Extracting partonomy from complexity

(~2300 words)

Complexity is defined in many ways, not all of them contradictory, but most staking out shifting conceptual boundaries. I follow Wikipedia in quoting complexity theorist Neil Johnson: “even among scientists, there is no unique definition of complexity.” There is, however, at least one conceptual structure that can be distilled from any complex system that, once distilled, is utterly straightforward to characterize and handle, and whose properties convey important, unique pieces of information about the system’s complexity. This structure is the system’s partonomy, and it represents the system’s parthood relationships. The partonomy of some systems is more ambiguous than others and thus more difficult to distill, but this difficulty has a habit of interacting in interesting ways with other renditions of complexity.

I’ll define what a partonomy is by reference to its depth, since a partonomy’s depth is its most complexity-relevant piece of information, which can be simply generalized as complex systems have deep partonomies. A partonomy’s depth counts the number of levels from the elementary objects in the system to the thing as a whole, where each level contains the smallest non-arbitrary aggregations of entities or parts from a lower level. More mathematically, a partonomy’s depth is the number of branch points between the root of a partonomy and its leaves, where a partonomy is [evidently] a mathematical rooted tree, with the whole system for its root and elementary objects for its leaves.

A picture is worth a thousand words, so I’ll render several partonomies here (created with some code you can find on my GitHub page)

Figure 1. The partonomy of a helium atom. The elementary objects used here are the elementary particles, including electrons (e), down-quarks (dq), and up-quarks (uq). The nucleus, represented by the node in the near-top center branching into four triplets, takes up most of the partonomy. This partonomy has a maximum depth of 3, with the paths up from the electrons having a depth of only 1.
Figure 2. The partonomy of the Protestant Bible. The elementary objects used here are the “books” of the Bible. The division between old and new testaments is clear. The large structure in the bottom center depicts the minor prophets and the group just to its right depicts the four gospels. This partonomy has a maximum depth of 4, although there are many paths up the partonomy with a depth of 3 and two paths with a depth of 2 (from Acts and Revelation).
Figure 3. The partonomy of Spain. The elementary objects used here are the provinces, and the only intermediate level is provided by the “autonomous communities” such as the largest, Castilla y León, left of center containing nine provinces. This partonomy has a maximum depth of 2, although there are several paths with a depth of 1.

Though I have highlighted (maximum) partonomic depth in these examples, there are clearly other metrics that could be computed from the partonomy such as average partonomic depth, average degree of branching at each node, amount of branching symmetry, etc.

Ambiguous partonomies

Partonomies, as mathematical rooted trees, are unambiguous, but distilling a partonomy from a system or object is often not an unambiguous process. Consider, for example, a bottle of wine, especially a bottle of chardonnay or pinot noir with its characteristically more tapered form, and try to construct a partonomy for it. If we start with some of the highest-level divisions and largest parts, we encounter the bulk of glass, the fluid wine and air trapped with it, the paper label, and the cork with the rest of the closure and capsule. The bulk of glass in particular is the most problematic, because while it encompasses three different parts—the container body, the neck, and the indented punt underneath—the transition between the first two is exquisitely seamless. If our partonomy were to reach down to the atoms, then somewhere between the body and the neck (also known as the “shoulder”), there would have to be two adjacent atoms in virtually identical material surroundings that would have to be assigned to two different branches of the partonomy, one through the body and the other through the neck.

Figure 4. A bottle of chardonnay.

The situation is made even worse if we choose to partonomically affiliate the glass neck more closely with the bottle closure than with the glass body. Then not only are two atomic neighbors in the glass separated in the partonomy, but one of them is more akin to several centimeters of cork than it is to its neighbor. There is simply no mathematical tree that does perfect justice to the parthood relationships in a bottle of wine, even though the various alternatives do capture different aspects of it: keeping all the atoms in the glass together creates a partonomy that privileges the materials and thus the manufacturing and recycling processes, and keeping the neck and the body separate creates a partonomy that privileges the function of the bottle as a product to be handled and consumed by people.

One thing I have noticed is that ambiguous partonomies are most common among highly optimized entities, especially when multiple trade-offs conflict. Between the body and neck of a wine bottle, these trade-offs are manufacturing simplicity and structural integrity, ergonomics, and practical volumetrics. On the other hand, first tries and prototypes tend to have rather unambiguous partonomies, but I have not performed the quantitative research to back this up, though if I did I’d probably want to ask questions about how human cognition bears on the result.

Co-existent partonomies

Sometimes, when it is difficult to find a single unambiguous partonomy for a system over some elementary objects, multiple unambiguous partonomies can provide a satisfactory resolution. We got a taste of that with the chardonnay, but that illustrates only one case. Imagine constructing a partonomy for human society, with individual people as the elementary objects. If we tried to find a single partonomy, then people could only belong to a single group, whether that group be a family, an organization, a village, or otherwise. This is clearly absurd, because most people participate in multiple groups simultaneously, and it is thus impossible to create a single unambiguous partonomy for human society. However, if we consider the offices and positions in groups that individual people hold instead, it may be possible to construct a single partonomy; though since the same individual would enter multiple times into that partonomy for every one of his or her offices and positions, we would in fact have multiple irreconcilable partonomies.

