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Packing and unpacking of 12 semi-regular Archimedean polyhedra
Psychosocial Implication in Polyhedral Animations in 3D (Part #4)
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In contrast with the 5 regular Platonic polyhedra, of similar interest is the set of 13 semi-regular polyhedra -- of which 12 can be uniquely configured around the 13th. This configuration is notably described in detail by Keith Critchlow (Order in Space: a design source book, 1969) where it is illustrated as follows. Critchlow specifically cites the inspiration of Buckminster Fuller (noted above)..
| 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 |
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| truncated tetrahedron (8 polygons: 3 / 6 sided) | |
| Arrangement of the 12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron | |
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| Rotation of cuboctahedral array of 12 Archimedean polyhedra (around an omitted 13th at the centre; totalling 984 edges, 558 vertices, 452 faces) Virtual reality variant (.wrl) |
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| Animation prepared with the aid of Stella Polyhedron Navigator Interactive 3D version |
The configuration immediately suggests possibilities of animating the relationships between the polyhedra in the array, most notably by animating them in a "pumping" motion of contraction to the common centre and expansion from it, as is evident from the following.
| Screen shots of animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre (without indication of the 13th at the centre: the truncated tetrahedron) | |
| Contextual cuboctahedron rendered partially transparent Video animation (.mov); virtual reality (.wrl; .x3d) | Wireframe version with all faces transparent Video animation (.mov); virtual reality (.wrl; .x3d) |
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| Animations prepared with the aid of Stella Polyhedron Navigator | |
This cuboctahedral configuration is especially significant given the importance associated with it by Buckminister Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979). He variously renamed it vector equilibirum and dymaxion, associating the expansion and contraction of the configuration with a fundamental jitterbug movement. This is comprehensively summarized by Fuller (Jitterbug: Symmetrical Contraction of Vector Equilibrium). [see videos: Vector Equilibrium: R. Buckminster Fuller; Buckminster Fuller's Jitterbug; Bucky's "Jitterbug": Vector Equilibrium].
The most comprehensive video is that presented to the American Mathematical Society by Joseph Clinton (R. Buckminster Fuller's Jitterbug: its fascination and some challenges, Synergetics Collaborative, 2006). A summary of the associated movements is provided by Robert W. Gray (The "Jitterbug" And Its Motion, 2001; The Jitterbug Motion, 2002). An earlier exercise discussed the tranaformations in some detail, with a mapping of many of them (Vector Equilibrium and its Transformation Pathways, 1980):
The cuboctahedron is the polyhedron obtained by bisecting the 12 edges and truncating the eight corners of the cube. It can also be developed, however, from the omnidirectional closest packing of spheres around one nuclear sphere. The centres of 12 such spheres define the 12 nodes of the cuboctahedron. As all spheres are the same size it can be seen that the length of the cuboctahedron's edges equal the distance from its centre to its 12 nodes. Thus the form can be considered to be a system of equal vectors which are in equilibrium -- a vector equilibirum -- where the outward radial thrust of the vectors from the centre is balanced by the circumferentially restraining chordal vectors. The explosive forces perfectly balance the implosive forces.
As explained, variants of the jitterbug are also distributed as an educational toy. As indicated by the videos, models of it have been presented as virtual reality animations, most notably by Bob Burkhardt (Jitterbug, 2008). The development of 3D animation now enables presentation of the dynamics of the jitterbug transformation of the cuboctahedral configuration of Archimedean polyhedra to be explored otherwise. This would offering a stimulus to the imagination which is otherwise constrained by the 2D representation (above) and access to physical models.
There is the further possibility of animating the lines of the cuboctahedral (jitterbug) array of Platonic polyhedra (as was done with the lines of the drilled truncated cube above). Given the use of the cuboctahedron to configure the Archimedean polyhedra, an even further possibility is to consider animating relations between those polyhedra and the Platonic polyhedra. This would offer a mapping facility composed of the following elements.
| Edges | Vertices | Faces | Totals | |
| Platonic polyhedra | 90 | 50 | 50 | 190 |
| Archimedean polyhedra | 984 | 558 | 452 | 1994 |
| Totals | 1074 | 608 | 502 | 2184 |
The question would then be what significance might be assoctiated with the transformations possible within such a complex, as partially discussed separately in relation to such patterns (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).
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