| bgcolor=#e7dcc3 colspan=2 | Order-4 square tiling honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | ||
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb | |
| Schläfli symbols | h ↔ | |
| Coxeter diagrams | ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Edge figure | ||
| Vertex figure | square tiling, | |
| Dual | Self-dual | |
| Coxeter groups | \overline{N}3 \overline{M}3 \widehat{RR}3 | |
| Properties | Regular, quasiregular |
The order-4 square tiling honeycomb has many reflective symmetry constructions: as a regular honeycomb, ↔ with alternating types (colors) of square tilings, and with 3 types (colors) of square tilings in a ratio of 2:1:1.
Two more half symmetry constructions with pyramidal domains have [4,4,1<sup>+</sup>,4] symmetry: ↔, and ↔ .
There are two high-index subgroups, both index 8: [4,4,4<sup>*</sup>] ↔ [(4,4,4,4,1<sup>+</sup>)], with a pyramidal fundamental domain: [((4,∞,4)),((4,∞,4))] or ; and [4,4<sup>*</sup>,4], with 4 orthogonal sets of ultra-parallel mirrors in an octahedral fundamental domain: .
The order-4 square tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling,, with infinite apeirogonal faces, and with all vertices on the ideal surface.
It contains and that tile 2-hypercycle surfaces, which are similar to these paracompact order-4 apeirogonal tilings :
The order-4 square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.
There are nine uniform honeycombs in the [4,4,4] Coxeter group family, including this regular form.
It is part of a sequence of honeycombs with a square tiling vertex figure:
It is part of a sequence of honeycombs with square tiling cells:
It is part of a sequence of quasiregular polychora and honeycombs:
| bgcolor=#e7dcc3 colspan=2 | Rectified order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | r or t1 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | cube | |
| Coxeter groups | \overline{N}3 \overline{M}3 | |
| Properties | Quasiregular or regular, depending on symmetry |
| bgcolor=#e7dcc3 colspan=2 | Truncated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t or t0,1 | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | square pyramid | |
| Coxeter groups | \overline{N}3 \overline{M}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Bitruncated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | 2t or t1,2 | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | tetragonal disphenoid | |
| Coxeter groups | 2 x \overline{N}3 \overline{M}3 \widehat{RR}3 | |
| Properties | Vertex-transitive, edge-transitive, cell-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | rr or t0,2 | |
| Coxeter diagrams | ||
| Cells | ||
| Faces | ||
| Vertex figure | triangular prism | |
| Coxeter groups | \overline{N}3 \overline{R}3 | |
| Properties | Vertex-transitive, edge-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | tr or t0,1,2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter groups | \overline{N}3 \overline{R}3 \overline{M}3 | |
| Properties | Vertex-transitive |
It is the same as the truncated square tiling honeycomb, .
| bgcolor=#e7dcc3 colspan=2 | Runcinated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t0,3 | |
| Coxeter diagrams | ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | square antiprism | |
| Coxeter groups | 2 x \overline{N}3 | |
| Properties | Vertex-transitive, edge-transitive |
| bgcolor=#e7dcc3 colspan=2 | Runcitruncated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t0,1,3 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | square pyramid | |
| Coxeter groups | \overline{N}3 | |
| Properties | Vertex-transitive |
The runcicantellated order-4 square tiling honeycomb is equivalent to the runcitruncated order-4 square tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Omnitruncated order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t0,1,2,3 | |
| Coxeter diagrams | ||
| Cells | ||
| Faces | ||
| Vertex figure | digonal disphenoid | |
| Coxeter groups | 2 x \overline{N}3 | |
| Properties | Vertex-transitive |
The alternated order-4 square tiling honeycomb is a lower-symmetry construction of the order-4 square tiling honeycomb itself.
The cantic order-4 square tiling honeycomb is a lower-symmetry construction of the truncated order-4 square tiling honeycomb.
The runcic order-4 square tiling honeycomb is a lower-symmetry construction of the order-3 square tiling honeycomb.
The runcicantic order-4 square tiling honeycomb is a lower-symmetry construction of the bitruncated order-4 square tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Quarter order-4 square tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | q | |
| Coxeter diagrams | ||
| Cells | ||
| Faces | ||
| Vertex figure | square antiprism | |
| Coxeter groups | \widehat{RR}3 | |
| Properties | Vertex-transitive, edge-transitive |