</a>Polyhedral empowerment: qeliciting the sparkle from networksq?

Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (Part #18)

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There is a fruitful play on the relationship between "dual" and "jewel" in that the wealth of civilization -- its highest values -- may be epitomized not so much by its precious stones (often set in symbolic "crowns", rings, or pendants) as by the dynamics of duality, so well mapped by morphing polyhedra. Hence also the association with sacred geometry as emblematic of the highest forms of integration and order.

It is also interesting to recognize that networks as such do not "sparkle"; it is the crystals, associated with the polyhedral forms into which networks may be folded, that enable the reflection and refraction of light for which they are so valued. Eliciting this capacity, in its most enhanced forms in gems and jewels, is dependent on symmetry (Steven Dutch, Symmetry, Crystals and Polyhedra, 1999). More specifically gems are most notably associated with the set of polyhedral forms that exhibit a degree of self-duality amongst themselves, namely the 5 Platonic forms: tetrahedron, cube/octahedron, icosahedron/dodecahedron. Enhancing their capacity to sparkle is achieved by appropriate cutting, facetting and polishing -- themselves offering interesting metaphors with respect to networks.

Crystals are:

• formed by linking of coordination polyhedra of which the most common are: 4-fold (tetrahedral), 6-fold (octahedral), 8-fold (cubic) and 12-fold (close-packed); more unusual forms are also possible.
• grouped into one of 7 crystal systems based on the symmetry of their atomic arrangements: cubic (isometric), tetragonal, hexagonal, rhombohedral (trigonal), orthorhombic, monoclinic and triclinic. The symmetry of the arrangement of atoms is associated with the regular geometric form of many (uncut) crystals.
• classified in 7 crystal systems in terms of the set of 14 Bravais lattices, namely categories of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices. In 2 dimensions there are five Bravais lattices. (rectangular, centered rectangular, hexagonal, and oblique); in 4 dimensions, there are 52 Bravais lattices (of these, 21 are primitive and 31 are centered).

As noted above, gems offer an interesting metaphor regarding processes of relevance to governance (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). In the above context, where polyhedral nets embody the necessary symmetry to enable them to "come alive" and "sparkle", what is to be said of the portions of the network that are appropriately configured to ensure that this process occurs?

• are the factions forming a polyhedrally empowered network to be understood as an appropriately configured array of facets, each important to this process?
• what is to be understood about how the polyhedral form is "cut" to expose the facets most capable of enhancing "sparkle"?
• given the etymological root, is "cut" then to be understood as related to "decision", and to decision-making capacity?
• what is to be understood by the necessary asymmetry required in a gem?

Given the way in which reflection and refraction work in gems, the existence of gem-like forms in which light is only reflected from the outer surface offers interesting metaphors regarding those in which light may only be reflected between the inner surfaces, not allowing it to pass out -- what might be termed "dark crystals". The issue might be related to the distinction between bonding ("within crystal") and bridging ("between crystals") (cf Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004; Y. Connie Yuan Geri Gay, Homophily of Network Ties and Bonding and Bridging Social Capital in Computer-Mediated Distributed Teams, 2006). This is partially played out, notably in academic circles, through what have been termed "mutual citation pacts".

Presumably such issues of reflection and refraction could be understood as being intimately related to the sense of image -- especially self-image -- and the importance attached to it by a person or a group.

Given the features of a network, people (or groups) may be variously identified with (and hold responsibility for):

• a node or vertex (perhaps representing a faction or a perspective)
• an edge or link (perhaps representing a challening or sympathetic relationship to another)
• a face (perhaps representing a coalition, minimally of three parties, but increasingly unstable beyond 6 unless appropriately set in a more complex polyhedral configuration)
• a great circle linking the nodes (perhaps to be understood as a theme or discussion thread) defining a plane of symmetry of a spherically symmetrical polyhedron
• an axis of symmetry perpendicular to any plane of symmetry (perhaps to be associated with cyclic forms of identity, as mentioned above)
• the polyhedral pattern as a whole, notably arising from the interlocking of the various thematic great circles, each of different orientation corresponding to different axes of symmetry (as discussed in Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998)

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