• DOI:10.1007/PL00000146
• Corpus ID: 13892502

A discrete form of the Beckman—Quarles theorem for rational eight-space

@article{Tyszka1999ADF,
  title={A discrete form of the Beckman—Quarles theorem for rational eight-space},
  author={Apoloniusz Tyszka},
  journal={aequationes mathematicae},
  year={1999},
  volume={62},
  pages={85-93}
}

• A. Tyszka
• Published 1 June 1999
• Mathematics
• aequationes mathematicae

Summary. Let
$ {\Bbb Q} $ be the field of rationals numbers. We prove that: (1) if
$ x,y \in {\Bbb R}^{n}\,(n>1) $ and
$ |x - y| $ is constructible by means of ruler and compass then there exists a finite set
$ S_{xy}\subseteq {\Bbb R}^{n} $ containing x and y such that each map from Sxy to
$ {\Bbb R}^{n} $ preserving unit distance preserves the distance between x and y, (2) if
$ x,y \in {\Bbb Q}^{8} $ then there exists a finite set
$ S_{xy} \subseteq {\Bbb Q}^{8} $ containing x and y such that each map from Sxy to
$ {\Bbb R}^{8} $ preserving unit distance preserves the distance between x and y.