Order-5 hexagonal tiling honeycomb

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell consists of 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 order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.

Symmetry
A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.

Images
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {&infin;,5}, with five apeirogonal faces meeting around every vertex.
 * H2 tiling 25i-1.png

Related polytopes and honeycombs
The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb.

The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔, with icosahedron and triangular tiling cells.

It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets:

It is also part of a sequence of regular polychora and honeycombs with icosahedral vertex figures:

Rectified order-5 hexagonal tiling honeycomb
The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.



It is similar to the 2D hyperbolic infinite-order square tiling, r{&infin;,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
 * H2 tiling 25i-2.png

Truncated order-5 hexagonal tiling honeycomb
The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.



Bitruncated order-5 hexagonal tiling honeycomb
The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.



Cantellated order-5 hexagonal tiling honeycomb
The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.



Cantitruncated order-5 hexagonal tiling honeycomb
The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.



Runcinated order-5 hexagonal tiling honeycomb
The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure.



Runcitruncated order-5 hexagonal tiling honeycomb
The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.



Runcicantellated order-5 hexagonal tiling honeycomb
The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.

Omnitruncated order-5 hexagonal tiling honeycomb
The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure.



Alternated order-5 hexagonal tiling honeycomb
The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔, has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.

Cantic order-5 hexagonal tiling honeycomb
The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔, has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.

Runcic order-5 hexagonal tiling honeycomb
The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔, has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.

Runcicantic order-5 hexagonal tiling honeycomb
The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔, has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.