# Hexagonal tiling honeycomb

Hexagonal tiling honeycomb

Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {6,3,3}
t{3,6,3}
2t{6,3,6}
2t{6,3[3]}
t{3[3,3]}
Coxeter diagrams

Cells {6,3}
Faces hexagon {6}
Edge figure triangle {3}
Vertex figure
tetrahedron {3,3}
Dual Order-6 tetrahedral honeycomb
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
${\displaystyle {\overline {Y}}_{3}}$, [3,6,3]
${\displaystyle {\overline {Z}}_{3}}$, [6,3,6]
${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
${\displaystyle {\overline {PP}}_{3}}$, [3[3,3]]
Properties Regular

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.[1]

## 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, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.

{6,3,3} {∞,3}
One hexagonal tiling cell of the hexagonal tiling honeycomb An 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: [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.

[6,3,3] family honeycombs
{6,3,3} r{6,3,3} t{6,3,3} rr{6,3,3} t0,3{6,3,3} tr{6,3,3} t0,1,3{6,3,3} t0,1,2,3{6,3,3}
{3,3,6} r{3,3,6} t{3,3,6} rr{3,3,6} 2t{3,3,6} tr{3,3,6} t0,1,3{3,3,6} t0,1,2,3{3,3,6}

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.

{p,3,3} honeycombs
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {∞,3,3}
Image
Coxeter diagrams
1
4
6
12
24
Cells
{p,3}

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{∞,3}

It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:

{6,3,p} honeycombs
Space H3
Form Paracompact Noncompact
Name {6,3,3} {6,3,4} {6,3,5} {6,3,6} {6,3,7} {6,3,8} ... {6,3,∞}
Coxeter

Image
Vertex
figure
{3,p}

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,∞}

### Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,3} or t1{6,3,3}
Coxeter diagrams
Cells {3,3}
r{6,3} or
Faces triangle {3}
hexagon {6}
Vertex figure
triangular prism
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
${\displaystyle {\overline {P}}_{3}}$, [3,3[3]]
Properties Vertex-transitive, edge-transitive

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.

Hexagonal tiling honeycomb
Rectified hexagonal tiling honeycomb
or
Related H2 tilings
Order-3 apeirogonal tiling
Triapeirogonal tiling
or

### Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,3} or t0,1{6,3,3}
Coxeter diagram
Cells {3,3}
t{6,3}
Faces triangle {3}
dodecagon {12}
Vertex figure
triangular pyramid
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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{∞,3} with apeirogonal and triangle faces:

### Bitruncated hexagonal tiling honeycomb

Bitruncated hexagonal tiling honeycomb
Bitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram
Cells t{3,3}
t{3,6}
Faces triangle {3}
hexagon {6}
Vertex figure
digonal disphenoid
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
${\displaystyle {\overline {P}}_{3}}$, [3,3[3]]
Properties Vertex-transitive

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

Cantellated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,3} or t0,2{6,3,3}
Coxeter diagram
Cells r{3,3}
rr{6,3}
{}×{3}
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure
wedge
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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

Cantitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram
Cells t{3,3}
tr{6,3}
{}×{3}
Faces triangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure
mirrored sphenoid
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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

Runcinated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,3}
Coxeter diagram
Cells {3,3}
{6,3}
{}×{6}
{}×{3}
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure
irregular triangular antiprism
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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

Runcitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,3{6,3,3}
Coxeter diagram
Cells rr{3,3}
{}x{3}
{}x{12}
t{6,3}
Faces triangle {3}
square {4}
dodecagon {12}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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

Runcicantellated hexagonal tiling honeycomb
runcitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,2,3{6,3,3}
Coxeter diagram
Cells t{3,3}
{}x{6}
rr{6,3}
Faces triangle {3}
square {4}
hexagon {6}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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

Omnitruncated hexagonal tiling honeycomb
Omnitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,3}
Coxeter diagram
Cells tr{3,3}
{}x{6}
{}x{12}
tr{6,3}
Faces square {4}
hexagon {6}
dodecagon {12}
Vertex figure
irregular tetrahedron
Coxeter groups ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties Vertex-transitive

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.