Conclusion

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #14)

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The irony of the presentation of this argument, on the occasion of worldwide media focus on the patterned movement of a truncated icosahedron in Kiev, was noted above -- as with the United Nations demonstration of its long-standing inability to elicit relevant intelligence in more creative response to ongoing massacres, most notably in the Middle East. The priority given to the application of the insights of mathematicians of different faiths to the development of ballistics was also noted, in contrast to the possibilities of taking more seriously the challenges of violent difference of belief (Mathematical Theology: future science of confidence in belief, 2011). This priority is of course also evident in relation to other security-related opportunities, most notably with respect to cyberwarfare.

The irony is all the greater in that technology has advanced to the point at which footballs (and the boots of players) are in process of being enhanced with computer chips to facilitate game pattern analysis. This is to be compared with the total confusion within the international community regarding the emergence of the global financial crisis and how it might best be managed -- seemingly in the absence of any comprehensible simulation of both remedial strategic possibilities and the implications of global strategic incompetence of a very high order.

Pattern-learning via game-playing: There is a case for recognizing the extent to which games are played as an unconscious psychosocial device for learning the patterns which are yet to be appropriately acquired in the handling of psychosocial relations. This point has been made with respect to the organization of popularly preferred styles of music, by contrasting the classical patterns implicit in current institutional organization with the modern music yet to acquire relevant organizational expression (Jacques Attali, Noise: the political economy of music, 1977). This suggests that the acquisition of insights from biomimicry and technomimicry could be extended to "ludomimicry" -- following the critical arguments of Roger Caillois (The Definition of Play and the Classification of Games, 2006) regarding those of Johan Huizinga (Homo Ludens: a study of the play-element in culture, 1950), as discussed by Andrew Brown (Agon, Alea, Mimicry and Ilinx, Embodied Knowers).

What is a global civilization endeavouring to learn by kicking around a truncated icosahedron -- whilst others are being kicked to death? How does such a game relate to the possibility outlined by James P. Carse (Finite and Infinite Games: a vision of life as play and possibility, 1987)? Curiously, although it has long been known that there can be no hexagon-pentagon polyhedron with less than 20 vertices, a classical theorem exists to show that polyhedra of any size can be created -- composed solely of hexagons and pentagons -- provided that there are exactly 12 pentagons (Thayer Watkins, Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: there must be exactly 12 pentagons).

What pattern dynamics is society exploring, especially in the Middle East, given that the relevant equation from that theorem excludes the possibility of a polyhedron with 22 vertices -- namely the number of players in association football, acclaimed as the world's most popular sport? If the argument above relates the Islamic faith with the 12 pentagons, what then is the implication of the relationship of Israel to the 20 hexagons in the truncated icosahedron? Why then should the number of hexagons in a hexagon-pentagon polyhedron be constrained to half the number of vertices minus 10 (namely with the number of vertices equal to twice the number of hexagons plus 20)?

Mathematical theology: What might the neglected discipline of mathematical theology contribute to reframing the relation of Judaism to Islam? Given the identification of Christianity with Jerusalem, how might a "third polygon" pattern be incorporated into a spherical geometry reframing of the cognitive challenge? Especially intriguing is the sense in which the "holiness" so powerfully associated with Jerusalem by the three faiths may imply an exceptionally elegant geometrical representation in which the three are configured together. This is consistent with the more general argument above that the distinctions between the faiths so important to the peoples of the world (including "science") may be strangely related to distinct cognitive patterning preferences -- represented to a degree by polygonal and polyhedral geometry and the possibilities of its configuration, possible in higher dimensions.

Global connectivity: Global civilization is currently characterized by erosion of confidence in various forms of authority with concerns widely expressed regarding the need for "confidence-building" -- but with little sense as to what that means. What is the nature of the connectivity  required and how can its emergence be facilitated? It is in this sense that the failure of mathematicians is so extraordinary -- given their insights into "connectivity" of the highest order, and their inability to "connect" with those concerned with "confidence" and "belief".

The disconnect, an "empathy deficit", might be compared to the challenge for those on the autism spectrum -- a recognized characteristic of mathematical excellence -- in comprehending the emotional dimension on which faith is based and which ensures cognitive engagement with a form of "psychoactive curvature" (Zippora Arzi-Gonczarowski, Self, Empathy, Manipulativity: Mathematical Connections between Higher Order Perception, Emotions, and Social Cognition,, 2001). It is this "empathy curvature" which is evoked by spherical geometry and its "global" implications in a faith-based society (cf Wikipedia entry on Empathizing-Systemizing theory).

