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Polyhedral meta-patterns of relationships?


Memetic Analogue to the 20 Amino Acids as vital to psychosocial life? (Part #7)

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It is of course very possible that a more insightful representation of the pattern of relationships encoded by polyhedra could be produced through further iterations (or may have already been) -- or simply one based on alternative design choices. For example, Keith Critchlow, has presented an arrangement of the 12 Archimedean polyhedra in their most regular pattern, as a cuboctahedron, around a 13th, the truncated tetrahedron (Order in Space, 1969, p. 39). This is presented in modified form below.

Archimedean polyhedra
(reproduced from Towards Polyhedral Global Governance complexifying oversimplistic strategic metaphors, 2008, and from 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
Polyhedron of Archimedean Polyhedra

The configuration around a truncated tetrahedron of the Archimedean polyhedra in the above map can be clarified to some degree by animation (on the right below). That on the left is an indication of a similar configuration around a tetrahedron. In each case that central polyhedron occupies a focal position in the earlier maps (above), despite the distortion of representation in two dimensions.

Configuration in three dimensions of Platonic and Archimedean polyhedra
Indication of 4 Platonic polyhedra configured
around tetrahedron 		Indication of 12 Archimedean polyhedra
configured around truncated tetrahedron
Animation of 4 Platonic polyhedra configured around tetrahedron Animation of 12 Archimedean polyhedra configured around truncated tetrahedron

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