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Amy C. Edmondson
A Fuller Explanation
Chapter 2, The Irrationality of Pi
pages 18 through 21

Finite Accounting System

"You cannot have a fraction of an energy event," philosophizes Bucky. Science's progressive subdivision ultimately reaches indivisible particles, which means that reality consists of whole numbers (1) of energy events. Therefore, we need models that will demonstrate the concept of structures consisting of discrete, or countable, units. In synergetics, area and volume are presented as quantities that can be counted, as opposed to measured and described as a continuum. Area is associated with some whole number of events on the surface, while volume is tallied as the number of events throughout a system. Chapter 8 will examine how sphere-packing models illustrate the association of volume and surface area with discrete units; Chapter 10 will discuss the concept of volume in general and its role in synergetics.

      We are not used to thinking of reality as submitting to a "finite accounting system," but in fact it does. Consider the phenomenon of light. Few things seem more continuous than the light that fills your living room when you turn on the switch, or the light that seems to fill the whole world on a sunny day at the beach. But we now know that even light consists of individual packages of energy, called photons. Although they could hardly be smaller or lighter, photons are nonetheless discrete events, countable in theory, even if there are always too many, jumping around too fast, to make that enumeration possible in practice. With its flawless illusion of continuity contradicting science's detection of constituent photons, light is a perfect phenomenon to symbolize the validity of a "finite accounting system." We could hypothesize a system in which volume was tallied in terms of the number of photons in a given space, and area, the number of photons found at the surface; however, it would be a fantastically impractical system. A more appropriate unit is called for. For now, however, we simply consider the qualitative implications of the discrete-events concept. A punctuated reality is hard to get used to.

Which Way Is "Up"?

Fuller proposed a revolution in modes of thinking and problem-solving, which above all else required a comprehensive approach, as will be discussed in Chapter 16. To Fuller, "comprehensive" means not leaving out anything—least of all humanity's important tool of language. A dictionary contains an inventory of 250,000 agreements, he explains, specific sounds developed as symbols for 250,000 nuances of experience. He saw this gradual accomplishinent as one of the most remarkable developments in the history of humanity, with its implied cooperative effort.

      One aspect of his revolution thus involves an effort to employ words accurately. Fuller's discourses on the subject tend to be quite humorous, almost (but not quite) concealing how deeply serious he was about the matter. Much of our language is absolutely stuck in "dark ages" thinking, he would lecture. Up and down, for instance. These two words are remnants of humanity's early perception of a flat earth; "there is no up and down in Universe!" exclaims Fuller. When we say "look down at the ground" or "I'm going downstairs" we reinforce an underlying sensory perception of a platform world. Neither the ground nor Australia can accurately be referred to as down; three hours after a man in California says that the astronauts are up in the sky, the shuttle is located in the direction of his feet. Up and down are simply not very precise on a spherical planet. The replacements? In and out. The radially organized systems of Universe have two basic directions: in toward the center and radially out in a plurality of directions. Airplanes go out to leave and back in to land on the earth's surface. We go in toward the center of the earth when we walk downstairs. The substitutions seem somewhat trivial at first, but again it is difficult to judge without trying them out. Experimenting with "in" and "out" can be truly reorienting; unexpectedly one does feel more like a part of a finite spherical system-an astronaut on "Spaceship Earth." (It's hard to keep at it for long however; up/down reflexes are powerful.)

      The sun does not go down, insists Bucky; how long are we going to keep lying to our children? First of all, we now know there is no up and down in the solar system and secondly, the sun is not actively touring around the earth. Rather, we are the travelers, and our language should reflect that knowledge. Oddly, the phraseology which gives the sun an active role ("darling, look at the beautiful sun going down") does seem to reinforce the erroneous conception of a yellow circle traveling across the sky. While we know this isn't the case, it often feels like the way things happen. This effect is not easily measured and probably varies from individual to individual.

      What does all this have to do with geometry? Remember that one of the goals of synergetics is to help coordinate our senses with reality, that is, to put us in touch with Universe, which to Fuller involves eradication of the erroneous vocabulary which keeps us locked into "dark-ages" thinking on a sensorial level. In short, we need to align our reflexes with our intellect. Instead of sunset and sunrise, reinforcing the sun's active role, Fuller suggests sunclipse and sunsight, which imply instead that our view of the sun has been obscured and that an obstacle has been removed, respectively. Who knows if people could adjust to such substitutions? It might be worth a try.

      The cube is another remnant of flat-earth days, and Fuller has a wealth of reasons to prefer another mathematical starting point, which will be discussed throughout this volume. (2) Our age-old dependence on squares and cubes is honored by an unfortunate verbal shorthand for "x to the second power" and "x to the third power." The expressions "x squared" and "x cubed" are so commonly used that most people assume these multiplication functions to have a true and exclusive relationship to squares and cubes. The shorthand "squared" is derived of course from the fact that a square can be subdivided by parallel lines, with "x" subdivisions along each edge, into "x to the second power" smaller squares. For example, a square with 2 "modular subdivisions" per edge contains 4 small squares, and similarly 3 subdivisions yield 9 squares, 4 yield 16, 5 yield 25, and so on (Fig. 2-1). Everyone is familiar with these diagrams, but what most people do not realize is that this result is not unique to squares. Triangles exhibit the same property, as also shown in Figure 2-1. Furthermore, explains Fuller, triangles take up only half the space, because every square divides into two triangles. Triangles in general, therefore, provide a more efficient diagram for the mathematical function of multiplying a number by itself. Nature is always most economical; therefore nature is not "squaring"; she is "trian-gling."

Thumbnail image: Triangling vs. Squaring
Fig. 2-1. "Triangling" versus "squaring."
Click on thumbnail for larger image.

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