Hexagonal tiling honeycomb

Hexagonal tiling honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic 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}
FacesHexagon {6}
Edge figureTriangle {3}
Vertex figure
tetrahedron {3,3}
Dual{3,3,6}
Coxeter groups{\bar{V}}_3, [6,3,3]
{\bar{Y}}_3, [3,6,3]
{\bar{Z}}_3, [6,3,6]
{\bar{VP}}_3, [6,3[3]]
{\bar{PP}}_3, [3[3,3]]
PropertiesRegular

In the field of hyperbolic geometry, the hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. 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 hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling of the plane 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 an tetrahedron. Thus, six hexagonal tilings meet at each vertex of this honeycomb, and four edges meet at each vertex.[1]

Images

Viewed in perspective outside of a Poincaré disk model, this 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 horocycle circumscribing vertices of apeirogonal faces.

{6,3,3} {,3}
One hexagonal tiling of this honeycomb 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

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb, {3,3,6}.

Polytopes and honeycombs with tetrahedral vertex figures

It is in a sequence with regular polychora: 5-cell {3,3,3}, tesseract {4,3,3}, 120-cell {5,3,3} of Euclidean 4-space, with tetrahedral vertex figures.

Polytopes and honeycombs with hexagonal tiling cells

It is a part of sequence of regular honeycombs of the form {6,3,p}, with hexagonal tiling cells:

Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{6,3,3} or t1{6,3,3}
Coxeter diagrams
Cells{3,3}
r{6,3}
FacesTriangle {3}
Hexagon {6}
Vertex figure
Triangular prism {}×{3}
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t1{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure.

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

Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt{6,3,3} or t0,1{6,3,3}
Coxeter diagram
Cells{3,3}
t{6,3}
FacesTriangle {3}
Dodecagon {12}
Vertex figure
tetrahedron
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a tetrahedral 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
TypeParacompact uniform honeycomb
Schläfli symbol2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram
Cellst{3,3}
t{3,6}
{}×{3}
FacesTriangle {3}
hexagon {6}
Vertex figure
tetrahedron
Coxeter groups{\bar{V}}_3, [6,3,3]
{\bar{P}}_3, [3,3[3]]
PropertiesVertex-transitive

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

Cantellated hexagonal tiling honeycomb

Cantellated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolrr{6,3,3} or t0,2{6,3,3}
Coxeter diagram
Cellsr{3,3}
rr{6,3}
FacesTriangle {3}
Square {4}
Hexagon {6}
Vertex figure
Irreg. triangular prism
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has octahedral and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

Cantitruncated hexagonal tiling honeycomb

Cantitruncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symboltr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram
Cellst{3,3}
tr{6,3}
FacesTriangle {3}
Square {4}
Hexagon {6}
Vertex figure
Irreg. tetrahedron
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has truncated tetrahedron and truncated trihexagonal tiling cells, with a tetrahedron vertex figure.

Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,3{6,3,3}
Coxeter diagram
Cells{3,3}
t0,2{6,3}
{}×{6}
{}×{3}
FacesTriangle {3}
Square {4}
Hexagon {6}
Vertex figure
Octahedron
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has tetrahedron, rhombitrihexagonal tiling hexagonal prism, triangular prism cells, with a octahedron vertex figure.

Runcitruncated hexagonal tiling honeycomb

Runcitruncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,3{6,3,3}
Coxeter diagram
Cellsrr{3,3}
{}x{3}
{}x{12}
t{6,3}
FacesTriangle {3}
Square {4}
Hexagon {6}
Dodecagon {12}
Vertex figure
quad-pyramid
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has cuboctahedron, Triangular prism, Dodecagonal prism, and truncated hexagonal tiling cells, with a quad-pyramid vertex figure.

Runcicantellated hexagonal tiling honeycomb

Runcicantellated hexagonal tiling honeycomb
runcitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,2,3{6,3,3}
Coxeter diagram
Cellst{3,3}
{}x{6}
rr{6,3}
FacesTriangle {3}
Square {4}
Hexagon {6}
Vertex figure
quad-pyramid
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has truncated tetrahedron, hexagonal prism, hexagonal prism, and rhombitrihexagonal tiling cells, with a quad-pyramid vertex figure.

Omnitruncated hexagonal tiling honeycomb

Omnitruncated hexagonal tiling honeycomb
Omnitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,2,3{6,3,3}
Coxeter diagram
Cellstr{3,3}
{}x{6}
{}x{12}
tr{6,3}
FacesSquare {4}
Hexagon {6}
Dodecagon {12}
Vertex figure
tetrahedron
Coxeter groups{\bar{V}}_3, [6,3,3]
PropertiesVertex-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 a quad-pyramid vertex figure.

See also

References

  1. Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
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