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Amy C. Edmondson
A Fuller Explanation
Chapter 15, From Geodesics to Tensegrity: The Invisible Made Visible
pages 235 through 240

Theory Behind Geodesic Structures: Summary

We can think of Fuller's task as a mathematical game confined to the rules of synergetics, in which the goal is to enclose as much volume as possible with the least amount of material. A solution must combine two geometric principles—integrating overall shape considerations with the requirements for local stability. Geometry tells us that the shape with the greatest volume/surface-area ratio is a sphere; synergetics challenges that sphere to materialize. "Since physics has found no continuums, we have had to clear up' what we mean by a sphere," writes Fuller (1023.11). This game calls for a solution we can actually build and touch, which means the best structural approximation of that elusive sphere.

      We thus have redefined the problem in terms of "operational mathematics." To get started, we might experiment with toothpicks, thus limiting the game to enclosures with identical struts. This brings us back to Fuller's "three prime structural Systems in Universe" —the inventory of self-stabilizing systems with equal vectors. Of the three—tetrahedron, octahedron, and icosahedron—the third has the most volume per toothpick (Chapter 5). An icosahedron is thus the best we can do with toothpicks, but of course it is not a satisfactory solution.

      How can the enclosure become more spherical and still be stable? The geometric logic continues: through systematic symmetrical subdivision of our best approximation—taking full advantage of the stability of triangles. It hardly needs to be restated: "If we want to have a structure, we have to have triangles" (610.12).

      In short, two simple principles taken together lead to a solution to the stated problem. Combine the advantageous shape of the icosa-hedron with the stability of triangles, and the geodesic dome almost materializes. The logic is as exquisite as it is simple.

      But we're missing a step. How can we develop the "checkerboard" tessellation illustrated above into a functional structure? The icosahedral edges can be divided into any number of segments. The greater the number, or "frequency," the more spherelike the end result will be. Then, to complete the transformation, each new vertex (superimposed on icosahedron faces) must be projected out to the surface of the imaginary sphere defined by the icosahedral vertices—and then interconnected by great-circle chords (Fig. 15-4). The nature of this projection accounts for the slight variation in shape and size of triangles; the farther away a chord is from an icosahedral vertex, the more it is stretched during the transformation from planar to spherical.

Vertex projected on to imaginary sphere
Fig. 15-4. Each new vertex is projected outward to the surface of an imaginary sphere defined by the original icosahedron vertices.
Click on thumbnail for larger image.

      The resulting omnitriangulated polyhedron has six triangles at everv vertex except for those located at the twelve vertices of the original icosahedron, which continue to join five triangles (Fig. 15-5). The system can have indefinitely many six-valent nodes, but the existence of exactly twelve five-valent vertices is a prerequisite to closure. Happily, there is also a considerable engineering advantage in having five or six struts leading out from each vertex: forces are instantly distributed in many directions—omniradially, Bucky might say—producing structures with unprecedented strength in relation to their weight.

4v icosahedron: transformation of Figure 15-3
Fig. 15-5. 4v icosahedron: transformation of Figure 15-3..
Click on thumbnail for larger image.

      This is the basic strategy behind the geodesic dome.

Geodesic Design in Nature

Fuller points out that an extremely high-frequency geodesic polyhedron provides the true model of physical systems which we interpret as spheres, as for example a soap bubble. The notion of a continuous surface equidistant from a central point is scientifically unacceptable, that is, inconsistent with physical reality; on some level of resolution all "spheres" consist of discrete quanta—untold numbers of energy events interconnected by an even greater number of vector-relationships, or forces. He has clarified this particular misconception countless times:

      535.11 Because spherical sensations are produced by polyhedral arrays of interferences identified as points approximately equidistant from a point at the approximate center, and because the mass-attractive or—repulsive relationships of all points with all others are most economically shown by chords and not arcs, the spherical array of points produces... very-high-frequency, omnitriangulated geodesic structures...

Our eyes cannot see individual molecules in the delicate transparent soap bubble, nor can we detect the chordal chemical attractions between molecules. Nevertheless they exist, explains Fuller, and it is our responsibility to understand and teach the truth about Universe. Once again, his goal is to provide tangible models of otherwise invisible phenomena.

