In this vein, it's interesting that there are plenty of tilted cubic sublattices in of the integer points Z^3 in 3-space, perhaps the simplest generated by the columns of (-1 2 2) ( 2 -1 2) ( 2 2 -1) . Cayley noticed that if F is any subfield of the reals and S is any n x n skew-symmetric (real) matrix over F, then the matrix A = (S - I)^(-1) (S + I) is an orthogonal matrix over F. Letting F = Q and applying this formula, A will be a rotation matrix over Q. Hence the columns of A are orthonormal. We can then multiply by an integer den(A) to clear the denominators, to get B = den(A) A whose columns will then generate a cubic lattice (i.e., their Z-linear combinations are the lattice points forming a set similar to the set Z^n of integer lattice points. E.g., in R^3 we could have ( 2 6 9) (-9 6 -2) ( 6 7 -6) (Is there an example where no two rows have the same set of absolute values?) —Dan Bill Gosper wrote: --- More generally, there is no threefold rotational symmetry. Julian's function can produce only gridpoints: ListPlot[{3, 1} # & /@ NestWhileList[{#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &@ {#[[1]], #[[2]] + Floor[381 #[[1]]/128]} &@ {#[[1]] - Floor[64 #[[2]]/127], #[[2]]} &, {2, 1}, # != -{1, 4} &], AspectRatio -> 1, Axes -> False, Frame -> True] gosper.org/trinsky3spiral.png And all the arguments to the Floors are merely rational. ---