Harmony-Seeking Computations: A Science of Non-Classical Dynamics Based on the Progressive Evolution of the Larger Whole

As we shall see, in all the examples I shall give, the essential thing, an essential feature of the process described, is that the new configuration which is created, does something to participate in, strengthen, extend, and enlarge the force or “presence”, of some larger configuration that stands in the world around it. In many cases, this larger configuration in the world around it, is not itself amenable to change during the “computation,” but it does nevertheless enter in as a crucial factor.

If a lane exists, connecting two villages, and the road between them is then asphalted one day, the new wider and harder road is primarily affecting the configuration of these two villages, a tract of land much larger than the lane itself; Indeed, the new road may (according to its position in the region), also affect and strengthen an even larger part of the region’s economic network. We all recognize this as a fundamentally useful act.

A musical note that is played will be most effective, according to the degree that it develops a melodic line, a rhythm, or a harmony. If it does all three simultaneously, it will be most effective of all. Once again, we easily recognize the usefulness of this act.

These are simple cases. Other cases, as we shall see, involve creation of the most highly organized and complex wholes; These have, so far, eluded us in science. Even the exciting and useful theories of bifurcation, symmetry breaking, chaos, and generative algorithms, have not yet deeply plumbed the meaning, or the origin, of true complexity.

All this is obvious, indeed hardly more than the most ordinary common sense. Yet, obvious or not, the extraordinary fact remains that this kind of adaptive process (the farmer building his fence with respect for the land) does not currently have an acknowledged part to play in present day theories of algorithms, in developmental biology, in architecture, not even in system theory. It is just not part of the mental models in our tool bag that we currently address and use.

And once we take this seriously, and do indeed add knowledge and perception of this type of transformation to our kit of tools, it will be natural, then, to contemplate the possibility of a type of computation which examines possible moves forward, from a given state of a system, tests them, and evaluates the extent to which each of these moves participates in and adds to the larger wholes in the world around it: then chooses the most effective, the most harmony creating, the most successfully whole-creating!

Such a process of computing, if it can be attained, would be enormously powerful, and powerful in its implications.

The essence of the whole-seeking computation, lies in the following. It creates new configurations, unknown configurations, and good ones, by taking off from a known configuration, but without (necessarily) requiring the input of human creativity. The process itself is creative.

It works like this. Consider a given configuration $\backslash mathcal\{C\}$. This configuration has certain features which are visible, and which, in the main define whatever whole, or wholeness we see in the configuration. But, in addition, in every configuration there are also traces, hints, of dim structures, not yet fully developed, but existing in a latent form, “between the lines” of the configuration. What happens in harmony-seeking computation, is that a process latches on to these latent structures, and enhances them, develops them. Sometimes what develops may be relatively small, with respect to the size of the entire configuration. At other times, very, very large structures may also be latent in a configuration. If the whole seeking computation identifies this latent larger whole, and strengthens it, so that what was before only barely visible, now becomes strong and easily visible, the configuration will seem, to an untrained eye, as having gone in a new direction all by itself. It is this process, that is the essence of all harmony-seeking computation.

The idea that every given geometric configuration might have its own “wholeness” seems odd at first, and even teeters on the edge of soft talk about holistic healing in California hot tubs. However, during the last decades, it has become clear to me that there is a real and definable, mathematical structure underlying any configuration, which captures, or “is”, the structure of the whole configuration, its global structure. Over the course of years I began to call this structure the wholeness. It is not uniquely defined, (mainly because the field is so difficult to describe in precise detail) but it is definite, and various attempts to represent this structure approximately all share something. There is a deeper, global “something” there.⁴

I first began to recognize this possibility in experiments undertaken at the Harvard Center for Cognitive Studies in the early 1960s. These experiments were inspired by the early 20^th-century gestalt psychologists -- Wertheimer, Koehler, Koffka and others -- who wrote extensively about the perception and cognition of wholeness and of wholes in the world around us.⁵ In the early 1960s, Huggins and I used various experimental techniques to find out how different perceivers see configurations.⁶ The essence of the technique was to ask people to group different patterns according to their degrees of similarity. It became clear that there are broadly two kinds of perceivers, those who see patterns holistically, and those who see the patterns according to analytical classifications. The perceivers who saw analytically were in the majority, about 85%. The perceivers who saw holistically were in the minority -- about 15%; these perceivers grouped patterns according to their overall figural character.

However, it was clear that the figural perception was somehow more real, while the analytical perception was arbitrary, and made according to arbitrary invented schemes. The figural quality, was inherent in the material. The ability to perceive figurally, was therefore of enormous importance.

I discovered, later, that it was possible to train people to perceive figurally; but it took a good deal of effort, to overcome what I believe is a culturally induced tendency to be analytical (perhaps understandable among Radcliffe students).

In broad terms, we may say that the global configuration is a structure of the main features of the configuration -- in particular when it is seen as a whole.⁷

During the same series of experiments, I made a significant discovery. The coherence of the different configurations could be measured, by counting the number of locally symmetric sub-configurations it contains.⁸ This measure gave results in very strong agreement with experimental results from cognition, memory, perception and so on,

Thus a configuration and its structure do depend, in some considerable degree, on the nesting of LOCAL SYMMETRIES and on the structure of the system of locally symmetric subsets which occur in it.

