| bgcolor=#e7dcc3 colspan=2 | Order-4 dodecahedral honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | ||
| Type | Hyperbolic regular honeycomb Uniform hyperbolic honeycomb | |
| Schläfli symbol | ||
| Coxeter diagram | ↔ | |
| Cells | (dodecahedron) | |
| Faces | (pentagon) | |
| Edge figure | (square) | |
| Vertex figure | octahedron | |
| Dual | Order-5 cubic honeycomb | |
| Coxeter group | ||
| Properties | Regular, Quasiregular honeycomb |
In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) 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.
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 (colors) 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 [5,3,4] Coxeter group family, including this regular form.
There are eleven uniform honeycombs in the bifurcating [5,3<sup>1,1</sup>] Coxeter group family, including this honeycomb in its alternated form. This construction can be represented by alternation (checkerboard) 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:
| bgcolor=#e7dcc3 colspan=2 | Rectified order-4 dodecahedral honeycomb | - | bgcolor=#ffffff align=center colspan=2 | --> |
|---|---|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |||
| Schläfli symbol | r r | |||
| Coxeter diagram | ↔ | |||
| Cells | ||||
| Faces | ||||
| Vertex figure | square prism | |||
| Coxeter group | \overline{BH}3 \overline{DH}3 | |||
| Properties | Vertex-transitive, edge-transitive |
There are four rectified compact regular honeycombs:
| bgcolor=#e7dcc3 colspan=2 | Truncated order-4 dodecahedral honeycomb | - | bgcolor=#ffffff align=center colspan=2 | --> |
|---|---|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |||
| Schläfli symbol | t t | |||
| Coxeter diagram | ↔ | |||
| Cells | ||||
| Faces | ||||
| Vertex figure | square pyramid | |||
| Coxeter group | \overline{BH}3 \overline{DH}3 | |||
| Properties | Vertex-transitive |
It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t with truncated pentagon and square faces:
| bgcolor=#e7dcc3 colspan=2 | Bitruncated order-4 dodecahedral honeycomb Bitruncated order-5 cubic honeycomb | - | bgcolor=#ffffff align=center colspan=2 | --> |
|---|---|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |||
| Schläfli symbol | 2t 2t | |||
| Coxeter diagram | ↔ | |||
| Cells | ||||
| Faces | ||||
| Vertex figure | digonal disphenoid | |||
| Coxeter group | \overline{BH}3 \overline{DH}3 | |||
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-4 dodecahedral honeycomb | |
|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |
| Schläfli symbol | rr rr | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | wedge | |
| Coxeter group | \overline{BH}3 \overline{DH}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-4 dodecahedral honeycomb | |
|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |
| Schläfli symbol | tr tr | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter group | \overline{BH}3 \overline{DH}3 | |
| Properties | Vertex-transitive |
The runcinated order-4 dodecahedral honeycomb is the same as the runcinated order-5 cubic honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Runcitruncated order-4 dodecahedral honeycomb | |
|---|---|---|
| Type | Uniform honeycombs in hyperbolic space | |
| Schläfli symbol | t0,1,3 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | ||
| Vertex figure | isosceles-trapezoidal pyramid | |
| Coxeter group | \overline{BH}3 | |
| Properties | Vertex-transitive |
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.