Scott, John Conway and I studied that question and several variants in the following paper: *Low-Dimensional Lattices V: Integral Coordinates for Integral Lattices <http://neilsloane.com/doc/Me151.pdf>*, J. H. Conway and N. J. A. Sloane, *Proc. Royal Soc. London, Series A*, 426 (1989), pp. 211-232. - available from http://neilsloane.com/doc/pub.html, item 151 Actually several other papers in this sequence are relevant to this math-fun thread - esp. Part II (item 146) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Jan 14, 2016 at 6:45 PM, Scott Huddleston < c.scott.huddleston@gmail.com> wrote:
What's the largest d required for an N-dimensional integer lattice to be a sublattice of Z^(N+d)? I believe E6 (N=6) requires d=2.
On Thu, Jan 14, 2016 at 3:21 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
x+y+z = 0 in Z^3 , perhaps?
WFL
On 1/14/16, Warren D Smith <warren.wds@gmail.com> wrote:
also, another issue is, you can have, say, a 12-dimensional sublattice (like K12) of Z^24. In other words it's integer, and it's N-dimensional, but the coordinates are only integer in some higher dimension D>N on which we restrict to some N-dimensional subspace.
.
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