Here is one step in Matisse's process of painting. In the first state, the canvas has a background pattern, the stripes of a piece of cloth on the chair, a dark mass where the woman's throat and scarf are to be; staring at us, is the large white area in the middle which is the woman's face, as it is yet to be. But this white area is a vivid latent center, which, even though unpainted as yet, draws attention to itself, draws us in. It is the most salient structure in State 1. So, of course, Matisse goes to that place; we already see his brush hovering over it, in the first photo. And he then makes a mark, the face itself, dark, which differentiates it the latent center, gives it character, and continues, and develops, this important latent center in the evolving structure.

CONTRAST and POSITIVE SPACE are the major transformations used by Matisse here, evident in the way he leaves the “things” – her hair and bodice – unpainted and white, and uses the dark paint to paint the “negative space” between them.

This example is taken from Brian Goodwin’s well known work on the morphogenesis of Acetabularia, a single-celled alga which takes the form of a stem with a cupped whorl at the end. Goodwin studied the growth of this form, and investigated and explained the series of transformations which generated the form of the completed alga.¹⁰

The key morphogenetic sequence that Goodwin described, illustrated in his five-diagram sequence above, showing the stem bulging and then making the whorl. It goes as follows:

5 to 6. The whorl formed around the top of the cylinder grows outward $\backslash xrightarrow\{\backslash text\{now\; turns\; into\}\}$ a full umbrella-like ring of small projections that ultimately turn into the umbrella shaped cap visible in the completed algae.

As Goodwin says, strongly and repeatedly in his writings on the Acetabularia example, this growth sequence is generated by natural physical phenomena. It has relatively little to do with DNA or genetic guidance. What is happening here is a naturally occurring and inevitable progression of morphological transformations that arise directly from the geometry and dynamics of the form itself, to generate the next geometry.¹¹

It is highly significant that a prominent biologist has begun taking the emphasis of morphogenesis away from the influence of DNA, and has begun to see the morphogenesis as an autonomous process in the geometry itself.

There is an interesting sequel to this story. When Brian first showed me the diagrammatic sequence illustrated above, I asked him if the progression from round-ended neck to flat-topped neck was correct. Why do you ask? he said. I told him that, looking at this purely from the point of view of structure-preserving transformations, I would have expected something different. It seemed to me very unlikely that a round topped structure could transform into a flat-topped structure. There is nothing like the flat top latent in the configuration.

There is, however, a ring latent in a hemispherical configuration, but further down the curve. If you have a rounded hill, or hemispherical end of a neck-like structure, there is, inherent in that configuration, a latent structure something like a shoulder, which is incipient near the top of the hill and just below it (see left-hand sketch below). In the right-hand photo below, we see such a structure on the prehistoric Wittenham Clumps near Oxford. It is, and evidently was, natural for the builders who made these mounds, a natural way to continue the smooth structure of a rounded hill. Somehow this point of inflexion in the curve of the hill is a natural development from the smooth curve.

If you are in doubt that it is “natural”, consider the two possible transformations of a hemispherical hill. One, A, just flattens the top by squashing it into a flat surface. The other, B, introduces a band below the top, with two points of inflection in the curve, as shown below. If you ask yourself which of these two transformations leaves the hill alone more, and preserves the global structure of the hill better, I think you will agree that it is B. It is in this sense, that B is more natural than A.

So I said to Brian, referring to his Acetabularia diagrams: If such a configuration were latent, and did then transform to create such a ring-like shoulder, then that shoulder, not the top, would be the most likely place where whorls and other irregularities might naturally form. So I asked him again, Are you sure that the real morphogenesis of Acetabularia doesn't go like that, rather than the way your diagram portrays it? Like this:

Brian laughed, and told me You are absolutely right: that is what actually happens. I just got the diagram wrong.

This story sheds interesting light on the way sophisticated harmony-seeking computations maybe able to help both observational biology, and theoretical biology, and structural modeling and simulation. We will come back to this later. The important point is, for now, that it is possible to make accurate statements about what structures are latent at a given phase of morphogenesis; and it is therefore possible to say, objectively, how things are likely to unfold.

Here is a view of the interiors of some Acetabularia caps, showing that within the whorl, there is indeed a remaining hillock, not a flat, just as my whole-seeking computation predicted. At the time I made the prediction, I had never seen enough detail of Acetabularia to know about this kind of cross-section. My guess arose solely from wholeness considerations, and from thought about what would be structure-preserving.

The left hand drawing shows a plan for this apartment building in Tokyo, done by a student. Though typical of many architectural projects in 1987 (the era of the project), the plan plainly lacks the wholeness-preserving quality of the right hand plan. The right hand plan shows the five-story apartment building which I subsequently built on the site.¹² The layout process for this design had about fifty structure-preserving steps, described in detail in the reference.¹³

The fact that the right hand plan (as built) complements and intensifies the wholeness of the site, is intuitively clear -- just looking at the ground plan alone. However, to emphasize what this really means, and to illustrate the deeper structures that are involved, I will describe some specific features of the global structure which existed in that place, and which were then solidified and complemented by the building that was placed there. Here are some of the structural features of the original site: The center formed by the Y-configuration of the fork; The curving nature of the two branches; The space between the forks, and its specific V-shape; The southern spot hit by the sun.

