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Upstaged by the crowd of oversized polyhedral toys, Bucky again resembles the small child we saw earlier, playing with his mother's necklace. But the words this time are more ambiguous: "There are only three basic structural systems in Universe."
Fuller long ago decided that science simply did not have a definition of structure, and took it upon himself to remedy the oversight. Science, Fuller explains, did not feel the need for a definition, because "structure" seemed to be self-evident. It holds its shape! Language, caught in the old world of "solids," did not keep up with science's evolving understanding of the true nature of matter. In a universe consisting entirely of fast-moving energy, we must ask how something holds its shape.
"Structure is defined as a locally regenerative pattern integrity of Universe" (606.01). A good starting point. Structure is also "a complex of events interacting to form a stable pattern." Similar, but more specific: the pattern consists of action, not things.
"Regenerative" is an important qualification, because of the transient nature of energy. The pattern, not the energy flowing through, has a certain degree of permanence. A structure must therefore be continually regenerating in order to be detected.
Structure is "local" because it is finite; it has a beginning and an end. "We cannot have a total structure of Universe" (606.01).
"Interacting" signifies the emphasis on relationships.
The phrase "complex of events" suggests an analogy to constellations, whose componentsthough spectacularly far apartare interrelated for some cosmic span of time, creating a set of relationships, in other words, a pattern. The seven stars of the Big Dipper are light-years apartthe epitome of nonsimultaneous energy events. They are only perceived as a meaningful pattern from a special vantage point, Spaceship Earth. The remoteness of individual atoms in any structure or substancenot to mention the distance between atomic constituentsprompted Fuller to write "one of the deeply impressive things about structures is that they cohere at all." There is nothing "solid" about structure.
What do all structures have in common that allows their coherence? Triangles. At the root of all stable complexes is nature's only self-stabilizing pattern.
No Fuller study is complete without an "inventory": a list, not of each and every "special case" example, but rather, of the types of categories. The task, then, is to combine our new working definition of structure with the earlier one of systems. That means triangles involved in a subdivision of Universe. The virtually unlimited variety of irregular triangulated enclosures are not to be included in this inventory; rather, we seek a list of symmetrical stable enclosures.
aComparison of results in third column:
0.2357 = 2 x 0.11785,
0.4363 = 3.70 x 0.11755.
And now we can sit back, for our task is already finished. Remember that only three systems can be made out of regular triangles: tetrahedron, octahedron, and icosahedron. These are the "three prime structural systems of Universe."
What can be learned through this kind of simplification? As in Euler's identification of vertices, edges, and faces, such categories organize the otherwise indigestible data to reveal new important features. An example is seen in Fuller's "structural quanta": the total material used for each of the three structural systems (easily measured in terms of number of edges) goes from six sticks to twelve to thirty. Chapter 10 will cover Fuller's ideas on the subject of volume in detail, but for now we can demonstrate an interesting fact while utilizing the traditional formulae of high-school geometry.
Going from the smallest to the largest structure, the volume increases, not only absolutely, but relative to the number of edges. In other words, the ratio of volume to structural investment is a significant variable, increasing with additional structural quanta. The same holds true for ratios of volume to surface-area, as will be seen below. For clarity, we adopt unit edge lengths for all three polyhedra. Appendix B shows each step of the calculations, but the relevant results are displayed in Table II.
The implications of this information are suggested by Fuller's summarizing statement:
The tetrahedron gives one unit of environment control per structural quantum. The octahedron gives two units of environment control per structural quantum. The icosahedron gives 3.7... . (612.10)
Fuller referred to six edges as a "structural quantum" because the total number of edges in each polyhedron is a multiple of six. "Environment control" simply refers to the ability to enclose and thereby regulate space.
Toward the goal of maximal enclosed space with minimal structural material (whether in terms of total strut length or surface area), designs based on the icosahedral end of the spectrum are advantageous. Hence Fuller's geodesic dome. For resistance to external loads, the tiny pointed tetrahedron is least vulnerable, for its concentrated structural elements resist buckling. The tetrahedron is all edges, enabling maximal structural resistance, and therefore highly applicable to truss design. (See Chapter 9.)
Three of Fuller's inventions, the geodesic dome, the Dymaxion Map, and the Octet Truss, stem directly from the above principles. All three will be discussed in detail later, as other relevant geometric principles are revealed.
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