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The isotropic vector matrix may be described as a regular distribution of VE's and octahedra which combine to fill all-space. Jitterbugging into and out of this ground state, the matrix seems to reach maximum disequilibrium (i.e. maximum expansion) when the contracting VE's and expanding octahedra both describe regular icosahedra. However, because regular icosahedra do not combine to fill all space, the contracting VE's and expanding octahedra cannot pass through their icosahedral phases simultaneously. In fact, at maximum expansion they both describe an irregular icosahedron which does combine, along with a complementary irregular tetrahedron, to fill all space.  The all-space filling irregular icosahedron is only 1.5 percent larger than the non-all-space filling regular icosahedron, an enticingly small difference. Eight of its twenty faces remain equilateral triangles (all angles 60°), while the remaining twelve, corresponding to six open square faces of the VE, are iso...

The Isotrophic Vector Matrix can be modeled as: vectors connecting the centers of closest-packed spheres; all-space-filling polyhedra; and as the space between the voids left by closest packed spheres. The space between closest packed spheres is a continuous web of concave octahedra and concave VEs.

If the jitterbug were modeled as a sphere collapsing to a concave octahedron, the counterclockwise-rotating triangles would transform from a convex spherical triangle to a flat concave triangle, while the clockwise-rotating triangles would simply invert themselves, exposing the concave surface of the same spherical triangle. This may be difficult to animate in SketchUp. But that's not what happens. The spheres transform into concave VEs, the concave VEs expand to spheres (or convex VEs), and the concave octahedra simply turn themselves inside out. I'm working on an illustration of this.

In the context of the isotropic vector matrix, the outer edges of the VE are doubled along with those of the eight tetrahedra which surround it. These extra edges my be recruited to serve as radial vectors, restoring the nucleus and halting the jitterbug at vector-equilibrium. However, because there are 24 edges and only 12 radial vectors, the radial vectors are now doubled, and can be be unfolded in the transformation illustrated below. The transformation is the same as would be achieved if each of the eighth tetrahedra comprising the VE were to turn themselves inside-out, pointing outward from, rather than sharing a common vertex at its geometric center.

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-rad...

I used the programming language, Ruby, and SketchUp Make 2017 ® to create the following animations demonstrating the geometrical transformation known to its discoverer, Richard Buckminster Fuller, as the Jitterbug . The jitterbug transformation is typically modeled as hinged vectors and rotating triangles. But as the vectors are meant to model the joining of sphere centers, I've here replaced the vertices with spheres. The doubling up of edges at the octahedron phase of the jitterbug is here represented by merging and diverging spheres. Different views and rotations reveal some interesting characteristics of the transformation. If the view rotation is synchronized with the rotation of the top and bottom triangles, the transformation resembles a pump with spheres orbiting around its equator. A different angle and without rotation, the spheres merge and diverge at right angles. Finally, the conventional view of the jitterbug, with its synchronously rotating triangles replaced...