The permutations of quadray (2,1,1,0) are similar to my system. Subtract one from each coordinate and you get (1,0,0,-1) and the twelve unique permutations are the vertices of a cuboctahedron in my system. But the permutations of (1,0,0,0) in quadrays are not the centers of closest packed spheres. They are something that requires fractions in my system. The idea is to use integers (mod a prime), and get only the centers of closest packed spheres.
By the way, is arithmetic remainder (as well as modulo) a prime a field?
Bucky wrote that the tetrahedron's four dimensions refer to the distances from the midpoint of the tetrahedron to the centers of the four faces. The advantage of that is to represent any size (positive or negative) tetrahedron anywhere with four coordinates. There is a way to transform from my system to quadrays just as there is a way to transform to XYZ. (or if the four coordinates do not sum to zero, to XYZT).
Cliff Nelson <A HREF="http://forum.swarthmore.edu/epigone/geometry-research/brydilyum">RBF electronic notebooks by Clifford J. Nelson