In the first video, an ideal polytope in hyperbolic three-space which deforms in time, and collapses (at the end of the video) to an ideal quadrilateral contained in a totally geodesic plane.

The polytope is pictured in the Klein (affine) model of hyperbolic space, and all the dihedral angles of the polytope (before the collapse) are right, except the top and bottom edges, which vary from 0 to π.

The second video below shows the same polytope, to which a projective transformation has been applied, to "rescale" in the vertical direction. The polytope does not collapse anymore, instead the deformation continues to a polytope in half-pipe space (pictured on the right) and then in Anti-de Sitter space.

One can glue copies of this polytope to construct a geometric transition from hyperbolic to Anti-de Sitter structures on a finite-volume three-manifold, with cone singularity on a link (the singularity arises from those edges whose dihedral angle is not right). A similar example, which gives a singularity along a θ-graph, is shown below: on the left, a video describing the collapse of a hyperbolic polytope to an ideal triangle, and on the right, a picture of the rescaled half-pipe limit.

In my joint paper with Stefano Riolo, "Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four", we prove the existence of similar phenomena in dimension four. The two above examples in three dimensions are somehow the "building blocks" of our four-dimensional construction, and more details are provided in Section 5 of our paper. See the ArXiv link of the paper if interested.

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