Implication of through approximation of strategic polyhedra to a sphere of "globality"

Embodying a Way Round Pointlessness? (Part #7)

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As a geometrical metaphor (as argued in the main paper), the sphere is readily held to be of more fundamental integrative significance than the circle. The arguments above relating to pi apply equally in relation to the sphere. Cognitively this necessarily offers a more encompassing "global" sense of fulfillment. The form epitomizes, to a greater degree, the strategic challenge of the "vision" of a multi-dimensional "holy grail" of global governance, as separately discussed (In Quest of Sustainability as Holy Grail of Global Governance, 2011).

As described above, one or more linear strategies are effectively challenged to engage and encompass the circularity of sustainability. In the case of the sphere, however, strategies are understood to be variously configured as a number distinct polygons. These together form the sides or surfaces of a polyhedron -- potentially symmetrically configured. Corresponding to the progressive approximation of a polygon to a circle, with the increase in the number of its edges (clarified by Fig. 2 and Fig. 3), the set of sides approximates to a surrounding sphere as the number of sides increases (as shown in Fig. 4). As with the circle, pi is necessarily the "x-factor" through which the "superficiality" of the variously configured polygons is reconciled -- to a degree -- with the "globality" by which sustainability is held to be characterized.

Fig. 4: Progressive approximation of increasingly complex regular polyhedra to a circumscribed sphere as indicated by surface area and by volume for the regular and semi-regular polyhedra up to 92 sides (derived from data in Stella Polyhedron Navigator)

Approximating globality: As with the case of the approximation to a circle, the question is then how many "strategic sides" fruitfully approximate to a sphere? Using the term "sides", of particular relevance to coherent global governance is the extent to which these polyhedral sides may be felt to be explicitly associated with mutually challenging political or ideological "sides". In this language, as the geometry illustrates, the smaller the number of sides (configured within the circumscribed sphere) the lower the ability to engage effectively with the globality implied by that sphere. There is then a case for exploring the number of sides -- characteristic of different polyhedra -- which could be held to approximate more "adequately" to the sphere, as was done in the case of the circle (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008).

As in the comparison of the polygons (in Fig. 1), Fig. 4 shows the greater relative capacity of the dodecahedron and iscosahedron to approximate to the sphere -- but only to the extent of 65-85%, depending on whether the comparison is made in terms of surface area or volume. It is only with the most complex of the semi-regular polyhedra, the 92-sided snub dodecahedron, that in excess of 90% approximation is achieved to globality.

Recovering functional insights from myth? This offers a degree of justification for the significance attributed to the association between the dodecahedron and the 12 Olympian Gods of Greek mythology, known as the Dodekatheon -- adapted as the Dii Consentes (also Dii Complices) in the pantheon of Ancient Rome, with its twelve major deities, six gods and six goddesses. This association might be understood systemically as an effort to reconcile the functional perspectives (or "sides") associated with each deity in order to give form to an elusive larger "global" system -- which otherwise readily escapes comprehension.

A similar argument has been made for other 12-fold patterns of articulation, as noted above (Eliciting a 12-fold Pattern of Generic Operational Insights, 2011). Particularly intriguing is the functional significance of the "relationships" between the gods as traditionally highlighted in numerous mythical tales, especially as indicative of requisite patterns of dialogue in governance (Enabling a 12-fold Pattern of Systemic Dialogue for Governance, 2011). Given the viability of these cultures and their governance over centuries -- exceeding those of current times -- a degree of prudence is appropriate in any deprecation of such implications out of hand.

It is curious, by contrast, that since those times little of functional significance is recorded regarding the individual relationships between, for example, the "12 Knights of the Round Table", the "12 Imams", or the "12 Apostles" (even at the Last Supper). This reinforces the case for "mining" myth for systemic insights, as argued by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999).

Requisite variety for global sustainability: Especially striking in the case of the graph in Fig. 4 is the extent to which the regular polyhedra (with fewer sides), including the dodecahedron, only partially approximate to the sphere -- to the extent that this is understood to imply a capacity to encompass globality. The greater capacity to do so, of the more complex semi-regular polyhedra, could be understood as indicating that a greater variety of functional "perspectives" and "viewpoints" is required to articulate that globality adequately -- a greater requisite complexity in cybernetic terms, in the light of the Law of Requisite Variety. What might be those contrasting insights? Does "semi-regular" suggest integration of "alternative" perspectives to complement "regular thinking"?

Patterning variety: Any such requirement would then raise questions of how these characteristic are to be "re-cognized" in practice and of how they might be fruitfully and comprehensibly configured together -- especially when each such contrasting perspective typically gives rise to a "side" in political and ideological dispute. Some examples have been explored separately (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008). Contiguous "sides" may of course be understood as potentially forming coalitions with sub-global preoccupations.

In approximating most closely to a sphere, the 92-sided snub dodecahedron is an example of such a configuration. It is oddly reminiscent of the 92 chemical elements found in nature and of the manner in which they can be configured in the periodic table of chemical elements. The possibility of exploiting this pattern to elicit functionally significant psychosocial relationships has been discussed separately (Tuning a Periodic Table of Religions, Epistemologies and Spirituality: including the sciences and other belief systems, 2007).

Given the fundamental importance attached to the "vision" metaphor, another approach is to explore the movement of light within facetted crystals as polyhedra -- the precious stones and jewels so fundamentally associated with value and insight. In order to maximize their effect in channelling light, these are cut into patterns of facets -- extending the applicability of the vision metaphor in the light of the optical possibilities of crystals, as separately discussed (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002).

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