Interplay of cognitive patterns in discourse on systemic change

Enhancing Strategic Discourse Systematically using Climate Metaphors (Part #8)

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This argument derives from the ongoing detection of large numbers of systemic loops interrelating world problems -- "loops" taking the form of 3, 4, 5, and more polygons with potential interlocking between them, as described previously (Eliciting Memorable Spheres and Polyhedra from Hyperspace: integrative connectivity of problems, strategies, themes, groups or people, 2015). It was suggested above that the drilled truncated cube of 64 edges offered a convenient 3D polyhedron onto which the 64 convex uniform polychora (in 4D) could be mapped, if only experimentally.

Cyclic transformation of any polyhedral mapping? A simple pattern of transformation of the drilled truncated cube is illustrated by the following. The images are indicative of a way of thinking about cubic order as introduced above. They suggest a way in which the sides of a (closed) cube might open and close in relation to the axes in 3 dimensions. The opening through the toroidal hole can be understood as enabling or precluding movement in either direction -- making for 6 directional possibilities.

Given the climate preoccupation were, the hole be suggestively framed by the classic BaGua circle of 8 trigrams possibly arranged in the two contrasting classic orders ("Earlier" and "Later"). Together these give 2 x 8 trigrams in each direction, namely a total of 48 for all 3 dimensions -- and understood as increasing to 64 with the 4th dimension. The 6 directions are usefully renascent of the 6 basic possibilities of change from any given condition, as encoded by the system of hexagrams of the I Ching.

6-phase cycle of axial transformations of drilled truncated cube
opening along z-axis only	opening along y-axis only	opening along x-axis only

The complementary roles of closing and opening have been explored by Orrin E. Klapp (Opening and Closing; strategies and information adaptation in society, 1978). By contrast, the following images explore perspectives through a diagonal of the above form (as oriented below). Rendering the octagonal faces transparent offers the image on the right (below).

Diagonal view of the drilled truncated cube
No faces transparent	All faces transparent	Octagonal faces transparent

Of particular interest are the striking contrasts along the diagonal (as above) as the degree of zooming is increased or decreased (as illustrated below). According to the coincidence of edges in a given perspective through the face-transparent structure, these take the form of symbols recalling successively those of each of the three Abrahamic religions: the trinitarian triangle, the cube at Mecca, and the Star of David. In the case of the first two, associated symbols emerge from the perspective by highlighting proportions with red dotted lines. The symbols derive from the interplay between the upright triangle on the proximate end of the diagonal, the inverted triangle at the opposite end, and the inner cube. Such indications reinforce the arguments for mathematical theology, as separately explored (Mathematical Theology: future science of confidence in belief, 2011).

In that spirit it is also appropriate to note the pattern of 12-foldness, both in the circumference (below) and in the inner cube (above).

Contrasting perspectives along one diagonal of the drilled truncated cube
Extreme of zooming in	Midway zoom	Extreme of zooming out

As a truncated cube, each triangular corner vertex can be tentatively associated with one of the 8 trigrams of the Ba Gua configuration of climate conditions (as argued above). In the following images, diagonally opposite corners are identically coloured. The distinctions between their patterns can be seen. As previously noted, the total number of edges coloured is 64, indicative of the pattern of conditions encoded by the 64 conditions of the I Ching.

Edge coloured views of drilled truncated cube to illustrate complementarity

Such images suggest the possibility of exploring the dynamics of such configurations as they might relate to the innovations of Nikola Tesla (Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014). Especially intrguing is the possibility discussed there of Imagining a method for adapting Tesla's insights to a psychosocial context. Most notably, with respect to the above images, is the degree to which these can be used to demonstrate and explain the psychosocial relevance of the principles of Tesla's rotating magnetic field.

