The Beckman-Quarles theorem for rational spaces
type: Item
instance of: scholarly article
title: The Beckman-Quarles theorem for rational spaces
zbMATH Open document ID: 1028.51015
DOI: 10.1016/S0012-365X(02)00535-6
author: Joseph Zaks
published in: Discrete Mathematics
publication date: 2003-05-25T00:00:00Z
review text: The Beckmann-Quarles theorem asserts that for \(d > 1\) every mapping from a \(d\)-dimensional real space \(R^d\) into itself preserving the unit distance (`unit-preserving', i.e. pairs of points with distance 1 will be mapped into pairs of points, again with distance 1) will preserve any distance and hence is an isometry. For the rational case \(Q^d\) the same proposition is not a foregone conclusion. For some numbers \(d\) the matter has already been settled; the author adds the proof for the case of dimensions \(d = 2 n^2-1\) for \(n > 2\). This proof is indeed versatile as it can also be applied to several other cases.
reviewed by: Johann Lang
Mathematics Subject Classification ID: 51K05
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 1915549
zbMATH Keywords: isometries
zbMATH Keywords: Beckmann-Quarles theorem
zbMATH Keywords: rational space
zbMATH Keywords: unit distance
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1915549