  We provide the first examples of geometric transition from hyperbolic to
anti-de Sitter structures in dimension four, in a fashion similar to Danciger's
three-dimensional examples. The main ingredient is a deformation of hyperbolic
4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a
3-dimensional ideal cuboctahedron. We show the existence of a similar family of
collapsing anti-de Sitter polytopes, and join the two deformations by means of
an opportune half-pipe orbifold structure. The desired examples of geometric
transition are then obtained by gluing copies of the polytope.
