| bgcolor=#e7dcc3 colspan=2 | Order-4 octahedral honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Perspective projection view within Poincaré disk model | |
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb | |
| Schläfli symbols | ||
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Edge figure | ||
| Vertex figure | square tiling, | |
| Dual | Square tiling honeycomb, | |
| Coxeter groups | \overline{R}3 \overline{O}3 | |
| Properties | Regular |
A half symmetry construction, [3,4,4,1<sup>+</sup>], exists as, with two alternating types (colors) of octahedral cells: ↔ .
A second half symmetry is [3,4,1<sup>+</sup>,4]: ↔ .
A higher index sub-symmetry, [3,4,4<sup>*</sup>], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .
This honeycomb contains and that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings and, respectively:
The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.
There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.
It is a part of a sequence of honeycombs with a square tiling vertex figure:
It a part of a sequence of regular polychora and honeycombs with octahedral cells:
| bgcolor=#e7dcc3 colspan=2 | Rectified order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | r or t1 | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | r | |
| Faces | ||
| Vertex figure | square prism | |
| Coxeter groups | \overline{R}3 \overline{O}3 | |
| Properties | Vertex-transitive, edge-transitive |
| bgcolor=#e7dcc3 colspan=2 | Truncated order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t or t0,1 | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | t | |
| Faces | ||
| Vertex figure | square pyramid | |
| Coxeter groups | \overline{R}3 \overline{O}3 | |
| Properties | Vertex-transitive |
The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | rr or t0,2 s2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | wedge | |
| Coxeter groups | \overline{R}3 \overline{O}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | tr or t0,1,2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter groups | \overline{R}3 \overline{O}3 | |
| Properties | Vertex-transitive |
The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Runcitruncated order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t0,1,3 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | square pyramid | |
| Coxeter groups | \overline{R}3 | |
| Properties | Vertex-transitive |
The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.
The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Snub order-4 octahedral honeycomb | |
|---|---|---|
| Type | Paracompact scaliform honeycomb | |
| Schläfli symbols | s | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | square tiling icosahedron square pyramid | |
| Faces | ||
| Vertex figure | ||
| Coxeter groups | [4,4,3<sup>+</sup>] [4<sup>1,1</sup>,3<sup>+</sup>] [(4,4,(3,3)<sup>+</sup>)] | |
| Properties | Vertex-transitive |