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Multiplication by Division
Multiplication occurs only through progressive fractionation of the original complex unity of the minimum structural systems of Universe: the tetrahedron. (100.102b)
Instead of starting with partspoints, straight lines, and planes-and then attempting to develop these inadequately definable parts into omnidirectional experience identities, we start with the whole system in which the initial "point"... inherently embraced all of its parameters... all the rules of operational procedure are always totally observed." (488.00)
Deeply impressed by Arthur Eddington's definition of science as "the systematic attempt to set in order the facts of experience," Fuller constantly sought meaningful organizations for groups of experiences or events. "Multiplication by division" is one such effort. ("Events" of course includes structures and almost anything else; in energetic Scenario Universe, things are events.)
Essentially, "multiplication by division" derives volume relations through the straightforward logic of direct observation, rather than by rote application of traditional formulae-which lead us through awkward values before revealing the underlying simple relationships. As will be seen below, this direct observation is accomplished by comparing given polyhedra to the unit-length tetrahedron.
Tetrahedron as Starting Point
Fuller's organizing strategy begins with the tetrahedron, because as the "topologically simplest structural system," it is a logical starting point. Consistent with his emphasis on "whole systems," the ultimate reference point in synergetics is "Universe." The tetrahedron thus acts as an appropriate "whole system" for the procedure described below, in that it is "the first finite unitarily conceptual subdivision of... Universe" (987.011b). More complicated systems are developed through subdivision of this tetrahedral starting point, so that a progression is contained within (and organized by) the whole:
In respect to such a scenario Universe multiplication is always accomplished only by progressively complex, but always rational, subdivisioning of the initially simplest structural system of Universe: the sizeless, timeless, generalized tetrahedron. (986.048b)
Onward! We seek to develop a variety of polyhedra through subdivision of the "whole" and in so doing provide the "experimental evidence" to verify the results shown in Table V. We imagine a single regular tetrahedron, and then bisect each edge to create the two-frequency tetrahedron shown in Figure 10-3. One by one, we remove a single-frequency tetrahedron from each of the four corners, unwrapping the hidden octahedron (Fig. 10-4). Choping off four unit tetrahedra subtracts four units of volume from the initial total of eight (the value determined earlier for a double-edge-length tetrahedron), indicating that the octahedral remainder has a volume of exactly four. In other words, an octahedron has four times the volume of a tetrahedron of the same edge length. We thus begin to derive the values in Table V.
Next, we split the octahedron in half, separating the two square-based pyramids. Two additional perpendicular slices divide each pyramid into quarters (Fig. 10-5), producing eight sections, or "octants," each with a volume of one-half (a volume of four, divided by eight, is equal to one-half). Each octant is an irregular tetrahedron with a unit-length equilateral base and three right-isoceles-triangle sides. The perpendicular corners of the eight octants meet at the octahedral center of gravity, an orthogonal relationship first noted in the previous chapter when IVM' vertices (body center) were added to the cells of the IVM.
To reconfirm the octahedron-tetrahedron volume relationship, we place an octant (with its equilateral face down) next to a regular tetrahedron on a flat surface and observe that the altitude of the octant is exactly half that of the tetrahedron. (A second octant can be put on top of the first to check: the height of both octants together is equal to the altitude of the tetrahedron, as shown in Figure 10-6.) An octant, therefore, has the same base and half the altitude as a regular tetrahedron, reconfirrning that its volume exactly half the volume of the tetrahedron.
In "Structure and Pattern Integrity" we discovered that a regular tetrahedron fits inside a cube, and subsequently we learned that each of the four "leftover" regions is equivalent to the portion of an octahedron from one face to its center of gravity (Chapter 9). We now take advantage of the recently disassembled octahedron for an experiment. Having equilateral triangles in common, octants can be superimposed on each face of a unit tetrahedron. One by one, four octants surround and thus obscure the tetrahedron. Lo and behold, a perfect cube emerges (Fig. 10-7). Four octants, with a combined volume of two units, have been added to the unit-volume tetrahedron, for a total volume of three. Compared again with the irregular volumes listed in Table V, generated by the unit-edge cube, these whole-number ratios for the cube and octahedron seem especially remarkable.
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