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

@article{Conway1996PackingLP,
title={Packing Lines, Planes, etc.: Packings in Grassmannian Spaces},
author={John H. Conway and Ronald H. Hardin and N. J. A. Sloane},
journal={Exp. Math.},
year={1996},
volume={5},
pages={139-159}
}
• Published 1 August 2002
• Computer Science, Mathematics
• Exp. Math.
We addressthe question: How should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The resuIts 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 ½(m–l)(m+2), which provides a (usually) lowerdimensional representation…
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## References

SHOWING 1-10 OF 85 REFERENCES
Sphere Packings, Lattices and Groups
• Mathematics, Computer Science
Grundlehren der mathematischen Wissenschaften
• 1988
The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to
Sphere packings, I
• T. Hales
• Mathematics, Computer Science
Discret. Comput. Geom.
• 1997
A program to prove the Kepler conjecture on sphere packings is described and it is shown that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.
The Cell Structures of Certain Lattices
• Physics
• 1991
The most important lattices in Euclidean space of dimension n≤ 8 are the lattices An (n≥ 2), Dn (n≥ 4), En (n = 6, 7, 8) and their duals. In this paper we determine the cell structures of all these
Equilateral point sets in elliptic geometry
• Mathematics
• 1966
This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , any
Equi-isoclinic subspaces of Euclidean spaces
• Mathematics
• 1973
This chapter provides an overview of the equi-isoclinic subspaces of Euclidean spaces. Two planes in Euclidean 4-space E 4 have two angles. They are the stationary values of the angle between the
The Closest Packing of Spherical Caps in n Dimensions
Let S n denote the “surface” of an n -dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O , so that the points P ( x
The grand tour: a tool for viewing multidimensional data
The grand tour is a method for viewing multivariate statistical data via orthogonal projections onto a sequence of two-dimensional subspaces and several specific types of sequences are tested for rapidity of becoming dense.
Discrete Non-Euclidean Geometry
Publisher Summary This chapter discusses discrete non-euclidean geometry with emphasis on inner product spaces, spherical geometry, elliptic geometry, and hyperbolic geometry. It describes discrete
BOUNDS FOR SYSTEMS OF LINES, AND JACOBI POLYNOMIALS
• Mathematics
• 1975
Bounds are obtained for the cardinality of sets of lines having a prescribed number of angles, both in real and in complex Euclidean n-space. Extremal sets provide combinatorial configurations with a
Hadamard matrices, Sequences, and Block Designs
• Mathematics
• 1992
One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X =