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Volume and Energy The inventory of twenty-four modules (sixteen A, eight B) indicate that the coupler has the same volume as the regular tetrahedron with its twenty-four A-quanta. Both have a tetrahedral volume of one. (Recall also that a coupler consists of two sixth-cube pyramids back to back, or 1/3 of a cube, and that in Chapter 10 the volume of a tetrahedron was shown to be 1/3 that of a cube.) Dissimilar in symmetry and shape, they are related by their shared unit volume, inviting comparison. The coupler is a different sort of minimum system: a semisymmetrical space filler, in contrast with the tetrahedron's origin as the minimum system of any kind, i.e., the first case of insideness and outsideness. Finally, we have to comment on the coupler's internal flexibility. The number of different ways to arrange eight Mites is greatly increased by the unexpected mirror symmetry. Since positive and negative Mites can switch places, we actually have a pool of sixteen from which to choose for each of the eight positions. A coupler might consist of four positive and four negative Mites, or all positive, all negative, or any of the possible combinations in between: (0-8, 1-7, 2-6, 3-5, 4-4,..., 8-0). Then, within each of these nine possible groups, the Mites can be switched around into four different arrangements. (A few of these combinations are shown in Figure 13-9.) The resulting 36 varieties of couplers all have the same outward shape, but in Fuller's view, their internal variations represent important distinctions in energy behavior: When we discover the many rearrangements within the uniquely asymmetric Coupler octahedra of volume one permitted by the unique self-interorientability of the
When we consider that each of the eight couplers which surround each nuclear coupler may consist of any of 36 different In other words, "energy" travels in one direction or another depending on the arrangement of the (oppositely biased) A's and B's. Synergetics thus accounts for nature's incredible variety and cornplexity despite its small number of different constituents. The secondary level of organization involves grouping the different couplers together and results in an explosion of potential variations. A- and B-modules can thus be rearranged into orderly octet configurations in myriad ways; clusters might appear quite chaotic in some locations and precisely ordered into whole octahedra and tetrahedra in others, while the overall space-filling matrix remains intact. As in genetics, a small number of simple constituents are able to generate a virtually unlimited repertoire of patterns. Review: All-Space Fillers
Dismantling the cube, we saw that each of its six inverted pyramids breaks down into four Mites. What does this tell us about A- and B-modules? 24 Mites yield a total of 72 modules, with twice as many A's and B's (or 48 : 24). This two-to-one ratio gets to the modular heart of the octahedron—tetrahedron prerequisite for space filling, as developed in the last chapter. Neither tetrahedra with only A's nor octahedra with equal numbers of A's and B's can qualify as space fillers. So far, so good. Thinking only in terms of A- and B-modules, what proportion of tetrahedra and octahedra would we need to satisfy the recipe of two A's for every B? The octahedron's 48 B's must co-occur with 96 A's; the octahedron itself supplies half of them, but 48 A-modules are still missing. Two tetrahedra will provide exactly the right number of A-modules to complete the formula, thus reconfirming the one-to-two octahedron—tetrahedron ratio discovered in the previous chapter. The 48 Mites in the rhombic dodecahedron are easily disected into 96 A's and 48 B's, or a total of 144 modules. With twice as many A's as B's, Fuller's "spheric" does not contradict the growing evidence of a general rule. A- and B-modules in the truncated octahedron—also a space filler—can be counted by recalling the numbers of internal single-frequency tetrahedra and octahedra determined in the last chapter. 32 tetrahedra, with 24 A-modules each, contribute 768 A's, while 16 octahedra consist of 768 A's and 768 B's, for a total of 1536 A's and 768 B's altogether (768 × 2 = 1536). We can begin to have confidence in the reliability of our 2:1 ratio, especially in view of the jump to much larger numbers. With a total of 2304 modules, this is the smallest truncated octahedron outlined by IVM vertices. Fuller's inventory of all-space fillers (954.10) lists the truncated octahedron with twice the linear dimensions and eight times the volume (consisting of 18,432 quanta modules) rather than the smaller version. The reason for this choice is not clear; however, the magnitude of these quantities hints at the complexity of his modular system—in terms of both the variety of systems that can be made from the modules and the number of rearrangements within those systems. From these numbers it is evident that, although the quanta modules help us to understand the conceptual essence of many polyhedral intertransformabilities, they are very impractical for hands-on experimentation. So many modules are needed to make complete polyhedra that this system does not offer an ideal strategy for model making. Instead, we might utilize the analysis to work out relationships on paper. From the above examples, we can see that the A- and B-modules get to the root of the two-tetrahedron-one-octahedron rule developed in the previous chapter. Because the earlier analysis depended on disassembling its two basic units, it was necessary to probe further to isolate the real quanta. A- and B-modules, which cannot be symmetrically subdivided, were isolated as the true quanta with which to measure and analyze related polyhedral systems. Impressed by the geometric significance of these modules, Fuller proposes that somewhere within this discovery lie secrets with far greater applicability than just to geometry: The From their energy associations to their remarkable symmetry, these modules synthesize much of Fuller's research. Significant relationships to physical phenomena may well reward continued investigation, for nature also deals with discrete quanta, creating endless variation through synergetic recombinations. Fuller reasoned that his geometnc quanta—the end result of a systematic and logical progression of steps—must relate to physical phenomena. The approach is typically Fuller's: assume significance until proven otherwise. In essence he suggests that tiny whole or discrete systems should replace irrational unending digits—somehow providing a comprehensive rational coordinate system. |

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