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Synergetics can be described as dynamic geometry. Its treatment of polyhedra as vector diagrams and emphasis on the changes and transformations in systems distinguishes Fuller's work from the traditional geometric approach. His conviction that mathematics ought to supply dynamic modelsin recognition of dynamic Universeled to a number of interesting discoveries. "Jitterbug" is the most striking example.
Twelve equiradius spheres pack tightly around one, as noted earlier, and if the nuclear sphere is removed the other twelve can shift slightly inward. A vector equilibrium thereby contracts into the triangulated icosahedron. What would this transformation from Chapter 8 look like in terms of vectors?
Following the example set by the transition from closepacked spheres to the isotropic vector matrix, we replace spheres with vectors. Twenty-four wooden dowels and twelve four-way rubber connectors are put together to make a vector equilibrium (Fig. 11-la). The model consists of eight triangles and six squares flexibly hinged together, and intentionally lacks the VE radial vectors, which would correspond to the now missing nuclear sphere. Therefore, since square windows collapse, we are back to the issue of stability.
In Chapter 5, we asked how many additional sticks were needed to stabilize each unstable system. For the VE, the answer was simple: six, one to brace each unstable square window. This time around we take a more open-minded approach to unstable systems and see where it leads. Suppose we don't stabilize the VE?
Instability enables motion. But what kind of motion? This new outlook inspires us to explore the ways in which flexible vector models change. The discoveries are remarkably satisfying (and the procedures are somewhat playful). In the case of the VE, the result is an elegant dance of symmetry that Bucky called the "jitterbug."
"Jitterbug" describes a transformation of the stick-model VE, in which all twelve vertices move toward the system's center at the same rate. The advantages of having an actual model on hand at this point are greater for the jitterbug than for any of the previous concepts. A verbal account of the transformation, no matter how precise, is inadequate; likewise for drawings. The wonder of the jitterbug lies in its motionfrom the unique equilibrium arrangement through various disorderly stages and on to new order. Jitterbug models tend to capture the attention of the most disinterested bystander; their dance is fascinating to watch. So find twenty-four sticks and twelve connectors and put them together; the model is guaranteed to intrigue.
Folding a Polyhedron
The model fascinates because so much seems to be happening at once. The vector-equilibrium starting point looks simple enough. Unstable if left to its own devices, the VE must be deliberately held openwith one triangle flat against a table top and two hands holding the opposite (top) triangle. Notice that the two triangles point in opposite directions, together forming a six-pointed star if you peer into the system from above (Fig. 11-1a).
Now, simply lower the top triangle toward its opposite triangle (i.e., toward the table) without allowing either one to rotate. The first surprise is that the equator seems to be twisting, despite your careful avoidance of rotation. If you pull the triangle back out and try it again, you will see that this equatorial twist can go in either direction; in fact the system can oscillate back and forth, going through the zero point, or equilibrium, every time.
Secondly, although you are only pushing on one direction (forcing the triangle toward the table) the entire system contracts symmetricallylike a round balloon slowly deflating. Apparently, the effects of your unidirectional force are omnidirectional. You can see and feel that although you push and pull along a single line the contraction and expansion are both uniformly spherical.
Focus on the square windows, because only squares can change; triangles hold their shape. As the top triangle approaches the bottom triangle, each square compresses slightly, becoming a fat diamond, and the dance has begun. The radius of the system is now slightly shorter than the length of its twenty-four edges (vector equilibrium no longer). The dance continues as the top triangle approaches the bottom triangle and the diamonds grow slightly narrower, reaching the point at which their width is exactly equal to the edge length (Fig. 11-lb). The VE has thus turned into an icosahedronat least the shape of an icosahedronbut we don't stop to add the six extra sticks across the diamond windows to complete the picture, for that would turn our jitterbug into a stable structure.
Instead, keep going: past the icosahedral stage, the diamonds, ever thinner, soon become narrow slits and finally snap shut (Fig. 11-1c, d). The dance comes to a halt in the form of an octahedron. Twenty-four sticks have come together into twelve pairs, creating a double-edge octahedron (Fig. 11-1d).
This contraction is continuous, and so there are countless slightly different stages, but only three are significant geometric shapes. The jitterbug is of interest primarily because of its surprising flow from one polyhedron into another; the emphasis is on its motion. However, we do want to be familiar with its geometric check-points.
First, the vector equilibrium. Unit-vector edges are balanced by unit-length radii. When the system contracts to the icosahedral position, the distances between each vertex and its five neighboring vertices are suddenly the sameunlike the VE, in which each vertex has only four nearest neighbors, each one the unit distance away, while two additional neighbors are approximately 1.414 units away, i.e., the length of the square's diagonal. To accommodate this surface equivalence, the radius must decrease from 1.0 to 0.9511. Nature will not compromise on these numbers: for all neighboring vertices to be separated by equal lengths, the radius must be shorter than that length. A perfect static balance is impossible; hence the dynamic, eternally fluctuating events of Universe.
The twelve vertices continue their inward journey until they land at the six corners of an octahedron; sets of dowels clamp together, grouping the twelve vertices into six pairs (Fig. 11-1d). The radius has decreased to 0.7071 times its original length, and that's the end.
But wait! There's another twist left in the jitterbug. Hold on to that top triangle, which has been so carefully kept from rotating until now, and deliberately start to twist it. (The triangle will only yield in one direction, depending on the direction of the jitterbug's initial twist.) If you turn it far enough (180 degrees) the entire system collapses into a flat two-frequency triangle spread out on the table (Fig. 11-2a). Then, fold in the three corner triangles, like petals of a flower, bringing their edges together to create the fourth and final stage: the minimum system of Universe (Fig. 11-2b).
It's a dense tetrahedron, with four parallel sticks for each of its six edges, three converged vertices at each of its four corners, and a radius of 0.6124. "Quadrivalent," says Bucky, and hence full of explosive potentialready to spring back out into "our friend the vector equilibrium."
This folded model also demonstrates the tetrahedron "turning itself inside out," for the three petals can be opened and flattened out again, and then folded back in the opposite direction, creating the mirror-image or "negative" tetrahedron. Fuller then reminds us that "unity is plural and at minimum two" and every system has an invisible negative counterpart. "Negative Universe is the complementary but invisible Universe" (351.00). Such digressions are unavoidable; for Fuller the implications of a model are always multifaceted, one observation plunging into another, layers upon layers, intertwined.
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