The Beckman-Quarles Theorem For Rational Spaces: Mappings Of Q^5 To Q^5 That Preserve Distance 1 (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2021)
Let Rd and Qd denote the real and the rational d-dimensional space, respectively, equipped with the usual Euclidean metric. For a real number, a mapping  where X is either Rd or Qd and is called - distance preserving  implies  , for all x,y in .Let G(Qd,a) denote the graph that has Qd  as its set of vertices, and where two vertices x and y are connected by edge if and only if  . Thus, G(Qd,1) is the unit distance graph. Let ω(G) denote the clique number of the graph G and let ω(d) denote ω(G(Qd, 1)).The Beckman-Quarles theorem [1] states that every unit- distance-preserving mapping from Rd into Rd is an isometry, provided d ≥ 2.The rational analogues of Beckman- Quarles theorem means that, for certain dimensions d, every unit- distance preserving mapping from Qd into Qd is an isometry.A few papers [2, 3, 4, 5, 6, 8,9,10 and 11] were written about rational analogues of this theorem, i.e, treating, for some values of  the property "Every unit- distance preserving mapping  is an isometry".The purpose of this thesis is to prove the following Theorem.Theorem 1:Every unit- distance preserving mapping f:→ is an isometry; moreover, dim(aff(f(L[5])))=5.