The plan for a new building, when injected into this site, does the following things which help to accentuate and strengthen these structural features of the place:

As these various features are consolidated, they form a coherent structure, in which each feature supports and helps the others. Marked among this coherence, is the presence of positive space throughout, even in a geometrically complex configuration; the fact that there are levels of scale within the structure; the focus of major centers to the structure, each with its own strength and beauty;

The process of building, and injecting, this structure into the configuration of the site, and designing and planning and building the entire five story-complex, I view altogether as a complex computation. It takes the initial configuration (the site) as its starting point, then transforms the configuration, by about 50 steps, finally ending with the configuration illustrated. Although the work of an architect of this kind would be conventionally viewed as an artistic act, hence as an arbitrary and “private” process, I believe this to be seriously in error. In the process illustrated here, each step is based on the configuration which is there at the moment before, and the latent structures which exist, and then makes one step forward, which specifically, and carefully, strengthens or supports, and evolves naturally out of, the structure which was there before.

On the next page there is a much simpler example – just the evolution of a sketch – but following the same kind of process. This drawing starts with a row of evenly spaced dots. Then, it goes like this:

In the drawing on the next page, through this very simple sequence of six steps, we see the growth of a coherent and sophisticated structure, from extraordinarily simple moves. The names given in parentheses at the end of each step (above), are references to the fifteen properties and transformations, defined in The Nature of Order.¹⁴ As we shall see later, these properties and transformations, play a fundamental role in the creation of all wholeness.

This example is instructive because it is complex enough to be interesting; and we get a vision, for the first time among these examples, that recursive use of these transformations on the emerging centers, and on the centers which the transformations themselves generate, is potentially a complete and very powerful process, perhaps one capable of generating all complex configurations in their totality.

However, there are still many years of work needed to generalize this harmony-seeking computation process in an effective and operational form for the broad range of problems we may wish to apply it to.

This house, built in Berkeley, California, demonstrates the effect of cyclical repetition of several hundred harmony-seeking computations. Starting with the site shown in the photograph above, two years later, the house shown below had been built on the site. The process is fully described in sixty pages of text in The Nature of Order, Book 2, The Process of Creating Life.¹⁵

Above is the plan of St. Mark's square, one of the most famous and beautiful public places in the world. This marvelous and highly complex structure grew, steadily, over a period of about a thousand years. What happened there, may be best understood as a series of harmony-seeking computations, carried out at intervals, with intense and deliberate care. The series of growth cycles shown below is historically accurate. I cannot be certain that the nature of the activity undertaken at each stage was exactly what I have indicated; but I am fairly certain that it must have gone roughly as I suggest. The fact that it is possible to define a coherent and rather simple paradigm for each cycle of activity, and that in their essentials they are all the same, is remarkable, and strongly suggests that what happened was indeed, something along the lines of the underlying computation I have proposed.¹⁶

In every cycle, there are three entities at work.

The procedure goes like this: Find the latent center which is most salient, seems most likely to strengthen the wholeness of the larger configuration. Act locally, in such a way that this latent center gets strengthened, and so that this strengthening helps, also, to strengthen the largest whole. Repeat this cycle ten times over a period of about 1000 years, (roughly once per century), and the result was St Mark's Square as we know it today. This rule, then, explains (or generates), the ten actual cycles of construction and improvement which occurred around St. Mark's in Venice from 600 AD to 1600 AD.

This is a very ordinary, daily event, at least for men.¹⁷ Trying to find a tie that looks just right with the rest of what you are wearing? Although the example seems homely, like the examples given earlier, it is no less a question of finding the particular thing which is most harmonious, and which does most to support the wholeness of the context into which the tie is going to be injected.

This is a rather ordinary tie. However, it comes from a well-known house of distinguished men's clothes in Germany, and does have some interest.

The quiet harmony which is achieved comes, in my view, from three features above all:

I suppose the man who chose the tie, was, in some half-conscious way, feeling these three facts; or perhaps they came to him, singly, as a direct awareness of the way in which the whole is made stronger and more resonant, when this transformation generated by injection of the tie, occurs. It is hard to view such an ordinary act as being a computation. It is even harder when we place it in the company of other results in cell biology, physics, and astrophysics.

But that is of course my point, exactly. I do believe this kind of computation is going on, in art, in science, and in daily life, and we have missed it, so far, simply because we have not yet given it a recognizable and graspable name.

In recent years, attention of cosmologists has focused on the truly enormous structures of galaxies and galactic clusters called filaments. Subsequently, then, even greater attention has been focused on the voids which seem to be encircled by these filaments. The filaments are hundreds of millions of light years long, and tens of million of light years thick. The voids are often a hundred millions of light years in diameter.

