The Poincaré homology sphere

This illusion (from the Chicago museum of illusions, and duplicated, I believe, in other similar museums in other cities) “almost” appears to give a tiling of \(\mathbf{R}^3\) by regular dodecahedra, which for a number of reasons is not possible. (It also looks remarkably like knot not.) Moving one’s point of view slightly, one can observe that the interior faces don’t quite match up. But can this be remedied with curved mirrors? That is, is there a way to shape these mirrors so that one can is looking inside the Poincaré dodecahedral space?