Thought hierarchy always manifests itself as a sequence of levels (hierarchical structures of systems), this layering is not as rigid as in the structural or systemic consideration. Hierarchical conversion is the key to understanding hierarchies. Convertibility of hierarchies provides a solid base for integrative studies. Basically, we observe that, treating something in a specific respect, we deal with its specific aspect. The same thing can be involved in many activities (processes, relations) and it may look quite differently in different circumstances, up to becoming almost unrecognizable. In the hierarchical approach, such different manifestations of the same hierarchy are called its different positions (in analogy to the possible positions of a musical chord).

However, the positions of a hierarchy are never arbitrary; they always reflect its overall organization. This means that any hierarchical structure or system is never imposed from outside as immediately given; the context can only favor one of the elements, and the rest of the layered construction unfolds itself according to the inner ties between the elements.

For example, imagine a crumpled net lying on the floor in a heap. If you pull one of the nodes, it will drag out the nodes immediately connected to it, and they will, in their turn, take out the nodes connected to them, and so on. In the end, you will have the net hanging down from the node you hold, with each node at its own height above the floor. You have produced a hierarchical structure. If you start with a different node, the result will be essentially the same, but the nodes will hang at some other distances from the floor, in a different order. Thus, varying the initial (topmost) element of the hierarchy, you produce different hierarchical structures.

Similarly, pulling up a point of a horizontal cord, you obtain a hierarchical structure ordering the points of the cord by their distance from the flat surface:

Pulling up a different point, you obtain a different ordering of the points:

This new hierarchical structure is yet another position (or another turn) of hierarchy. To understand why the idea of rotation is invoked, consider another example. In the simplest hierarchy, there are two elements and one link between them. The two possible positions of such a trivial hierarchy can be pictured as

and

Note that the link from A to B is of a different kind as compared to the link from B to A, which is stressed by the notation. The example of a triadic hierarchy gives even stronger impression of rotation:

Of course, such simple examples do not convey the whole spectrum of hierarchical convertibility. However, they illustrate how a hierarchically organized thing can turn its different aspects to the world, changing as well as remaining the same in the same time. In addition, the above examples of the net, and the rope, demonstrate yet another important feature of refolding: to get to a specific turn of the hierarchy, the original structure must first be folded to some neutral state, and then unfolded, starting from a single element that represents the hierarchy in this particular position (new hierarchical structure). In the discrete case these two operations are not as obvious, but they still have to be involved.

Unfolding Hierarchies

The logic of unfolding is based on the relativity of distinction between the elements and links. Thus, in the scheme

,

the link → can be considered as an element M mediating the connection of A to B:

.

As a result, one obtains three levels of hierarchy instead of the initial two. Any link between the neighboring levels can thus be represented by an intermediate level, and the hierarchy will unfold itself again and again. This is an example of qualitative infinity inherent in any hierarchy.

Once again, one must observe that the quality of links between the elements and levels in any hierarchical structure depends on the way of unfolding, and similar structures may represent quite different positions of hierarchy. There are numerous examples in modern mathematics, where the same notion (e.g. a set) can be introduced in the context of very different conceptualizations (like the number theory or the categorial approach), with all the properties preserved, but in a different sense. Sometimes, this difference can become apparent, like in the case of Riemann and Lebesgue integrals, which coincide in the non-singular domain, but can lead to different results for singular integrands.

Despite of its apparent difficulty, hierarchical unfolding is quite common in our everyday life. Thus, when we first meet somebody, we usually pay attention to some particular details of the person's appearance or behavior, and our further acquaintance with that individual proceeds through extension and moderation of this primary impression. Similarly, to develop a large project, we split it into relatively independent stages, which can further be split into even smaller subtasks.

In nature, hierarchical unfolding is often associated with a fluctuation, a violation of symmetry, or "bifurcation" (in the sense of the catastrophe theory). In any case, this is a natural process, co-relating a thing with its environment.

Folding Hierarchies

The inverse process of folding a hierarchical structure treats every indirect (mediated) link

as a direct link of a different type:

.

Intuitively, this corresponds to the common figure of reasoning that, if two things are related through some other thing, they are related. The focus shifts from the mediation of connection (its mechanism) to the connection itself, since, in many applications, we do not need to know about the details, as soon as we get the overall result.

Folding is a transition from one hierarchical structure to another structure, which is simpler than the original in certain respects. In our everyday life, a typical example of hierarchical folding is provided by learning, when a complex action is first performed operation by operation, but it gradually folds into a single operation that does not require conscious control of the intermediate steps.

In principle, a hierarchy can be folded into a single element; commonly, however, the process of folding stops at some level, with following unfolding in another direction. The "neutral" state, to which the hierarchy becomes folded, can therefore be complex enough, and there can be a hierarchy of such neutral states.

Multidimensional Structures

In a hierarchy (idiarchy), any element, or link, is a hierarchy itself, and it can be unfolded in its own way, regardless of the current position of the parent hierarchy. Thus, the scheme could become something like

Since any part of the hierarchy is connected to any other part, schemes like that always imply missing links, which can be restored in different ways. For instance, one could consider parallel unfolding of each of the primary levels:

Quite often, however, there is no parallel development of different levels. Thus, the hierarchical structure of the lower level (as the result of its unfolding) can be represented by one of the higher level elements; the rest of higher level development is only indirectly related to the lower level structures:

There are many directions of unfolding a hierarchy, and the number of dimensions in the resulting hierarchical structure can grow to infinity. Nevertheless, all the possible unfoldings (positions) of a hierarchy are determined by the hierarchy as a whole and, in that sense, they are contained in it. Every individual thing, at every moment, is in infinitely many relations with the rest of the world, in every one of which it is represented by a specific hierarchical structure. In human activity, that infinity is normally handled using the idea of convertibility, applied to the hierarchy of admissible rotations of hierarchy: at any instance, we only see a particular turn (the topmost element), with the rest serving to enrich it with inner complexity.

Sequencing

Saying that the levels of hierarchy represent the stages of its history, we assume that any development can be considered as a sequence of distinct phases. However, the very way of distinction depends on the level of detail, and those considering three stages may be as right as those who distinguish twenty. The process of development is hierarchical itself. Each phase of development can be "split" into many smaller phases, and so on without limit. Conversely, minor changes can be merged in larger units, thus providing a grosser scale for the whole process. Such folding can merge phases in different combinations, and the resulting higher level sequences will be different:

unfolds into

which folds to

or

or

etc.

This is a special case of conversion of hierarchies, which makes them exhibit quite different hierarchical structures and systems (the different positions of hierarchy), remaining the same integrity. Each of the possible positions corresponds to a possible route of development.