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

  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.},
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… 
Combinatorial constructions of packings in Grassmannian spaces
  • Tao Zhang, G. Ge
  • Computer Science, Mathematics
    Des. Codes Cryptogr.
  • 2018
This paper gives a general construction of equiangular lines, and gives three constructions of optimal packings in Grassmannian spaces based on difference sets and Latin squares, and obtains many new classes of optimal Grassmanian packings.
Packing Planes in Four Dimensions and Other Mysteries
An embedding theorem is shown 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 the authors' packings are optimal.
Game of Sloanes: best known packings in complex projective space
It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield
This report presents a numerical method for finding good packings on spheres, in projective spaces, and in Grassmannian manifolds equipped with various metrics. In each case, producing a good packing
The optimal packing of eight points in the real projective plane
The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and from this graph an exact expression for the minimum distance of eight optimally packed points in the real projective plane is recovered.
A Notion of Optimal Packings of Subspaces with Mixed-Rank and Solutions
We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to
Packings in real projective spaces
A computer-assisted proof of the optimality of a particular 6-packing in $\mathbb{R}\mathbf{P}^3$, a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and local optimality certificates for two infinite families of packings are provided.
Constructing Subspace Packings from Other Packings
The desiderata when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identity like the projections onto
Constructing Packings in Grassmannian Manifolds via Alternating Projection
The alternating projection method can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini–Study distance and can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.
Equiangular lines and subspaces in Euclidean spaces
It is proved that for every fixed angle θ and n sufficiently large, there are at most 2 n − 2 lines in R n with common angleθ, and this is achievable only for θ = arccos ⁡ 1 3 .


Sphere Packings, Lattices and Groups
  • J. Conway, N. Sloane
  • 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
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
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
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 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
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 =