A discrete form of the Beckman-Quarles Theorem for rational spaces

Joseph Zaks

doi:10.1007/s00022-001-8581-1

A discrete form of the Beckman-Quarles Theorem for rational spaces

Joseph Zaks

Department of Mathematics

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

Abstract

The Beckman-Quarles Theorem states that every unit-preserving mapping of R^d into R^d is an isometry. A. Tyszka proved that every unit preserving mapping of Q⁸ into Q⁸ is an isometry. We extend the last result by showing that every unit-preserving mapping from Q ^d into Q^d is an isometry for all even d of the form d = 4k(k + 1), k ≥ 1, as well as for all odd d of the form 2k² - 1 which are complete squares, k ≥ 2.

Original language	English
Pages (from-to)	199-205
Number of pages	7
Journal	Journal of Geometry
Volume	72
Issue number	1-2
DOIs	https://doi.org/10.1007/s00022-001-8581-1
State	Published - 2001

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s00022-001-8581-1

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abstract = "The Beckman-Quarles Theorem states that every unit-preserving mapping of Rd into Rd is an isometry. A. Tyszka proved that every unit preserving mapping of Q8 into Q8 is an isometry. We extend the last result by showing that every unit-preserving mapping from Q d into Qd is an isometry for all even d of the form d = 4k(k + 1), k ≥ 1, as well as for all odd d of the form 2k2 - 1 which are complete squares, k ≥ 2.",

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A discrete form of the Beckman-Quarles Theorem for rational spaces

Abstract

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