R Hansen says: >Thus, it isn't a mapping of volume like the cube.
Kirby is right, the holes are octagons that can be subdivided into (non regular) tetrahedrons.
But your word choice (mapping of volume) resonates with my comment about coordinates. Looking at something basic like the wikipedia page on tetrahedrons shows all the math there is based on rectilinear coordinates. I would guess that generally the rule for the various programs that compute tet meshes and what-not, but that's just a guess. When the rubber hits the road (computing) the little cubes win.
I don't really see this as cubic space *versus* tet space, its both...and... If you are interested in tensegrity structures (a B.Fuller coined word, which Google spell check is not recognizing!) tetrahedrons are obviously going to be important, and all these facts Kirby brings up will be relevant. In other pursuits too, I'm sure.
But it seems the underlying math will be generally be what we are all used to, all that conventional stuff. Or, prove me wrong, I'm no expert here, just a guess. But that's why I said I don't see this as a whole "branch" of mathematics, the way Kirby put it.
Assigning unit volume = 1 unit sided tetrahedron may make some calculations easier, so why not?