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Amy C. Edmondson
A Fuller Explanation
Chapter 8, Tales Told by the Spheres: Closest Packing
pages 100 through 102


Tales Told by the Spheres:
Closest Packing

Much has been written over the years by mathematicians and scientists about the problem of "closepacking" equiradius spheres. It's not a subject that the rest of humanity has tended to get excited about; however, the orderly patterns revealed by these packings are unexpectedly fascinating. Closepacking equiradius spheres might at first sound like the type of abstract mathematical game Fuller railed against; after all, there's no such thing as a sphere. But if nature exhibits no examples of pure spheres—that is, no perfectly continuous surfaces equidistant from one center—we can still discuss the concept of a spherical domain. Imagine various approximations of the model, such as a soap bubble or, less fragile, a Ping Pong ball. The concept of multiple equiradius spheres turns out to be quite useful, providing a superb tool with which to investigate the properties of space. Let's look into some of the reasons why.

Equilibrium: Equalization of Distances

The connection to equilibrium is perhaps the most important reason to experiment with sphere packing. A sphere is defined as the locus of all points at a given distance from a central point; consequently, in an array of tangent spheres, their centers will be separated by a uniform distance. The configuration developed in the previous chapter to represent vector equilibrium—requiring equal lengths in all directions—can be created quite simply with the aid of this model. If one sphere is completely surrounded by a number of spheres of the same size, the distances between the internal sphere's center and the centers of all surrounding spheres are necessarily the same as the distance between the centers of adjacent external spheres, provided all spheres are in contact with each other. The resulting cluster is shown in Figure 8-1, along with a cross-section of the packing to illustrate that the distances are the same. Closepacked spheres automatically set up an array of evenly spaced points.' Equal distances represent balanced forces: ergo, equilibrium.

Closed packing of spheres creates an array of equally spaced points in space
Fig. 8-1
Click on thumbnail for larger image.

Symmetry versus Specificity of Form

A sphere is the form of "omnisymmetry" in spatial reality. Symmetry describes the degree to which a system can be rearranged without detectable change. The sphere's shape presents no corners, no angles—in short, no landmarks—by which to detect rotation or reflection. Its very shapelessness enables us to explore the shape of space. Furthermore, the total absence of angular form makes the precisely sculpted shapes generated by packing the identical "shapeless" units together all the more surprising. It is easy to see that individual spheres, as omnisymmetrical forms with neither surface angles nor specific facets to mold the form of clusters, cannot determine through their own shape the overall shape of packings. In conclusion, we are not so much interested in the ("nondemonstrable") spheres themselves, as in using sphere-packing as a medium through which spatial constraints can take visible shape.

Organization of Identical Units

Finally, the standard model of an atom is spherical: packets of energy are spinning so rapidly about a tiny nucleus that the atom can be considered occupying a spherical domain. (In fact, the orbit of any object spinning in all directions defines a sphere.) We can therefore pack spheres together in the hope of learning about atomic and molecular aggregations. To state the problem more generally, the organization of identical units is an important theme in biology and chemistry. All sorts of units—such as atoms, molecules, cells, DNA nucleotides—must be organized to function cooperatively in structures far more complex than the individual units themselves. Spatial constraints are responsible for much of the superb organization of biological phenomena—allowing and encouraging certain configurations while prohibiting others. Yet, despite the influential role of space, scientific thought does not as a matter of course take this into consideration.

      Sphere-packing can be thought of as a method of blindly gathering evidence; we experiment with these identical units without knowing the outcomes, and space enters in to direct traffic. The resulting configurations are absolutely reliable. We are thereby able to observe the shape of space, manifesting itself through the innocent spheres.

New Level of Focus

Despite our discipline of viewing whole systems, we have reached a point at which we must zoom in to look closely at certain details of Fuller's Synergetics. The sections called "Closest Packing of Spheres" contain some of the most difficult passages in his book, rendering Fuller's observations inaccessible without considerable perseverance. Not only is the description hard to follow, but these patterns seem to elude application. It is therefore especially important to understand the logic behind Fuller's use of sphere-packing in an investigation of nature's coordinate system. Otherwise, it will be difficult to see how these details fit back into the big picture. Even though the immediate goal of this text is to clarify the configurations described by Fuller, a list of results, no matter how clear, is not likely to be interesting unless the premise behind the search is understood. At this point, the reader may even have thought of further reasons to add to the ones stated above, for there are many dimensions of this issue. However, as it probably remains difficult to predict or visualize the patterns themselves, we bravely proceed.

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