Single Comment

CECV doesn't suggest whether the word Xyrixa has some meaning.  Since we
now know that this geometry can be viewed as the rhombic dodecahedron
tiling, perhaps a name based on that is appropriate.  A clever name
lacking, I'll use RD for the rest of this message.

On RD geometry: it's been noted that it can be viewed as the standard 3D
cubic tiling, but using the cubic diagonal as the RD orthogonal (and
restricting to one color of the cubic 2-coloring, say black).  Charles's
RD bishop direction seems to correspond to the cubic dabbabah.  Perhaps
another interesting direction is the cubic triagonal, which must skip over
one white cell.  This direction corresponds to passing through a degree-3
vertex of the RD.  (The RD orthogonal passes through the faces, and
Charles's RD bishop direction passes through the degree-4 vertices.)  I'm
not quite sure, but I think passing through the edges of the RDs would
correspond to a (2,1,1)-leap in the cubic tiling.  I haven't thought yet
about: do these directions have meaning in the hex-plane parts of RD
geometry?