# 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} }

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…

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- 1999

Summary. We prove that:¶(1) if
$ x,y\in {\Bbb R}^{n}\ (n>1) $ and
$ |x-y| $ is an algebraic number then there exists a finite set
$ S_{xy}\subseteq {\Bbb R}^{n} $ containing x and y such that each…

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