A discrete form of the BeckmanQuarles theorem for rational eightspace
Abstract
Let Q denote the field of rational numbers. Let F \subseteq R is a euclidean field. We prove that: (1) if x,y \in F^n (n>1) and xy is constructible by means of ruler and compass then there exists a finite set S(x,y) \subseteq F^n containing x and y such that each map from S(x,y) to R^n preserving unit distance preserves the distance between x and y, (2) if x,y \in Q^8 then there exists a finite set S(x,y) \subseteq Q^8 containing x and y such that each map from S(x,y) to R^8 preserving unit distance preserves the distance between x and y.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1999
 DOI:
 10.48550/arXiv.math/9906001
 arXiv:
 arXiv:math/9906001
 Bibcode:
 1999math......6001T
 Keywords:

 Metric Geometry;
 51M05 (Primary);
 05C12 (Secondary)
 EPrint:
 added Remark 4 by Joseph Zaks, to appear in Aequationes Math