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Having settled on the most symmetrical and dense sphere packing, to faithfully present the characteristics of space, we are ready to explore, the shapes and "periodicities" observed by Fuller. The reliable precision of these patterns indicates that they are molded by space, not by imposed design.
What are these patterns that captured Fuller's attention so long ago (and kept it for fifty years)?
We start very simply with the phenomenon of "triangular numbers," a sequence of numbers in which each successive term is equal to the previous term plus the number of terms so far. These numbers can be generated by triangular collections of balls, arranged as in a rack of billiard balls. The total number of balls in each triangle, in a series of progressively larger groups, is a triangular number. Figure 8-8 shows the first five groups. The first member of the sequence is "1"; the second is obtained by adding two to the first, to get "3"; the third, by adding three to the other two, to get "6"; and so on. Each successive number is generated by the addition of a row with one more ball than the last. The sequence of numbers thus generated (1, 3, 6, 10, 15, 21, 28,...) is specified by (n ² - n)/2 for n = 2, 3, 4, ... . (n = 1 corresponds to 0, which is notstrictly speakingmodeled by a triangle).
The formula (n ² - n)/2 might appear to be a more difficult way to obtain these values (and certainly for small groups its easier just to count balls), but of course for the twentieth or even the ninth triangular number, it's far more direct to subtract 10 from 10 ² and divide that by 2 to get 45 than to draw rows and rows (nine rows) of circles! Mathematics presents a shortcutotherwise known as a generalized principle.
So far we have a numerical progression (n ² - n )/2 that happens to be modeled by triangular clusters. We might have chosen to discuss a variety of other sequences, for example, the numbers generated by n ²: 1,4,9, 16,25,..., which could be labeled "square numbers" because they are geometrically represented by square clusters. But we have a specific motivation for paying attention to triangular numbers, because of one especially significant characteristic: the n th term is the number of relationships between n items, for n = 1, 2, 3,4,.... For example suppose that for any given number of people, we wish to know how many telephone lines are required to link everyone to everyone else by a private line. The answer is the triangular number corresponding to a number of rows one less than the number of people. Two people require only one line; three require three; four require six (as portrayed in Figure 3-1), and n require (n ² - n )/2 private lines. It is no longer far-fetched to imagine applications for this formula. The real lesson from triangular numbers is that significant algebraic expressions, such as (n ² -n )/2, the number of relationships between n events, can be represented geometrically. That much established, we proceed to the next development.
We can stack triangular arrays of decreasing size, creating tetrahedral clusters. (We could thereby isolate a sequence of "tetrahedral numbers.") But let's go back to the very beginning.
One sphere alone is completely free to move, and closepacking is of course not an issue. Suppose we have two billiard balls tangent to each other. If the only requirement is that the two balls stay in contact, we shall observe that they are free to roll around each other's entire surfaces (Fig. 8-9a). We then introduce a third ball, allowing it to roll in any direction while touching at least one of the first two balls. It eventually rolls into the valley between the other two, establishing a naturally stable triangle (Fig. 8-9b). At this point we require all three balls to stay in contact, and discover that they are still able to roll, but only inward or outwar4 (toward or away from the triangle's center, in tandem) "like a rubber doughnut," to use Bucky's words. The freedom of motion of each sphere is thus considerably more limitedspinning about one specific horizontal axis instead of unrestrained motion in every direction.
A fourth ball rolls across the surfaces and lands comfortably in the triangular nest. Suddenly, all four balls are locked into place, unable to roll or move in any direction (Fig. 8-9c). This is the first stable arrangement, with the requisite minimum of four. The tetrahedron is once again at the root of our investigation.
At this point, it may be illuminating to construct some of these structures, for example with Styrofoam balls and toothpicks, or small plastic beads and glue. One particularly satisfying demonstration involves bringing four spheres together and trying to create a square. It is easy to feel how unstable that arrangement is: the balls gravitate naturally toward the tight tetrahedral cluster. Fuller placed considerable emphasis on the benefits of hands-on construction to gain thorough familiarity.
The theme of Fuller's tetrahedral sphere packings is the presence or absence of nuclei. The word "nucleus" evokes the image of a central ball spatially surrounded by other balls, which is exactly the way Fuller uses it for the VE packings. However, his observations about tetrahedral patterns are based on a somewhat different approach. Most of the inpenetrability of the sphere-packing sections in Synergetics can be removed with one simple clarification: a "nucleus" in VE packings is defined as a ball at the geometrical center of the whole cluster, whereas a "nucleus" in tetrahedral stacks is a ball at the exact center of an individual planar layer.
Start with the four-ball tetrahedron developed above, which consists of a fourth ball added to a triangle of three others. Next, the simple four-ball tetrahedron is placed on top of a flat six-ball triangular base, creating a tetrahedron with three balls per edge (Fig. 8-10). There are three layers, with ten balls altogethersix plus three plus one. Throughout his sphere-packing studies, Fuller uses the number of tangency points per edge (in other words, the number of spaces between spheres along an edge of the cluster, rather than the number of spheres) for the assignment of frequency. The four-ball tetrahedron is thus "one-frequency" (as is appropriate for the first case), and the next case, the ten-ball tetrahedron, is "two-frequency." We can visualize that each sphere-cluster polyhedron corresponds to a line drawing (or toothpick structure) in which the spheres' centers locate vertices which are interconnected by lines (or toothpick edges) through tangency points. Recalling that "frequency" is defined as the number of modular subdivisions, the justification for Fuller's frequency assignment is evident from this translation, because the number of subdivisions (or line segments) per edge corresponds to the number of spaces between spheres, rather than to the spheres themselves, which correspond directly to the vertices (Fig. 8-10).
Triangular clusters, each with one more row than the last, are stacked to create larger and larger tetrahedral packings. We place the ten-ball (two-frequency) tetrahedron on top of the next triangular base, which itself consists of ten balls, to get a three-frequency tetrahedron, with twenty spheres altogether (Fig. 8-10). We have thus begun a list of values that might be called tetrahedral numbers: 4, 10, 20, followed by 20 plus the additional triangular layer of 15, to total 35 (Fig. 8-11). The progression can be continued indefinitely.
Fuller calls our attention to this periodicity (or pattern) within the system: starting from the top, the pattern is Y-N-N-Y-N-N-Y...(notice that "N" can stand for "nest" as well as for "no"):
415.55 Nucleus and Nestable Configurations in Tetrahedra: In any number of successive planar layers of tetrahedrally organized sphere packings, every third triangular layer has a sphere at its centroid (nucleus.)
Fuller's yes-no-no pattern describes the presence of nuclei in certain layers of a pyramid; he does not ask which pyramids of gradually increasing frequency contain an overall nucleus (at the center of gravity.) This is a subject open for further exploration, which we leave for the time being as we continue to explore Fuller's material.
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