the "jitterbug"

R. Buckminster Fuller called "vector equilibrium" (VE) a set of 12 vectors in the space defined by the center and the 12 vertices of a cuboctahedron (it is the only spatial configuration in which the length the polyhedral edges is equal to that of the radial distance from its center of gravity to any vertex); the angles between each vector and its four "neighbors" are all 60° and the vectors are opposite by pairs. The VE is obviously related to the CCP (cubic close packing of spheres).
"Jitterbug" is the name given by Fuller to a transformation of the VE stick-model in which the 12 vertices move symmetrically. The jitterbug transforms smoothly a cuboctahedron into a regular octahedron with an intermediate icosahedral shape; thus it appears as a unifying motion between 4-fold (octahedral) and 5-fold (icosahedral) polyhedral symmetries.

the jitterbug motion

At first glance the motion of this fascinating object seems complex but it is quiet simple when you observe the motion of one of the eight triangles. It is a radial displacement combined with a rotation around the 3-fold symmetry axis of the triangle (a "screwing"). Two triangles move symmetrically along and around each of the four 3-fold symmetry axes of the octahedron.
Since the triangle does not change in size during the motion, its 3 vertices move on the surface of a cylinder, and symmetry considerations constrain each vertex of the jitterbug to remain in a plane. Thus two opposite vertices of the jitterbug move on symmetrical portions of an ellipse (there are 12/2=6 ellipses, each one of them is the intersection of two cylinders and goes through 4 vertices of the octahedron).

animation

Around the animated jitterbug are the four remarkable positions: cuboctahedron, octahedron, icosahedron and the less known dodecahedron.
The distance of the triangles from the center is given by  e×Sqrt[2/3]×Cos[a]  where e is the edge's length and a the angle of the rotation.
The values of a for the four positions are respectively 0°, 60°, around 22.24° and around 37.76° (Robert W. Gray, more in the references).

references: •  A Fuller Explanation  (chapter 11) by Amy C. Edmondson
•  "Jitterbug defines polyhedra" and "The Jitterbug Motion": web pages by Robert W. Gray
•  jitterbug applet by Bob Burkhardt  -  jitterbug animations by Adrian Rossiter

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French 		convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects April 2005
updated 30-04-2005