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Systems and Synergy
Fuller's mathematical explorations seem to fly out in many directions at once, but they share a common starting point in the concept of systems. Derived from the Greek for "putting together," the word system means any group of interrelated elements involved in a collective entity. If that sounds vague, it's meant to. The theme is widely encompassing.
Long ago, secluded in his room experimenting with toothpicks or ping-pong balls or whatever available material seemed likely to reveal nature's secrets, Fuller began to see a persistent message of interdependence. He was later to discover the precisely descriptive word "synergy," but even without that lexical advantage a sense of interacting parts increasingly dominated his vision. More like the poets and artists of his generation than the scientists, he was drawn to relationships rather than objects.
By stating that Fuller looked at systems, we learn very little, especially in view of the word's current popularity. We have transportation systems and systems analysts, stereo systems and even skin-care systems, all conspiring to diminish the precision and usefulness of the word. But let us enter into the spirit of Bucky's half-century search and abandon our twentieth-century sophistication in order to rediscover the obvious en route to the surprising and complex. Much can be gained, for alongside our era's growing consciousness of systems and interdependence is also its ever-increasing specialization. Individuals are encouraged to narrow their focus, precluding a comprehensive vision and inhibiting curiosity. How-things-work questions are reserved for children; as adults we are afraid to step outside our expertise. Furthermore, we are quite likely to have decided that we have no use for mathematics at all by the time we reach high school. Both factorsspecialization and avoidance of mathematicscause some aspects of Fuller's synergetics to seem dense while others seem oddly simple. However, the novelty of his approach serves to give us new insights, so bear with Bucky as he "discovers the world by himself'; (1) his enthusiasm may be contagious.
Fuller was unafraid to appear naïve. He announced his observations with equal fervor for the simplest ("only the triangle holds its shape") and the very complex (the surface angles of the planar rhombic triacontahedron correspond exactly to the central angles of the icosahedron's fifteen great circles), but on every level he was conscious of systems.
A system, says Bucky, is a "conceivable entity" dividing Universe into two parts: the inside and the outside of the system. That's it (except, of course, for the part of Universe doing the dividing; he demands precision). A system is anything that has "insideness and outsideness." Is this notion too simple to deserve our further attention? In fact, as is typical of Fuller's experimental procedure, this is where the fun starts. We begin with a statement almost absurdly general, and ask what must necessarily follow. At this point in Fuller's lectures the mathematics sneaks in, but in his books the subject is apt to make a less subtle entrance! (Half-page sentences sprinkled with polysyllabic words of his own invention have discouraged many a reader.) The math does not have to be intimidating; it's simply a more precise analysis of our definition of system.
So far a system must have an inside and an outside. That sounds easy; he means something we can point to. But is that trivial after all? Let's look at the mathematical words: what are the basic elements necessary for insideness and outsideness, i.e., the minimum requirements for existence?
Assuming we can imagine an element that doesn't itself have any substance (the Greeks' dimensionless "point"), let's begin with two of them. There now exists a region between the two pointsalbeit quite an unmanageable region as it lacks any other boundaries. The same is true for three points, creating a triangular "betweenness," no matter how the three are arranged (so long as they are not in a straight line). In mathematics, any three noncollinear points define a plane; they also define a unique circle.
Suddenly with the introduction of a fourth point, we have an entirely new situation. We can put that fourth point anywhere we choose, except in the same plane as the first three, and we invariably divide space into two sections: that which is inside the four-point system and that which is outside. Unwittingly, we have created the minimum system. (Similarly, mathematics requires exactly four noncoplanar points to define a sphere.) Any material can demonstrate this proceduresmall marshmallows and toothpicks will do the trick, or pipecleaner segments inserted into plastic straws. The mathematical statement is unaltered by our choice: a minimum of four corners is required for existence.
What else must be true? Let's look at the connections between the four corners. Between two points there is only one link; add a third for a total of three links, inevitably forming a triangle (see if you can make something else!). Now, bring in a fourth point and count the number of interconnections. By joining a to b, b to C, C to d, d to a, a to C, and finally b to d (Fig. 3-1), we exhaust all the possibilities with six connections, or edges in geometrical terminology. Edges join vertices, and together they generate windows called faces.
This minimum system was given the name tetrahedron (four sides) by the Greeks, after the four triangular faces created by the set of four vertices and their six edges (Fig. 3-1). Fuller deplored the Greek nomenclature, which refers exclusively to the number of facesthe very elements that don't exist. ("There are no solids, no continuous surfaces... only energy event complexes [and] relationships." (2) ) However, he did not fully develop a satisfactory alternative, so we shall have to work with the time-honored convention. What we lose in accurate description of physical reality, we gain in clarity and consistency.
The tetrahedron shows up frequently in this exploration. This and other recurring patterns seem coincidental or magical at first, but soon come to be anticipatedendless demonstrations of the order inherent in space. The process is typical of synergetics: we stumble into the tetrahedron by asking the most elementary questionwhat is the simplest way to enclose space?and later, everywhere we look, there it is again, an inescapable consequence of a spectrum of geometric procedures.
The straightforward logic of our first encounter with the tetrahedron drives us to wonder if it displays any other unique properties. It turns out to be a reliable sort of minimum module or "quantum," as Fuller points out in myriad ways. Not the least impressive involves counting the edges of all regular, semiregular, and triangulated geodesic polyhedra (from the simple cube to the more complex rhombic dodecahedron to the vast array of geodesics.) The resulting numbers are all multiples of the tetrahedron's six. We can therefore take apart any polyhedral system in these categories and reassemble its edges into some number of complete tetrahedra. Even though we are not yet familiar with these other polyhedra, the observation stands as a representative example of the surprising whole-number relationships which make our investigation increasingly alluring.
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