster of small dots to its right, similar to the previous image.
A square with a central dot and a cluster of small dots to its right, similar to the previous image.
A square with a central dot and small dots at the corners.
A square with a central dot and small dots at the corners.
Two overlapping squares, each with a central dot.
Two overlapping squares, each with a central dot.
A square with a central dot and several diagonal lines crossing it, representing a transformation.
A square with a central dot and several diagonal lines crossing it, representing a transformation.

Upper row: Good transformations of the square with a dot

Lower row: Bad transformations of the square with a dot

A preference for movement towards the structure-preserving transformation is almost exactly what we have seen in the examples of chapter 1. Throughout nature, we see a continuous smooth unfolding of the wholeness which preserves structure at every moment, even when it seems to be introducing new structure. That is what happens even when a bullet shatters a piece of glass (page 31). It is what happens when a seed grows into a plant. It is what happens when a wave breaks or a river meanders.

Here are some more examples of structure-preserving transformations. At the top of the page, I take one of the transformed versions of the square: the square with a dot in the middle. I make further marks to transform this figure further. Again, these marks may be structure-preserving or not. The three in the top row are

structure-preserving. The two in the second row are not structure-preserving. The transformations in the first row, even though they bring in new structure and open up new directions, preserve and enhance the wholeness of the square with the dot. The transformations in the second row also bring in new structure, but they do it in a way which violates the structure of the square with the dot. Its structure is weakened or destroyed.

The idea of structure-preserving transformations is quite general. If we are faced with any configuration at all — simple or complex — and we are asked to modify it by adding elements or making changes, we can distinguish between types of additions and changes which preserve or enhance the structure and types which weaken or destroy the structure.

It is the structure-preserving transformations which give us the key to the creation of wholeness. Look at the situation (below) where two very similar trees are standing close together (first diagram). If I string a hammock between them, this is a structure-preserving transformation. The wholeness of the two trees with the hammock is similar to the wholeness of the two trees without the hammock (second diagram). Another structure-preserving transformation occurs if I put a single bench around one of the trees (third diagram). However, this transformation is slightly less structure-preserving, since it introduces an asymmetry that was not there before, and changes the larger wholeness substantially.

A series of sketches showing two trees, a hammock, and a bench, illustrating transformations that preserve or destroy wholeness.
A series of sketches showing two trees, a hammock, and a bench, illustrating transformations that preserve or destroy wholeness.

Two trees; two trees plus hammock; two trees with bench around one of them.

Putting in a hammock leaves the wholeness of the two trees intact; putting a single round bench around one of the trees leaves it somewhat less intact.

Plan 1: A first possible site plan, rather conventional in character, which is NOT structure preserving. The plan shows a rectangular building footprint with a central courtyard, situated on a triangular lot. A street runs along the bottom edge of the lot.
Plan 1: A first possible site plan, rather conventional in character, which is NOT structure preserving. The plan shows a rectangular building footprint with a central courtyard, situated on a triangular lot. A street runs along the bottom edge of the lot.

Plan 1: A first possible site plan, rather conventional in character, which is NOT structure preserving. Although this plan follows typical design character for a typical building in the 1970s or 1980s, the placing of the volumes, the badly formed exterior space, and the lack of structure-preserving done to the two streets and to the sunshine in the south are all negative.

Plan 2, as built: A site plan which IS structure-preserving. The plan shows a more complex, angular building footprint that follows the triangular shape of the lot, preserving the existing structure. A street runs along the bottom edge of the lot.
Plan 2, as built: A site plan which IS structure-preserving. The plan shows a more complex, angular building footprint that follows the triangular shape of the lot, preserving the existing structure. A street runs along the bottom edge of the lot.

Plan 2, as built: A site plan which IS structure-preserving. It shows the unusual configuration caused by the fork, and two bent streets.

A photograph of a multi-story apartment building in Tokyo. The building has a unique, angular design that fits into a narrow street. A sign with Japanese characters is visible on the side of the building. A van is parked on the street in front of the building.
A photograph of a multi-story apartment building in Tokyo. The building has a unique, angular design that fits into a narrow street. A sign with Japanese characters is visible on the side of the building. A van is parked on the street in front of the building.

The view of our apartment building in Tokyo after completion. It kept the character of the neighborhood alive because it was structure-preserving in so many ways.

To explain the point with a complex, full-scale example from architecture, I give the ex-

ample of an apartment building I built in 1987. It was built at an acute-angled fork in a busy Tokyo street. The fork had an unusual angle; both streets were (and are) narrow. I show two possible plans for the building, considered while it was in the earliest design process. One of them, highly conventional from the point of view of architectural planning, circa 1970–80, and done as an exercise by someone in my office, is made of several rectangular volumes arranged to fill the site as nearly as possible. It is not structure-preserving. The other, following the street contours as they are, forms a volume which was unusual by the standards of 1987; but it is more structure-preserving. It enhances the spatial volumes of the two streets. The second plan is also more structure-preserving for the neighborhood as a whole. It is the plan which we subsequently built. The photograph to the right of the plans shows the apartment building when it was finished.

On this page, I give a second similar example of real built things, but they are much more modest in scale. This shows how the same principle affects even the smallest things in the environment: the following two everyday illustrations from the Berkeley hills show how ordinary this process is. The photographs are of two mail boxes on a street near my house. The first, on the left, is very simple. The person needed a mailbox, put it on a stick, and let the grass grow around it. It is beautifully structure-preserving and sensitive.

A photograph of a mailbox on a grassy hillside. The mailbox is a small white box on a wooden post. The hillside is covered in green grass and has a set of stone steps leading up it. A metal railing is visible on the right side of the steps. The background shows more greenery and trees.
A photograph of a mailbox on a grassy hillside. The mailbox is a small white box on a wooden post. The hillside is covered in green grass and has a set of stone steps leading up it. A metal railing is visible on the right side of the steps. The background shows more greenery and trees.

Mailbox which is structure-preserving. The landscape, steps, grass, and their wholeness are preserved by the insertion of the mailbox.

In contrast, on the right, is another mailbox, from a house further down the street. It is almost the same kind of mailbox. You see that the owner of this mailbox has built a kind of pyramidal structure under the mailbox, evidently trying to make it “nice.” In our language, you might say that this person was trying to make a STRONG CENTER. Should he not get some brownie points, then? No. The center he created has too little to do with the context of the situation where he created it. The reason is that, compared with the first one, this center has less relation to the grass, flowers, and driveway around it. It did not arise as naturally from the wholeness of its location. Thus it is a more isolated, more self-aggrandizing center, exaggerated and less helpful to its context. It seems a bit overblown. And it seems overblown because it is less structure-preserving than the first mailbox.

As these examples suggest, examples of structure-preserving and structure-destroying transformations are visible all around us.

The difference between the two types of cases plays a fundamental role in architecture and in the evolution of all living structure.

A photograph of a mailbox on a concrete structure. The mailbox is a small white box on a post. It is situated on a concrete base that has been built up, creating a pyramidal shape. The base is made of concrete blocks and has some plants growing around it. A driveway is visible in the background.
A photograph of a mailbox on a concrete structure. The mailbox is a small white box on a post. It is situated on a concrete base that has been built up, creating a pyramidal shape. The base is made of concrete blocks and has some plants growing around it. A driveway is visible in the background.

Mailbox which is not structure-preserving. The center which is created under the mailbox does not arise naturally from the surrounding wholeness.