
Mapping crystal structures in a 4 IVM frameworkThe isotropic vector matrix defines tetrahedral and octahedral voids, which may in turn be occupied by alternate IVMs. The tetrahedral voids define two alternate IVMs, and the octahedral voids a complement to the currently active one  for a total of 4 IVMs in all. By selecting combinations of these interpenetrating IVMs, we define some of the lattices typically encountered by crystallographers. 

The rhombic dodecahedron provides a useful reference system for anchoring all four IVMs to a single shape.[1] The short diagonals of the rhombic faces terminate in tetrahedral voids, while the long diagonals terminate in the octahedral voids. In this paper, I am using Russell KasmanChu's nomenclature, identifying the 4 IVMs as compementary pairs: A and B, C and D.[2] These are shown in the animated GIF at right. Each pair of IVMs is equivalent to the simple cubic lattice, and these two lattices interpenetrate to define the bodycentered cubic lattice. 

In the graphic below we see IVMs C + D combining to form a simple cubic lattice.
Note that the IVMs serve to position the lattice points, at which point the specific chemistries will determine the actual bonding configurations of the atoms in question. The purpose of the 4 IVMs framework is to suggest a convenient schema for categorizing various commonly encountered lattices. See KasmanChu's paper for more details. The bodycentered cubic lattice (bcc) is derived from 2 interpenetrating simple cubic lattices i.e. starting with (C+D) as shown above, we add IVMs (A+B): VRML views [1] Fuller uses this approach in Synergetics in his discussions of Coupler axes interconnecting IVM sphere centers and interspheric voids ( 954.40, 954.70). [2] The ccp and hcp family of structures derived from interpenetrating lattices by Russell KasmanChu For further reading
