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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...