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Conceptual and Real Systems
Notice that these geometrical systems are purely conceptual: so far they exist only in our mind, as sets of relationships. They can be lent substance by any number of materials, as for example the toothpicks and marshmallows mentioned above. However, the essence of a system is independent of the choice of materials: six sticks will create a tetrahedron whether we use wood or metal. Similarly, four Ping Pong balls or four people constitute a tetrahedral system. The tetrahedron, being a conceptual entity, is "sizeless and timeless." Thus Fuller writes in Synergetics, Size is always a special-case experience." (515.14) Size belongs to a different category of parameters than vertices, edges and facesthose which only relate to actual constructs, such as color, temperature, and duration.
Does he take this concern with terminology too far? His justification is twofold, encompassing first a deep conviction that words influence the shape of our thinking, and secondly faith in the power of accurate models in problem-solving. For humanity to solve its complex problems, he was convinced that vocabulary and other models had to be absolutely precise. So Fuller's concern that we recognize conceptual systems as sizeless sets of relationships capable of being physically embodied is an essential part of his geometry of thinking. In either form, the emphasis is on the relationships.
We begin to see a basis for the phenomenon of vastly different properties exhibited by systems with identical constituents. One notable example is the soft grey graphite of pencils in contrast to sparkling impenetrable diamonds, both consisting exclusively of carbon atoms. Geometry alone accounts for their differences. We shall see how later, but as always our attention will be on shape and valency (numbers of connections) rather than substance.
Let's get back to our starting point. Any subdivision of Universe constitutes a system. We have found the simplest example and learned the mathematical terms; our next step is to look at more complex systems. It does not stretch our definition to discuss some very elaborate forms, such as a school, or even a crocodile (both have the requisite boundary). For that matter, our entire planet is a systemunimaginably intricate but still finite. This line of thought, together with our geometry lesson, suggests an approach to problem-solving: a "whole Systems" view that demands consideration of the influence of every move on its entire system. Such an approach, which prohibits short-term or piecemeal solutions to long-term problems, may sound simplistic or vague at first; however, the method is based on the assumption of rigorous analytical procedures.
In Synergetics, Fuller introduces the concept with a deliberately simple example, which provides an analogy for more complex situations. A fictitious child draws on the ground with a stick, announcing that he has made a triangle. Then Bucky himself intervenes to point out to the child that he has created four trianglesnot just onebecause "operational mathematics" requires that a triangle must be inscribed on something in order to exist. Whether on a piece of paper or on the surface of the earth, that something is always a system, with an inside and an outside. Unwittingly, the child has divided the earth's surface into two areas. Both regions are bounded by three arcs, and therefore both qualify as spherical triangles, (3) despite the fact that one is small and tangible and the other covers most of the earth's surface. We are not used to thinking in these terms, philosophizes Fuller, but we must begin to really think about what we're doing.
Hold on, says the child, that's only two triangles! Why did you say there were four? Well, Bucky continues, concave and convex (3) are not the same; when you delineated two concave triangles on the outside surface, you also created two convex spherical trianglesone very small and the other very largeon the inside. But I didn't mean to make four triangles, protests the bewildered child. That doesn't matter, his teacher replies; you are still responsible for them.
His story can be considered a parable; its purpose is as much to encourage a sort of holistic morality as to make a mathematical statement. The message: tunnel vision is obsolete. As human beings, we cannot afford to ignore the effect of our actions on the rest of a system while working on an isolated part. Rather we must become responsible for whole systems. We didn't mean to make four trianglesor indeed, to make "the big mess of pollution" (814.01).
As playful as this example seems, it calls our attention to what Fuller perceives as a dangerous "bias on one side of the line" inherent in traditional mathematics. He points out that our grade-school geometry lessons involve concepts defined as bounded by certain linesa triangle is an area bounded by three lines, for examplethus excusing us from paying attention to its environment. Once a figure is delineated, we no longer have to consider the rest of the system. This narrow approach, Fuller argues, instills in us at an impressionable age a deep bias toward our side of the line; we see and feel an unshakable correctness about our side's way of "carrying on." On the other hand, "Operational geometry invalidates all bias" (811.04). It forces us to remain aware of all sides. In Fuller's opinion, being taught in the first place that all four triangles are "equally valid" would significantly influence our later thinking and planning.
One consequence of this approach is Fuller's realization that "unity is inherently plural" (400.08). "Oneness" is impossible, he explains, for any identifiable system divides Universe into two parts, and requires a minimum of six relationships to do so. Furthermore, as illustrated in the above parable, all "operations" produce a plurality of experiences, and awareness itselfwithout which there can be no lifeimplies the existence of "otherness." Ergo, "Unity is plural and at minimum two." (4)
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