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The icosahedron's crooked position within an inscribing octahedron defines specific leftover spacesix pockets of empty territory between the icosahedron and its octahedral embrace (Fig. 11-5). The symmetry of the six identical pockets is such that each can be split in half, producing two equivalent irregular tetrahedra, which then further divide into two mirror-image halves. This final thin tetrahedron is Fuller's "S-quanta module." It is a volumetric unit that describes the degree to which the icosahedron is out of phase with the IVM. Just as grade-school "long division" introduces the arithmetic remainder, the process of dividing an octahedron by an icosahedron requires a geometric remainderthe S-module. Chapter 13 will describe Fuller's A- and B-modules, volumetric counterparts of the S-module; taken altogether these quantum units comprise Fuller's finite accounting system. Finally, Figure 11-5 indicates the golden-section ratio between different edge lengths of the S-module.
Icosahedron and Rhombic Dodecahedron
A pattern emerges. The out-of-phase relationship between icosahedron and different IVM polyhedra appears to involve the golden section. We test the pattern on the rhombic dodecahedron. Do its twelve faces correspond to the twelve vertices of the icosahedron? It turns out that the two shapes exhibit an interesting relationship, with the icosahedron fitting inside the rhombic dodecahedron, predictably in a skew position. Its vertices impinge on the rhombic faces slightly off center, dividing the long diagonal of each diamond into two unequal segmentsthe longer again 1.618 times the length of the shorter (Fig. 11-6). Ever reliable, the golden section reinforces our awareness of the underlying order in space.
Finally, recall the pentagonal faces of the icosahedron's dual; the fivefold symmetry of this dodecahedron is right out in the open. Furthermore, the pentagon is a prime source of golden section ratios (see Loeb's "Contribution to Synergetics"). The pentagonal dodecahedron is of course also out of phase with the IVM, for its symmetry scheme is the same as that of its dual, the icosahedron.
Back to the jitterbug. Fuller proposes that this fluid transition from stage to stage is best described as four-dimensional:
The vector-equilibrium model displays four-dimensional hexagonal central cross section... . (966.04)
Four-dimensionality evolves in omnisymmetric equality of radial and chordal rates of convergence and divergence... . (966.02)
First of all, radial and chordal equivalence produces four distinct planes, and secondly, the jitterbug contraction operates around four independent axes. Let's see how this works.
Triangles hold their shape, and therefore an equivalent model to the stick jitterbug described above can be built out of eight cardboard triangles hinged together with strong tape. The advantage in this case of "solid" triangles is that it makes Fuller's "four-dimensional" assignment easier to understand (Fig. 11-7).
The eight triangles operate in pairs. Diametrically opposite triangles remain aligned, rotating synchronously about a common axis as they together approach the jitterbug center at a constant rate. The separate pairs thus rotate around four different axes, displaying simultaneous motion in four distinct directions.
"Push straight toward the table; don't let either triangle rotate": Bucky emphasizes the simplicity of the task. He then feigns surprise at the subsequent twisting at the equator. The entire system converges symmetrically, despite the unidirectional force. He calls this behavior four-dimensionality, referring to the four independent directions of rotation.
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