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Amy C. Edmondson
A Fuller Explanation
Chapter 6, Angular Topology
pages 68 through 70

Topology and Vectors

Fuller has declared his scope: "Synergetics consists of topology combined with vectorial geometry" (201.01). Topology in essence analyzes numbers of elements. (Euler's law is topological, involving neither symmetry nor size.) And now we must again think about vectors, for they are the key to this combination.

      Vectors provide an ideal tool for representing velocity, force, and other energetic phenomena. As you may recall, the concept is actually quite simple—despite its lack of popularity among high-school students. A vector is a line with both specific length and angular orientation. It's the ultimate simplification of actions or forces, presenting only the two most relevant bits of information: magnitude and direction. Mathematics defines this tool and the accompanying rules for its manipulation, just as it defines the set of real numbers and the rules for addition and multiplication. The mechanism as a whole enables us to predict the results of complex interactions of forces and bodies in motion.

      Surveying classical geometry, Fuller decided that "there was nothing to identify time, and nature has time, so I'd like to get that in there." Vectors seemed to provide the solution, "I liked vectors. A vector represents a real event of nature... . I wondered if I couldn't draw up a geometry of vectors; that would mean having the elements of experience." (2)

      Back to synergetics: What is meant by a combination of topology and vectorial geometry? And how does it fit into the search for nature's coordinate system"? By viewing polyhedra as vector diagrams, Fuller integrates the two subjects (vectors and topology) in a deliberate attempt to develop one comprehensive format to accommodate both the inherent shape of space and the behavior of physical phenomena. Polyhedra with. vectors as edges necessarily incorporate both shape and size.

Vector Polyhedra

The spectrum of possible forms of polyhedra is certainly informative about the shape of space; polyhedra are systems of symmetry made visible. Any configuration allowed by space can be demonstrated by vertices and edges, and, as noted earlier, experimentation quickly reveals that the variety of possible forms is limited by spatial constraints. Furthermore, the shape of space is fundamental to the events of nature. Fuller believed that mathematics, the science of structure and pattern, should be based on these principles.

      So Fuller coined the rubric "angular topology" to express what he saw as the principal characteristic of synergetics: integration of the static concepts of geometry with energetic reality. These may not have been the words he used back then, but the desire for such a system dates back to Bucky's early school days. Or at least that's how the story goes. Such myths evolved over time to convey the spirit of the child's inquisitive confusion through concise anecdotes. Fuller's lectures and writings incorporate a full repertoire of autobiographical moments in which the young Bucky has startlingly complete and rich realizations. The process is beautifully described by Hugh Kenner, who relieves us of the burden of asking, "Did that really happen, just like that, one morning"?

      Not that he deceives. He mythologizes, a normal work of the mind... to embrace multitudinous perceptions, making thousands of separate statements about different things [into] summarizing statements...

      What a myth squeezes out is linear time, reducing all the fumblings and sortings of years to an illuminative instant. We can see why Bucky needs myth. The vision that possesses him eludes linearity... The myth is anecdotal.(3)

      We return to the geometry lesson: the grade-school teacher has put a drawing on the blackboard and said, "This is a cube." Young Bucky wonders aloud, "How big is it, how much does it weigh, what is its temperature, how long does it last"? The teacher says, "Don't be fresh," and "You're not getting into the spirit of mathematics." Again, the implication is that mathematics does not deal with real things, but only with absurd constructs and arbitrary rules.

      It's not hard to accept the message behind the story—that something about the teacher's lesson was profoundly disturbing to the child. It seemed to Bucky that mathematics was a serious enterprise and it should limit itself to "experimentally demonstrable" phenomena. That meant no fooling around in the fringe area of sizeless points, infinite planes, and weightless cubes. We reconsider these early musings in the new context of "angular topology" to see where they led, and recall that Fuller was later to attribute mathematics' lack of popularity to the perfectly natural discomfort people felt with those elusive concepts. Explanations ought to be in terms of tangible experience.

      But then what are we to make of such claims as "Synergetics permits conceptual modeling of the fourth and fifth arithmetic powers; that is, fourth- and fifth-dimensional aggregations of points or spheres in an entirely rational coordinate system that is congruent with all the experimentally harvested data of astrophysics and molecular physics..." (202.01)? Under the heading "Angular Topology," this statement is found too early in Fuller's Synergetics to be easily understood. One might wonder if a page was left out of that particular copy; but the root of the confusion is not that easily located. With some additional background material, we can begin to understand Fuller's assertion. The word "dimension" has been lurking behind the scenes in this entire discussion, and must now be brought out into the open.

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