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 surface in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, 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 (the horosphere incident with edge midpoints). 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, {&infin;,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.

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: [6,3,3],  [3,6,3],  [6,3,6],  [6,3[3]] and [3[3,3]], having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are, , , and , representing different types (colors) 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 [6,3,3] 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 {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} 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 {6,3,p}, which are each composed of hexagonal tiling cells:

Rectified hexagonal tiling honeycomb
The rectified hexagonal tiling honeycomb, t1{6,3,3}, 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{6,3,3}, 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{&infin;,3} with apeirogonal and triangle faces:
 * H2 tiling 23i-3.png

Bitruncated hexagonal tiling honeycomb
The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.



Cantellated hexagonal tiling honeycomb
The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, 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{6,3,3}, 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{6,3,3}, 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{6,3,3}, 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{6,3,3}, 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{6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.