| bgcolor=#e7dcc3 colspan=2 | Order-6 hexagonal tiling honeycomb | |
|---|---|---|
Perspective projection view from center of Poincaré disk model | ||
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb | |
| Schläfli symbol | ||
| Coxeter diagram | ↔ ↔ | |
| Cells | ||
| Faces | ||
| Edge figure | ||
| Vertex figure | or | |
| Dual | Self-dual | |
| Coxeter group | \overline{Z}3 \overline{VP}3 | |
| Properties | Regular, quasiregular |
The Schläfli symbol of the hexagonal tiling honeycomb is . Since that of the hexagonal tiling of the plane is, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.[1]
The order-6 hexagonal 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 the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively):
The order-6 hexagonal tiling honeycomb has a half-symmetry construction: .
It also has an index-6 subgroup, [6,3<sup>*</sup>,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: .
The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.
There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.
This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: ↔ .
The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells:
It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:
| bgcolor=#e7dcc3 colspan=2 | Rectified order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | r or t1 | |
| Coxeter diagrams | ↔ ↔ ↔ ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | hexagonal prism | |
| Coxeter groups | \overline{Z}3 \overline{VP}3 \overline{PP}3 | |
| Properties | Vertex-transitive, edge-transitive |
it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q, ↔ .
It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r with infinite apeirogonal faces, and with all vertices on the ideal surface.
The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:
It is also part of a matrix of 3-dimensional quarter honeycombs: q
| bgcolor=#e7dcc3 colspan=2 | Truncated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | t or t0,1 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | hexagonal pyramid | |
| Coxeter groups | \overline{Z}3 \overline{VP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Bitruncated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | bt or t1,2 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | tetrahedron | |
| Coxeter groups | 2 x \overline{Z}3 \overline{VP}3 \overline{V}3 | |
| Properties | Regular |
The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . It contains hexagonal tiling facets, with a tetrahedron vertex figure.
| bgcolor=#e7dcc3 colspan=2 | Cantellated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | rr or t0,2 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | wedge | |
| Coxeter groups | \overline{Z}3 \overline{VP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | tr or t0,1,2 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | mirrored sphenoid | |
| Coxeter groups | \overline{Z}3 \overline{VP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Runcinated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | t0,3 | |
| Coxeter diagram | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | triangular antiprism | |
| Coxeter groups | 2 x \overline{Z}3 | |
| Properties | Vertex-transitive, edge-transitive |
It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr, with square and hexagonal faces:
| bgcolor=#e7dcc3 colspan=2 | Runcitruncated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | t0,1,3 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | ||
| Vertex figure | isosceles-trapezoidal pyramid | |
| Coxeter groups | \overline{Z}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Omnitruncated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbol | t0,1,2,3 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | ||
| Vertex figure | phyllic disphenoid | |
| Coxeter groups | 2 x \overline{Z}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Alternated order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | h | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | hexagonal tiling | |
| Coxeter groups | \overline{VP}3 | |
| Properties | Regular, quasiregular |
The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, ↔ . It contains triangular tiling facets in a hexagonal tiling vertex figure.
| bgcolor=#e7dcc3 colspan=2 | Cantic order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | h2 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | triangular prism | |
| Coxeter groups | \overline{VP}3 | |
| Properties | Vertex-transitive, edge-transitive |
The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, ↔, with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure.
| bgcolor=#e7dcc3 colspan=2 | Runcic order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | h3 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | triangular cupola | |
| Coxeter groups | \overline{VP}3 | |
| Properties | Vertex-transitive |
| bgcolor=#e7dcc3 colspan=2 | Runcicantic order-6 hexagonal tiling honeycomb | |
|---|---|---|
| Type | Paracompact uniform honeycomb | |
| Schläfli symbols | h2,3 | |
| Coxeter diagrams | ↔ | |
| Cells | ||
| Faces | ||
| Vertex figure | rectangular pyramid | |
| Coxeter groups | \overline{VP}3 | |
| Properties | Vertex-transitive |