Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
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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.

Images

320px

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}
	240px
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.

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.

{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 subgroups 1 4 6 12 24 File:Cdel tet4 1111.png File:Cdel tetinfin 1111.png 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
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							Error creating thumbnail: File with dimensions greater than 25 MP
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
320px
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	60px 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. The half-symmetry construction alternate two types of tetrahedra.

Hexagonal tiling honeycomb	Rectified hexagonal tiling honeycomb or
200px	200px
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	60px 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	60px 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	60px 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	60px 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	60px 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	60px 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	60px 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	60px 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

• 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 [1] [2]
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [3]

• Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III