Hexagonal tiling honeycomb


In the field of hyperbolic geometry, the hexagonal tiling honeycomb is 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 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 is, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is, the vertex figure of this honeycomb is a tetrahedron. Thus, six hexagonal tilings meet at each vertex of this honeycomb, and four edges meet at each vertex.

Images

Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere. In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, of H2, with horocycles circumscribing vertices of apeirogonal faces.
One hexagonal tiling cell of the hexagonal tiling honeycombAn order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: , , , ] and ], having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: ; or ; ; all of these are isomorphic to ]. The ringed Coxeter diagrams are,,, and, representing different types of hexagonal tilings in the Wythoff construction.

Related polytopes and honeycombs

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
It is one of 15 uniform paracompact honeycombs in the Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.
It is part of a sequence of regular polychora, which include the 5-cell, tesseract, and 120-cell of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
It is also part of a sequence of regular honeycombs of the form, which are each composed of hexagonal tiling cells:

Rectified hexagonal tiling honeycomb

The rectified hexagonal tiling honeycomb, t1, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternates two types of tetrahedra.

Truncated hexagonal tiling honeycomb

The truncated hexagonal tiling honeycomb, t0,1, has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t with apeirogonal and triangle faces:

Bitruncated hexagonal tiling honeycomb

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2, has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

Cantellated hexagonal tiling honeycomb

The cantellated hexagonal tiling honeycomb, t0,2, has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

Cantitruncated hexagonal tiling honeycomb

The cantitruncated hexagonal tiling honeycomb, t0,1,2, has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

Runcinated hexagonal tiling honeycomb

The runcinated hexagonal tiling honeycomb, t0,3, has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

Runcitruncated hexagonal tiling honeycomb

The runcitruncated hexagonal tiling honeycomb, t0,1,3, has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Runcicantellated hexagonal tiling honeycomb

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3, has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Omnitruncated hexagonal tiling honeycomb

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.