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The volume of the VE can be quickly determined through direct observation. Recall that twelve unit-length radit outline eight regular tetrahedra and six half octahedra. This fact, combined with our newly generated volume data, provides a conclusive value for the vector equilibrium: six half octahedra, each with the tetrahedron volume of two, plus eight unit-tetrahedra yields a total tetrahedron volume of twenty: 12 + 8 = 20. This simple breakdown supplies further evidence of a natural order of precise volume relationships, and it is especially reassuring that a unit-length vector equilibriumthe conceptual foundation of Fuller's energetic mathematicsfalls into place with its own whole-number volume ratio.
To begin with, our recently disassembled octahedron must be put back together. Octants are thus lifted away from the composite cube shown in Figure 10-6, and their right-angled comers are again turned inward to meet at the octahedron's center. Next, we divide two tetrahedra into quarters. Each tetrahedron yields four shallow pyramids, encompassing the region from an outside face to the center of gravity (Fig. 10-8). As before, the equilateral base of each shallow pyramid allows the quarter tetrahedra to fit directly onto an octahedral face, and as soon as eight quarter tetrahedra are attached to the octahedron, a rhombic dodecahedron emerges (Fig. 9-12). Two units of volume have been added to the octahedral four, for a total of exactly six.
Looking at the shape of the rhombic dodecahedron, with its strange angles and facets, the perfection of this whole-number volume relationship seems particularly remarkable. Because both shape and topological characteristics of the tetrahedron and rhombic dodecahedron appear utterly dissimilar, the discovery of this orderly relationship between the two polyhedra adds significantly to our growing sense of an underlying spatial order.
Again, the considerable fiexibihty of the IVM framework enables us to plot all of the above polyhedra with IVM and IVM' vertices. The rationale for using the unit-diagonal cube also applies to the rhombic dodecahedron, which arises naturally out of the interaction of IVM and IVM' cells. Recall that every octahedron in the matrix is surrounded by quarter tetrahedra, thereby defining rhombic dodecahedra with unit-length diagonals.
We discovered many shared syrnmetries in the previous chapter by dissecting the IVM, and we now develop the significance of these observations with the discovery of the rational volume relationships inherent in this framework. Bucky was not the first to discover these ratios, but he may have been their most visible spokesman. He brought this esoteric information to the attention of countless packed lecture halls, as one of the more satisfying indications of the fallibility of our coordinate system. (Such sublime disclosures by nature must not go unheralded!) These volume ratios provided Fuller with a powerful source of confidence in the legitimacy of pursuing syn-ergetics, and indeed their significance is worth our serious consideration. What accounts for the lack of attention paid to these simple mathematical facts? Loeb offers the following explanation:
When these relations are derived with the aid of the usual formulae for the volume of a pyramid, V = 1/3Ah, a good many irrational numbers are involved, and the simple integral ratios emerge almost incidentally. Somehow, these simple integral values of the volume ratios of common solids are not part of our scientific culture, and a lack of familiarity with them frequently leads to unnecessarily cumbersome computations. It appears that a bias of our culture to orthogonal Cartesian coordinates has obscured these relations. [My (A.C.E.) italics.] (2)
Multiplication by Division
Bisecting the edges, as before, we take special note of the tetrahedron's square cross-section. This fourfold symmetry was a significant factor in previous discussions, notably in the sphere-packing demonstration, in which two sets of spheres came together at 90 degrees and unexpectedly produced a tetrahedron. However, this aspect of the tetrahedron is easily overlooked; as a triangular pyramid, with its preponderance of 60-degree angles, this shape is easily perceived as a completely triangular affair.
Suppose we ask the following question: how much of our cheese tetrahedron would be chopped off when the newly exposed surface (created by the slice) is a perfect square? Without a certain amount of previous exposure to the tetrahedron, your reaction would probably be that the slicing-plane could never be square. It might be a very small triangle, or any number of larger triangles as you position the knife closer to the base. But a square?!
Wait. The tetrahedron has four faces. Do they somehow outline a square? Anyone who has read this far knows the answer, but countless students challenged to find that hidden square have been stuck. Handicapped by the perpendicular bias of mathematics, they are unable to find the square cross-section in the exact center of the tetrahedron, whichonce seenis unmistakable (Fig. 10-9a).
The square in Figure 10-9a is parallel to and between two opposing edges, which themselves are perpendicular to each other. Delineating the square cross-sections corresponding to each of the three sets of opposite edges, this aspect of the tetrahedron's symmetry is exhausted, and the two-frequency subdivision is complete. A total of twelve new edges outline the octahedron, and by now this relationship is quite familiar (Fig. 10-9b).
We continue inward. This time, bisect and interconnect the edges of the octahedron. The process is equivalent to Loeb's "degenerate truncation" and outlines the edges of a vector equilibrium hiding inside the regular octahedron (Fig. 10-l0b). We could continue, by joining the midpoints of VE edges, to produce a "rhombicuboctahedron"; however, Fuller's sequence comes to an end at the vector equilibrium. The final lines, which are one-quarter the length of the edges of the original tetrahedron and the smallest vectors in the model, are thus designated as unit vectors.
We quickly run through the sequence in reverse to review the geometric relationships. A four-frequency tetrahedron, in which each edge is equivalent to four unit vectors, is the smallest tetrahedron to contain a complete VE in its center, and so it acts as the ultimate "whole system" (Fig. 10-11). The review starts in the center: a unit-length half octahedron is tacked onto each square face of the nuclear VE, thereby forming a two-frequency octahedron, which in turn has two-frequency tetrahedra added to four of its eight faces to create the large tetrahedron. Figure 10-10 illustrates this transition, and Figure 10-11 shows the complete four-frequency tetrahedron and its implied hierarchical system, employing progressively thicker lines to emphasize the three different polyhedra.
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