## Abstract

A. Tyszka [8] proved that every unit-distance preserving map ping of Q^{8} to Q^{8} is an isometry. The purpose of this paper is to extend this property to all even d, d ≥ 6, by using the result of Zaks [10]. A mapping f: Q^{d}-+ Q^{d} is called unit-distance preserving if llx - Yll = 1 implies llf (x) - f(y) ll = 1. The Beckman-Quarles Theorem ([1], see also [2]) states that every unit-distance preserving mapping of R^{d} to R^{d}, d ≥ 2, is an isometry. Tyszka [8] (see also [6, 7]) proved that every unit-distance preserving mapping of the rational 8-space Q^{8} to Q^{8} is an isometry. Zaks [10] extended Tyszka's result for unit-distance preserving mappings f: Q^{d-}+ Q^{d}, for all even d of the form d = 4k(k + 1), as well as for all odd d of the form 2k^{2} - 1 which are also complete squares. The purpose of this paper is to extend the results of Tyszka and Zaks to all even dimensions d, d≥6.

Original language | English |
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Title of host publication | Discrete Geometry |

Subtitle of host publication | In Honor of W. Kuperberg's 60th Birthday |

Publisher | CRC Press |

Pages | 193-200 |

Number of pages | 8 |

ISBN (Electronic) | 9780203911211 |

ISBN (Print) | 9780824709686 |

DOIs | |

State | Published - 1 Jan 2003 |

### Bibliographical note

Publisher Copyright:© 2003 by Taylor & Francis Group, LLC.

## ASJC Scopus subject areas

- General Mathematics