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 Amy C. EdmondsonA Fuller Explanation Chapter 6, Angular Topologypages 65 through 67

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Angular Topology

Our study so far has primarily examined the conceptual foundation of synergetics. Except for occasional reference to volume and symmetry, the emphasis has been on numbers of elements rather than on shape. It's now time to look at the rest of the picture. Fuller's appreciation of the MIT definition of mathematics ("the science of structure and pattern in general") led him to ponder the appropriate tools and methods. "Science" is a systematic endeavor, requiring exact procedures for its description of structure and pattern.

A coordinate system describes the shape and location of a body in space by specifying the position of a sufficient number of that body's components. But a position can only be specified by its relationship to some other known location, or coordinate-system origin. In essence, mathematics functions by locating points relative to an agreed-upon frame of reference, such that the mathematician can say there is a point here and a point there and they are related by this kind of trajectory, and so on, until there is enough information to describe the entire system. Fuller points out that this information can be broken down into two aspects: shape and size.

What does "shape" consist of? "Shape is exclusively angular" (240.55): a simple but powerful observation. It's easy to envision identical shapes of completely different scale: for example, an equilateral triangle is a precisely defined concept, yet it contains no indication of size. It may be two miles or two centimeters in edge length, but its angles must be 60 degrees. Shape is influenced only by angle, and "an angle is an angle independent of the length of its sides" (516.02). The word "triangle" by itself (without further modification) does not describe a specific shape but rather a concept—three interrelated events without specific length or angle.

What does size consist of? Measurement, or dimension. In synergetics, these parameters are always expressed in terms of "frequency." The word is aptly applied, serving as a reminder of the role of time. Fuller dwells on the point: every real system ("special case") involves time and duration. (1) Real Systems are events, and it takes time for an event to occur. He bases his objection to purely static concepts in mathematics on the fact that they are incompatible with twentieth-century scientific thought:

Since the measure of light's relative swiftness, which is far from instantaneous, the classical concepts of instant Universe and the mathematicians' instant lines have become both inadequate and invalid for inclusion in synergetics. (201.02)

Since Einstein, Bucky reminds us, we can no longer think in terms of an "instant Universe," that is, a single-frame picture. Because even light has been found to have measurable speed, every aspect of physical Universe from the smallest tetrahedron to life itself involves the passage of time. Quite simply, "it takes time to get from here to there."

Frequency and Size

He insists upon nothing more adamantly than this distinction—between real ("experimentally demonstrable") phenomena and imaginary concepts. "Size" relates to real, time-dependent systems, whereas "shape," influenced only by angle and therefore independent of time, is a factor in both real and conceptual systems. ["Angles are... independent of size. Size is always special-case experience" (515.14).]

But how does "frequency" apply to size and length? Frequency connotes number: the number of times a repeating phenomenon occurs within a specified interval—ordinarily an interval of time, but Fuller extends the concept to include space. Length is measured in synergetics in terms of frequency to underline the fact that the "distance from here to there" involves time and can be specified in terms of number: number of footsteps across the room, or number of heartbeats during that interval, number of water molecules in a tube, number of inches, number of photons, number of somethings. The choice of increment depends on what is being measured, but frequency (and hence size) is inescapably a function of time and number.

Units of Measurement

Fuller explains frequency as subdivisions of the whole, suggesting another advantage of the term: it provides a built-in reminder that there is no absolute or single correct unit of measurement; rather,distance is measured relative to arbitrarily devised units. It is not a minor challenge to perceive distance this way; our conventional units—like inches or miles-are such an integral part of awareness that they seem a priori elements of size. The teacher in Fuller will not let us accept such useful conventions blindly, and so he employs tools such as "frequency" to keep us on our toes-aware of the nature of distance.

Time and Repetition: Frequency versus Continuum

Just as length cannot exist without time, there also could be no awareness without time. Time, inseparable from all other phenomena, cannot be isolated. "Time is experience" (529.01).

The concept of time is inextricably tied to awareness; appropriately it is measured in terms of the frequency of detectable repeating events. Periods of daylight reliably alternating with darkness gave us a unit we call a "day." Heartbeats might have defined the "second," planting the awareness of that tiny increment in long-ago human beings. The predictable repetition of days growing longer and shorter with their accompanying weather changes defined a "year." To conceive of time requires repetition.

However, the limits of perception prevent recognition of the periodicity in very high-frequency patterns such as light waves or repeating molecules in a toothpick. If repetition is too frequent, we perceive a continuum rather than segmented events. Fuller's use of "frequency" to specify size draws attention to the nonexistence of continuums. Here, as always, his goal is to develop a mathematical language which accurately represents reality.

Shape and size are thus replaced by angle and frequency.

Fuller's principle of design covariables summarizes by stating that two factors are responsible for all variation. "Angle and frequency modulation exclusively define all experiences, which events altogether constitute Universe" (208.00). In short, "structure and pattern in general" are described completely by only two parameters: angle and frequency—another way of saying that the differences between systems are entirely accounted for by changes in angle and length. Again the goal of such simplification is the demystification of mathematics.

Remember that Fuller's overall goal was to isolate "nature's coordinate system"—by which he meant the simplest and most efficient reference system to describe the events of nature. We gradually narrow in on his solution.

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