      Of all possible solutions, a high-frequency triangulated shell with icosahedral symmetry provides the most efficient method of enclosing space with a minimum of material and effort. Accordingly, nature relies on this elegant design in many situations calling for protective enclosures, regardless of scale. Examples include the small sea creatures called Radiolaria, (4) the fibrous web of the eye's cornea, and the protein shell of many viruses. (5) We looked into the structure of spherical viruses in Chapter 8 and can now go into greater detail based on our increased familiarity with geodesic theory.

      The design problem is familiar by now: tiny amounts of genetic material must be protected by a tough protein shell. As nature is scrupulously efficient, the choice is clear. A "spherical" distribution of protein molecules will satisfy the basic criteria in terms of conserving material relative to volume, while icosahedral symmetry will provide the most even distribution. The natural balance sought by chemical forces leads to approximately equivalent spans, which is geometrically accomplished by high-frequency icosahedral systems. In short, the structure of viruses is a product of nature's eternal tendency toward equilibrium. Geometry imposes the rules.

      Reassuringly, observations (with the electron microscope) of isometric virus shells have consistently revealed icosahedral designs. Dr. Aaron Klug, who first observed the geodesic structuring of viruses, wrote to Fuller in 1962 telling him of this discovery. Bucky, delighted by the news, immediately wrote back with the formula for the number of nodes on a shell (10f ² + 2, varying according to frequency) as confirmation of Klug's hypothesis. Klug answered that the values obtained from this equation proved consistent with the virus research, and thereby provided Fuller with one of his most valued anecdotes—a prime example of nature's economic elegance—which enriched many lectures in subsequent years.

      Insufficient awareness of spatial constraints causes these structural similarities to be perceived as "coincidence." As geodesic domes were utilized worldwide 15 years before electron microscopy enabled detection of virus capsids in 1962, the resemblance of the tiny biological forms to large architectural structures seemed to many quite extraordinary and improbable. Science does not as a rule take into consideration the active role of space; however, such awareness can assist the prediction of unknown structures based on their functional demands.

      Another fascinating example was contributed to the inventory of geodesic structuring by scientists at General Dynamics working on the problem of rocket reentry, who wrote to Fuller describing their results and enclosing photographs. The experiment involved two hemispheres of thin-sheet titanium, precisely machined to achieve consistent shell thickness. The diameter of one was exactly an inch greater than that of the other, so that when the larger was placed over the smaller, a half-inch hemispherical cavity separated the two shells. Their bases were sealed together to create a double-shell dome, and the air was then pumped out of the intervening space to create a vacuum. Atmospheric pressure outside the dome caused its thin titanium sheet to buckle in toward the vacuum. The hemisphere "dimpled" in a "pure icosahedral pattern," as Fuller recalls in a 1975 lecture. Like the virus, it had no choice! The titanium sheet experienced an automatic reaction based on the shape of space; caving in most efficiently required a symmetrical distribution of dimples. The frequency of this pattern, that is, the number of dimples per icosahedral "edge," was found to be inversely proportional to the shell thickness: a thicker shell produced fewer dimples and vice versa.

      Consider the above progression. We started with a mathematical puzzle; geometry laid out the rules and led to a solution, and it turned out that nature had been playing the same game all along. That is essentially how Fuller describes his experience in developing the geodesic dome:

      I did not copy nature's structural patterns. ... I began to explore structure and develop it in pure mathematical principle, out of which the patterns emerged in pure principle and developed themselves in pure principle. I then... applied them to practical tasks. The reappearance of [geodesic] structures in scientists' findings at various levels of inquiry confirms the mathematical coordinating system employed by nature. (203.09)

The principles behind the geodesic dome are not new; they are eternal laws of nature. The application of these geometric facts to a building system is new. Fuller is quick to explain,

      Though... similar in patternings to... flies' eyes, geodesic structuring is true invention... Flies' eyes do not provide human-dwelling precedent or man-occupiable... structures. (640.01)

Invention can be defined as the novel application of generalized principles. Chapter 16 will explore the concept in more detail.

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