Fifteen transformations

Let us now consider the fifteen properties, not merely as results of structure-preserving transformations, but as the names of particular types of structure-preserving transformations themselves.

In general, all the geometric properties identified in Book 1 are also associated with dynamic transformations which will inject these geometric properties into the system of centers of any emerging, growing whole.

Levels of scale

The levels-of-scale transformation introduces intermediate-sized centers to fill out the hierarchy of scales that exist in a given wholeness. In this case, some zone that has been loosely distinguished, is differentiated further into smaller parts. This can happen so that these new parts are similar in size to one another, but one level smaller than the center which is being differentiated. In another application of this transformation, a large center is made more coherent and distinct by the introduction of smaller parts, which then act together with the large center to form a recognizable and distinctive hierarchy.

Strong center

The strong-center transformation is the most fundamental transformation of all. Any weak center which exists is made more emphatic by this transformation. It may be more strongly differentiated, more strongly defined, more strongly integrated by virtue of its differences, or more sharply drawn and distinguished. Of course, all the transformations help, in some form, to achieve this fundamental goal. However, the transformation itself, in a primitive form, acts to give weight and definition and distinction and centeredness, to any weak center which has begun to crystallize in any given field.

Alternating repetition

The alternating-repetition transformation generates a repeating pattern of similar entities, within a previously undifferentiated field. The way it works, though, it simultaneously generates a second pattern of repeating centers, interlocking and alternating with the first. This transformation is the most basic way that a large system may be given a structure as a repeating field of many repeating smaller entities.

Positive space

The positive-space transformation makes strong positive space by creating new centers in the space between other centers, thus strengthening and shaping spaces between the other centers that are not yet centers themselves. This is one of the most powerful of the fifteen transformations.

Good shape

The good-shape transformation takes an existing center or system of centers (often formed by earlier application of the alternating-repetition transformation). The transformation intensifies the products of the alternating repetition, by strengthening them, making them more distinctive — and this is done by applying the good-shape transformation in the weakly existing centers in such a way that any loosely formed shape which exists in the space, is made more marked, stronger, by giving more life to the centers within the shape. The effect is to make a more beautiful, more living, shape.

Local symmetry

The local-symmetry transformation strengthens a center (or system of centers) by making the center (or each center in the system) have an internal axis of symmetry. The symmetries induced are only local, and do not extend beyond the limits of the center, and may sometimes even be used only to strengthen or “symmetrize” the kernel of the center. This is often the way in which an emerging center first receives its strength. Shortly after an entity is differentiated and made to stand out from its ground the symmetry transformation then sets it up as a strong center in its own right.

Deep interlock

The deep-interlock transformation takes an existing structure, especially in its boundary zones, and weaves the distinct opposing parts at the boundary into a tighter, less separated, union by physically creating connections in which part of one enters into the other, and vice versa. This imbrication of the boundary cements the whole (the structure plus its context); the transformation helps to unify the growing whole. It would be unusual for this transformation to happen at the outset of a differentiating process.

Contrast

The contrast transformation is a kind of sharpening which occurs. In a system where two types of centers occur the transformation works to increase the distinction between the two kinds; it separates them more sharply from one another, thus creating a field of more strongly contrasting entities. The contrast may be achieved by color, darkness, polarity, to by other physical characteristics. The polarity of the two, generates a more well-knit system as a whole in which the two kinds of centers can complement each other better.

Gradient

The gradient transformation creates transitions of size and character. In response to an uneven, or non-homogeneous field, certain aspects of size, shape, weight, darkness, spacing, are made to vary systematically — thus introducing coherence of a new kind into an almost random-like field of structure. The gradient transformation thus begins to create structure where none was visible before. In other cases, a simple polarity or position, or axis, engenders a gradient, and the inner parts and centers are then given features which vary systematically according to the gradient. In this case the gradient transformation can have a very large, global, field effect within an extended zone. It has a surprising ability to order complex and inchoate structure, without greatly bending to changing circumstance.

Roughness

In the course of making positive space, strong centers, local symmetries, or alternating repetition, it is often necessary to introduce or pack in irregular variants of repeating centers, to make things work out. The roughness transformation uses intentional irregularity to find the most regular fit possible for a given configuration, and one which permits things to work out successfully and simply in the large. It is of enormous importance. Wholeness would not be possible without it.

Echoes

The echoes transformation applies procedures, angles, and shapes and shape-character of certain repeating centers to other centers in the field, thus generating a widespread family resemblance among different centers and so strongly unifying the whole.

Void

The void transformation is at work getting rid of garbage. Areas which are relatively undifferentiated, and which do not need their differentiation, are cleaned out and made more homogeneous, and defined by a boundary zone which is attached, surrounded, by more differentiated structure. The transformation also preserves an imitation of the greater undifferentiated void.

Simplicity

The simplicity transformation, like the void transformation, also cleans, simplifies. However, it works by removing unwanted centers, differences, and other kinds of complexity, throughout the structure, where the void does it by creating a single homogeneous zone in one place. The simplicity transformation gets rid of unnecessary structure by reducing it.

Not-separateness

The not-separateness transformation may be thought of as a kind of knitting. In applying this transformation to an existing object or system of centers, modifications are made to the centers and their surroundings so that the center gains more of the subtle substance from its surroundings; and at the same time the surroundings gain more of the substance inherent in the center. The effect is that the two are brought closer together, forming a more indissoluble unity. All in all, the purpose of the transformation is to unify, to knit together, to create a texture in which the separateness of any given entity is reduced.
The way the not-separateness transformation most typically works is somewhat similar to the effect of the color transformation called mutual embedding. When operating on two major areas, A and B, that are differentiated from one another, the transformation takes pieces of A and copies them inside B, and takes pieces of B and copies them within A. The result is that A and B become more associated, more allied, more united, and less distinct from one another.
The not-separateness transformation may occur early or late in the differentiation of a structure. Essentially this transformation binds the entity which is being created and its surroundings more tightly. This may be accomplished by a variety of specific means including echoes, deep interlock, boundaries and so on. However, the overall unification of an entity and its surroundings, is a transformation in which the two distinct entities (a center and its context) are made more connected, more similar, more different, more interlocked, more reminiscent of each other, more complementary, more distinct, less distinct, and more united. The transformation stretches them apart and binds them together, making inside and outside less distinguishable.

The inherent limitations of space have the effect that, for purely mathematical and geometrical reasons, there are only a certain small number of ways that a given wholeness can be extended, while preserving its essential structure.

Although I cannot claim to give a rigorous proof that the fifteen transformations are the only ways to extend and conceive a given wholeness, I believe that this is true, and that a more sophisticated mathematical treatment will one day be able to show why it is true.

The fifteen transformations form a coherent system. We have in them, a limited palette of transformations which may be made to act on a given system. These are the fifteen most basic ways in which structure-preserving transformations can be made to occur. Every differentiating process is accomplished, in a structure-preserving way, by successive application of these fifteen transformations. The range of possible sequences and combinations, and the range of results which can be achieved by this type of differentiation, is amazingly rich and varied.
We see now that the fifteen properties are not merely observable end-products of structure-preserving transformations. They provide the base transformations from which, in practice, all structure-preserving transformations are made.

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