Dense packings of equal spheres in a cube
type: Item
instance of: scholarly article
title: Dense packings of equal spheres in a cube
zbMATH Open document ID: 1053.52023
published in: The Electronic Journal of Combinatorics
publication date: 2004-10-13T00:00:00Z
full work available at URL: https://eudml.org/doc/124057
full work available at URL: v11i1r33.html
review text: Summary: We describe an adaptation of the billiard algorithm for finding dense packings of equal spheres inside a domain of the Euclidean space. In order to improve the convergence of this stochastic algorithm, we introduce systematic perturbations in it. We apply this perturbed billiard algorithm in the case of \(n\) spheres in a cube and display all the optimal and best known packings up to \(n=32\). We improve the previous record packings for all \(11\leq n\leq 26\) except \(n=13,14,18\). We prove the existence of the displayed packings for \(n=11,12,15, 17,20,21,22, 26,32\), by constructing them explicitly. For example, the graph of the conjectured optimal packing of twenty-two spheres is composed of five octahedrons and four isolated points. We also conjecture that the minimum distance \(d_n\) between spheres centers of the optimal packings is constant in the range \(29\leq n\leq 32\).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 2107475
zbMATH Keywords: maximum separation problem
zbMATH Keywords: disk packing problem
zbMATH Keywords: billiard algorithm
author: Thierry Gensane
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2107475