We address the question: How should $N$ $n$-dimensional subspaces of
$m$-dimensional Euclidean space be arranged so that they are as far
apart as possible? The results of extensive computations for modest
values of $N, n,m$ are described, as well as a reformulation of the
problem that was suggested by these computations. The reformulation
gives a way to describe $n$-dimensional subspaces of $m$-space as
points on a sphere in dimension $\half(m-1) (m+2)$, which provides a
(usually) lower-dimensional representation than the Plücker
embedding, and leads to a proof that many of the new packings are
optimal. The results have applications to the graphical display of
multi-dimensional data via Asimov's grand tour method.