I think I'd already come across something like this, for 3-space onto 2-space anyway, when wondering how wangle projective transformations from a Euclidean geometric algebra. I'm trying to leave it to DW to investigate, though I must admit to not quite understanding NJAS here --- rotations are compact, so if you can project arbitrarily close to the lattice, why can't you hit it exactly? Incidentally, Veit Elser points out that the 6-cube projects into a rhombic triacontahedron, not an icosahedron as I earlier claimed (groan!). Fred Lunnon On 7/24/11, N. J. A. Sloane <njas@research.att.com> wrote:

Fred, Did you see A Note on Projecting the Cubic Lattice on my home page, about 6 items from the top? Also arXiv: 1004.3072

We show that given any (n-1)-dimensional lattice L, there is a vector v in Z^n such that the projection of Z^n onto v^perp is arbitrarily close to L.

Not quite your problem, but in the same spirit.

Best regards Neil

Home page: http://www.research.att.com/~njas/