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Amy C. Edmondson
A Fuller Explanation
Chapter 8, Tales Told by the Spheres: Closest Packing
pages 120 through 121

Further Discoveries: Nests

Throughout our investigation, we note the recurrence of a limited inventory of polyhedral shapes. This is perhaps revealed most dramatically by the behavior of closepacked spheres. A striking example is found in the results of an extra sphere placed in the central nest of flat triangular clusters. The first case is already quite familiar: a sphere placed in the minimum triangle of three spheres produces a regular tetrahedron, assuming all four spheres are the same size. A half octahedron is born out of a sphere nesting in a square group of four spheres, and we might reproduce any number of familiar shapes by putting equiradius balls together, but here we confine ourselves to the triangular clusters, for the scientific method tells us that the strength of an experiment often rests in drawing boundaries. Results thus obtained may lead to broader generalizations about related questions.

      Remember our observation that every "N," in Fuller's Y-N-N pattern, represents a layer with a nest, since any triangular cluster without a central ball has a central nest. The center of a ball added to the first nest becomes the fourth vertex of a regular tetrahedron; the next group, six balls, also has a central nest, so a seventh ball is put in the space. Magically, the shape thus created is a significant one: the semisymmetrical tetrahedral pyramid, which is exactly one-eighth of the regular octahedron (Fig. 8-16). To understand what is meant by "one-eighth" of an octahedron, imagine that we slice a regular octahedron made out of "firm cheese," to use one of Bucky's images, in half—creating two square-based pyramids (Bgyptian style). Then, cut both halves into quarters, to get eight octants, each with an equilateral triangle base and three isosceles-triangle side faces (45° -45° -90°). The octahedral central angle is 90 degrees, a fact which will prove especially significant in the next chapter.

      The seventh ball is simply placed in the nest of the six-ball triangular layer to complete the four vertices of an "octant." Nothing in our sphere-packing investigation thus far would lead us to predict the appearance of this important shape, which is already a significant part of our polyhedral inventory. (Such unpredictability is getting to be a pattern.) So we proceed to the next case.

Shallow pyramid of fifteen balls in a plane and one more at the center

Fig. 8-16
Click on thumbnail for larger image.

      The ten-ball triangle has a nucleus (1: yes; 3: no; 6: no; 10: yes;...) and therefore no nest. So we skip to the fifteen-ball cluster and, as before, drop a sixteenth ball in the central nest. We are no longer surprised to discover that the resulting shallow pyramid is also a very special shape: one-quarter of a regular tetrahedron—a portion encompassing the volume from the tetrahedron's center of gravity out to any one of its four faces (Fig. 8-16). The tetrahedron's central angle, 109.471 degrees (109° 28' 16"), seems so irregular that the sense of coincidence is underlined.

      With fifteen balls in the plane and a sixteenth in the center, this pyramid is quite shallow—and in fact, as a section of the minimum system, it is Fuller's terminal case. For each of the first five triangular numbers without nuclei, a sphere placed in the central nest forms an important shape in the VE-octet framework, an idea that we shall explore in greater detail in the next chapter. We thus have come to the end of this particular experiment, with the conclusion that closepacked spheres automatically yield many significant geometric shapes.

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