Quest for a bigger "tent": It is ironic that one of the most significant contributors to the development of spherical geometry, fundamental to the argument above, was Omar Khayyam -- from a country whose bombardment is currently envisaged with the use of ballistic skills to whose development he could be said to have contributed. Curiously Islam and Judaism share in the fundamental metaphorical significance they attach to the protective role of a tent and to its construction -- a metaphor inherited by Christianity to some degree. Omar Khayyam valued the inspiration of his tent-making tradition, as discussed separately (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? Strategic implication in encompassing nothing and coming to naught, 2011). Given the counter-intuitive challenge of "stitching" a football, as noted above, it is appropriate to note his self-reflexive use of the metaphor: Khayyám, who stitched the tents of science.

What "higher dimensionality", offering connectivity and coherence, would facilitate the evident incoherence and rivalry between faiths at a "lower dimensionality"? (cf. Stephen Prothero, God is Not One: the eight rival religions than run the world and why their differences matter, 2011; Thomas Sowell, A Conflict of visions: ideological origins of political struggles, 1987). How might such tent-making now be understood (Towards higher dimensional "tent-making"? 2011). Renowned for his skills in both mathematics and poetry, Omar Khayyam exemplifies the insight where characterized as "empathy curvature".

Strangely the nature of the "nothing" which a tent usefully encompasses is echoed both in the football, so enthusiastically kicked around globally, and in enthusiasm for the spirit of the game and its outcome, as might be detected in one of the quatrains of his Rubáiyát:

And if the Wine you drink, the Lip you press,
End in the Nothing all Things end in -- Yes --
Then fancy while Thou art, Thou art but what
Thou shalt be -- Nothing -- Thou shalt not be less.

At a time when the culture of Omar Khayyam is being associated with Nazism (Israeli foreign minister Lieberman compares Iran to Nazi Germany, The Guardian, 28 June 2012), the mysteriously dangerous relationship between "thoughtlessness" and "nothingness" merits further consideration, as can be variously argued (cf. Unthought as Cognitive Foundation of Global Civilization, 2012; Exploring the Hidden Mysteries of Oxfam's Doughnut, 2012; Swastika as Dynamic Pattern Underlying Psychosocial Power Processes, 2012).

The poetic poignancy of "nothing" for the individual is now echoed in the collective experience of collapse of confidence, historic deficits, and the failure of decision-making -- so recently exemplified by the Earth Summit. Reframing of remedies in terms of global missile-enhanced "security" then merely conflates the emptiness of political rhetoric, and the annihilation it engenders, with the nothingness of that insight -- as conventionally understood. It is the cognitive implication of the spherical geometry capable of encompassing and containing "nothing" that then merits attention. Football offers a good reminder.

Facilitating imagination: The argument has been presented using relatively sophisticated computer applications -- although readily accessible. In highlighting the need for a computer-assisted pattern language, the point was made that the Stella Polyhedron Navigator offers numerous facilities vital to education of spherical imagination, as previously argued (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008). With the rapid development of mobile applications on smartphones, it is not difficult to predict that such facilities will soon enhance social networking interactivity in surprising ways (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008).

Whilst such software also enables paper construction of polyehdra for educational purposes, it was also stressed that many "counter-intuitive" aspects of the argument regarding requisite connectivity can be developed using a marker pen and an ordinary football.

Replicating restrictive territoriality through copyright: It is appropriate to conclude with a reference to the regrettable tendency for the most remarkable creativity with respect to relevant spherical innovation to be marked by a counter-productive preoccupation with intellectual copyright -- suggesting a tendency to replicate in higher dimensions the problematic patterns so evident with regard to two-dimensional territory. Noteworthy examples relate to the legacy of R. Buckminster Fuller and Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994) and the tragic legal dispute between Stan Tenen and Dan Winter (Future Coping Strategies: beyond the constraints of proprietary metaphors, 1992). The only merit this offers is the motivation to explore alternatives by ignoring those based on such misappropriation of the cultural heritage of a global knowledge-based civilization.

Afterthought: Relevance of Goldberg Polyhedra as duals of Geodesic Icosahedra

The argument above, focused as it is on the truncated icosahedron, merits amendment and development in the light of the so-called Goldberg polyhedra. These are a convex polyhedra made from hexagons and pentagons. They are dual polyhedra of geodesic spheres. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. Their variety can can be explored interactively on the website Dual Geodesic Icosahedra. (Dana Mackenzie, Goldberg Variations: new shapes for molecular cages - flat hexagons and pentagons come together in new twist on old polyhedra, Science News, 14 February 2014). They have aroused interest as being of relevance to new understanding of viral capsids, potentially including the coronavirus (Guang Hu, Extended Goldberg polyhedra, â-*MATCH Communications in Mathematical and in Computer Chemistry, 59, 2008, 3).

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