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Amy C. Edmondson
A Fuller Explanation
Chapter 15, From Geodesics to Tensegrity: The Invisible Made Visible
pages 232 through 234


From Geodesic to Tensegrity:
The Invisible Made Visible

"A geodesic is the most economical relationship between any two events" (702.01). Fuller's definition immediately calls to mind great circles, which provide the shortest routes between two events dn a spherical system. Actually, clarifies Fuller, the general case of "most economical relationship" is necessarily a great circle. "It is a special case in geodesics which finds that a seemingly straight line is the shortest distance between any two points in a plane." In other words, a given area may be such a small portion of a spherical system that it appears flat; however, because all identifiable experiences belong to systems, "great-circle segment" and "geodesic" are interchangeable in synergetics.

      Already familiar with the theory of great circles and their polyhedral symmetries, we can apply theory to practice. A little experimentation uncovers two important discoveries, demonstrating again why the discipline of building models was essential to Fuller's mathematical exploration.

      The first discovery is easily visualized without actually building a model. Imagine a metal sphere and a wire ring just large enough to fit around the widest girth of the sphere. This circular ring qualifies as a great circle and therefore can delineate the shortest route between any two points on the sphere. However, there is a practical problem; the ring slides off the sphere. It may seem like a strange observation, but Fuller tended to investigate unusual aspects of his subject matter, and frequently such seemingly whimsical sidetracks have proved fruitful.

      Bring in a second wire circle of the same diameter. As we recall from the last chapter, a pair of great circles intersect at two points 180 degrees apart. To keep the two wire loops on the sphere, they are tied together at both crossing points. The effort fails, for the two circles are free to spin around their common axis, and as soon as they line up both circles slide off the sphere together. Try again. A third great circle is placed anywhere on the sphere except through the intersection of the other two. Whether arranged randornly or symmetrically (as a spherical octahedron), when intersections are tied together all three circles are immobilized. Triangulation creates a stable cage. "Not until we have three noncommonly polarized, great-circle bands providing ornnitriangulation... do we have the great circles acting structurally to self-interstabilize..." (706.20).

      Three differently oriented great circles is the minimum for a stable model. The discovery may seem trivial at first, but we have no indication from design that the stability of a "three-way grid" is widely understood in our culture. On the other hand, Southeast Asians have utilized this principle for thousands of years. Fuller points out that a vital need for strong baskets led them long ago to discover that a triangulated weave stays rigidly in place, whereas the two-way weave used by other cultures is easily distorted. (1) Southeast Asian children today still play with a reed sphere consisting of six interwoven great circles—perhaps the oldest known toy (Fig. 15-1). Characterized by icosahedral symmetry, the ancient design utilizes the inherent stability of a three-way grid to make a lightweight and virtually indestructible ball out of delicate reeds. A modern toy has yet to improve upon its simplicity and durability.

Southeast Asian toy reed ball
Fig. 15-1
Click on thumbnail for larger image.

      The second discovery, which should be experienced to be fully appreciated, will nevertheless not come as a surprise at this point. Experimenting with wire models, Fuller found that the more great circles, the stronger the sphere. That much is self-evident, but the degree to which their strength increased far exceeded his expectations. This was not the kind of linear relationship exhibited by ordinary structures; rather the increasing rigidity of his great-circle models (from the minimum three to the icosahedral 31) could only be called synergetic.

      We skip directly to the strongest model, provided by last chapter's limit case, 31 great circles. Triangulated geodesic arcs produce an extremely sturdy, lightweight enclosure out of thin wire. Let's look closely at the pattern; our study is made considerably less complicated by isolating the LCD as described in the preceding chapter. Accordingly, we study only one of the 120 triangles framed by the icosahedron's 15 great circles, and observe that it is asymmetrically subdivided by the 6- and 12-great-circle patterns (Fig. 15-2). Notice the variations in arc length and surface angie~made more obvious by viewing this small region out of context. The overall three-way grid incorporates longer and shorter arcs, thereby subdividing the sphere's surface into triangles of very different shape and size.

LCD of the icosahedron subdivided
Fig. 15-2. LCD of the icosahedron is asymmetrically subdivided by the 31-great-circle pattern.
Click on thumbnail for larger image.

      The load distribution and resulting strength of these wire models is a function of symmetry; the longer the arc, the more vulnerable it is to stress. (2) It is clear that the most advantageous system would have all arcs as close to the same length as possible, but how can we improve upon the symmetry of the 31-circle pattern?

      A second problem with the arrangement is that as a limit case, it does not present a logical course for further subdivision. To build progressively larger models with sufficient strength, we must find a way to generate more and more great-circle segments, or "higher frequency" in synergetics terihinology. Both problems are solved by Fuller's next step.

Four-frequency (4v) triangles superimposed on each face of the icosahedron
Fig. 15-3. Four-frequency (4v) triangles superimposed on each face of the icosahedron.
Click on thumbnail for larger image.

      To explore other methods of developing large multifaceted enclosures, we go back to the system that already has the greatest number of equivalent regular faces, the icosahedron. A three-way grid of evenly spaced lines (imagine a triangular checkerboard) divides the icosahedron's equilateral triangles into as many smaller triangles as desired (Fig. 15-3). However, this subdivision does not yet lead to an effective design strategy, for if neighboring triangles of a structural system lie in the same plane, the enclosure will deflect in and out, like a trampoline, in reaction to an applied load. Unless vertices are reinforced using rigid joints—in which case, the system is functionally equivalent to the original icosahedron—the advantages of triangulation are lost. It is thus clear that a convex polyhedral enclosure with more than twenty triangles cannot consist exclusively of equilateral faces. Adjacent triangles must differ slightly in shape and size to allow angles around each six-valent vertex to add up to less than 360 degrees, as necessary for continuous convexity. (3) Fortunately, this unavoidable variation among chord lengths can be far less than that of the 31-great-circle pattern. To understand how these irregular triangles are generated, we back up and review the problem as a whole.

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