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Amy C. Edmondson
A Fuller Explanation
Chapter 14, Cosmic Railroad Tracks: Great Circles
pages 219 through 222

Operational Mathematics

Fuller made a remarkable discovery about great-circle patterns that is responsible for their great significance in his mathematics. This discovery involves an intricate relationship between central and surface angles and could so easily be missed that one cannot help reflecting on the intuition that led Fuller to such an insight. As with other aspects of his "operational" method, the demonstration relies on readily available materials, but its significance extends to structuring in nature. This is a particularly satisfying exercise, and readers are encouraged to make Bucky's discovery themselves by following the simple instructions. Rather than a "plane," Bucky starts with a real system, a "finite piece of paper" (831.01). Using a compass, draw four circles with diameter of approximately 6 inches, and then cut them out with scissors. Fold each one. Then fold the resulting sernicirdes into thirds, as shown in Figure 14-13. The section labeled A is folded toward you, while C is folded back, producing a Z-shaped cross-section.

Creating a 'bowtie'
Fig. 14-13
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      Making sure that all creases are sharp, unfold the paper to obtain a circle with three intersecting diameters clearly marked by fold lines (Fig. 14-13). The circle is thus divided into six equilateral triangles. The process of sweeping out a circle with a compass insures that all radil are equal, and because at one point in the procedure all six pie-slices are piled up together, we know that the central angles must all be exactly the same. 360 degrees divided among six equal segments yields 60-degree angles. First-hand experience has confirmed both the constant radius and sixty-degree angles, and therefore the presence of equilateral triangles with arc segments at their outer edges is experientially proved.

      One fold (line AB) faces you; the other two folds are facing away. Bringing point A to point B, we create one of Fuller's "bowties" (Fig. 14-13, bottom). A bobby pin straddling the seam keeps the bowtie together: two unit-length regular tetrahedra joined by an edge. We repeat the procedure three times, producing four bowties in all. It is then apparent that two of them can be placed seam to seam and pinned together with two more bobby pins, to produce four tetrahedra surrounding a half-octahedral cavity. The other two bow-ties are similarly paired, and finally the two pairs are connected along their congruent fold lines with four more bobby pins. A complete paper sphere emerges (Fig. 14-14). This strange procedure has created a very familiar pattern: four continuous great circles, or a spherical VE. The only materials required are four paper circles and twelve bobby pins.

Four bowties create the four great circles of the VE
Fig. 14-14. Four bowties create the four great circles of the VE.
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      What has happened? Four separate paper circles have been folded, bent, transformed into bowties, and pinned together without paying any attention to converging angles. No special jig is required to line up adjacent bowties and insure that consecutive great-circle arcs are collinear. Folded edges are simply brought together, and four continuous great circles magically reappear, as if the original paper circles were still intact. Looking only at the finished model, it appears that we had to cleverly cut slits in the paper circles in just the right places to allow the four circles to pass through each other. The procedure is reminiscent of the magic trick in which a hankerchief is cut into many tiny pieces and thrown randomly into a hat, only to reappear intact.

      A spherical VE can be constructed through this simple folding exercise because of the specific interplay of its surface and central angles. Remarkable numerical cooperation is required to allow adjacent central angles to fold out of flat circular disks, while automatically generating correct surface angles. That four great circles will submit to this bowtie operation is not at all obvious from studying the whole pattern, and even less so in the case of other, far more complicated models.

Conservation of Angle

Physics tells us that a beam of light directed toward a mirror at some angle from the left will bounce away from the mirror making the same angle on the right. The angle of incidence is equal to the angle of reflection. Fuller points out that the same is true for the great-circle models, if we think of the paths as trajectories.

      Great circles maintain the illusion of being continuous bands, argues Fuller; however the bowtie procedure reveals the truth about these patterns, and in so doing illustrates an aspect of energy-event reality. A great-circle path may look continuous, but what really happens is that as soon as a trajectory (or great-circle arc) meets an obstacle, in the form of another great-circle event, they collide and the course of both trajectories is necessarily altered. Both paths are forced to bounce back, just like a ball bouncing off a wall.

      Here's the fun part. Because the intersection of two great circles provide two pairs of equal angles, their paths mimic the classical collision in physics. The same angular situation results from two overlapping great circles as from an idealized "energy-event" collision; these symmetrical patterns can therefore be thought of as the paths taken by billiard balls on the surface of a spherical pool table. If you didn't see the collisions, you might think that two balls went through the same point at the same time; however, their true paths are bowties.

      Imagine a great-circle wall constructed vertically out from the surface of a sphere. If a ball traveling parallel to the sphere's surface (describing another great circle) hit the wall and bounced back, it would bounce inwardly off that wall at the same angle that it would have traversed the great-circle line had the wall not been there..." (901.13; Fig. 14-15a). Adjacent bow-ties therefore produce collinear great-circle arcs, because the angle made by the paper disk "bouncing back" from a bobby-pin collision point is the same angle the circle would have made on the other side if the bobby pin had not been there. Fuller continues, "and it would bounce angularly off successively encountered walls in a similar-triangle manner...(901.13). Hence the completed bowties. The image of the colliding ball and great-circle wall makes it easier to understand why Fuller interpreted great circles as the trajectories of energy events, and explains why he was convinced that these models have physical significance. His bowtie analogy is consistent with the classical collision model.

Ball bouncing and 'Local holding pattern:' figure-eight loop
Fig. 14-15. (a) Ball bounces back from the wall at the same angle it would have made with the wall on the other side had the wall not been there. (b) "Local holding pattern:" figure-eight loop.
Click on thumbnail for larger image.

      Consider the paper model described above. Soon after an energy event meets its first collision, the new trajectory meets a second obstacle and its course is again altered. Conservation of angle determines the new heading once again, and the process repeats until, at the sixth collision, the great-circle path comes back upon itself, completing the bowtie loop (Fig. 14-15b).

      Successive arc segments of one energy event form figure-eight loops, or "local holding patterns" (455.05), which lie side by side and appear to be continuous great circles. Construction paper shows us how it works, and Fuller tells us that this is what happens in physical reality as well. Discrete energy events form local citcuits, just as discrete paper circles form bowties; they only look continuous.

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