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Amy C. Edmondson
A Fuller Explanation
Chapter 8, Tales Told by the Spheres: Closest Packing
pages 121 through 125


The essence of precession, to Fuller, is 90°. And indeed, the counter-intuitive or mysterious thing about the behavior of gyroscopes (and other examples of precession in physics) is the resultant motion in a direction ninety degrees away from that of an applied force. For example if a downward force is imposed at the north point of a gyroscope spinning clockwise, it will tilt toward the east: 90 degrees away from the direction one intuitively expects. Fuller's "interprecessing" involves two systems "precessing" together, which he uses to mean oriented at 90 degrees with respect to each other. In his sphere-packing studies, "interprecessing" reveals subtle facets of symmetry which might otherwise go unnoticed.

Interprecessing of two pair of spheres in closest packing creates a tetrahedron

Fig. 8-17
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      We start with the simplest case: two identical pairs of tangent spheres, parallel to each other and separated by some distance. Rotate one of the two-ball sets 90 degrees, and then move the two pairs toward each other until they meet in the middle, so that the midpoints, or tangency poffits, of the two pairs are as close together as possible. (Notice that without the 90-degree twist, the result of bringing the parallel pairs together would be a square-unstable and not closepacked.) What is the result of this simplest case of interprecessing? A tetrahedron, of course (Fig. 8-17).

      In retrospect, the answer appears obvious, for the initial condition of four spheres-the necessary ingredients of a tetrahedron-gives it away. However, the experiment highlights the 90-degree symmetry of the tetrahedron, which is otherwise obscured by thepredominance of triangles and 60-degree angles. Rather than elaborating on the tetrahedron's right-angle symmetry here, we shall allow subsequent demonstrations to further illustrate this orthogonal characteristic. (See especially Chapters 9 and 10.)

Tetrahedron as chef's hat
Fig. 8-18a
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      Take two identical sets of sixty spheres, closepacked as shown in Figure 8-18a. Their irregular trapezoidal shape eludes immediate identification. That they do not seem to be a part of our familiar group of shapes is confirmed by numerous experiments in which participants are given these two pieces and asked to put them together in some way that seems correct. Countless false moves involve bringing similar faces directly toward each other, and again and again, the identical halves are put together in unsatisfying and incorrect ways. The correct solution is rarely discovered by the uninitiated-but once seen is unmistakable. This problem (simple, after the fact) is initially challenging because it is so hard to get beyond the natural assumption that the two halves must approach each other directly-as if one half were approaching its own reflection in a mirror. What actually has to occur of course is that one half rotates 90 degrees with respect to the other (interprecessing) and the two rectangles mesh together perfectly, at right angles. "Wow!" Bucky would exclaim, apparently as surprised as his audience. The surprise is genuine in a sense, for the result is visually striking even if one already knows the answer: a perfect tetrahedron, eight balls per edge, or seven-frequency (Fig. 8-18b.)

Chef's hat back to tetrahedron
Fig 8-18b
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      Along the same lines, we now look at 60-degree twists. An especially pleasing example involves two simplest triangles, of three spheres each. The triangles face each other directly; then one rotates 60 degrees before pushing them together, and the result is an octahedron. The six spheres are precisely situated as octahedron vertices, framing eight triangles. (Refer back to Fig. 8-7.)

Two eighth octahedron create minimum stable cube
Fig. 8-19
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      Next, let's take two eighth-octahedron seven-ball sets (the six-ball triangle with a seventh ball in the central nest). The two triangular bases of each cluster face each other, and then one is rotated 60 degrees, allowing the triangles to come together as a six-pointed star, and suddenly the fourteen balls become a cube! This is the minimum stable cube formed out of spheres (Fig. 8-19). Eight spheres alone, positioned as the eight corners of the cube, are not closepacked, and that configuration would therefore be unstable, as the spheres have a tendency to roll into the unoccupied valleys.

      In fact, just as a floppy toothpick cube needed six extra diagonal sticks (the six edges of a tetrahedron) to stabilize the square faces, eight balls also require an additional six, to complete a stable cube. Thus we have fourteen balls altogether: a parallel to the fourteen topological parameters (vertices plus edges plus faces) of the tetrahedron. The cube in every stable form seems to be based on an implied tetrahedron.

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