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 lower-symmetry construction of index 120, [6,(3,5)<sup>*</sup>], 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 â , 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, t<sub>1</sub>{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, t<sub>0,1</sub>{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.
The bitruncated order-5 hexagonal tiling honeycomb, t<sub>1,2</sub>{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.
The cantellated order-5 hexagonal tiling honeycomb, t<sub>0,2</sub>{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.
The cantitruncated order-5 hexagonal tiling honeycomb, t<sub>0,1,2</sub>{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, t<sub>0,3</sub>{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, t<sub>0,1,3</sub>{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, t<sub>0,1,2,3</sub>{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}, â , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.
The cantic order-5 hexagonal tiling honeycomb, h<sub>2</sub>{6,3,5}, â , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.
The runcic order-5 hexagonal tiling honeycomb, h<sub>3</sub>{6,3,5}, â , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.
The runcicantic order-5 hexagonal tiling honeycomb, h<sub>2,3</sub>{6,3,5}, â , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.