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Amy C. Edmondson
A Fuller Explanation
Chapter 1, Return to Modelability
pages 6 through 9

Operational Mathematics

We can imagine the young Bucky, an enthusiastic misfit sensing that he is alone in his skepticism about the fundamental premises of geometry. ("Does no one see what I see? Does no one else sense the terrible problems that lie ahead if we follow these absurd premises to their logical ends?") While his grade-school classmates were apparently content to go right along with the teacher's strange games without complaint, Bucky was astonished by the implausible new concepts.

      Bucky would tell us that he tried, constantly, to accept the rules—be a good student, make his family proud, submit to and even excel at the illogical activities—but somehow his efforts at model behavior were always thwarted. He just couldn't help pointing out that the teacher's "straight line" was not at all straight, but rather slightly curved and definitely fragmented. Perplexed by her lack of accuracy, Bucky saw a trail of powdery chalk dust left on the blackboard, a trace of the motion of her hand, and it seemed quite unlike her words.

      Bucky was virtually blind until he got his first eyeglasses at the age of five, so he had truly experienced life without this primary sense. Now he was insatiably curious about the visual patterns around him. An "infinite straight line"? He would turn toward the window, thoughtfully pondering where that "infinite line" stopped. "Out the window and over the hill and on and on it goes"; it didn't seem right somehow. Bucky, childishly earnest even in his eighties, would tell this story, explaining that he didn't mean to be fresh, he just couldn't help wondering if that teacher really knew what she was talking about.

      Much later, fascinated by Eddington's (5) definition of science as the systematic attempt to set in order the facts of experience, Fuller had a plan. It must be possible to develop a mathematical system consistent with experience. He concluded that humanity had been on the wrong track all these years.

Experimental Evidence

It seemed to Fuller that mathematicians arbitrarily invent impossible concepts, decide rules for their interaction, and then memorize the whole game. But what did he propose as an alternative?

      Starting from scratch. Mathematical principles must be derived from experience. Start with real things, observe, record, and then deduce. Working with demonstrable (as opposed to impossible) concepts, the resulting generalizations would reflect and apply to the world in which we live. It seemed highly likely that such an experimental approach would lead to a comprehensive and entirely rational set of principles that represented actual phenomena. Furthermore, Bucky suspected that such an inventory would relate to metaphysical as well as physical structure.

      Fuller decided that to begin this process of rethinking mathematics he had to ask some very basic questions. What does exist? What are the characteristics of existence? He proposed that science's understanding of reality should be incorporated into new models to replace the no longer appropriate cubes and other "solids" that had kept mathematicians deliberately divorced from reality since the days of ancient Greece.

      To begin with, there are no "solids"; matter consists exclusively of energy. "Things" are actually events—transient arrangements of frenetically vibrating atomic motion. It's almost unthinkable, but perhaps if we get our vocabulary and models to be more consistent with an energy-event reality, humanity can be retuned to become comfortable with the invisible discoveries of science. In short, simplifies Fuller, we have the option to tell children the truth about nature in the first place.

      What if we do go along with the rule that mathematics cannot define and depend upon a concept that cannot be demonstrated; where does that leave us? Fuller saw inconsistencies even in the notion of an "imaginary straight line," for imagination relies on experience ("image-ination" he would say) to construct its images. Therefore an "operational mathematics" must rely on concepts that correspond to reality.

      Bucky pulls us back into the turn-of-the-century schoolhouse of his childhood. The teacher stood at the blackboard, made a little dot, and said, "This is a point; it doesn't exist." ("So she wiped that out.") Then she drew a whole string of them and called it a "line." Having no thickness, it couldn't exist either. Next she made a raft out of these lines and came up with a "plane." "I'm sorry to say it didn't exist either," sighs Bucky. She then stacked them together and got a "cube," and suddenly that existed. Telling the story, Bucky scratches his head as if still puzzled seventy years later: "I couldn't believe it; how did she get existence out of nonexistence to the fourth power? So I asked, 'How old is it'? She said, 'don't be naughty.' ... It was an absolute ghost cube." (6)

      Instead of a dimensionless "point," Fuller proposes the widely applicable "energy event." Every identifiable experience is an energy event, he summarizes, and many are small enough to be considered "points," such as a small deposit of chalk dust. An aggregate of events too distant to be differentiated from one another can also be treated as a point. Consider for example a plastic bag of oranges carried by a pedestrian and viewed from the top of the Empire State Building, or a star—consisting of immense numbers of speeding particles—appearing as a tiny dot of negligible size despite having an actual diameter far greater than that of the earth. (7)

      The mathematician's "straight line," defined as having length but no width, simply cannot be demonstrated. All physical "lines" upon closer inspection are actually wavelike or fragmented trajectories: even a "line of sight" is a wave phenomenon, insists Fuller; "physics has found no straight lines." (2) But forces exist, and they pull or push in a line, which can be modeled by a vector. We shall explore vectors and their role in synergetics in great detail in later chapters. Finally, the "continuous plane" with no thickness must be replaced by a mesh of energy events interrelated by fine networks of tiny vectors. To Fuller, these adjustments were crucial, for mathematicians' games with continuous planes and sizeless points were ultimately irrelevant diversions; however enticing the intellectual pursuit might be, they do not lead to a better understanding of how nature works.

      The essential nature of the above revolution is semantic and can easily seem trivial. The difficulty in evaluating the impact of such changes lies in the subtlety of the effect of words and the images they produce. Only through experimenting with Fuller's substitutions for some period of time can we judge the merits of mathematical terminology that reflects science's new understanding of reality.

      Back to the starting point! Nothing can be accepted as self-evident; a new mathematics must be derived though "operational" procedure. Fuller decided that through sufficient observation of both naturally occurring and experimentally derived phenomena without reference to a specific framework, nature's own coordinate system might emerge. He sought a body of generalizations describing the way patterns are organized and able to cohere over time. We shall see how these principles can be discerned both in deliberate experiments with various materials and by recording existing natural patterns.

      Bucky's grade-school skepticism was thus the beginning of a lifelong search for "nature's coordinate system." After rejecting traditional academia through his dramatic departure from Harvard's freshman class in 1914, (8) he began an independent exploration of mathematics which he was to pursue for the rest of his life. A sort of philosophical geometry gradually began to take shape, consisting of a rich body of facts and principles, some new and others newly considered. Synergetic geometry is tied together as one cohesive system by the unmistakable presence of Bucky Fuller in conventional and bizarre observations alike. Including expositions called "Tetrahedron Discovers Itself and Universe," "Life," "Cosmic Hierarchy," "Complex of Jitterbugs," and the more conventional "Closest Packing of Spheres," Synergetics does not fall neatly into any preconceived category. This volume will attempt to clarify these multifaceted (or polyhedral) observations, which together constitute synergetic geometry.

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