  How should you choose a good set of (say) 48 planes in four dimensions? More
generally, how do you find packings in Grassmannian spaces? In this article I
give a brief introduction to the work that I have been doing on this problem in
collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains
and P. W. Shor. We have found many nice examples of specific packings (70
4-spaces in 8-space, for instance), several general constructions, and an
embedding theorem which shows that a packing in Grassmannian space G(m,n) is a
subset of a sphere in R^D, where D = (m+2)(m-1)/2, and leads to a proof that
many of our packings are optimal. There are a number of interesting unsolved
problems.
