Order-4 dodecahedral honeycomb

In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

Description

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.

It has a half symmetry construction,, with two types of dodecahedra in the Wythoff construction. ↔.

A view of the order-4 dodecahedral honeycomb under the Beltrami-Klein model

There are four regular compact honeycombs in 3D hyperbolic space:
There are fifteen uniform honeycombs in the Coxeter group family, including this regular form.
There are eleven uniform honeycombs in the bifurcating Coxeter group family, including this honeycomb in its alternated form.
This construction can be represented by alternation with two colors of dodecahedral cells.
This honeycomb is also related to the 16-cell, cubic honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures:
This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

The rectified order-4 dodecahedral honeycomb,, has alternating octahedron and icosidodecahedron cells, with a square prism vertex figure.

There are four rectified compact regular honeycombs:

The truncated order-4 dodecahedral honeycomb,, has octahedron and truncated dodecahedron cells, with a square pyramid vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t with truncated pentagon and square faces:

The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb,, has truncated octahedron and truncated icosahedron cells, with a digonal disphenoid vertex figure.

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

The cantitruncated order-4 dodecahedral honeycomb,, has truncated icosidodecahedron, truncated octahedron, and cube cells, with a mirrored sphenoid vertex figure.

The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb.

The runcicantellated order-4 dodecahedral honeycomb is the same as the runcitruncated order-5 cubic honeycomb.

The omnitruncated order-4 dodecahedral honeycomb is the same as the omnitruncated order-5 cubic honeycomb.