A discrete form of the Beckman-Quarles theorem for rational eight-space
Apoloniusz Tyszka
Abstract
Let Q denote the field of rational numbers. Let F ⊂eq R is a euclidean field. We prove that: (1) if x,y ∈ Fn (n>1) and |x-y| is constructible by means of ruler and compass then there exists a finite set S(x,y) ⊂eq Fn containing x and y such that each map from S(x,y) to Rn preserving unit distance preserves the distance between x and y, (2) if x,y ∈ Q8 then there exists a finite set S(x,y) ⊂eq Q8 containing x and y such that each map from S(x,y) to R8 preserving unit distance preserves the distance between x and y.
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