The kissing number of the regular pentagon
type: Item
instance of: scholarly article
title: The kissing number of the regular pentagon
zbMATH Open document ID: 1007.52010
DOI: 10.1016/S0012-365X(01)00460-5
author: Likuan Zhao
author: Junqin Xu
published in: Discrete Mathematics
publication date: 2002-08-29T00:00:00Z
review text: The kissing number (or Newton number) of a convex disc \(D\) is the maximum number of congruent copies of \(D\) that can be arranged so that each touches \(D\) but no two of them overlap. The kissing number of a regular \(n\)-gon has been determined for all \(n\neq 5\) by \textit{K. Böröczky} [Period. Math. Hung. 1, 113-119 (1971; Zbl 0231.52008)] and later again by \textit{L. Zhao} [Discrete Math. 188, No. 1-3, 293-296 (1998; Zbl 0958.05022)]. Here the authors consider the case \(n=5\) and want to prove that the kissing number of the regular pentagon is equal to \(6\). Unfortunately, their proof does not seem to be valid, since it is based on the following assertion: ``Let \(AOB\) be the triangle spanned by the centers of three mutually touching congruent regular pentagons. Then the angle at \(O\) is at least \(54\) degrees.'' This is not true, since it is not difficult to find three such pentagons with the corresponding angle measuring even less than \(53\) degrees. The main point is that the authors erroneously believe that the minimum angle is attained if either the three pentagons have a common vertex or the two pentagons centered at \(A\) and \(B\) have a common side. On the other hand, the kissing number of the regular pentagon is actually equal to \(6\), as has already been proved previously by the reviewer [Period. Math. Hung. 4, 315-328 (1973; Zbl 0283.52002)].
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 52C15
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1792435
zbMATH Keywords: kissing number
zbMATH Keywords: Newton number
zbMATH Keywords: regular pentagon
MaRDI profile type: Publication
full work available at URL: s0012-365x(01)00460-5
OpenAlex ID: W4213123559
description: scientific article; zbMATH DE number 1792435