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Amy C. Edmondson
A Fuller Explanation
Chapter 10, Multiplication by Division: In Search of Cosmic Hierarchy
pages 157 through 158

Cosmic Hierarchy (of Nuclear Event Patternings)

"The Cosmic Hierarchy is comprised of the tetrahedron's intertransformable interrelationships" (100.403b). Fuller's curious description is now clear, for we have become familiar with most of these "intertransformable relationships" and how they fit into the IVM context—as well as with the simple operations that transform one shape into another.

      The order of this polyhedral hierarchy is determined by complexity, from least to most. It is worth noting that the ladder can extend inward indefinitely by progressive subdivision; edges can be continually bisected to generate higher frequency systems. Notice how convenient it is to have "conceptuality independent of size." We do not have to specify the size of the initial tetrahedron, for Fuller's use of frequency to designate length provides a means to specify the system's geometric characteristics precisely, without recourse to "special case" examples. Ratios remain consistent; like conceptuality, they are independent of size.

      In summation, "cosmic hierarchy" pertains to volume ratios as well as to complexity and frequency, and its relationships are uncovered through "multiplication by division." In this way, Fuller describes the order inherent in space.

Volume Reconsidered

Fuller attaches considerable significance to volume ratios and cosmic hierarchy. The subject suggests a number of philosophical implications, concerning both reasons why these orderly relationships go unnoticed and also the potential benefits of a mathematics that emphasizes systems and the relationships of parts to wholes. The first aspect is best summarized by Loeb in his "Contribution":

      Uncritical acceptance of geometrical formulas as fundamental laws, particularly in systems that do not naturally fit orthogonal Cartesian coordinates, frequently leads to unnecessarily clumsy calculations and tends to obscure fundamental relationships. It is well to avoid instilling too rigid a faith in the orthogonal system into students of tender and impressionable age! (3)

      Overdependence on the cube is the culprit. Fuller attributes much of our attatchment to this building block to an understandable desire for "monological" solutions, or single answers to complex questions. Blissfully unaware of the "inherent complementarity" of Universe, he cautions, humanity naturally sought one "building block" with which to understand space, a single unit to be the basis of all mathematics. Even without considering the concept of "inherent complementarity," there are significant advantages to using the tetrahedron instead of the cube as the basic unit in quantifying volume. Bucky would complete his argument by reminding us that the cube is inefficient. Having demonstrated the respective volumes of this traditional shape compared to nature's minimum system, he would summarize by saying, "if you use cubes, you use three times as much space as necessary." And, once again, "Nature is always most economical."

      Finally, he hypothesizes that the irrational volumes of simple polyhedra, inherent in cubic accounting, tended to reduce the importance of these basic systems in the eyes of mathematicians. Shapes with such troublesome volumes could not possibly be relevant to natural order:

      Though almost all the involved geometries were long well known, they had always been quantized in terms of the cube as volumetric unity...; this method produced such a disarray of irrational fraction values as to imply that the other polyhedra were only side-show geometric freaks or, at best, "interesting aesthetic objets d'art." (454.02)

The exclusive adoption of the cube thus served to inhibit sustained serious attention to the other polyhedra.

      Back out to the big picture. To understand Universe, Fuller argued, we must think in terms of the synergetic principles governing the relationship of parts to whole systems. Multiplication by division is one of many exercises to encourage the development of a habitual orientation toward solving problems in context. If our early mathematics training encourages us to isolate and consider parts separately, rather than as components of a larger system, then, Fuller thought, our natural inclination throughout life would be to view problems myopically.

      In Fuller's view, we have been blinded to a whole family of rational order by an initial (90-degree) wrong turn—itself a result of humanity's early perception of an up-down platform Earth.

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