Comprehension of elusive connectivity and coherence for sustainable governance

Engaging with Elusive Connectivity and Coherence (Part #9)

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It is appropriate to recall that the focus here on the interlinking of three rings emerged from consideration of issues of seemingly incommensurable three-foldness in governance (Triangulation of Incommensurable Concepts for Global Configuration, 2011). This is most notably evident between the mindsets, priorities and language of right-wing, left-wing and centre in political systems, or between academia, business and government, as reflected in the current preoccupation of the Triple Helix model. In the language of classical mechanics it could be understood as the challenging dynamics of the three-body problem and its integrative comprehension.

The traditional theological preoccupation with any trinity can then be understood as another language through which this cognitive challenge to coherence has been explored. One insightful clarification of the challenge is itself articulated through a triangle of colours in mathematicaal terms by Ron Atkin (

Implications of polygonal configurations for comprehension of complexity: Of potential relevance to this argument is the attention given to developing an appreciation of nine-foldness in children, as described by Sheryl Nichols Morris (Nine: Observing Nine with Young Children (SNAP: Scaffolding for Numerical Synapses: Awakening Curiosity in the Numbers One to Ten. 2013).

The challenge of acquiring such insight -- beyond the binary -- is remarkably articulated using a triangle, from the mathematical perspective of q-analysis by Ron Atkin (Multidimensional Man; can man live in 3-dimensional space?, 1981). Atkin illustrates the challenge of comprehension in relation to experience constrained "within" the geometry of a triangle -- especially with regard to the perspective necessary to comprehend the geometry of the triangle as a whole, as separately summarized (Comprehension: Social organization determined by incommunicability of insights).

The constraints are a consequence of being trapped in a form of tunnel vision "within" the lines (of argument) configured as communication traffic around the point (especially when the lines are constructed as cylinders). This inhibits the acquisition of any sense of perspective. The cognitive significance of this approach is well-illustrated by visual sensitivity to colours resulting from the three primary hues (red, green and blue). These may be represented on a simple triangle. Here the vertices (O-simplexes) represent the primary hues, the sides are twofold combinations (1-simplexes), and the combination of the three hues makes the central white (2-simplex).

Challenge of recognizing a higher-order perspective schematic used by Ron Atkin
	0-dimension vision: Red, Green or Blue 1-dimension vision: Yellow (=Red/Green) Purple (=Red/Blue); or Turquoise (=Blue/Green) 2-dimension vision: White (=Red/Green/Blue)

The relevance of q-analysis has been more recently clarified (Konstantin Y. Degtiarev, Q-Analysis and Human Mental Models: a conceptual framework for complexity estimate of simplicial complex in psychological space. 2011; Jacky Legrand, How far can Q-analysis go into social systems understanding?).

Divinity -- as a metaphor? The Saiber study (2013) has the merit of framing the quest with the respect for subtlety associated with divinity -- despite the various topological formalisms to which reference is made. As a quest, the sustainability for "governance" implied by that comprehension (as it might be most generally understood) can similarly be compared with that for the Holy Grail (In Quest of Sustainability as Holy Grail of Global Governance, 2011; Larry Miller, Sustainability: the New Holy Grail, Political Christian, 29 December 2010). Aside from the traditionally articulated dangers of "grasping" and seeking to possess it, that form lends itself to comprehension through catastrophe theory (Interrelating Cognitive Catastrophes in a Grail-chalice Proto-model, 2006). Fundamentally the issue may be one of perspective and perception.

