Aha — interesting. (I was recently reminded that the Z^2 lattice has the same kind of similarity property.) If you start with the Hurwitz integral quaternions J, then those ever-finer copies of the D_4 lattice ought to also be subrings of the entire skew-field of quaternions (which I'll call H). So if we took the union of all those ever-finer copies, the whole thing would be what, J ⊗ Z_2^oo ??? (Where Z_2^oo denotes the ring of rational numbers whose denominator is a power of 2.) —Dan Adam Goucher wrote: ----- Something I noticed about the D_4 lattice is that the union of the lattice together with its deep holes is geometrically similar to the original D_4 lattice, scaled by 1/sqrt(2). This means that if you take the 4-dimensional torus R^4 / D_4 and start greedily placing points to be as far away as possible from the nearest point, you'll end up progressively filling ever-finer copies of the D_4 lattice. (This also works on the more conventional torus R^4 / Z^4, because after the first two points are emplaced they'll form D_4 / Z^4.) -----