Packing Lines, Planes, etc.: Packings in Grassmannian Spaces

J. H. Conway 
Department of Mathematics 
Princeton University, Princeton, NJ 08544 

R. H. Hardin and N. J. A. Sloane(*)
Mathematical Sciences Research Center 
AT&T Bell Laboratories 
Murray Hill, NJ 07974 

(*) Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com

April 12, 1996 


ABSTRACT

This paper addresses 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 (m-1) (m+2)/2, which provides a (usually)
lower-dimensional representation than the Pluecker 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.




This paper was published (in a somewhat different form) in
Experimental Math., 5 (1996), 139-159.


For the full version see
http://NeilSloane.com/doc/grass.pdf (pdf) or
http://NeilSloane.com/doc/grass.ps (ps)


Abstract of a related talk:


Packings in Grassmann Manifolds; or, Displaying Multi-Dimensioanal Data

Neil J. A. Sloane
AT&T Shannon Lab
Florham Park, New Jersey

How should you place 18 planes through the origin in
Euclidean 4-space so that they are as far apart as possible?
More generally, how do you place N points on the Grassmann
manifold G(m,n) so they are as far apart as possible?
I will discuss bounds and constructions, including several
infinite families of optimal packings. There are surprising
connections with quantum codes, group theory and spherical
t-designs (and of course statistics, for the
graphical representation of multi-dimensional data).

This is based on joint work with Calderbank, Conway,
Hardin, Nebe, Rains, Shor, Nebe.
For more information see my home page, NeilSloane.com.