  The hyperbolic dodecahedral space of Weber and Seifert has a natural
non-positively curved cubulation obtained by subdividing the dodecahedron into
cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular
cover with the property that the canonical hypersurfaces made up of the
mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover
in which the associated cubulation is special. We also describe the natural
cubulation and covers of the spherical dodecahedral space (aka Poincar\'e
homology sphere).