We could have a partonomy for voting jurisdictions, collecting people into a tree by virtue of their status as voters in local, regional, and national elections. We could have a partonomy for all for-profits and non-profits, collecting people into trees by management and subsidiary relationships. We could have a partonomy for all the classes in departments in a university in a university system. We could have a partonomy for families collecting them into, of course, family trees. In this view, one of the primary developments of modern civilization has been to increase the number of partonomies required for disambiguation in the human social system, directly reflecting the increase over time in average number of group memberships per person.

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Figure 5. An urban development designed with a single tree, from A City is Not a Tree.

In cities, there exist partonomies for the drainage system, the transportation system, the electrical system, and many others. This fact is the centerpiece of one of the most influential essays on urban planning from the twentieth century: Christopher Alexander’s A City is Not a Tree, wherein he explicitly denies the ability to fit a city into a single unambiguous partonomy. He further states that cities that have been designed and built as trees tend to be unlivable, and I tend to agree. I also think Alexander’s observation is a special case of what I said at the end of the last section: that highly optimized entities rarely submit to a single unambiguous partonomy, and cities that have developed by being lived in for centuries count as a highly optimized entities.

A final example enters with cognitive systems. Imagine an entity with an unambiguous partonomy that reaches down to the atoms, or what I’ll just go ahead and call an atomic partonomy. If it contains parthood relationships that are evident visually, then a sighted human that encounters it may use those relationships within its cognitive representation. Indeed this is a popular interpretation of the function of the repeated layers within human visual cortex: that they iteratively group together over the pixels of retinal ganglion cells. We can thus posit that our human has a visual partonomy for that entity, and that this visual partonomy will have much the same structure as the atomic partonomy, at least at the highest levels. The similarity breaks down deeper in the partonomies because the human visual system is not remotely capable of resolving individual atoms, nor does it have enough bandwidth to account for each of them.

The remainder after extracting partonomy

A system or object’s partonomy (or partonomies) captures quite a bit of its complexity and character, but it very specifically does not and cannot capture all of it. The easiest way to explore what it misses is to consider non-identical entities with identical partonomies. Some examples are rather trivial: if the elementary objects in a partonomy are macroscopic, then changing their color or other visual properties preserves the partonomy they construct together. I can give less trivial examples that depend on chirality, spatial perturbations, type of associations, and external context.

Chirality: the bone structures in my right and left hands have the same partonomic breakdown, but the hands themselves are the mirror images of each other rather than being identical. If you had just the partonomy of the bones of one of my hands, you wouldn’t be able to tell which of the two was used to construct it. Spatial perturbations: a covalent solid (e.g. diamond) has the same partonomic structure hot as it does cold, because the difference between it being hot and cold comes from differential amounts of jiggling of the atomic bonds, not from any amount of reconfiguration of them. As soon as the bonds start reconfiguring with enough heat, it stops being diamond. Type of association: to continue with diamond, an infinite diamond lattice and an infinite plane of graphite (diamond and graphite are both forms of pure carbon) have the same partonomy, which collects all of the carbon atoms (if we assign them as elementary objects) together one level up as the carbon chunk as a whole. External context: a square made with sticks and a diamond made with the same sticks have the same partonomy, because the difference between them depends on the relative orientation of the observer or other entities not participating in the partonomy itself over the four sticks.

Case studies

1. Galactic partonomies

What is the partonomy of the Milky Way Galaxy? If we retain celestial bodies (entities existing by virtue of gravitational compaction, thereby rounding them into spheres, e.g. stars, planets, moons) as our elementary objects, then we can build a partonomy that iteratively collects entities in their gravitational interactions: Earth and Moon come together as the Earth-Moon system, which comes together with the other planetary systems to produce the solar system, which does not definitely participate in any aggregations until the Orion Arm of the Milky Way, which finally comes together with the other arms as the Milky Way itself.

Incidentally, partonomies and partonomic depth allow us to hew away a little at the Copernican principle. In terms of spatiotemporal extent and mass, humans and our biosphere are a tiny speck in the Milky Way. In terms of partonomic depth, however, even just our bodies put it to shame. The partonomies of our bodies contain at least a dozen levels, whereas the Milky Way can barely scrape half of that. Unlike other things that set us apart, like the beauty of our biosphere or the human faculty of language or the diversity of terrestrial species, the partonomic depth of life on Earth is a simple, objective physical measure that loudly proclaims a unique position for us in the Universe.

2. Designing a garden

Imagine we have a large expanse of undeveloped land and that we have to build a garden in it while optimizing for different things. First, let’s optimize for area; quite clearly we will use the entirety of our undeveloped land for the garden. Let’s optimize for height; then we will probably select a few species like redwoods or eucalyptuses and plant the garden in whatever microclimate they favor. Let’s optimize for diversity; then we will get as many different species as we possibly can, planting one or a few of each any which way.

Now let’s consider optimizing for partonomic depth. What will we plant and perhaps more importantly how will we plant it? The simplest way to create a level in the partonomy is to segregate the garden into two sections, setting them apart perhaps by type of plant—flowers over here, saplings over there—or even more simply by putting an empty tract between the sections, of short grass, gravel, or otherwise. The process can be repeated with each section, dividing each into two or more subsections which can be further divided. What results is often startlingly aesthetic, and it was created with almost no gardening expertise whatsoever. What I seek to impress is that deepening or otherwise directly manipulating the partonomy of a system you are responsible for is a powerful design tool, universally applicable and independent of and in addition to any domain-specific tools.

Figure 6. A garden with a partonomy about 4 levels deep. e.g. whole garden > middle section > single flower bed > group of identical flower plants > one flower plant