The image above is one of the early pictures of the distribution of galaxies and galaxy clusters which clearly shows the voids and filaments.¹⁹ From my point of view, one of the remarkable things about the voids is that they appear to be convex, and quasi spherical. To me that implies the presence of some active principle pushing outwards. In terms of the language of wholeness, the existence of THE VOID (a general structure existing at all scales, and not specific to astrophysics and cosmology), and of thick BOUNDARIES, and of the scale ratio given by LEVELS OF SCALE, is to be expected in all wholeness, since these are three of the fifteen main ways in which coherent structure typically occurs.

Further, the voids seem to be close packed, like soap bubbles, and like soap bubbles, where three bubbles meet, there is a linear film where the soap solution is concentrated. In just this way, the galaxies are spread out in linear filaments. The so called great wall is 500 Mpc long, 200 Mpc wide, and 15 light-years thick. (1500 light-years long, 600 light-years across (band) and 15 light-years thick. Filaments are typically 70 to 150 Mpc in length. (200 to 500 light-years). Voids are 10 to 150 Mpc (30 to 500 light-years in diameter).

The artists 3D sketch below tries to show how these ring-like strands lie in 3D on the surface of the voids.

The picture on the next page is a simulation by George Lake and Tom Quinn, showing the structure more clearly.²⁰ Looking at this structure, from the point of view of harmony-seeking computation, one of the structure's most unexpected features is the ratio (filament width to void diameter), about 1 to 10, or one order of magnitude. Why is this interesting? Because it so closely parallels our own cognitive and intuitive sense of the relative dimension of a typical ring's thickness and the diameters of the space that it surrounds. After all, the filaments could have been much thinner compared with the diameter of the voids, or could have been fatter compared with the voids. In the absence of any current theory which might give us a predictive model of the relative size (unless one fits the parameters just to get the result we observe), it is worth speculating whether the universe itself is making a wholeness-seeking computation, and that the result of this computation is the system of voids and filaments we see.

Next time you are in the bath, try spitting in the bathwater. Very often you will see that the spit moves in swirls on the surface of the water, and once again the roughly circular swirling ring-shapes that are formed have a diameter to thickness ratio in the range of about one order of magnitude. I do not believe the ring ratio merely fits a cognitive niche. Rather, I suspect there are, deep in geometry of space, reasons why ringlike structures with this kind of ratio occur. In current jargon, rings of this particular ratio might be viewed as attractors in some phase space. However, this is too easy. The tendency for such structures to occur might be shared over a very wide range of circumstances, and the voids and ring-filaments in the cosmos might form in shapes similar to the spit in the bath, not for dynamic reasons having to do with physics, but for mathematical reasons having to do with the structure of space itself.

The actual physical diameter of one of these voids, is unimaginably huge, about 100 million light years. It is in miles, $186,000\; *\; 3600\; *\; 365\; *\; 24\; *\; 100,000,000\; =\; 5865696\; *\; 10^6\; *\; 10^8\; \backslash approx\; 6\; *\; 10^6\; *\; 10^6\; *\; 10^8\; =\; 6\; *\; 10^\{20\}$ miles. A jet plane flying at 600 mph, would take a $10^\{14\}$ years to cross this void – something like 5000 times the age of the universe itself. I say this only to emphasize the truly huge size of the voids that we are talking about, and in particular to draw attention to the fact that if it is that huge, the ring thickness could be almost anything, and the ratio of ring thickness to ring diameter could have a large range of possible values.

There is no a-priori reason for the ratio (ring diameter/ring thickness) to be in the range of 10 to 1 (one order of magnitude). It could just as easily be 100 or 1000 or 1,000,000 – i.e. 2 or 3 or 6 orders of magnitude. Why is it on the order of one? Stated differently, what possible reason could there be that the rings around such a colossally enormous, almost impossibly huge-to-our-understanding, structure in the universe should have roughly the same geometric proportion as the gold wedding ring on a woman's finger or a rubber band. Is this not pushing coincidence a bit?

But if we accept that the harmony-seeking computation is based on transformations that are (for whatever reason) largely congruent with human cognition and mentality, then we may not be silly to consider such a process as an explanatory process even in the most outlandish structures.

This supposes then, that there may be mathematical reasons which generate circles or ring-like filaments whose thickness is just about one order of magnitude smaller than their diameter. There is, at the moment, no astrophysical theory that would predict this result – at least none that I know of – but from the point of view of harmony-seeking computation that I am speaking about, there is a possibility that the cognitive simplicity of this particular ratio for a ring-structure, could suggest that it might have to be so for mathematical or quasi mathematical reasons.

Is this strange? Yes indeed, to our ears it is very strange. Is it more strange than Pauli's exclusion principle, when it was first put forward? I do not think so. Thus a purely geometric harmony-seeking computation, might explain at least one aspect of the structure better than any current physical theory. Does it seem unlikely? Yes, perhaps. But in the absence of better computations, I do not think it can be considered entirely useless. At the very least, some reason needs to be found why such processes, congruent with cognition, might occur in structures at the scale of the universe and at the scale of human beings, and the ring-ratios tend to be the same.

