Exploring a Cubic Subspace



In my paper Flowers and Steps in the Boolean Lattice of Hexagrams I explore how change can be represented within the formal framework of the Boolean lattice. On this page we shall consider a concrete example, the particular change from Binding Greatness as the primary hexagram, to Gnawing Through as the related, or resulting hexagram. I first show the algebraic definition of this change, and then demonstrate how this generates a cubic subspace of potential. The resulting cube actually encodes a number of distinct changes. Using an animation, I show how any given change within the cube represents a particular flow of energy through the cube. I also provide a detailed narrative interpretation of this flow. A second change is also animated to demonstrate the multiplicity of change within a single structure. Finally, I show how these different flows within a structure can be represented formally using labelling.

The reader should be aware that some of the following discussion involves what you might call non-classical interpretations of the hexagrams. These interpretation arise from the various formal properties explored on this site.

The Basic Cubic Change

The example of change being considered here has the traditional numerical representation of 799687, so there are three changing lines: the 9 in the 2nd place, the 9 in the 3rd place, and the 6 in the 4th place. This is represented algebraically with the following equation:

    • x =

This says that Binding Greatness is transformed into Gnawing Through as the result of multiplying it by the energy of , Resonating with Arousal. The general form of this equation, meaning that the Primary figure multiplied by the change Quanta gives the Related figure, is:

    1. P x Q = R

This change, in turn, gives rise to the cube of potential shown below.

Building a Cube

In general, a change with n transforming lines will give rise to an n-dimensional subspace. In the concrete example here, a change with 3 transforming lines gives rise to a cubic (3-dimensional) subspace of potential.

The top hexagram of the cube is formed as the energetic union of the primary and the related hexagrams:

    • v =

The general form of this equation, which says that the top of the sublattice is the energetic union of the primary and related figures, is:

    1. P v R = TOP

The bottom hexagram of the cube is formed as the energetic separation of the primary and the related hexagrams:

    • & =

The general form for this equation, which says that the bottom of the sublattice is the energetic separation of the primary and related figures, is:

    1. P &= BOT

Then, the set of hexagrams between the top and the bottom, including the primary and related hexagrams, is given by the function:

    1. sub(P, Q) = {G in L : P & R G P v R} where R = P x Q

This says that the sublattice for a primary hexagram P and a change quanta Q is the set of gua G from the complete lattice L, such that G falls between the TOP and BOT elements generated from P and R.

Defining the sublattice in this way gives a static structure, and there are 7 values for P other than for which Q gives the same cube. In concrete terms, the eight distinct changes represented by the following set of equations, would all generate the same cube:

  1. x =
  2. x =
  3. x =
  4. x =
  1. x =
  2. x =
  3. x =
  4. x =

Again, it is possible to express this concrete example in more general terms.

    1. For all P, Q in L, if G is in sub(P, Q) then G x Q is also in sub(P, Q)

This simply says that if a gua is in a subspace defined for a particular energy Q, then so is its related figure. In fact, this gives us an interesting technique for exploring and categorizing the Forest of Change in terms of the unfolding of individual perspectives, which I shall return to at a later date.

The Character of the Places

I have shown in a number of previous papers that the trigrams give the basic definition of cubic space; see, for example, The Yijing as a Symbolic Language for Abstraction. The cubic trigram lattice is shown on the right. Clearly, this is structurally identical to the hexagram cube shown above and, as such, the trigrams present a natural way of characterizing the places in any cube.

Firstly, notice the arrangement of the trigrams within the structure and compare this with the hexagram cube constructed above. This diagram is repeated below with the locus of change highlighted in red. This makes the parallel between the trigram lattice and cubical hexagram lattice clear. We can see that by picking out the scope of the changing lines in the hexagram cube, the trigrams can be identified. This gives us a method for naming the places in the hexagram lattice:

The place in the structure is .

The place in the structure is .

The place in the structure is .

The place in the structure is .

The place in the structure is .

The place in the structure is .

The place in the structure is .

The place in the structure is .

Once the places are named, we have a way of describing the different changes within any cubic structure. So, for example, continuing with the example, we could describe the change represented by the equation x = as the change from Open to Bound . In fact, any of the changes within a cubic sublattice will be described by a transition from T to ~T; that is, from one trigram to its Boolean opposite.

This description also clarifies the idea of figure and ground, or foreground and background, in a change. Here, the lines in red are the foreground of the change, those aspects of the situation that are in focus and transforming. The lines in black are unchanging in this context, these form the background for the change. Just as we can characterise the foreground transition in terms of movement from one trigram to another, so we can characterise the background in terms of a trigram. When we look at the background, the black lines, we see yang, yin, yang - this gives .

