Smallest maximal snakes of translates of convex domains
type: Item
instance of: scholarly article
title: Smallest maximal snakes of translates of convex domains
zbMATH Open document ID: 0841.52010
DOI: 10.1007/BF01265297
published in: Geometriae Dedicata
publication date: 1995-10-17T00:00:00Z
review text: A maximal snake of size \(n\) is a set \((K_1,\dots,K_n)\) of non-overlapping translates of a convex domain \(K\) in the plane, such that \(K_i\) touches \(K_j\) if and only if \(|i -j|= 1\) and no translate of \(K\) can touch \(K_1\) or \(K_n\) without intersecting an additional \(K_i\). The authors prove that the size of the smallest maximal snake is 11 if \(K\) is a parallelogram and 10 otherwise, thereby generalizing the corresponding theorem on circular disks. For the proof, some properties of Minkowski planes are studied in detail, and a modification of the usual notion of angles in such planes is introduced. For these angles, a partial ordering and a partial addition are defined. The angles of equilateral triangles turn out to be of particular importance.
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 806197
zbMATH Keywords: Minkowski plane
zbMATH Keywords: maximal snake
author: Károly jun. Böröczky
author: Valeriu Soltan
reviewed by: Johann Linhart
MaRDI profile type: Publication
cites work: On the smallest limited snake of unit disks
cites work: On a conjecture of H. Hadwiger
full work available at URL: https://doi.org/10.1007/bf01265297
OpenAlex ID: W2003227982
description: scientific article; zbMATH DE number 806197