Isometric selfmaps of Euclidean spaces

What is it about?

A short and simple new proof of a basic fact concerning at least two-dimensional Euclidean spaces.

Featured Image

Why is it important?

Juhász, Rozália: Another proof of the Beckman-Quarles theorem. Advances in Geometry 15(2015), no. 4, 519-521. The theorem in the title asserts that for n>1, if a self-map \tau of the n-dimensional Euclidean space preserves some nonzero distance, then it is an isometry. Our proof is elementary and uniform for all dimensions: a generalization of the Moser graph is used to show that under the assumption of the theorem there exist a nonzero distance d such that \tau preserves both d and d+1, and then the existence of such a distance d is shown to imply that \tau preserves every distance.