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Volume and Phase Changes
The jitterbug exhibits a total transformation of shape and size without any change in material. Nothing is added or taken away, but the system's characteristics are profoundly altered by rearrangement of the parts. The concept is reminiscent of the differences between ice, water, and water vapor, all consisting exclusively of H20 molecules. When a childwhose model-making experience is limited to "building blocks"first learns that rearrangement of the constituents is responsible for these profound changes, the idea is not easily accepted. Fuller maintains that experience with models like the jitterbug would better prepare a child for the lessons of science. Chemistry's invisible phase changes would seem perfectly logical, as they would be consistent with first-hand experience. And indeed, the jitterbug's floppy, flexible behavior as VE is parallel to that of a gas: the dense tetrahedral configuration with its motion totally restrained is more like the "solid" phase. Same stuff, radically different properties. He has a point. These dynamic models inspire a different kind of conceptualizing.
At the zero point, twenty-four wooden dowels and twelve rubber connectors embrace a volume of twenty, as we recall from "Multiplication by Division." After contracting and twisting through the jitterbug, the bundle of sticks encloses a single unit of volume, one tetrahedron. The system has thus gone from twenty tetrahedron volumes to one, with a stop at four, in the octahedron.
The icosahedron however refuses to cooperate. Its volume of approximately 18.51 is not as appealing as the whole-number ratios shared by the other stopping points. Jitterbug now introduces a rationale. The icosahedron is a phase that falls in between octahedron and vector equilibrium, rather than a definitive stopping point in the flow. The jitterbug is a continuous transformation through countless transitional stages, both regular and not, and at certain intervals an ordered polyhedron emerges. Found when the jitterbug is simply open as far as possible, the cuboctahedron is definitive, absolute zero. The octahedron clicks into place when six pairs of vertices suddenly come together. No ambiguity at either point. The icosahedral stage on the other hand is always approximate; we have to eye the distances between vertices, guessing whether or not they are equal to one. The dance does not stop naturally at this point; we just recognize the familiar shape along the way from VE to octahedron. It is thus a transient phase of the jitterbugwith no reason to stop and rest, no choice but to continue.
Similarly, the icosahedral vertices fall in between nodes of the IVM, the omnisymmetrical framework that outlines most of the symmetrical geometric shapes. One result of being out of phase with this matrix (which is also defined by closepacked spheres) is that the icosahedron's frequency cannot be increased by surrounding it with additional layers of spheres, or vectors. It cannot grow modularly; the initial choice of size, or frequency, is final. To change the frequency, a new model must be built from scratch. The icosahedron is thus restricted to single-layer construction. "The icosahedron must collapse to exist," explains Bucky; it always "behaves independently" of the other polyhedra:
The icosahedron goes out of rational tunability due to its radius being too little to permit it having the same-size nuclear sphere, therefore putting it in a different frequency system. (461.05)
Accordingly, its volume does not fit into the "cosmic hierarchy" of rational systems.
Single Layer versus IVM
What are the consequences of the icosahedron's "independence" of the cosmic hierarchy? As a collapsed VE, it is always a shell, a single-layer construction:
The icosahedron, in order to contract, must be a single-layer affair. You could not have two adjacent layers of vector equilibria and then have them colapse to become the icosahedron... . So you can only have this contraction in a single-layer of the vector equilibrium, and it has to be an outside layer, remote from other layers... . It may have as high a frequency as nature may require. The center is vacant. (456.20-1).
Accordingly, as we recall from Chapter 8, the design chosen by nature for many protective shells involves icosahedral symmetryfrom the microscopic virus cap sid to larger (visible with an ordinary microscope) radiolaria, the cornea of an eye, and a plethora of other elegant creations.
"Trans-Universe" versus "Locally Operative"
"The vector-equilibrium railroad tracks are trans-Universe, but the icosahedron is a locally operative system" (458.12). Fuller's ambiguous and somewhat mystical declaration becomes almost straightforward after the jitterbug demonstration. Vector equilibrium is incorporated into an infinitely extending network. Conceptual and timeless, VE is everywhere; it is the balance of forces at the root of all phenomena. The icosahedron, on the other hand, is always a special-case collapse, an aberration in the omnisymmetrical frame of reference. Aberrations are finite, local, inescapably stuck in time, as we recall from "Angular Topology." The icosahedron is thus fundamentally different from VE, whichwith its timeless perfectionpermeates all of Universe.
Fivefold symmetry dominates the icosahedron, distinguishing it once again from the cosmic hierarchy, with three-, four-, and sixfold rotational symmetries. This is another sign of the icosahedron's nonconformism. It's full of fives: to begin with, the obvious five triangles around each vertex, determining the symmetry about each of its long axes. Then, its thirty edges fall into five sets of six orthogonal edges, that is, three parallel pairs of mutually perpendicular edges. Figure 11-3a highlights the five distinct sets of orthogonal edges. (1) (The edges can also be grouped into sets of five parallel edges embracing the equator, in six different directions.) Joining the mid-points of the six edges of one set displayed in Figure 11-3a, we discover an octahedron hiding insideimplicit in the icosahedral symmetryin one of five possible orientations (Fig. 11-3b). The icosahedron may be out of phase with the rest of the IVM family, but it displays many significant relationships to these other shapes, which, being unexpected, are all the more fascinating to uncover. A few examples will be given below, and there's always room for further exploration. Just as in our earlier development of the cosmic hierarchy, we investigate how various shapes fit inside each other, and thereby learn about similarities in shape, volume, and valency.
Whereas the cosmic-hierarchy relationships are consistently straightforward and balanced (just bisect edges and connect midpoints to generate the next shape), whenever the icosahedron is introduced, more intricate connections emerge. We therefore have to look somewhat harder to find these new relationships which high-light the icosahedron's transitional role in the hierarchy.
We saw how the octahedron emerges out of the arrangement of icosahedral edges, on the inside, and now we reverse the situation. The icosahedron can be oriented so that eight of its twenty faces are coplanar with and flush against the eight faces of a surrounding octahedron, while the twelve icosahedral vertices are located on its twelve edges. However, the icosahedron must sit in a skew (or twisted) position, with its vertices intersecting the octahedral edges off center, dividing each edge into two segments, the longer 1.618 times the length of the shorter. This asymmetry means that there are two distinct orientations of the icosahedron inside the octahedron-positive and negative, as shown in Figure 11-4a, b.
The ratio 1.618 to 1, known as the "golden section," might have played a prominent role in synergetics, for it shows up frequently (especially in relation to the icosahedron); however, Fuller rarely mentions this intriguing number. Accordingly, this text will not spend time exploring the famous ratio, whichas a source of fascination to geometers for milleniaenjoys considerable press already. (Specifically, for the role played by the golden section in the icosahedron, see Loeb's "Contribution to Synergetics" and its "Addendum" in Synergetics 2, both Section G. (2) ) Fuller has a different method of coping with such relationships; rather than describing certain comparisons and their numerical values, he employs geometric "modules"a holistic way of describing geometry with geometry.
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