Three points about the voids. First, there is a ratio of void diameter to wall diameter, consistent with the LEVELS OF SCALE property. Not larger, not smaller. Second, we should try and find data about the evolution of voids (by examining different red shifts, hence different eras, to find out how the structure of voids has evolved and to try and find observational data about the ring thickness at different eras. Third, we should make a more precise prediction, based on the harmony-seeking computation, and test it against observations.

The examples on pages 4 to 30 all have the fact (most of the cases), or the possibility, of some kind of harmony-seeking at work. In all cases, there is some kind of structure-preserving process at work, mainly geometric in nature, sometime intellectually graspable, sometimes not so clear. Apparently, as a result of this structure-preserving process, at each step the generated configuration puts the starting configuration in a slightly more harmonious, more coherent, state than it was in before.

Most important, if we choose to do so, we may regard ALL these examples -- both physical processes, simulations, and actual paper and pencil exercises -- as computations, each of which takes its system forward, step by step, from an existing state $W\_1$ to a state $W\_2$, and then to $W\_3,\; \backslash dots,\; W\_i,\; W\_\{i+1\}$, and so forth, and as it goes increasing structure and increasing harmony and wholeness in the system. These steps are carefully calculated, normal in many spheres of human experience and in many natural systems. Some of them are done by human instinct; some are done by animal instinct; some are done by adherence to a certain tribal sensibility; some are done by concern for ecology. Many occur naturally (without human intervention) as part of the behavior of some complex physical system.

In a generic sense they are all computations. But in nearly all these cases they are not easy to deal with as computations recognizable in the present way we understand calculations or algorithms. This does not make them trivial, it makes them fascinating. There are reasons to believe that a fundamental kind of process is at work in all these cases. If we can examine these computations, and begin to understand or extract the underlying way they work, all of them together, and if we succeed in getting the gist of this computation, we may find a way towards a powerful new way of computing that is guided by emerging harmony, and by a motion towards harmony.

I view these examples as computations, and state it that way now, because, after twenty years of work, it has slowly become likely, to me, that they all embody some kind of single process, or function, which has the capacity, starting with an arbitrary configuration, to reach a better and more harmonious configuration. Although the idea of an SP-transformation is, for the moment, still somewhat loosely defined, there is considerable similarity between the different examples in the way they work, and in what they accomplish. I conclude that it is reasonable then, to think of the examples as exemplars of the operation of a single type of mathematical function, or computation, at work.

However, to learn to see these examples as embodiments of a single type of computation, we must now work harder to try and be specific about what this computation might be, how it might work, and how, above all, we can put it to work, in cases where we seek better organization, better harmony, better deep adaptation and coherence for a configuration.

An important experimental result. Apart from the structural explanations which have been given already, my own cognitive experiments show strong agreement among observers as to comparative judgments regarding what steps are structure-preserving and what steps are not.

A step is a harmony-seeking step, if it preserves or extends or deepens structure. When asked to judge different possible steps, starting from a given configuration, in terms of the degree to which the steps are SP (structure-preserving), or not, people seem to agree – based on their intuitions, or cognition. This alone strongly suggests that the quality of “being SP” is to some degree objective, and gives a strong hint that the phenomenon reflects an underlying physical reality.

It is also true that if we ask people taking in part in such an experiment, to make (in their own chosen terms) any kind of structural diagram in which they try to capture the wholeness of the thing they are looking at, before judging which steps forward from it are structure-preserving and which are not, the level of agreement between observers in their ability to judge what is SP and what is not, goes up strongly. People who have performed such a private diagram-making process, even though their diagrams are not alike, will then agree more strongly with one another about what steps forward from that structure are SP. This occurs even though the making of a diagram, which describes the structure, is a private matter, and people are given no special instructions about how to do it, and the different diagrams people make are unlike. It seems to be enough, that after attempting to represent the structure, they can then judge more reliably what is structure-preserving and what is not. This, once more, is evidence that the phenomenon of SP is real and objective as a phenomenon, with objective structural content.

How may we formulate, mathematically, the character of the harmony-seeking steps which occur in the examples?

In each case there is a whole, $\backslash mathcal\{W\}$, and within the whole a latent center which is being modified, transformed, shaped, or reshaped, by a certain step. This latent center is the focus of the transformation, and the latent center sets the boundary of the geometrical and physical transformations that are then actively being undertaken. Let's call this focal latent center $\backslash mathcal\{L\}$.

At the same time, there is a larger whole, often an order of magnitude bigger than $\backslash mathcal\{L\}$. The transformation which is structure-preserving, preserves the structure of the $\backslash mathcal\{W\}$, and to do so modifies the $\backslash mathcal\{L\}$, and modifies it in relation to the whole context around it. Thus the output from this step is a modification geometrically within $\backslash mathcal\{L\}$, but it is a function of both $\backslash mathcal\{L\}$ and $\backslash mathcal\{W\}$. In addition, there is a sense in which $\backslash mathcal\{L\}$ is being fitted to $\backslash mathcal\{W\}$, it is being made to fit $\backslash mathcal\{W\}$, to be congruent with $\backslash mathcal\{W\}$, adapted to $\backslash mathcal\{W\}$, harmonious with $\backslash mathcal\{W\}$. Further, in order to modify $\backslash mathcal\{L\}$ in this way, new centers are going to be created within and around $\backslash mathcal\{L\}$. We may refer to these new centers as $\backslash mathcal\{N\}\_i$, and since there may be several of them, we may think of them as $\backslash mathcal\{N\}\_1,\; \backslash mathcal\{N\}\_2,\; \backslash mathcal\{N\}\_3$, etc.

