Mappings preserving some geometrical figures

We introduce new characterizations of linear isometries. More precisely, we prove that if a one-to-one mapping f : Rⁿ →Rⁿ (n > 1) maps the periphery of every regular triangle (quadrilateral or hexagon) of side length a > 0 onto the periphery of a figure of the same type with side length b > 0, then there exists a linear isometry I : Rⁿ →Rⁿ up to translation such that f(x) = (b/a) I(x). This revised version was published online in June 2006 with corrections to the Cover Date.