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. 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.
Related tilings
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 :
Symmetry
The order-6 hexagonal tiling honeycomb has a half-symmetry construction:.It also has an index-6 subgroup,, 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 List of [regular polytopes#Tessellations of hyperbolic 3-space|regular hyperbolic honeycomb] in 3-space, and one of eleven paracompact honeycombs in 3-space.There are nine uniform honeycombs in the 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, 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, ↔.
It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r with infinite apeirogonal faces, and with all vertices on the ideal surface.
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
Truncated order-6 hexagonal tiling honeycomb
The truncated order-6 hexagonal tiling honeycomb, t0,1, 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, 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, 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, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr, with square and hexagonal faces: