Read here about the flexible cuboctahedron that Fuller called the jitterbug, and how it relates to tensegrity studies.
Jitterbug is the name of flexible cuboctahedron that is amenable to specific manipulations that instantiate rotational symmetries and formation of other polyhedra including the icosahedron, octahedron and tetrahedron.
The jitterbug transformation was discovered by Fuller and named, by him, after a 1920's dance.
The Jitterbug as a Tensegrity
Gerald de Jong worked with Richard Hawkins to express Fuller's Jitterbug transformation in terms of a tensegrity. de Jong was motivated to create this model by some aspects of the classic Jitterbug that troubled him. The Jitterbug was obviously an outline of a Vector Equilibrium, but since that shape required radial vectors and a nucleus, both lacking in the jitterbug. Also, the Jitterbug consisted of eight conventional triangles, while tensegrity triangles seemed more in keeping with Fuller's vision of isolated tension and compression in structure.
de Jong's solution shows a nuclear vertex in the Vector Equilibrium phase of the jitterbug, and ends as Fuller's jitterbug ends, folded into an octahedron.
It begins with a tensegrity tetrahedron, with its six compression members and twenty four tension members. Each of the four faces are ringed with tension lines, and each of the tetrahedron's points also has a triangle holding it together. In the tetrahedron phase, the tension and compression member are indistinguishable, but as the transformation progresses, they separate.
If each of the struts are rotated in the same direction on axes that go through the tetrahedron's center, the entire tetrahedron can transform into its brother tetrahedron, with points being replaced by faces and faces being replaced by points. To illustrate this, de jong built what he calleed a "tensejit clock" image in the 3DV.EXE DOS viewing program, where 12 o'clock is one tetrahedron and 6 o'clock is its twin. The two ways of transforming appear on the paths via 3 o'clock and 9 o'clock. This jitterbug can tranform without stretching its tension members if the compression members are allowed to compress and re-expand. Not only that, but such a constant-tension-member jitterbug can function in a allspace-filling array, as it transforms one tetrahedron to the other within a bounding cube.
Hawkins and de jong created animations of eight of these bounding cubes joined into a big cube with the tetrahedra arranged so that their edges make up a nucleated Vector Equilibrium structure. As the tensegrity jitterbug transforms in each of the sub-cubes, the compression members eventually become parallel to each other and move further to create the alternate form, which is a central octahedron with a tetrahedron on each face - the 'duotet'.
The animations are hosted here: http://www.newciv.org/Synergetic_Geometry/tjbug.htm
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