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Tools of the Trade
Whether or not the thought of high-school geometry class stirs unpleasant memories, chances are that most of the actual material is long forgotten. Moreover, few of geometry's more pleasing properties are taught, leaving volumes of elegant transformations in the realm of esoteric knowledge. Lack of exposure to these age-old discoveries is the primary barrier to understanding synergetics.
By now familiar with Fuller's underlying assumptions, we shall take time out to introduce some background material. The origins of humanity's fascination with geometry can be traced back four thousand years, to the Babylonian and Egyptian civilizations; two millennia later, geometry flourished in ancient Greece, and its development continues today. Yet most of us know almost nothing about the accumulated findings of this long search. Familiarity with some of these geometric shapes and transformations will ease the rest of the journey into the intricacies of synergetics.
A little experimentation with basic geometric forms and procedures reveals the important role of space itself. The work of other thinkers reinforces the fundamental premise of synergetics: space has shapeall structures are formed according to spatial symmetries and constraints. It turns out that the number of symmetrical arrangements allowed by space is surprisingly limited; perpetual (synergetic) interaction of relatively few patterns accounts for the seemingly endless variety of form.
Even for readers whose background in geometry is already strong, reviewing it can be enjoyable and perhaps even illuminating. There are so many significant connections among polyhedral shapes and operations that even experienced geometers continually discover new ones.
Bucky's delight in a new-found truth never lost its intensity. He promptly adopted each discovery into his growing synergetics inventory. If, in his enthusiasm, he appeared to be taking credit for age-old discoveries, let usrather than judgingtry to enter into the spirit of his search. And if egotism seemed to have gone hand in hand with enthusiasm, it is because both grew out of his constant willingness to see everything as if for the first time. Both were part of the whole system Bucky.
Our most rewarding course is to immerse ourselves in the geometry. We shall first set the stage with some of the Greek basics, and then move on to the work of Arthur L. Loeb of Harvard Universitywhose "Contribution to Synergetics" (1) provides an analytical counterpart to the 800 pages of Fuller's more intuitive approachbefore moving on to the thought-provoking twists of Fuller's mathematical thinking.
As this thinking is encompassing and holistic, it would be counterproductive to scrutinize isolated parts of Fuller's geometry. Synergetics must be critically examined as a whole system before judging its contribution to mathematics. The emphasis (and certainly the excitement) in Fuller's ongoing research was in the inherent "omni-interaccommodation"that is, uncovered principles are never contradictory but rather augment each others' significance. Fuller's developing inventory was thus continually strengthened, becoming more and more integral to his thinking. He was especially gratified, in his later years, to receive numerous letters from noted scientists, reinforcing his discoveries with their observations from research in other fields.
Bucky's guiding purpose in developing synergetics was to reacquaint us with Universe. Such eagerness is compelling. ("Beautiful, beautiful bubbles . . . ," "eternally regenerative Universe," "...ecstasy in discovering nature's beautiful agreement.") We do him and ourselves a disservice to expect his approach to fit comfortably into the traditional scholarly framework. Unacknowledged by academic institutions for most of his life, he felt unencumbered by their conventions. (But we are not, and this volume will attempt to locate sources wherever possible.)
Onward! Let's take another look at geometry; the journey itself is fascinating, as well as full of useful tools for understanding Fuller's work.
The Platonic polyhedra, as their name suggests, have been around almost two millenia. However, despite the fact that the five shapes (also called regular polyhedra) are well known, few people are aware of what exactly defines this groupand fewer still of the implications about space itself.
The requirements seem lenient at first. They are two: the faces of a polyhedron must be identical, and the same number of them must meet at each vertex. The tetrahedron fits, with its four triangles and four equivalent (three-valent) vertices. (2) (Notice the roots of the word equivalent.) From these two criteria alone, one might suspect the existence of many more regular polyhedra.
Let's study the possibilities step by step, beginning with the simplest polygon (fewest sides) and the smallest number of edges meeting at each vertex. In keeping with Fuller's use of vectors as edges, this study will be confined to "straight" edges. It is quickly apparent that a minimum of three edges must meet at each vertex of a polyhedron, for if vertices join only two straight edges, the resulting array is necessanly planar. This lower limit can also be expressed in terms of faces, for we can readily visualize that a corner needs at least three polygons in order to hold water-which is another way of saying the inside is separated from the outside, as specified by Fuller's definition of system.
The minimal polygon is a triangle. Three triangles around one vertex form a pyramid, the base of which automatically creates a fourth triangular face. As all corners and all faces are identical, the first regular polyhedrona tetrahedronis completed after one step (Fig. 4-1a).
Next, a second regular polyhedron can be started by surrounding one vertex with four triangles, resulting in the traditional square-based pyramid (Fig. 4-1b). But, as our specifications for regularity indicate that all vertices must connect four edges, an additional edge is required at each of the four vertices around the pyramid's base, bringing the total to twelve edges. By connecting the four dangling edges (Fig. 4-1c), we introduce a sixth vertex, which is also surrounded by four triangles. The result is an octahedron, with eight triangular faces and six four-valent vertices. Both criteria for regular polyhedra are satisfied, and so the octahedron is added to our list.
As the procedure is thus far simple and successful, analogy suggests the next step. Five triangles around one vertex form a shallow pyramid (Fig. 4-2a). Paying attention as before to nothing but the two rules, we continue to employ triangular faces while making sure that five of them surround each corner. The structure essentially builds itself in that there is only one possible outcomeand we don't even have to know what it is to be able to finish the task. The icosahedron, with twenty (in Greek, "icosa") triangles and twelve five-valent corners, is indeed regular (Fig. 4-2b).
Once more then. Bring six triangles together at a corner. But wait! Six 60-degree angles add up to 360 degrees, or the whole plane (Fig.4-2c). An unprecedented result, this indicates that we could surround vertices with six equilateral triangles indefinitely and never force the collection to curve around to close itself off. A space-enclosing system is therefore unattainable with exclusively six-valent vertices. Is it possible we have exhausted the possibilities for regular polyhedra out of triangles? We simply cannot have fewer than three or more than five triangles around all vertices and create a closed finite system. We thus encounter a first upper limit.
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