In a cubic change both the foreground and the background are trigrams. In general, if there are n lines in the foreground of the change, there will be 6-n lines in the background of the change. Thus, for example, in a planar change, the focus of the change, the figure, is a bigram (two lines), whilst the background is a quadragram (four lines).

Note that this technique only describes the two end points of the change. For a complete description of the change we need to consider the actual path that the energy takes through the cube.

Paths Through the Potential

We have seen that each change generates a structure that also encodes a number of other changes. As shown by equations (i) to (viii) above, any hexagram in the structure can be taken as the primary hexagram, and the resulting change is contained within the structure. So, given that a sublattice defines a number of distinct changes as equivalent, what can we do to differentiate those distinct changes within the structure? We have seen a way of naming the different changes, but the complete answer lies in defining a direction of energy flow through the structure.

We shall begin by reconsidering our example, classically represented as 799687, and described by the equation:

    • x =

From Open to Bound

The animation on the right shows the change for 799687 flowing through the cube. In this example, has the most change energy and this flows through the structure in a number of parallel paths until is reached, which has no remaining change energy.

It should be clear from the animation how the energy flows in directed, acyclic paths from the primary to the related hexagram: all paths diverge from the primary hexagram and converge again at the related hexagram. For a formal analysis of this flow, see the section describing labelling below. For the moment, an informal understanding is sufficient.

Adding a Narrative

In addition to the formal, structural description of the change, it is important to consider the content, or interpretation of the resulting structures in ways analogous to the traditional interpretation of change. This parallels the distinction between form and content in logic. The form of an argument is the abstract structure of the reasoning without reference to any particular individuals, properties or relations. Conversely, the content of an argument involves reference to the specific individuals and the properties and relationships being reasoned about.

In this work, the abstract lattice structures are the form: like the hexagrams and trigrams themselves, these are independent of any particular concrete situation. The structures represent energetic relations between the aspects of a complex situation. On the other hand, the content reflects the actual circumstances under consideration. Just as many particular arguments will fit a given formal pattern of reasoning, so different actual circumstances can be understood through a particular structure. This understanding is reached through the addition of a narrative for a structure.

Form and content, structure and narrative, are the yin and yang of this analysis of change. In the following discussion, the proposed narrative is kept reasonably general, for an actual reading, this would be made more specific, relating it directly to the given circumstances being explored.

A narrative can be composed of a number of distinct elements. The first aspect of interpretation is a description based directly on the equation for the change. In this case we have x = which gives the main idea for the situation:

Binding the Great Energy when Resonating with Arousal requires Determined Effort

Binding the Great Energy comes from the primary hexagram : the lower trigram, the manifest energy of the circumstances, is the pure yang Creative, but the energy of Heaven's pattern is Binding, controlling the situation. Resonating with Arousal comes from the change quanta hexagram : here the upper trigram, the determining pattern, is Arousing, and the circumstance, the lower trigram, Gentle, resonates with this.1 Finally, Determined Effort comes from the traditional meaning of the related hexagram , "gnawing through".

The flow of change starts from the primary hexagram. This gives us the initial aspect of the interpretation of the flow itself: with a finite, limited bit width, it is necessary to focus ones energy most efficiently to get the most out of the available resources.

It is also interesting to consider the structure itself. Notice that half of the hexagrams have Fire in the upper trigram, and half have Binding, whilst the lower trigram has one of the four following energies: . This suggests two general themes - half of the structure is about understanding energies and the other half is about binding and controlling those energies.

Now consider the initial changes, the first steps in the flow of change. The following three hexagrams describe the aspects of the change most closely connected to the primary energy. Working around the lattice structure, starting directly beneath the primary hexagram:

When the third yang opens, the centre of the change, in the heart's line, a Sacrifice is required; not all activities can be maintained, with limited resources there is a choice of what to advance, of what to give up. Sacrifice involves being open to the binding. What forms the spark in the fire in this context, is what needs to be sacrificed!

The lowest changing yang in the second place brings Illuminated Binding: this suggests the importance of seeing the nature of the required pattern of restriction with clarity. This will illuminate the choice of what to sacrifice2.

The top changing line, yin closing to yang in the fourth place, strengthening the mind's place, takes us to the upper pole of the cubic subspace, Great Possession. The peak of energy in this cube's context is focused in the place of spirit; this will help give the strength to make the sacrifice.

The next wave of change creates further aspects that need to be integrated into the situation. These aspects are more closely connected to the resulting energy, and bring the paths of change towards convergence.

At the yin end of the cube's axis, as the common continuance of Sacrifice and Illuminated Binding, is the lower pole of this cubic subspace. In this context we shall know it as the Gate3, and it shows that to resolve this transformation it will be necessary to reconcile both ends of the cycle.

