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.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
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.
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.
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:
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{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
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.
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.
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.
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.
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.
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.
The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral 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.
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.
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.
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.
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.