Sometimes I get a request for a canonical lesson plan, one that will capture the flavor and style of Synergetics, by which they mean the Bucky stuff.
What I'm coming to on that score is the four random walkers starting from the same lamp post in the CCP (=IVM), and wandering for t time cycles.
The four randomly arrived at balls define the corners of a tetrahedron which, upon having its six edge lengths get run through our volume computer, will turn out to always have a whole number volume. In tetravolumes, that is. Four CCP balls define our D-for-diameter-edged tetrahedron of volume one.
In order to calculate the random walks, we use Quadrays as syntactic sugar. The IVM ball packing is their sweet spot, which is why they're "IVM coordinates" by some accounts (including mine), in contrast to XYZ.
In order to calculate the sixth edge lengths, we simply perform vector subtraction between adjacent corners. Quadrays have essentially the same vector algebra as XYZ when it comes to adding, subtracting, and scaling.
Finally, in order to calculate the tetrahedron's volume, we use Gerald de Jong's formula, even though he has lost his derivation. There's no denying it works well.
Six edge lengths go in, fanning out from any apex and circuiting the opposite base, and the tetravolume comes out, natively, with no need for a modifying constant.
The corresponding XYZ volume is computed accordingly, as IVM volume times 1/S3 (S3 being the Synergetics Constant for converting volumes).
In sum, we needed to learn what the IVM was, and to visualize movement within it as a process of hopping in one of twelve directions, by distance D, at each turn to play. Then we needed to absorb the concept of tetravolumes.
Getting whole number tetravolumes for the tetrahedra helps shock us into a mindset that might be open to the concentric hierarchy, wherein those rhombic dodecahedral cells around each sphere, each have a volume of six.
All of the above, along with figurate and polyhedral numbers more generally, including kissing point counts, form our IVM-XYZ bridge over troubled waters, the C.P. Snow chasm.
