On mappings of ℚd to ℚd that preserve distances 1 and √2 and the Beckman-Quarles Theorem

Joseph Zaks

doi:10.1007/s00022-004-1660-3

On mappings of ℚ^d to ℚ^d that preserve distances 1 and √2 and the Beckman-Quarles Theorem

Joseph Zaks

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Benz proved that every mapping f : ℚ^d → ℚ^d that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping f : ℚ^d → ℚ^d that preserves the distances 1 and √2 is an isometry, provided d ≥ 5.

Original language	English
Pages (from-to)	195-203
Number of pages	9
Journal	Journal of Geometry
Volume	82
Issue number	1-2
DOIs	https://doi.org/10.1007/s00022-004-1660-3
State	Published - Aug 2005

Keywords

Beckman-Quarles Theorem
Distance preserving mappings
Isometry

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.1007/s00022-004-1660-3

Cite this

@article{de8e6674893045f1a38fdc3c83e29f18,

title = "On mappings of ℚd to ℚd that preserve distances 1 and √2 and the Beckman-Quarles Theorem",

abstract = "Benz proved that every mapping f : ℚd → ℚd that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping f : ℚd → ℚd that preserves the distances 1 and √2 is an isometry, provided d ≥ 5.",

keywords = "Beckman-Quarles Theorem, Distance preserving mappings, Isometry",

author = "Joseph Zaks",

year = "2005",

month = aug,

doi = "10.1007/s00022-004-1660-3",

language = "English",

volume = "82",

pages = "195--203",

journal = "Journal of Geometry",

issn = "0047-2468",

publisher = "Birkhauser Verlag Basel",

number = "1-2",

}

TY - JOUR

T1 - On mappings of ℚd to ℚd that preserve distances 1 and √2 and the Beckman-Quarles Theorem

AU - Zaks, Joseph

PY - 2005/8

Y1 - 2005/8

N2 - Benz proved that every mapping f : ℚd → ℚd that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping f : ℚd → ℚd that preserves the distances 1 and √2 is an isometry, provided d ≥ 5.

AB - Benz proved that every mapping f : ℚd → ℚd that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping f : ℚd → ℚd that preserves the distances 1 and √2 is an isometry, provided d ≥ 5.

KW - Beckman-Quarles Theorem

KW - Distance preserving mappings

KW - Isometry

UR - https://www.scopus.com/pages/publications/26044432221

U2 - 10.1007/s00022-004-1660-3

DO - 10.1007/s00022-004-1660-3

M3 - Article

AN - SCOPUS:26044432221

SN - 0047-2468

VL - 82

SP - 195

EP - 203

JO - Journal of Geometry

JF - Journal of Geometry

IS - 1-2

ER -