Advertisement

Periodica Mathematica Hungarica

, Volume 72, Issue 2, pp 252–257 | Cite as

An analogue of a theorem of van der Waerden, and its application to two-distance preserving mappings

  • Victor Alexandrov
Article

Abstract

A theorem of van der Waerden reads that an equilateral pentagon in Euclidean 3-space \({\mathbb {E}}^3\) with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. We prove the following many-dimensional analogue of this theorem: for \(n\geqslant 2,\) every n-dimensional cross-polytope in \({\mathbb {E}}^{2n-2}\) with all diagonals of the same length and all edges of the same length necessarily lies in \({\mathbb {E}}^n\) and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.

Keywords

Euclidean space Pentagon Cross-polytope Cayley–Menger determinant Beckman–Quarles theorem 

Mathematics Subject Classification

52B11 52B70 52C25 51K05 

References

  1. 1.
    F.S. Beckman, D.A. Quarles, On isometries of Euclidean spaces. Proc. Am. Math. Soc. 4(5), 810–815 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    K. Bezdek, R. Connelly, Two-distance preserving functions from Euclidean space. Period. Math. Hung. 39(1–3), 185–200 (1999)MathSciNetzbMATHGoogle Scholar
  3. 3.
    D.E. Blair, T. Konno, Discrete torsion and its application for a generalized van der Waerden’s theorem. Proc. Japan Acad. A 87(10), 209–214 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L.M. Blumenthal, Theory and Applications of Distance Geometry (Clarendon Press, Oxford, 1953)zbMATHGoogle Scholar
  5. 5.
    O. Bottema, Pentagons with equal sides and equal angles. Geom. Dedicata 2, 189–191 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    H.S.M. Coxeter, Regular Polytopes (Methuen, London, 1948)zbMATHGoogle Scholar
  7. 7.
    B.V. Dekster, Nonisometric distance 1 preserving mapping \(E^2\rightarrow E^6\). Arch. Math. 45(3), 282–283 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ch. Frayer, Ch. Schafhauser, Alpha-regular stick unknots. J. Knot Theory Ramif. 21(6), (2012). doi: 10.1142/S0218216512500599
  9. 9.
    A.V. Kuz’minykh, Mappings preserving unit distance. Sib. Math. J. 20(3), 417–421 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. O’Hara, The configuration space of equilateral and equiangular hexagons. Osaka J. Math. 50(2), 477–489 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Th.M. Rassias, On the Aleksandrov problem for isometric mappings. Appl. Anal. Discrete Math. 1(1), 18–28 (2007)Google Scholar
  12. 12.
    B.L. van der Waerden, Ein Satz über raümliche Fünfecke. Elem. Math. 25(4), 73–78 (1970)MathSciNetzbMATHGoogle Scholar
  13. 13.
    J. Zaks, On mappings of \(\mathbb{Q}^d\) to \(\mathbb{Q}^d\) that preserve distances 1 and \(\sqrt{2}\) and the Beckman–Quarles theorem. J. Geom. 82(1–2), 195–203 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

Personalised recommendations