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The Heart of the Matter:
A- and B-Quanta Modules
Minimum system, triangulation, equilibrium of vectors, closest-packed spheres, and space-filling: the path toward the isotropic vector matrix has many origins, any of which can yield the unique omnisymmetrical array. The result of traveling all of these routes is a powerful awareness of the interplay of octahedral and tetrahedral symmetries in space. However, the more thoroughly we search the IVM, the greater the intricacy of these "intertransformabilities"calling for a new level of analysis to keep track of our discoveries.
IVM got us down to the basics. Even the cube, mathematics' conventional building-block, reduces to octahedron-tetrahedron components. However, tetrahedra and octahedra are not true structural quanta, for it was often necessary to break them apart into subcomponents in order to build other polyhedra. Our task is therefore still unfinished. We have yet to get to the heart of the tetrahedron and octahedron.
Necessity thus leads us to Fuller's A- and B-quanta modules. (1)
The missing element in our IVM analysis is a way to handle redundancy. Symmetrythe degree to which a system looks exactly the same in different orientationsis a kind of structural redundancy. The tetrahedron, the octahedron, and their various combinations all have a high degree of symmetry, and now we intend to get to the root of it. How? By subdividing symmetrically until we can go no further. As long as a system exhibits some degree of symmetry, it can be divided into identical subunits, which can be put together to recreate the original system. Ergo, the system was redundant. Through progressive subdivision, we can locate the minimum sub-unit of any system. This final asymmetrical moduleor "least common denominator" (LCD)contains the geometrical data needed to reconstruct the whole system. We find the LCD by subdividing a polyhedron until we reach the limit case, that is, a module that can no longer be split into similar units.
Let's start again with our highly symmetrical friend, the tetrahedron. The fact that its four faces are equivalent presents the first opportunity for subdivisionresulting in four equal parts. Each quarter tetrahedron encompasses the region from the center of gravity (cg) to a face (Fig. 13-la). It is evident from the threefold symmetry of these shallow pyramids that each can be sliced into three identical pieces, as if it were a triangular pie (Fig. 13-lb). The resulting pie pieceslong thin tetrahedra stretching from the apex of the quarter tetrahedron (cg of regular tetrahedron) out to an original unit-length edgeare quite irregular, and so the process is almost complete. However, one type of symmetry conspicuously remains. Each sliver can be split in half to produce two mirror-image parts: a right- and a left-handed version with identical angles and lengths (Fig. 13-lc). And suddenly we have come to the end. There is no possible way to divide that final product into equal parts; the shape thus generated is the limit case. Fuller calls this asymmetrical tetrahedron the "A-quanta module."
A-quanta modules contain the complete geometric ingredients needed to create a regular tetrahedron. We need twenty-four A-modules (twelve positive and twelve negative) to make a tetrahedron, but one module alone supplies the information. The A-module, representing the volumetric essence of the tetrahedron, introduces a new kind of building block.
Having ascertained the minimum unit of the minimum system, we must not forget about inherent complementarity. The above analysis is repeated. A regular octahedron splits into eight equivalent pyramids (octants) (each of which divides into six equal pieces), as did the quarter tetrahedra, yielding forty-eight asymmetrical minimum units (LCDs) of the octahedron (Fig. 13-2a). That would have been the end of the story except for a subtle catch, an overlooked redundancy.
The eighth-octahedron has twice the altitude of the quarter tetrahedron, but both pyramids have the same (equilateral triangle) base. The quarter tetrahedron therefore fits right inside the octant, occupying exactly half the available volume (Fig. 13-2b). The remaining volume is pure octahedrona hat-shaped wedge that accounts for the shape difference between the two pyramids. This concave triangular lid can be subdivided into six equal irregular tetrahedra (three positive, three negative) and these are called B-quanta modules. They are generated by the LCD of the octahedron after that of the tetrahedron is taken away. As long as the octahedron's asymmetrical unit contained a complete A-module within its boundaries, the unit was redundant. By removing the A-module which had occupied half the volume of the octahedron's LCD, we finally achieve a second true modular quantum (Fig. 13-2c). Thinner and more pointed than the A's, the B-quanta have a very different shape but precisely the same volume. Neither A nor B can be made from the other; they are fundamentally distinct, complementary equivolume modules.
The LCD of the octahedron (1/48) consists of one A and one B, while the LCD of the tetrahedron (1/24) is simply an A-module. The analysis is complete: we have broken down our IVM constituents into their essential characteristics. We can go no further.
However, the field is now wide open for experimentation. Armed with the IVM quantum units, we can anticipate a tremendous range of combinations and permutations, or rearrangements. Having systematically analyzed our basic systems, we now start to put their essential quanta back together in order to further understand the relationships between polyhedra. We have thus developed a far more sophisticated (and specific) framework with which to explore polyhedral intertransformability. Fuller takes it a step further. (Why stop with polyhedra?)
If you are willing to go along with the physicists, recognizing complementarity, then you will see that tetrahedra plus octahedraand their common constituents, the unit-volume A- and B-Quanta Modulesprovide a satisfactory way for both physical and metaphysical, generalized cosmic accounting of all human experience. (950.34)
We also observe considerable multiplication of complexity with the new framework created by subdivision. As Bucky would have it: multiplication by division.
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