Mapping hexagrams otherwise: The representation in the left below constitutes an alternative to the conventional columnar representations of hexagrams in an effort to explore other mnemonic possibilities. The order follows that used in the lower right hand corner of the table in the Richard Wilhelm translation of the I Ching. Although this order is used in the Wikipedia entry on the I Ching, it is not the order of the classical King Wen sequence. The lower, internal triangle, corresponds here to the lower trigram (in which the horizontal line corresponds to the lowest in the columnar hexagram representation). The upper lines correspond to the upper trigram (in which the central vertical line is the uppermost in the columnar hexagram representation). The convention of complete and broken line is maintained.

Mapping of I Ching hexagram encoding (each presented as an animation of 8 rows of 8 columnar variants)
Triangular representation	Mapping onto double triangle pattern of Star of David

In the earlier exploration noted above (Sustainability through Magically Dancing Patterns 8x8, 9x9, 19x19 -- I Ching, Tao Te Ching / T'ai Hsüan Ching, Wéiqí, 2008), using one particular triangular design convention, it is therefore also of interest to explore the configuration of the 6 lines of the I Ching hexagram into a double triangle consistent with such traditional symbolic use. The following table **** was presented in that document (as Table 6: Double triangular representation of hexagrams: Star of David)

This array has been presented in an experimental animation in a separate paper (Mapping of I Ching hexagram coding onto Star of David, 2008). With respect to higher dimensional ordering (discussed below), a natural group of transformations of hexagrams has been discovered during an investigation of the six-dimensional affine space over the two-element field by S. H. Cullinane (Geometry of the I Ching. 2006)

Dynamic between polytopes: However, given the emphasis here on time and dynamics in relation to systemic change, it is appropriate to understand that 3D form as implying a dynamic associated with the relations between the polygons (or polytopes) defining that form -- especially if aspects of that form are associated with the 64 polychora. The apparent invariance of the form is perhaps then to be better understood as akin to that of a standing wave.

Of further interest is the manner in which the drilled truncated cube might be integrated as an intermediary form in the dynamics associated with the cube nested with the other Platonic polyhedra within the rhombic triacontahedron (as illustrated above) -- and perhaps even to be considered as framing that complex within the inner cube.

Given the polygon manipulation capacities demonstrated by Pieter Huybers with the CORDIN application, the question is how to enable reflection on the role the polygonal components of the drilled truncated cube (as a striking example) might play in relation to the other polyhedra in that complex. Are the dynamics then to be understood in systemic terms -- as a complexification of the cognitive pumping cycle suggested above, perhaps in accordance with insights from enantiodromia, as discussed separately (Psychosocial Energy from Polarization -- within a Cyclic Pattern of Enantiodromia, 2007)?

The following images offer additional ways of reflecting in visual terms on the systemic interplay of forms within a process of collective discourse in relation to systemic change and governance. Again it is appropriate to emphasize that the 64 distinctions it is sought to map can be understood as folding into and out of understandings of "climate" as might be generally understood. Climate metaphors *****

In addition to how these images may be indicative of the operation of some kind of cognitive pump or motor, the dynamic of the assembly of disparate polygons in the configurative phase of the cycles shown is a reminder of the above-mentioned challenge of configuring disparate systemic loops of problems and strategies.

Screen shots of polygonal dynamics associated with the drilled truncated cube
Exploding	Unfolding

Requisite morphogenesis of cognitive models? The dynamics of unfolding and enfolding in the animations above also offer a visual illustration of the argument made earlier with respect to the quadrilemma. In particular phases in the cycle, it may be variously asserted (but not definitively) that the structure:

is a cube is not-a-cube

is both a cube and not-a-cube is neither a cube nor not-a-cube

Is there a case for recognizing any cognitive or strategic model in such terms, thereby responding to restrictive tendencies to over-determination when a degree of flexibility is required? **** Thom semiophysics

In that light, are the geometrical possibilities of morphing forms of potential relevance to how any cognitive model could lend itself to fruitful transformation, as suggested by the following screen shots and their associated animations?

• Morph duals by tilting to rectify
• Morph duals by tilting triangles
• Morph duals by tilting quads
• Morph duals by truncation

Morphing duals of drilled truncated cube by contrasting methods (click on screen shots for separate animations****)
by tilting to rectify	by tilting triangles	by tilting quads	by truncation

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