To a topologist the cube and dodecahedron are the same. Take the dodecahedron and fold along the black edges to identify pairs of faces. The result is a 3-sphere and the black edges have become the Borromean rings. By removing the rings we have a space homeomorphic to the complement of the rings.Now shrink the black edges in the dodecahedron to get a rhombic dodecahedron with 6 special points. Remove the points and identify pairs of rhombs to get another model for the complement of the Borromean rings. By choosing a suitable hyperbolic metric the 3 points go to infinity.

Here is a view from inside the dodecahedral model. The faces have been removed, edges have become beams, and the beams corresponding to the folding axes are coloured. The image is taken from the video Not Knot.

Here is Escher's view of the hyperbolic plane. Click on it to get a large version.