Original Paper

Journal of Geometry

, Volume 82, Issue 1, pp 195-203

First online:

On mappings of \(\mathbb{Q}^d \) to \(\mathbb{Q}^d \) that preserve distances 1 and \(\sqrt 2 \) and the Beckman-Quarles Theorem

  • Joseph ZaksAffiliated withDepartment of Mathematics, University of Haifa Email author 

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract.

Benz proved that every mapping \(f:\mathbb{Q}^d \to \mathbb{Q}^d \) that preserves the distances 1 and 2 is an isometry, provided d ≥ 5. We prove that every mapping \(f:\mathbb{Q}^d \to \mathbb{Q}^d \) that preserves the distances 1 and \(\sqrt 2 \) is an isometry, provided d ≥ 5.

Mathematics Subject Classification (2000).

51M05 46B04

Key words.

Distance preserving mappings isometry Beckman-Quarles Theorem