Flexible cross-polytopes in spaces of constant curvature

We construct self-intersecting flexible cross-polytopes in the spaces of constant curvature, that is, in Euclidean spaces $\mathbb E^n$, spheres $\mathbb S^n$, and Lobachevsky spaces $\Lambda ^n$ of all dimensions $n$. In dimensions $n\ge5$, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces $\mathbb E^n$, $\mathbb S^n$, and $\Lambda ^n$. For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.