To undertake this transformation, in such a way that it is indeed a structure-preserving transformation, the various $\backslash mathcal\{N\}\_i$ are to be generated by fifteen generic types of transformation acting on $\backslash mathcal\{L\}$. These are the fifteen principal center-creating and structure-preserving transformations listed on pages 2-4:

STRONG CENTERS
LEVELS OF SCALE
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

These transformations are enumerated and defined in considerable detail in chapter 2 of The Nature of Order, Book 2 (The Process of Creating Life),²¹ and in chapters 5 and 6 of The Nature of Order, Book 1 (The Phenomenon of Life).²²

It has nowadays become almost commonplace, to explain how geese fly in formation²⁴, how ants accomplish complex tasks together²⁵, or how slime mold (apparently an uncoordinated aggregate of cells), is able to move coherently as a whole²⁶. In most cases, the explanation is, in some form or other, that the individual “cells” copy the behavior or action of their nearest neighbors; in many cases this ultra-simple rule of action explains considerably coherent movements of the larger body of cells.²⁷ In other more complicated cases, the atomic actions of individual cells are coupled: and the coupling helps more complex forms of behavior to occur in the group.²⁸ In either case, the resultant aggregate seems to be acting as a whole.

However, to be sure we are talking sense when we say these things, and to avoid exaggerating what we seem to have discovered under the rubric of emergence, it is important to analyze – with very great care – just what we mean by “acting as a whole,” in these kinds of cases.

When health, or wholeness, or harmony exists in a part of the world, what is under discussion is always the relationship of a given system, to the larger world beyond that system. So the issue is not merely, whether a group of elements act together. What is important above all, is that when the elements are grouped together to form a system, the resulting system either does or does not act in such a way as to heal, or sustain, or improve, the coherence and health of the yet larger system around it some part of the world outside and beyond the group – some system of which this group of elements is a part.

The emergence phenomenon is a two-fold relationship, between a set of elements and a group they form. The harmony phenomenon is a three-fold relationship, between a set of elements, the group they form, and the helpfulness of the ensuing group to the world beyond the group. Thus:

One of the most familiar analyses of the evolution of natural phenomena and configurations, is the one that Ian Stewart and Martin Golubitsky have popularized: The phenomenon of symmetry breaking in geometry, not only in systems of equations and variables.³⁵

Indeed, even in these cases of symmetry reduction, what we actually see, if we look closely, is symmetry elaboration. The infinite Euclidean 3D-space, does indeed lose the simple symmetries, but at the same time, new local symmetries appear, and are generated.

This is a better picture of how harmony-seeking computations work. The failure to see it, comes, I think, from the fact that in recent years the symmetry structure of the plane, or of the 3D-continuum, have been viewed too simply through the symmetries of the whole. It has not been sufficiently clear that there are, hidden in the plane, or in the continuum, an infinite number of symmetries of smaller local sets, all over the place, and that there are, in the world, an infinity of systems of smaller and smaller symmetries which occur in these nested sets. What happens when the whole evolves under the harmony-seeking computations I envisage, is that many of these smaller symmetries are generated, or strengthened, thus giving a hierarchical nesting of local symmetries at different levels of scale.

What I have said in the forgoing, can be simply summarized. Consider the modern masters: Mandelbrot, Goodwin, Stewart, Prusinkiewicz, Gordon, Nakaya, Reynolds. We must, and do, take off our hats to them. But if we examine the contributions made to complexity theory, by these modern masters, we see that while making enormous strides, and placing our discipline on its first legs, it must also be said that each one of them has fallen slightly short in one all-important respect. All of them, at least all that I know, have tried to explain complex emergence, as a product of coupled interactions among local events. Yet, when one examines in very painstaking detail, what is actually going on, and what is emerging, it turns out in every case that there is some (sometimes small -- but not small enough to be overlooked) aspect of the emerging whole that cannot be properly explained by this approach. Though they aspire to reach the whole that emerges, through analytical means, in fact it is just the whole -- the real whole in the world that makes us marvel in the first place -- that is propelled by a second process, which is whole-oriented. I have been at pains to show that this whole-seeking or harmony-seeking process is not teleological, not goal-seeking. It comes about, because of a new type of operation, which is performed on the structure that exists, and which then brings to fruition a larger, unexpected, and unanticipated new structure of wholeness in each case. The existence of such a computation, and its operation in virtually every creative process in nature and in art -- this is the real creativity of the universe at work.

I believe there is a realistic chance that this creative kind of emergence, where the whole inspires the emergent structure, and gives it direction can, at least in part, be encompassed by a computable formula, by a rational and attainable, though very new, mathematical formulation.

