Flexible cross-polytopes in spaces of constant curvature

We construct self-intersected flexible cross-polytopes in the spaces of constant curvature, that is, the Euclidean spaces, the spheres, and the Lobachevsky spaces of all dimensions. In dimensions greater than or equal to 5, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces of constant curvature. For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.