Generating vectors of the face-centered cubic lattice

Question

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by angles (I think it would take three angles to completely describe it but I'm not sure on this either.)

Answer 1

Here you go. A3 or D3, called face centered cubic

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html

Note that an extra dimension is required to avoid square roots in the basis coordinates. So, the three basis vectors are in $\mathbb R^4.$

$$ \langle 1,-1,0,0 \rangle $$ $$ \langle 0,1,-1,0 \rangle $$ $$ \langle 0,0, 1,-1 \rangle $$

EEEEEEEDDDDDDIIIIIIIITTTTTTTTTT: Here is a somewhat cleaner answer, after I checked it in Schiemann's reduction. The lattice you want is the one I asked about, all integer points in $\mathbb R^3$ such that $$ x+y+z \; \; \; \; \mbox{is even.} $$

A basis can be given in the same dimension by $$ \langle 0,1,1 \rangle $$ $$ \langle 1,0,1 \rangle $$ $$ \langle 1,1, 0 \rangle $$ Notice that the squared lengths are 2, and the inner products are 1. So the matrix of inner products, the Gram matrix is

$$ \left( \begin{array}{rrr} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right) $$ The twelve lattice points nearest the origin, in this lattice, are $$ (0,1,1), (1,0,1), (1,1,0), $$ $$ (0,-1,1), (-1,0,1), (-1,1,0), $$ $$ (0,1,-1), (1,0,-1), (1,-1,0), $$ $$ (0,-1,-1), (-1,0,-1), (-1,-1,0) $$ for kissing number 12, as they are all the same distance from the origin, $\sqrt 2.$