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	{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]]
Properties	Regular

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

Subgroup relations

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.

11 paracompact regular honeycombs  {6,3,3} {6,3,4} {6,3,5} {6,3,6} {4,4,3} {4,4,4} {3,3,6} {4,3,6} {5,3,6} {3,6,3} {3,4,4}

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}.

[6,3,3] family honeycombs  {6,3,3} t1{6,3,3} t0,1{6,3,3} t0,2{6,3,3} t0,3{6,3,3} t0,1,2{6,3,3} t0,1,3{6,3,3} t0,1,2,3{6,3,3} {3,3,6} t1{3,3,6} t0,1{3,3,6} t0,2{3,3,6} t1,2{3,3,6} t0,1,2{3,3,6} t0,1,3{3,3,6} t0,1,2,3{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.

{p,3,3} honeycombs  Space S^3 H^3 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}

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:

{6,3,p} honeycombs  {6,3,p} Space H^3 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}
Faces	Triangle {3} Hexagon {6}
Vertex figure	Triangular prism {}×{3}
Coxeter groups	${\bar{V}}_3$, [6,3,3]
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.

Hexagonal tiling honeycomb	Rectified hexagonal tiling honeycomb
Related H^2 tilings
Order-3 apeirogonal tiling	Triapeirogonal tiling

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	tetrahedron
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The truncated hexagonal tiling honeycomb, t_0,1{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
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} {}×{3}
Faces	Triangle {3} hexagon {6}
Vertex figure	tetrahedron
Coxeter groups	${\bar{V}}_3$, [6,3,3] ${\bar{P}}_3$, [3,3^[3]]
Properties	Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,2{6,3,3}, has truncated tetrahedra and hexagonal tiling cells, with a tetrahedral 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}
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure	Irreg. triangular prism
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The cantellated hexagonal tiling honeycomb, t_0,2{6,3,3}, has octahedral and rhombitrihexagonal tiling cells, with a triangular prism 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}
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure	Irreg. tetrahedron
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t_0,1,2{6,3,3}, has truncated tetrahedron and truncated trihexagonal tiling cells, with a tetrahedron vertex figure.

Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,3}
Coxeter diagram
Cells	{3,3} t0,2{6,3} {}×{6} {}×{3}
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure	Octahedron
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The runcinated hexagonal tiling honeycomb, t_0,3{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
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} Hexagon {6} Dodecagon {12}
Vertex figure	quad-pyramid
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t_0,1,3{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
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	quad-pyramid
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,3{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
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	tetrahedron
Coxeter groups	${\bar{V}}_3$, [6,3,3]
Properties	Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,3{6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with a quad-pyramid vertex figure.

See also

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

References

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

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130.