aequationes mathematicae

, Volume 59, Issue 1, pp 124–133

# Discrete versions of the Beckman—Quarles theorem

• A. Tyszka

DOI: 10.1007/PL00000119

Tyszka, A. Aequ. math. (2000) 59: 124. doi:10.1007/PL00000119

## 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 map from Sxy to $${\Bbb R}^{n}$$ preserving all unit distances preserves the distance between x and y,¶(2) only algebraic distances $$|x-y|$$ have the property from item (1),¶(3) if $$X_{1},X_{2},\dots ,X_{m}\in {\Bbb R}^{n}\ (n>1)$$ lie on some affine hyperplane then there exists a finite set $$L(X_{1},X_{2},\dots ,X_{m}) \subseteq {\Bbb R}^{n}$$ containing $$X_{1},X_{2},\dots ,X_{m}$$ such that each map from $$L(X_{1},X_{2},\dots ,X_{m})$$ to $${\Bbb R}^{n}$$ preserving all unit distances preserves the property that $$X_{1},X_{2},\dots ,X_{m}$$ lie on some affine hyperplane,¶(4) if $$J,K,L,M \in {\Bbb R}^{n}\ (n>1)$$ and $$|JK|=|LM|\,(|JK|T < |LM|)$$ then there exists a finite set $$C_{JKLM}\subseteq {\Bbb R}^{n}$$> containing J,K,L,M such that any map $$f : C_{JKLM} \rightarrow {\Bbb R}^{n}$$ that preserves unit distances satisfies $$|f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)| < |f(L)f(M)|)$$.

Keywords. Discrete form of the Beckman—Quarles theorem, finite rigid unit-distance graph.