This hinges, I think, on the construction of a formalism in which nested systems of symmetries and centers, are acted on by the transformations I have described, and where the future (the (t+1)-trajectory in phase space) is determined by the latent structures already present at time t, where these latent structures, too, are couched and described in terms of symmetries and centers.

In natural systems we may expect, and in human-created urban systems we may decree, that when things are "OK", harmony-seeking computations will be occurring, and must define the step-by-step unfolding.

Here are two almost purely natural systems. We see in the morphological character of the structures portrayed, that these are generated by harmony-seeking computations. What we see here are two structures, which are products of structure preserving transformations. One can trace a history of SP-transformations as the origin of what we see in the morphology.

Here are two mixed systems, including limited man-made portions. They are almost purely natural systems, but include parts undertaken by human processes which include human versions of SP transformations (the boat, the gardens, the farm-villa). We see in the morphological character of the mixed structures portrayed, that these are still generated by harmony-seeking computations. What we see here are two structures, which are products of structure-preserving transformations. One can trace a history of SP-transformations as the origin of what we see in the morphology.

Here we see two structures mainly dominated by humanness (not by nature in the raw), where harmony-seeking computations dominate the process and its resulting morphology. The situations seem primitive, because the technically advanced civilization we have become, has -- for the moment -- reduced ability to perform these computations. We therefore see fewer and fewer harmonious results or examples of true harmony in our surroundings.

Example 21. Look around you, wherever you are today, in almost any city: Bradford, London, Birmingham, Atlanta, Stockholm, Tokyo, Moscow, Addis Ababa, Cape Town, Singapore, Lyon, Madrid, Santiago, Mumbai.

The destruction you see is caused by the absence of harmony seeking computations in the process that has produced these environments today.

Example 22. Somerville. A Housing Project For 200 Apartments In Boston, In Which The Project Plan And Form Are to be Generated By Harmony-Seeking Computations.

Cycle 1. The process starts with a triangular brown-field site in Somerville, 5.5 acres in area, between a railroad line, a bike path, and an existing neighborhood → The computation then identifies latent centers in the site: the bike path, along the south side, Warwick avenue along the west, and Lowell Street in the south east corner.

Cycle 2. latent centers in the site: the bike path, along the south side, Warwick avenue along the west, and Lowell Street in the south east corner. → The computation then identifies a connection and pedestrian precinct generates a curved line, more or less a median through the triangle of the site, but curving slightly and leading to stairs at the east end where there is a 20 foot rise to Lowell street.

Cycle 3. Consideration of density suggests that the environment can become wholesome and enjoyable, only if entirely pedestrian, thus connecting this new area to the surrounding neighborhood and making it useful.

Cycle 4. The articulated pedestrian spine now has, as its most obvious latent centers, the areas on either side of it. To place 200 dwellings in this limited area, and to do it in such a way that people have pleasant gardens, for their own use, and for the neighbors who may like to enjoy these gardens, leads to a form of organization where houses are laid in long thin strips around useful open spaces. The gardens should, typically, have a diameter on the order of some 100 feet; the buildings in strips should have a depth of some 25 feet; the building height would then be 2 and 3 stories.

Cycle 5. The abstract and schematized grid-like array, is characterized by two parameters, courtyard diameter, and building thickness. However, its regularity is not essential or even good: indeed it needs to be adapted to the site boundaries in such a way as to generate coherent courtyards and pedestrian space, while leaving building volumes simple. This transformation requires the use of ROUGHNESS, LOCAL SYMMETRIES, DEEP INTERLOCK, NOT SEPARATENESS, GOOD SHAPE and INNER CALM. Application of these transformations in concert achieves the necessary computation at this stage.

Cycle 6. More careful adaptation. The schematic arrangement of the previous cycle, now gives way to a series of shapes which pay more detailed attention to each individual garden, as a shape in itself, using GOOD SHAPE, LOCAL SYMMETRIES, POSITIVE SPACE, THE VOID, and ALTERNATING REPETITION, so that the whole is coherent, and feels like one thing.

In addition, the boundary where this neighborhood meets and abuts other neighborhoods is modified by BOUNDARIES, DEEP INTERLOCK, and ALTERNATING REPETITION, so that it becomes a thickened semi permeable membrane, capable of allowing people who want to talk a stroll, to pass in and out comfortably, yet also maintaining a certain privacy for the interior of the neighborhood.

In the plan above we see the beginning and the framework, for a further harmony-seeking process. The plan shows 200 households on a five acre site, each house potentially unique, and where the pattern of gardens, walkways and road access have been laid down so as to protect the harmony of the adjacent neighborhood, and project the immediate environment for five hundred people.

One aspect of the generated structure for Somerville is highly significant. If we examine the structure which has been generated by the harmony-seeking computations that were used, we see that each part, though similar in broad structural character to others, is unique. That comes about because the application of the computation, to even slightly differing contexts, will inevitably produce morphologically different results.

For example, at the largest level of scale, each of the fifteen or so courtyard gardens has a slightly different shape and configuration. That occurs because the application of the principal transformations (POSITIVE SPACE, LOCAL SYMMETRIES, ECHOES and GRADIENTS, ALTERNATING REPETITION and CONTRAST), generates a different configuration for each one according to its starting point – that is, according to its context.

