Incommensurable cognitive patterns and their symbolism

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #13)

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The (questionable) assumption of this argument is that effective choice by a group of a primary symbol -- as on its flag -- reflects the distinctive identity of that group and its engagement with life. The focus here is on the 5-fold in relation to the 6-fold. Similar arguments -- and analogous animations -- could be explored with respect to other psychosocial relations that are intractable and incommensurable in "two-dimensions", perhaps 3 vs 4, 4 vs 5, 6 vs 8, etc.

Recognizing contrasting systems of belief: The 5 and 6-pointed stars on which this exploration is based are instances of a set of star polygons which have not as yet been formally defined. Only the most regular ones have been studied in any depth. They can be represented as a sequence of stellations of a convex regular core polygon. Wikipedia offers  convenient images of stellation models of associated polyhedra (see stellations of: octahedron, dodecahedron, icosahedron, cuboctahedron and icosidodecahedron). The image below makes clear that the above argument relates only to first two instances in the second diagonal line. Other stars in the diagram are potentially variously significant to other belief systems.

Regular star polygons (reproduced from Wikipedia entry)

Embodiment of vital insights by the "other": In an intractable conflict it is interesting to explore how their "reconciliation" in a three-dimensional configuration would seem to highlight other number-based patterns in an unexpected manner. In the truncated icosahedron, it is the "Islamic" 5-fold which effectively represents the pattern of "12 tribes" so important to the traditions of Israel. There are 12 pentagons in which the Islamic Star has been embedded. Is this characteristic of the sense of problematic "otherness" -- as psychologists would tend to claim? What might the pattern of 20 hexagonal -- with the embedded Star of David -- be recognized as representing as vital within Islamic tradition? Rather than explore this possibility, the question it raises is how the numbers associated with truncated icosahedron (faces, edges, vertices, etc) might interweave elements of both traditions.

Failure of systemic connectivity: The widespread importance of 12-fold pattern, notably characteristic of jury systems and the like, is noted separately  (Checklist of 12-fold Principles, Plans, Symbols and Concepts, 2011). Their operational implications have also been explored (Enabling a 12-fold Pattern of Systemic Dialogue for Governance, 2011; Eliciting a 12-fold Pattern of Generic Operational Insights: Recognition of memory constraints on collective strategic comprehension, 2011). The disconnect between the 5-fold and 6-fold in the Middle East is an indication of a failure of systemic connectivity and its requisite underlying conceptual organization.

Polyocular strategic vision: Whether in that particular case, or in the case of other concept schemes which could be mapped into a set of polygons in two dimensions, each polygon can be understood as having a particular systemic function. In effect each operates as a distinctive "eye" -- an organ of strategic "vision". Configured in three dimensions, it is more understandable that each relates to a subsystem with which a form of "subunderstanding" is associated -- as with dependence on the right or left eye when both are required for depth perception through stereoscopic vision.

A 12-fold vision, implicit in the justification of jury size, is then to be understood as enabling "polyocurlar vision" as articulated by Magoroh Maruyama (Polyocular Vision or Subunderstanding? Organization Studies, 2004). Assumptions relating to a singular vision, or a single strategic sense, can then be usefully challenged (Cyclopean Vision vs Poly-sensual Engagement, 2006). How does each of the 12 both understand and not understand -- as implied by "blindspots" and "horizon effects" within the vision metaphor?

Polyhedra as systems -- of belief: Also intriguing is the sense in which configurations of pentagons around hexagons, or hexagons around pentagons (as presented above) -- implying extended zones of discourse and systemic preoccupation -- may overlap or interlock within the three-dimensional structure. It is appropriate to note the arguments of R. Buckminster Fuller by which polyhedra, especially the regular polyhedra, are to be understood as systems. This reinforces the subsystemic status of individual polygons, or clusters of polygons, through which any three-dimensional structure is configured.

The case of the Middle East offers a further possibility for exploration -- given the role of Christianity, most notably in relation to Jerusalem. This suggests the need to identify polyhedra which combine polygons representative of Islam and Israel with those distinctively representative of Christianity -- perhaps a triangle.

