PCAPPC

The rhombic dodecahedron is basically a cube turned inside out and has been shown mathematically to be a projection of the tesseract (the four-dimensional cube) into three-dimensional space. It is, like the cube, an all-space-filling polyhedron and directly corresponds to the closest packing of spheres, enclosing the sphere and the space surrounding it. Rotations of the rhombic dodecahedra demonstrate its projection of the tesseract and show the potential of the isotropic vector matrix as a model of four-dimensional space.

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.