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Multiplication by Division:
In Search of Cosmic Hierarchy
It may seem that we have strayed from Fuller's "operational mathematics" while investigating the symmetrical properties of various polyhedra in the previous chapter. Recall that "operational" indicates an emphasis on procedure and experience: what to do to develop and transform models or systems. "Multiplication by division" brings us back to experience, introducing an operational strategy, which will add new meaning to Fuller's term "intertransformabilities." We thus elaborate on the shared symmetries among shapes while discovering new transformations from one to another, and this time previous experience allows us to anticipate results.
Multiplication by division describes Bucky's journey through our expanding polyhedral inventory. Previous exposure to both Loeb's work and the IVM sets the stage, making us so familiar with these shapes that additional results can be immediately placed in context. The transformations explored in this chapter occur within the IVM frame of reference, adding volume relationships to our accumulated information about topology and symmetry. You may be surprised to find that many statements seem obvious at this point; resist the temptation to dismiss them as trivial. Appreciate instead the implicationwhich is that we cannot take a wrong turn. Each step is inherently tied to the shape of space; we can only uncover what is already there.
The use of ratio is an inherent part of quantifying volume, and yet not everyone is aware of the implicit comparison. As with measuring distance, our conventional units can seem like a priori aspects of volume.
Once again, Fuller calls our attention to Avagadro's discovery that a given volume of any gas, subject to identical conditions of temperature and pressure, always contains the same number of molecules. "Suddenly we have volume clearly identified with number," declares Bucky.
Actually, volume is intrinsically related to number. When we ask, "what is the volume of that swimming pool"? we expect an answer expressed in terms of some number of "cubic feet." What this answer tells us is how many cubes with an edge-length of one foot could fit into the pool. Whether the situation calls for feet, inches, or centimeters, a cube of unit edge length is conventionally employed as one unit. The word "volume" may evoke an image of a continuum; however, it is quantified in terms of discrete quanta. We so uniformly express spatial quantity in terms of cubes that we are simply not aware of the invisible framework of "ghost cubes" incorporated into our concept of volume. We conceptualize space cubically: length, width, and depth seem absolutely fundamental directions. Again Bucky points out that this conceptual cube is a remnant of flat-earth thinking. Myopic in cosmic terms, humanity readily adopted the orthogonal box as the correct shape with which to segment space.
Results: Volume Ratios
Volume has to be measured relative to something, so why not experiment with the tetrahedron? We are so used to using the cube, the suggestion seems blasphemous-a violation of basic laws of volume. Nevertheless, given our growing list of the tetrahedron's unique properties, such an experiment might be worthwhile.
Accordingly, we allow a tetrahedron of unit-edge length to be called one unit of volume. The results are astonishingly rewarding. Perfect whole-number values describe the volumes of most of the polyhedra covered so far, and all of those contained within the IVM and IVM' combined. (Some exceptions are found in transition shapesthose that fall in between IVM verticesas we shall see in the next chapter.) In contrast, the volumes of these familiar polyhedra, relative to a cube as the unit shape, are strangely cumbersome valuesoften irrational (never-ending) decimal fractions. Table V displays the results, which we shall derive below. It compares the volume ratios generated by three different polyhedra successively adopted as one unit of volume. Five different systems are compared first with the unit-edge cube, then with the unit-diagonal cube, and finally with the unit-edge tetrahedron.
Remember that these polyhedra arose as a consequence of spatial symmetry; we simply located vertices within the unique isometric array of vectors. Recalling this origin, it is again clear that the various shapes and sizes of the polyhedra in question are not the product of deliberate design. Their whole-number volume ratios are not contrived; we stumble onto them after the fact.
Why investigate two different cubes? Primarily, to demonstrate that neither choice yields the elegant results disclosed by the tetrahedron. All factors considered, the unit-diagonal cube is a better choice, for it arises naturally out of the IVM networkthat is, out of the shape of space! Further justification for this choice will be developed below.
Imagine building a cube out of the requisite twelve struts. The topological recipe, which simply calls for three-valent vertices and four-valent faces, in no way indicates precisely what the finished product should look like. Without deliberate shaping into a perpendicular form by a knowing hand, the configuration does not favor any particular surface angles. Which version of this hexahedron of quadrilaterals is the desired result? This ambiguity must somehow be resolved. Chapter 5 revealed that an orthogonal "cube"as defined by mathematicscannot be reliably created without diagonal braces. (Refer back to Fig. 5-3.) Compare this experience with building an octahedron out of the same twelve structs. Following the recipe of four-valent vertices and three-valent faces. the octahedron builds itself. The interior shape is precisely specified by its topology; in other words, the procedure leaves no room for choice. In order for the cube to have that kind of integrity, or exactitude, six face diagonals must be inserted. An inscribed tetrahedron solves the problem in a single step.
The above comparison reinforces our previous experience of how the right-angled cube fits into spatial symmetry. Fuller's "operational mathematics" prescribes learning by procedure: pick up a box of toothpicks (pre-cut unit-vector models) and start building. Space will let you know what works. The satisfaction gained from feeling the cube hold its shape draws attention to its supporting diagonal members, and thus unit-vector diagonals seem an appropriate choice for comparing volumes. As volume is always a matter of ratio, we want to insure that our comparisons make sense, in this case, remain consistent with space's isotropic vector matrix. That the vector-diagonal cube is contained within this hierarchy adds to the advantages established by its stability.
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