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12-fold Modalities for heavy duty global governance?


Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors (Part #9)


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Geometrically again, the most coherent and compact such configuration is termed the closest packing of 12 spheres:

  • each of the 12 being tangential to the sphere of the central, global identique -- the inner sphere, thus "internalized"
  • each of the 12 touching some neighbouring spheres -- as complementary modes essential to the management of the whole system

Aside from the geometrical niceties, what is to be understood as the cognitive significance of these 12 distinct but complementary modes? There are a number of pointers:

Of particular interest in any adaptation or development of Arthur Young's thinking is the complexity he was addressing as developer of the Bell helicopter. He was concerned with the distinct but complementary functions required of the "governor" of the helicopter, namely the pilot. These functions exceed in complexity and interrelationship those normally associated with other "piloting" metaphors that have been an inspiration to governance (eg driving a car, steering a vessel, piloting a conventional aircraft).

The significance of these cognitive challenges might be further understood through a recent study by George Lakoff with Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000). Other leads have been explored elsewhere (Functional Complementarity of Higher Order Questions: psycho-social sustainability modelled by coordinated movement, 2004) notably in the light of the categories of non-western cultres (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002).

A potentially valuable means of understanding and configuring 12 such modalities is through the Archimedean polyhedra, as envisage elsewhere (Union of Intelligible Associations: remembering dynamic identity through a dodecameral mind, 2005). In an earlier exploration of Patterns of Conceptual Integration (Annex 18: Polygons and polyhedra, 1984), the 13 distinct Archimedean polyhedra were described as similar arrangements of regular, convex polygons of two or more different kinds meet at each vertex of the polyhedron [which can itself be circumscribed by a tetrahedron, with 4 common faces]. Such semi-regular polyhedra are defined by the fact that all their vertices lie on a circumscribing sphere.

Keith Critchlow (Order in Space, 1969) configures 12 of them, within their circumscribing spheres, in a closest packing configuration around the circumscribing sphere of the 13th -- a truncated tetrahedron -- as shown below. The truncated tetrahedron is the only semi-regular solid with 12 independent axes passing through its vertices from its centre. Removal of the central sphere allows the 12 other spheres to close into a more compact icosahedral configuration.

In his description of Omnidirectional Closest Packing, R Buckminster Fuller (Synergetics: Explorations in the geometry of thinking, 1975) notes:

In omnidirectional closest packing of equiradius spheres around a nuclear sphere, 12 spheres will always symmetrically and intertangentially surround one sphere with each sphere tangent to its immediate neighbors. We may then close-pack another symmetrical layer of identical spheres surrounding the original 13. The spheres of this outer layer are also tangent to all of their immediate neighbors. This second layer totals 42 spheres. If we apply a third layer of equiradius spheres, we find that they, too, compact symmetrically and tangentially. The number of spheres in the third layer is 92.

Archimedean polyhedra
(Union of Intelligible Associations: remembering dynamic identity through a dodecameral mind, 2005)
Successive truncations of octahedron
2, 3, 4-fold symmetry
Successive truncations of icosahedron
2, 3, 5-fold symmetry
  1. truncated octahedron (14 polygons: 4 / 6 sided)
  2. cuboctahedron / vector equilibrium (14: 3 / 4)
  3. truncated cuboctahedron (26: 4 / 6 / 8)
  4. snub cube (38: 3 / 4)
  5. rhombicuboctahedron (26: 3 / 4)
  6. truncated cube / hexahedron(14: 3 / 8)
  1. truncated icosahedron (32 polygons: 5 / 6 sided)
  2. icosidodecahedron (32: 3 / 5)
  3. truncated icosidodecahedron (62: 4 / 5 / 10)
  4. snub dodecahedron (92: 3 / 5)
  5. rhombicosidodecahedron (62: 3 / 4 / 5)
  6. truncated dodecahedron (32: 3 / 10)
truncated tetrahedron (8 polygons: 3 / 6 sided)
Arrangement of the 12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron (from Keith Critchlow, Order in Space, 1969, p. 39). Arrows indicate the succession of truncations from 1 to 6 in each case. (Clicking on a polyhedron links to a spinning image)
Polyhedron of Archimedean Polyhedra
truncated dodecahedron icosidodecahedron truncated icosidodecahedron truncated octahedron rhombicosidodecahedron rhombicuboctahedron truncated tetrahedron truncated cuboctahedron truncated icosahedron cuboctahedron truncated cube snub cube

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