Order-6 hexagonal tiling honeycomb

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the triangular tiling is {3,6}, the vertex figure of this honeycomb is a triangular tiling. Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.

Related tilings
The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {&infin;,&infin;}, with infinite apeirogonal faces, and with all vertices on the ideal surface.
 * H2 tiling 2ii-4.png

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):
 * H2 tiling 23i-6.png H2-I-3-dual.svg

Symmetry


The order-6 hexagonal tiling honeycomb has a half-symmetry construction:.

It also has an index-6 subgroup, [6,3*,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:.

Related polytopes and honeycombs
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:

Rectified order-6 hexagonal tiling honeycomb
The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure.

it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔.



It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{&infin;,&infin;} with infinite apeirogonal faces, and with all vertices on the ideal surface.
 * H2 tiling 2ii-2.png

Related honeycombs
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{2p,4,2q}

Truncated order-6 hexagonal tiling honeycomb
The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.



Bitruncated order-6 hexagonal tiling honeycomb
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.



Cantellated order-6 hexagonal tiling honeycomb
The cantellated order-6 hexagonal tiling honeycomb, t0,2{6,3,6}, has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.



Cantitruncated order-6 hexagonal tiling honeycomb
The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2{6,3,6}, has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure.



Runcinated order-6 hexagonal tiling honeycomb
The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3,6}, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.



It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces:
 * H2 tiling 266-5.png

Runcitruncated order-6 hexagonal tiling honeycomb
The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3{6,3,6}, has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.



Omnitruncated order-6 hexagonal tiling honeycomb
The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6}, has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure.



Alternated order-6 hexagonal tiling honeycomb
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.

Cantic order-6 hexagonal tiling honeycomb
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.

Runcic order-6 hexagonal tiling honeycomb
The runcic hexagonal tiling honeycomb, h3{6,3,6},, or , has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure.

Runicantic order-6 hexagonal tiling honeycomb
The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6},, or , contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure.