N. J. A. Sloane: How to Pack Lines, Planes, 3-Spaces, Etc.
How should 16 laser beams passing through a single point
be arranged so as to make the
angle between any two of the beams as large as possibe?
You have a table of data with 4 columns and you want to project it
onto the screen in 48 different ways - which set of 48 planes
in 4-space should you use for the projections?
More generally, the Grassmannian space G(m,n)
is the space of all n-dimensional subspaces of m-dimensional
Euclidean space.
The problem is to find the best packing of N of these subspaces.
In other words, choose N points in G(m,n)
so that the minimal distance between any two of them is as
large as possible.
How is the distance between two n-spaces defined?
For n=1 we define it to be the sine of the smaller of the
two angles between the two lines.
In general the distance between two n-spaces is the
the square root of the sum of the squares of the sines of
the n principal angles between the two spaces.
A selection of the best packings we have found in dimensions
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16.
Note that in the higher dimensions we only made a few runs
for each set of parameters - in some cases only one.
The spaces are named after
Hermann Guenther Grassmann (1809-1877), professor at the gymnasium
in Stettin, whose picture can be seen here.
The papers:
J. H. Conway, R. H. Hardin and N. J. A. Sloane,
Packing Lines, Planes, etc., Packings in Grassmannian Spaces,
Experimental Mathematics, Vol. 5 (1996), 139-159.
A Family of Optimal Packings in Grassmannian Manifolds
[postscript,
pdf],
P. W. Shor and N. J. A. Sloane,
J. Algebraic Combinatorics,
7 (1998),
pp. 157-163.
A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces
[postscript,
pdf],
A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane,
J. Algebraic Combinatorics,
9 (1999),
pp. 129-140.
(This paper discusses
the connections with quantum error correcting codes.
For more about this see
Quantum Error Correction Via Codes Over GF(4)
[postscript,
pdf],
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane,
IEEE Trans. Information Theory,
44 (1998),
pp. 1369-1387.)
Packing Planes in Four Dimensions and Other Mysteries
[postscript,
pdf],
N. J. A. Sloane,
Algebraic Combinatorics and Related Topics (Yamagata 1997),
E. Bannai, M. Harada and M. Ozeki (editors),
Yamagata University, Faculty of Science, Department of Mathematics,
1998, 1999.