What are all the best sphere packings in low dimensions?
type: Item
instance of: scholarly article
title: What are all the best sphere packings in low dimensions?
zbMATH Open document ID: 0844.52013
DOI: 10.1007/BF02574051
author: N. J. A. Sloane
author: J. H. Conway
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131369
review text: This remarkable paper has been brewing for a long time (as has this review), partly because the questions asked are not easily formulated and partly because the results depend on certain as yet unproved postulates. The goal of the paper is to describe what may be all the tight packings (of nonoverlapping equal spheres) in dimension \(\leq 10\) and to give partial information about higher dimension. But which packings are tight. Well, those that are not loose. Suppose we can dissect the space of the packing into finitely many polyhedral pieces (possibly infinite) in such a way that the center of each center (of the spheres in the packing) lies in the interior of some piece, and there are also some empty pieces containing no centers. Then if we can rearrange the nonempty pieces into another dissection in which the centers are at least as far apart as they were originally then the packing is called (by the authors) loose. Packings that are tight in this sense certainly have the highest possible density. In 10 sections devoted in turn to dimension 1 to 10 the authors investigate tight packings. Some of the results they obtain depend on Postulate \(n\): A tight \(n\)-dimensional packing fibers over some tight \(2^k\)-dimensional packing, where \(2^k\) is the largest power of 2 strictly less than \(n\). (An \(n\)-dimensional packing \(P_n\) fibers over an \(m\)-dimensional packing \(P_m\) if \(P_n\) can be decomposed into sets -- or layers -- of points lying in parallel \(m\)-dimensional spaces, each of which is a packing of type \(P_m\).) They believe (but they cannot prove) Postulate \(n\) is true for \(2 \leq n \leq 8\); but Postulate 9 requires modification and Postulate 10 is false. In their concluding remarks they state: ``We have achieved what seems very likely to be a complete description of all the tightest packings in up to 9 dimensions, perhaps also in 10 dimensions. However, there seems to be little point in carrying these detailed arguments much further. \(\d…
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 764233
zbMATH Keywords: best sphere packings
zbMATH Keywords: low dimensions
zbMATH Keywords: tight packings
MaRDI profile type: Publication
cites work: The construction of perfect and extreme forms II
cites work: Binary codes with a minimum distance of four (Corresp.)
cites work: THE ARITHMETIC MINIMA OF POSITIVE QUADRATIC FORMS (I)
cites work: On the enumeration of self-dual codes
cites work: Q4023809
cites work: Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others
cites work: An Extreme Duodenary Form
cites work: Q5813620
cites work: Q3316129
cites work: Q5554102
cites work: The sphere packing problem
cites work: The status of the Kepler conjecture
cites work: ON THE SPHERE PACKING PROBLEM AND THE PROOF OF KEPLER'S CONJECTURE
cites work: Five Dimensional Non-Lattice Sphere Packings
cites work: Six and Seven Dimensional Non-Lattice Sphere Packings
cites work: New sphere packings in dimensions 9-15
cites work: Sphere Packings and Error-Correcting Codes
cites work: A new bound on the local density of sphere packings
cites work: Q5568974
description: scientific article; zbMATH DE number 764233