Order-4 dodecahedral honeycomb

The dihedral angle of a regular dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space.

However in hyperbolic space a properly-scaled regular dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.

The cantellated order-4 dodecahedral honeycomb, , has rhombicosidodecahedron, cuboctahedron, and cube cells, with a wedge vertex figure.

The runcitruncated order-4 dodecahedral honeycomb, , has truncated dodecahedron, rhombicuboctahedron, decagonal prism, and cube cells, with an isosceles-trapezoidal pyramid vertex figure.

It can be seen as analogous to the 2D hyperbolic order-4 pentagonal tiling , {5,4}
It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling , r{5,4}