As noted by Saiber (2013), regarding perception of the three "giri" by the Pilgrim in Dante's Paradisio:

It (or "they", as we will consider below: parvemi or parvermi ?) appear in the form of (or looked like to him) three giri (circles/discs, spheres/balls, tori, cylinders, spirals, ellipsoids, or other round things?), di tre colori (each a different color, or each containing three colors?), and d'una contenenza (all the same size, all occupying a single space, all contained within a single space, and/or all of the same substance?). One of the giri appears as a reflection of another, come iri da iri (resembling a double rainbow, or merely an analogy for generation?), and the third giro appears as a flame (parea foco) breathed or breathed forth in equal measure (si spiri) by the first two (quinci e quindi). (pp. 207-8)

The definition of the form itself is elusive:

What begins to become apparent is that the actual shape of the giri could be one or more of these "roundnesses" simultaneously, especially if the giri were in motion. A ring (circle) spinning on its edge, for example, could look like a sphere revolving about its center, like a wheel or a disc. And depending on the figures' configuration, other "roundnesses"' could emerge to the eye (p. 242)

Dynamics? Their movement is a factor, raising the question as to the possible nature of that movement:

If we read Dante's giri as "roundnesses in motion", what are the possible ways in which roundnesses could move? The poet certainly does not say anything specific about motion; nor does he leave any clues. Each giro , if flat like a circle or disc, could be turning around its own central axis like a pinwheel or frisbee; or it could be spinning on an edge like a coin dropped on a table. If the giri are spheres, then they could look like tennis balls with topspin.... Furthermore, the giri could also be partaking of multiple types of circular motion.... Many medieval theologians held that God's thought moved circularly (human thought linearly, and angelic thought in a spiral) and that to contemplate God we too needed to set our mind moving in a circle. (pp. 245-7) [emphasis added]

There is a provocative irony to framing the spiral as characteristic of angelic thought, given the current secular preoccupation with the Triple Helix model of innovation (Reality of the demonic and the angelic by comparison with secular understanding of existence, 2016).

Three-dimensionality? Little consideration has however been traditionally given to comprehension in three-dimensions, although it is noted that one the most sustained meditations on circular geometry as a key to understanding God is the work of Nicholas of Cusa (David Albertson, Mathematical Theologies: Nicholas of Cusa and the Legacy of Thierry of Chartres, 2014, especially pp. 243-52), As the study notes:

Most commentaries on the giri's configuration have imagined them to be arranged in a planar format. Only a few have theorized how the giri could, instead, be arranged using spherical geometry.... It is not clear, in fact, from Dante's description of the giri how he intended them to be linked; but they had to be united/unified in some way.... The most commonly-imagined arrangement of the giri, however, has been of three intertwined circles, which as knots, are actually occupying three dimensions, but when seen face on, can seem flat....The final section of this study is a topological and combinatorial analysis of the layout of three intertwining circles we find particularly compelling vis-a-vis Dante's description of the Trinity: the round figures arranged in a triangular format with rotational and reflective symmetry. (pp. 255-260) [emphasis added]

Comprehension? Appropriate to this argument, the analysis uses "vertigo" to name the challenge to comprehension:

When we think of all the possible ways three circles could be linked, as we have done so far in this essay, a kind of vertigo begins to set in. If we narrow our focus to just a single arrangement -- that of three, linked circles in a triangular format, it turns out that there is a limit to how many ways they could be configured: there are only sixty-four. These sixty-four patterns of "links", as topologists call them, can be sorted into five categories. Of these five categories, there are two... that provide us with particularly intriguing models for Dante's Trinity: the Borromean rings and the (3,3)-torus link (p. 260) [emphasis added; see overlapping circles grid]

Transience? Given both the potential significance of movement and the 64 patterns, there is a case for recognizing that, rather than being "permanent", the elusive unity implied may involve a high degree of transience, as extensively explored in the argument of Raymond Tallis (Of Time and Lamentation: reflections on transience, 2017).

The pattern of 64 is of course characteristic of the Chinese classic, the I Ching, in which such completion is but one phase in a cycle (Transformation Metaphors: derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997). It is in this sense that one animation to be explored would have the three linked circles cycling through all 64 phases. Aspects of the case are presented separately (Reframing the Dynamics of Engaging with Otherness: triadic correspondences between Topology, Kama Sutra and I Ching, 2011).

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