It follows that there is the intriguing possibility of not only associating the expression of faith with a polygon -- ironically to be caricatured as "putting a face" on each. Going further, there is the possibility of associating the integrity of that faith as a "closed system" with a regular polyhedron, potentially as follows::

• tetrahedron (4 triangular faces)
• hexahedron / cube (6 square faces)
• octahedron (8 triangular faces)
• dodecahedron (12 pentagonal faces)
• icosahedron (20 triangular faces)

These could then each be understood as the vehicle or container for the respective faith -- when unchallenged by any "other". The mutual challenge of the main faiths is then a form of "geometrical challenge" between the regular polyhedra. In the case of the Middle East, primarily between the dodecahedron and the icosahedron, say. This symbolic mapping could be readily deprecated as simplistic but this would preclude more complex understandings arising from the possibilities in which polyhedra morph into each other dynamically to express particular insights under certain conditions.

If polygons and polyhedra can be usefully understood as representative of faiths -- if only as mnemonic mappings -- this then raises the possibility of a useful relationship between any periodic table of polytopes (polygons and polyhedra) and any periodic table of religions, as separately discussed (Tuning a Periodic Table of Religions, Epistemologies and Spirituality -- including the sciences and other belief systems, 2007). Within such a framework an atheistic belief system might be appropriately, or provocatively associated with the nobles gases.

"Resonance"? Given the role of resonance in enabling the biology of living structures, is there the intriguing possibility that a process akin to resonance may play a vital role with respect to the viability and sustainability of psychosocial systems -- and to "belief"? Is there some intuition that the currently much-challenged "bond" vital to "confidence" in any psychosocial system -- "my word is my bond" -- has characteristics which could benefit from the insightful consideration of the dynamics of bonds in chemical molecules? How might "confidence-building" then be understood?

Of relevance to this possibility is the extensively articulated description of one process of the Arab Spring by Gaston Gordillo (Resonance and the Egyptian Revolution, Space and Politics, 6 February 2011), variously indicating:

• What has coalesced as a powerful, unstoppable force on the streets of Egypt is resonance: the assertive collective empathy created by multitudes fighting for the control of space... Resonance is what gives life to this human rhizome and the source of its power.
• Everybody feels the resonance reverberating from Egypt and is trying to make sense of it, to name it. But the words seem inadequate, partial, incomplete: enthusiasm, energy, passion, anger, contagion, electrifying, domino effect. These terms name features of resonance but miss its salience as a physical, affective, political force made up of living bodies.
• Resonance has been conceptually invisible for so long because it involves the most immanent, physical, taken-for-granted dimensions of social life: bodies and space, modulated by the same temporal pulsation.
• ... a defining feature of resonance is that it does not stand still. It is mobile and expands, affecting more and more bodies. This is why so many reporters use metaphors of contagion to explain its expansive force.
• Resonance shakes bodies, even foreign bodies, and makes them act out of empathy. And because it reverberates and is contagious, resonance can travel long distances, spreading outwards from its original node.
• Resonance, in short, forces us to look at wider, complex, ever shifting and fluid topographies of unrest that connect and affect distant and seemingly disconnected geographies.
• What is most unfathomable about resonance is its power, a power that has fueled all the revolutions of human history. Resonance can erode and destroy the most powerful of states, especially when it affects the bodies of those with orders to shoot.
• Resonance... is not a metaphor. The power of affectation that these bodies create is as material as the forms of resonance that are studied in physics and travel through air, water, and solids. Understanding political resonances indeed requires developing a physics of politics. Unlike standard physics, which seeks to find predictability in motion, a physics of political resonance involves the shifting patterns of movement by rhizomes of striving bodies coming together and spreading in unpredictable ways. Resonance unfolds in the realm of political contingency yet through well-defined patterns.

Football illustration of questionable attribution of "positive" and "negative" -- of relevance to Middle East perceptions (given that the standard football is manufactured with a variety of panel colourings and markings)
5-fold as "negative" / 6-fold as "positive"	6-fold as "negative" / 5-fold as "positive"

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