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Degrees of Freedom
The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus effecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space?
Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors.
In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hubtypically about a half inch wide and 3 inches longmust be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any directionup, down, back or forthwithout budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b).
This experiment is quite rewarding to experiencewell worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4)
What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5)
The twelve vectors needed to restrain a body can also be omnidirectional, instead of the basically planar organization of the spoke wheel. Fuller takes us through a similar sequence in Synergetics, which starts with a ball attached to one string. The ball is free to swing around in every direction; the only restraint is on the radial "sweepout" distance. The ball's motion is thus free to describe a spherical domain. The addition of a second string restricts the ball to motion within a circular arc in a single plane (Fig. 7-11a). A third string allows the ball to swing only back and forth, in a linear path. The ball can always be pushed slightly out of place, no matter how taut the three strings (Fig. 7-11b). And just as, in our search for the minimum system, a fourth event suddenly created insideness and outsideness, by adding a fourth string to the ball, its position is suddenly fixed. ("Four-dimensionality" again.) In their most symmetrical array, the four strings go to the four vertices of an imaginary tetrahedron, and are therefore separated by approximately 109.47 degrees, the tetrahedron's central angle (Fig. 7-11c).
But of course that's not the end of the story; the ball is still free to twist in place. To prevent this slight rotation, three strings must be attached to each location of the original four (Fig. 7-11d). This result is related to the fact that three is the minimum number of coordinates needed to specify the location of a point in space, with reference to the origin of a coordinate system.
The whole picture is falling into place. Every physical body has four basic sides, or four comers: two points alone are only collinear, and three are only coplanar; not until there are four comers can the property of spatial existence be recognized. As a result, any physical body must be held at four noncoplanar points, with three restraints at each point, in order to be stabilized (Fig. 7-11d).
This result suddenly ties in to the earlier discussion of pattern integrity. Triangles are necessary for stability. Therefore, while the ball was seemingly held in place by four restraints, it could still rotate locally because the locus of each individual restraint could not be stable without triangulationsubsequently provided by the addition of three strings per locus.
Bucky explains the situation further. Consider the ball with the original four restraints. The strings impinging on the ball create four vertices without supplying the necessary six edges to stabilize their position with respect to each other. They essentially form an unstable quadrilateral rather than a stable tetrahedron. The lesson is the same. There are four fundamental corners in every system, and each must be triangulated: 4 × 3 = 12. Thus there are twelve degrees of freedom, tetrahedrally organized. Twelve is a frequently recurring number in synergetics, a fundamental part of space and geometry, as we shall see again and again.
The above procedure describes Fuller's own interpretation of "degrees of freedom," which must be distinguished from Loeb's analysis of the concept, as briefly explained in Chapter 5.
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