By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone four-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds $M_t$ interpolating between two cusped hyperbolic four-manifolds $M_0$ and $M_1$.

This is a joint work with Bruno Martelli.