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Background: Closepacking
Packing spheres together with a minimum of interstitial space is a problem that still presents a challenge to mathematicians. In the same way, there is only one closest-packing arrangement of spheres in the plane: each sphere must be in contact with six others (Fig. 8-3). A second identical layer can be placed on top of the first, with its spheres all landing in nests created by three neighbors on the first layer. To achieve our goal of packing spheres as close together as possible, we have thus far had no choice as to the next step. A third layer however can be superimposed on the second in one of two different ways to maintain maximum density. The spheres of the third layer can either be placed directly above the
Instead of trying to imagine indefinitely large planar expanses, we focus on a small portion of the closepacking—the arrangement surrounding a single sphere. We start with one ball on a table; six others closepack around the first, and find themselves exactly tangent to each other. As with the pennies, there is no choice as to the number of spheres in the cluster. The flat hexagon of spheres creates six separate "nests" of three spheres each, as seen in Figure 8-3, but additional spheres of the same size, sitting in any one of the six nests, partially block adjacent nests. As a result, there is only room for a ball in every other nest, allowing a total of three nesting spheres on the hexagonal cluster. These three balls (by magic or else by inherent spatial properties) rest exactly tangent to each other, a perfect equilateral triangle (Fig. 8-5a, top). The arrangement has neither leftover space nor crowding; all three spheres on top are tightly shoved into nests, and—because the planar group (six around one) are clearly as close together as possible—all ten spheres are convincingly closest-packed.
We can then flip the whole package over and repeat the procedure on the other side. We again have two choices: the second three-ball addition can be placed either directly above the three on the bottom (meaning both the top and bottom triangles are pointing the same way), or it can be oriented the opposite way (Fig. 8-5a, b). The former choice (hexagonal) outlines the vertices of a polyhedron in which the squares are adjacent to other squares (in three pairs meeting at the "equator"), as are six of the eight triangles (Fig. 8-5a). The latter choice (cubic) consistently alternates triangles and squares, so that squares are entirely framed by neighboring triangles—and triangles by squares (Fig. 8-5b). The latter arrangement-"our friend the vector equilibrium" as Bucky says—is the more symmetrical of the two choices, and is therefore used as the basis of Fuller's subsequent sphere-packing studies. |

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