Analysis of the Jitterbug Transformation

By Greg -

The Jitterbug transformation can be reduced to the rotation of a triangle inscribed in a cube. If the rotation is constant around the vector described by the cubic diagonal, and the triangle's vertices are constrained to follow the x, y, and z planes defined by the cube, the triangle will move along the vector in an inverse-square relationship to its rotation. Consequently, the jitterbug seems to "bounce" between phases.


Eight triangles, four rotating clockwise and four rotating counter-clockwise, comprise the jitterbug which collapses into a regular octahedron with each "bounce."


The three phases of the jitterbug transformation are associated with three polyhedra. The first is the cuboctahedron, or truncated cube, which Richard Buckminster Fuller called "the vector equilibrium." All vectors, both the edge vectors and radial vectors, are the same length, and may thus be thought of as the vector representation of the closest-packing of equi-radius spheres around a nucleus. The shape may also be conceived as eight regular tetrahedra edge-bonded around a common vertex, as shown in the following illustration.


When the radial vectors are removed, the shape collapses symetrically to form an octahedron.

Midway between the vector-equilibrium and octahedron phases, the jitterbug describes an icosahedron.


Perhaps counter-intuitively, it is at the icosahedron phase that the jitterbug occupies the most space.

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