Still more exciting though, is that the same quality of uniqueness and subtle differentiation, continues to smaller and smaller scales. If we examine the drawings, we shall see that the detailed configuration of terraces, entrances, paths, lawns, stairs and archways, produces unique results in each part of the larger whole, and in each part of the individual gardens. This is not because of a shallow desire to make each thing different for its own sake (sometimes the driving force behind the more commercial post modern developments). It occurs because the effect of harmony seeking computations, on only slightly different starting conditions, is to generate entirely new and different configurations, but all members of a fairly simple family.

I do not think this uniqueness of every place is visible to such a considerable extent, in such simulations as the L-system generation for trees and plants.⁴⁴ L-systems create an approximation to harmonious structure of this kind; but it is not deep. The reason is, that the system and its adaptations (as it has presently been worked out) are too elementary. Harmony-seeking computations, by comparison, have the potential to do better: but the algorithms are far from perfect: but they have the potential to do better because they rely on fifteen kinds of transformations, acting in concert and in parallel: and that allows each place to develop its own peculiar nature in an appropriate and inevitable fashion.

There is a single all-important conclusion to be drawn from all these studies. Atomistic, bottom-up, computations cannot adequately describe what is really happening in the world, and – further – do not describe those especially important processes which heal the world, which bring order into configurations in the land, in nature, in buildings deeply adapted to the land, and also deeply adapted in their internal structure.

The issue is recursive. It is not only large-scale configurations that need to benefit from this insight. The idea that every configuration, at every level, is working to help the coherence of some larger configuration in which it is embedded, runs up and down the ladder of scales, and must do so in any living world. A successful computation will emulate this upwards- and downwards-reaching process. Purely bottom-up forms of calculation, not invoking this principle, will always remain too sterile to be real or profound.

I am not only proposing that we consider these many real world systems as computational processes. I am saying, that we need to find out how they work, and how, in particular, they work to allow the unfolding of structure, under the impact of wholeness. That will require a form of representation which is new. It has been sketched, in broad brushstrokes, in Books 1 and 2 of The Nature Of Order. We now need to work at finding ways of describing this kind of unfolding, in more well-defined mathematical and computational terms.

That will be a very long job. My colleagues and I have, in the last few years, gained intuitive and form insight into the nature of the harmony-seeking computations. But we are far from understanding them in detail. That enormous task must now be undertaken, hopefully by many dedicated scientists together.

I must emphasize that the phenomena I am talking about cannot successfully be grouped under what is loosely called ‘emergence.’ The emergence of wholes does not come about by autonomous processes which happen to aggregate themselves to form wholes. Rather they are processes which contain specific whole-seeking processes: these are the structure-preserving transformations I have described.

In all these real-world examples, there is a common phenomenon. We may describe it by saying that the steps of the computation are SP (structure-preserving)-transformations which follow this scheme. Each SP-transformation operates on one wholeness, to produce another wholeness. It does it in such a way as to preserve or embellish or enhance the global structure of the first wholeness.

The mathematical description of an SP-transformation is not yet fully known. However, there is abundant evidence to show that the concept of being SP, of being structure-preserving, is well-defined and objective, in the sense that different observers largely agree among different possible transformations of a given whole, which ones are more SP and which are less so.

A new science of harmony-seeking computation and SP-transformations can make amenable to computation, phenomena which are, for the moment, altogether beyond the reach of currently available computational methods. It will help to open doors to the global quality of harmony, figural goodness, ecological health and structural coherence as a computable feature of configurations,

One of first practical items on the agenda, is to provide well-defined and precise versions of the fifteen transformations in The Nature Of Order. Though easy to state, this is a remarkably difficult task for three reasons. The fifteen properties, though defined with some level of precision, remain somewhat elusive. Defining computable operations which can induce these properties in arbitrary configurations, is a challenging task. Further, it is difficult to define them as transformations, since this presupposes a language of configurations that is amenable to the transformations. And, finally, some of the transformations are easier than others to define operationally in sufficiently concrete terms. For example, LOCAL SYMMETRIES, BOUNDARIES and LEVELS

OF SCALE are relatively easy. POSITIVE SPACE and ECHOES are harder. SIMPLICITY AND INNER CALM and NOT SEPARATENESS are among the most difficult. However, I am fairly sure that the task of dealing with all fifteen transformations can be accomplished by a small team, in the next five years, and that should then open the door to a full fledged, though elementary, version of a first-draft science of harmony-seeking computations. First steps have already been taken.⁴⁶

I hope the idea of harmony-seeking computation may then sit alongside other methods as a new tool in an armory of well-founded alternative computational techniques to be used when appropriate. It is likely to be appropriate whenever a computational task is defined more by issues of adaptation, health, wholeness, and wellness, with reference to the position some system in some still larger whole, or perhaps even by a desire for beauty, or life, or elegance.

All these might one day play a key role in very general kinds of computation. Science, architecture, biology, ecology, physics, and cosmology may all be the better for it.