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Progressive subdivision has left us with legitimate geometric quanta: final packages which cannot be split into equal halves. In a general sense, the model parallels science's search for the ultimate aspects of reality. Physics probes deeper and deeper into matter, breaking it down into ever-smaller constituentscells, molecules, atoms, sub-atomic particlesultimately seeking a package of energy (quarks?) which cannot be split apart. Fuller probes similarly into his geometry, hoping to gain insights about Universe itself.
Fuller's profound faith in the significance of reliable patternscoupled with his unflagging determination to find nature's coordinate system-led him to draw many parallels between synergetics and nature. Most of these are both suggestive and undeveloped; he planted his seeds, left the cultivation for posterity, and went on with his search. Certain that science can be modeled, Fuller felt a great responsibility to pursue what he saw as an ever more relevant investigation; if patterns are to emerge, sufficient data must be collected. Toward the goal of clarifying the patterns observed by Fuller, our strategy might be to forge ahead: cover as much ground as possible and worry about significance later. However, in his coverage of intriguing geometric properties, Fuller often immediately assigns connections to physical phenomena. We can neither ignore nor confirm such speculation, and so for now, we merely record and file away. The respective energy "valving" (ability to direct, store, control) properties of A- and B-modules is a typical example.
Once again, the issue is based on "operational" procedure. Models of Fuller's two tetrahedral quanta are constructed out of paper. The process starts with a planar "net" (a flat pattern piece which can be folded up and taped together to make a specific polyhedron). Consider for example the regular tetrahedron. Its four equilateral-triangle faces can be generated by folding one (double-size) triangle along lines that connect mid-edge points (Fig. 13-3a). This ability to form a tetrahedron by folding one triangle is not to be taken for granted; the situation that allows all four faces of a generic tetrahedron to fit inside a triangular framerequiring exactly supplementary angles out of the infinite possibilitiesis an exception. We are not surprised by Figure 13-3a for we expect such exceptional cooperation from the uniquely symmetrical regular tetrahedron.
The surprise is that the asymmetrical A-module unfolds into one planar triangle: "an asymmetrical triangle with three different edge sizes, yet with the rare property of folding up into a whole irregular tetrahedron" (914.01). This unusual property makes it a kind of pure form. The B-module, on the other hand, will always fold out into four separate planar triangles no matter which vertex you start with; its net will never fit into one triangular frame. Figure 13-3b compares the two nets.
Fuller connects this geometric property with "energy." A-modules are thus said to concentrate or hold energy, while B's release or distribute. This conclusion is based on the fact that "energy bounces around in A's working toward the narrowest vertex," only able to escape at a "twist vertex exit" (921.15). Comparing pattern pieces, A's planar net offers three escapes; the jagged four-triangle complex of B offers twice that number (Fig. 13-3b). It is a somewhat bizarre observation to begin with, and characteristically it led to the following ambiguous assignment of meaning: a vertex, or non-180-degree junction, in a planar net represents disorderly energy-escaping properties. Hence in A-modules, energy is seen as contained, able to bounce around inside the net without many available exits, whereas in B-modules, energy is quickly released.
We can conceive of this energy in many ways-as light beams, as bouncing electrons, or even as billiard balls for a more tangible image. All three qualify as "energy events," and having a specific image in mind makes it easier to think about the different "energy-holding" characteristics. To understand and evaluate Fuller's assertion, we go along with his use of "energy," for the word covers a great deal of territory afready and his usage is internally consistent. Fuller calls our attention to a geometric property that we may not have otherwise noticed, and with respect to this phenomenon of planar nets there is no doubt as to the difference between A- and B-modules. What is the significance of this distinction? What are we to conclude about the orderly contained A versus the disorderly sprawling B? In terms of physical Universe, a judgement probably cannot be made. However, for the purposes of this text and of continuing to explore the geometric interactions of the two quanta, we adopt Fuller's energy assignment: A's conserve; B's dissipate. It provides a consistent reference system with which to classify the two quanta and their subsequent interactions. Furthermore, two basic modules exhibiting the same volume and different energy characteristics provide an even more attractive model of "fundamental complementarity": equivalent weight or importance, opposite charge. Sound familiar? The parallels are tantalizing.
Next we apply our LCD analysis to the IVM. The procedurecalling for progressive subdivision in search of the minimum repeating unitcomes to an end with a unit that can no longer be symmetrically divided. As before, we seek the smallest system that can be duplicated to recreate the whole IVMa microcosm, containing all the ingredients of the macro-array.
Having already split tetrahedra into equal quarters and octahedra into eighths, we skip directly to a unit consisting of a quarter tetrahedron and an eighth-octahedron back to back, sharing an equilateral-triangle face. This unit, which connects the geometrical centers (or cg's) of any adjacent tetrahedron and octahedron in the IVM, exhibits the same threefold symmetry as its two triangular pyramids taken separately. Final subdivision thus yields six equivalent asymmetrical tetrahedra: three positive and three negative (Fig. 13-4).
As the smallest repeating unit of the IVM, this system is the minimum space filler, thus inspiring Fuller's term "Mite" (Minimum space-filling Tetrahedron.) A collection of A-modules cannot fill space; they can only make regular tetrahedra. The skinny irregular B's cannot even create a symmetrical polyhedron by themselves-let alone fill space. The Mite is the LCD of the omnisymmetrical space-filling matrix, and therefore is the minimum case, the first all-space filler:
954.09 We find the Mite tetrahedron... to be the smallest, simplest, geometrically possible (volume, field, or charge), allspace.filling module of the isotropic vector matrix of Universe.
Knowing that the Mite encompasses the asymmetrical units of both the tetrahedron and octahedron, we can identify its constituent A- and B-modules. The tetrahedron contributes an A-module, while the adjacent octahedron adds both another A (the mirror image of the first) and a B, for a total of three equivolume Modules. The positive and negative A's are in balance, and the solo B may be either positive or negative, thereby determining the sign of the whole Mite. Like A- and B-modules, Mites come in one of two possible orientations (Fig. 13-5a).
We cannot fail to comment on another 2:1 ratio just displayed by the minimum space-filler. As suggested by this discovery, it turns out that there must be two A-modules for every B-module in any space-filling polyhedron. We shall see how this rule applies to our familiar candidates below, and once again, Loeb's "Contribution" provides a more thorough analysis of the phenomenon. Appropriate examples will be cited throughout this chapter, but readers are encouraged to turn to the back of Synergetics for Loeb's report. (2)
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