| bgcolor=#e7dcc3 colspan=2 | Order-6 cubic honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Perspective projection view within Poincaré disk model | |
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb | |
| Schläfli symbol | ||
| Coxeter diagram | ↔ ↔ | |
| Cells | ||
| Faces | ||
| Edge figure | ||
| Vertex figure | triangular tiling | |
| Coxeter group | \overline{BV}3 \overline{BP}3 | |
| Dual | Order-4 hexagonal tiling honeycomb | |
| Properties | Regular, quasiregular |
A half-symmetry construction of the order-6 cubic honeycomb exists as, with two alternating types (colors) of cubic cells. This construction has Coxeter-Dynkin diagram ↔ .
Another lower-symmetry construction, [4,3<sup>*</sup>,6], of index 6, exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram .
This honeycomb contains that tile 2-hypercycle surfaces, similar to the paracompact order-3 apeirogonal tiling, :
The order-6 cubic honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
It has a related alternation honeycomb, represented by ↔ . This alternated form has hexagonal tiling and tetrahedron cells.
There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including the order-6 cubic honeycomb itself.
The order-6 cubic honeycomb is part of a sequence of regular polychora and honeycombs with cubic cells.
It is also part of a sequence of honeycombs with triangular tiling vertex figures.
| bgcolor=#e7dcc3 colspan=2 | Rectified order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | r or t1 | |
| Coxeter diagrams | ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | hexagonal prism | |
| Coxeter groups | \overline{BV}3 \overline{DV}3 \overline{BP}3 \overline{DP}3 | |
| Properties | Vertex-transitive, edge-transitive |
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r, alternating apeirogonal and square faces:
| bgcolor=#e7dcc3 colspan=2 | Truncated order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t or t0,1 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | hexagonal pyramid | |
| Coxeter groups | \overline{BV}3 \overline{BP}3 | |
| Properties | Vertex-transitive |
It is similar to the 2D hyperbolic truncated infinite-order square tiling, t, with apeirogonal and octagonal (truncated square) faces:
The bitruncated order-6 cubic honeycomb is the same as the bitruncated order-4 hexagonal tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | rr or t0,2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | wedge | |
| Coxeter groups | \overline{BV}3 \overline{BP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | tr or t0,1,2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter groups | \overline{BV}3 \overline{BP}3 | |
| Properties | Vertex-transitive |
The runcinated order-6 cubic honeycomb is the same as the runcinated order-4 hexagonal tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | t0,1,3 | |
| Coxeter diagrams | ||
| Cells | ||
| Faces | ||
| Vertex figure | isosceles-trapezoidal pyramid | |
| Coxeter groups | \overline{BV}3 | |
| Properties | Vertex-transitive |
The runcicantellated order-6 cubic honeycomb is the same as the runcitruncated order-4 hexagonal tiling honeycomb.
The omnitruncated order-6 cubic honeycomb is the same as the omnitruncated order-4 hexagonal tiling honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Alternated order-6 cubic honeycomb | - | bgcolor=#ffffff align=center colspan=2 | --> |
|---|---|---|---|---|
| Type | Paracompact uniform honeycomb Semiregular honeycomb | |||
| Schläfli symbol | h | |||
| Coxeter diagram | ↔ ↔ ↔ ↔ | |||
| Cells | ||||
| Faces | ||||
| Vertex figure | trihexagonal tiling | |||
| Coxeter group | \overline{DV}3 \overline{DP}3 | |||
| Properties | Vertex-transitive, edge-transitive, quasiregular |
A half-symmetry construction from the form exists, with two alternating types (colors) of triangular tiling cells. This form has Coxeter-Dynkin diagram ↔ . Another lower-symmetry form of index 6, [4,3<sup>*</sup>,6], exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram .
The alternated order-6 cubic honeycomb is part of a series of quasiregular polychora and honeycombs.
It also has 3 related forms: the cantic order-6 cubic honeycomb, h2, ; the runcic order-6 cubic honeycomb, h3, ; and the runcicantic order-6 cubic honeycomb, h2,3, .
| bgcolor=#e7dcc3 colspan=2 | Cantic order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | h2 | |
| Coxeter diagram | ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | rectangular pyramid | |
| Coxeter group | \overline{DV}3 \overline{DP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Runcic order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | h3 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | triangular cupola | |
| Coxeter group | \overline{DV}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Runcicantic order-6 cubic honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | h2,3 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter group | \overline{DV}3 | |
| Properties | Vertex-transitive |