The common onward flow of Sacrifice and Great Possession is Polarization, this shows an unavoidable side-effect of working to bind and control the great yang energy. Opposition - yang energy naturally expands and flows and cannot be restrained without creating a reaction.

The common onward flow of Great Possession and Illuminating Binding is Clarity, seeing clearly what is required and acting softly through the yin. The energies of heaven and earth are made coherent, ideals and circumstances must move together as one.

Finally, the flow of change through the cube converges:

From Clarity, Polarization and the Gate, completing the cube at Determined Effort. This is the requirement to complete the change: the Arousing energy from the earth must add to the Fire of heaven, strengthening and maintaining it.

The narrative given here is intentionally kept general. Reread it now, with the idea of someone working to control a compulsive habitual behaviour.

From Fire to Water

We have seen above that this equation generates a cube which also contains a number of equivalent equations representing different changes. The difference between the individual changes encoded within any particular structure is defined in terms of the direction of energy flow through the structure.

The animation on the right shows the change for 796987 flowing through the same cube. This has the equation:

    • x =

In this case, the change starts in and flows through the structure to . I shall not present a detailed description of this flow, but you can contrast it with the animation for 799687 shown above.

Formalizing Flow as Labelling

It can be seen from the set of 8 equations given above that the two equations of change animated here generate the same cube: if you inspect the hexagrams in the two animations shown above, it is clear that both of these cubes are structurally identical. However, the energy flows through them in different directions. This is demonstrated informally by the animation.

We can represent this flow formally using a numerical labelling function. A labelled hexagram will be of the form N-G, where N is the numerical label and G is the gua. A concrete example is 3-, which shows the hexagram Illuminated Binding labelled with the number 3. Because we are interested in the flow of change through a structure, the labelling function that I shall present here notates the distance of each hexagram in the sublattice from a particular starting point, where that starting point is the primary hexagram of the equation under consideration.

Beginning with the basic set of hexagrams defined by the sub function, we can see that the two sets of hexagrams generated by the changes described above are, as expected, identical (the formal definition of this function is given above):

    • sub(, ) = sub(, ) = { }

If we then label the set, assigning the number of steps from the primary hexagram to each hexagram in the set, we get the following results. This clearly shows the differences between the two flows of change:

    • lab(, sub(, )) = {2- 1- 0- 1- 2- 1- 2- 3-}
    • lab(, sub(, )) = {2- 1- 2- 3- 2- 1- 0- 1-}

Compare the labels applied to the hexagrams to the order of the energy flow in the animations above. The order defined by the labels matches the order in which the hexagrams are highlighted in the respective animations. So, by labelling the elements in the sublattice according to their difference from the primary hexagram, we can represent the direction of energy flow through the structure. This, therefore, encodes the difference between the individual changes within identical structures.

The formal definition of the lab function is:

    1. lab(P, S) = {ea(P x G)-G : G in S}

In this function, ea(P x G) is the label; this is the absolute energy level (ea) of the difference between the given hexagram G and the primary hexagram P.

Interpretation Notes

Some aspects of the discussion above involve what might be called non-classical interpretations of the hexagrams. These interpretations are derived from the various formal systems of energetic change described on this site. The following notes give a few details of these differences, with links to those aspects of the energy systems that I believe support the interpretations. These are not intended to replace traditional meanings, but are suggested as supplemental possibilities.

1: I would like to make the suggestion that the traditional interpretations of Persevering and Conjoining could be reversed. That is, interpret as Conjoining Energies. In the trigrams, the Strong yang energy bursting forth above, is matched Gently below. The two trigrams are complementary, suggesting the name Resonating with Arousal. In contrast, considered through the trigrams, seems more like Persevering: Binding the circumstances to create duration, in the context of an Open pattern. Also, both are placed adjacently as the second and third steps respectively in the three yang coherent wave sequence. This sequence is about the effort of raising three adjacent yang lines up through the spectrum. As the first step, is about the initial impetus to elevate the energy, whilst as the next step, is about the perseverance necessary to continue that elevation.

2: Although different to the usual meaning of Grace or Adornment, the interpretation used here draws directly on the images of the trigrams: the pattern of Binding active in the situation can be Illuminated by consideration of the circumstances. A similar idea is expressed in Wilhelm's commentary on the hexagram. An alternative interpretation for this hexagram comes from taking a similar angle to the interpretation of : the energy of the Fire is Bound and controlled. This is a delicate situation, Fire itself is the energetic union of Binding and Arousal, thus Binding Fire is binding an already bound energy.

3: Here, this is called The Gate. For supporting material see: i) - the Receptive facet of the Gate's own unfolding, where this hexagram takes the first and last steps of the canonical threaded lattice progression and links them through energetic unification; and ii) - the two yang coherent wave sequence, where itserves the same linking role, connecting the first and last steps in a natural sequence.