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Hexagonal tiling honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	6,3,3 t3,6,3 2t6,3,6 2t6,3[3] t3[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	V¯3displaystyle bar V_3, [6,3,3] Y¯3displaystyle bar Y_3, [3,6,3] Z¯3displaystyle bar Z_3, [6,3,6] VP¯3displaystyle bar VP_3, [6,3[3]] PP¯3displaystyle 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	r6,3,3	t6,3,3	rr6,3,3	t0,36,3,3	tr6,3,3	t0,1,36,3,3	t0,1,2,36,3,3
3,3,6	r3,3,6	t3,3,6	rr3,3,6	2t3,3,6	tr3,3,6	t0,1,33,3,6	t0,1,2,33,3,6

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	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 [ v t e ]
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	r6,3,3 or t16,3,3
Coxeter diagrams	↔
Cells	3,3 r6,3 or
Faces	Triangle 3 Hexagon 6
Vertex figure	Triangular prism ×3
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t₁6,3,3, 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
Related H2 tilings
Order-3 apeirogonal tiling	Triapeirogonal tiling or

Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t6,3,3 or t0,16,3,3
Coxeter diagram
Cells	3,3 t6,3
Faces	Triangle 3 Dodecagon 12
Vertex figure	tetrahedron
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The truncated hexagonal tiling honeycomb, t_0,16,3,3, 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	2t6,3,3 or t1,26,3,3
Coxeter diagram	↔
Cells	t3,3 t3,6
Faces	Triangle 3 hexagon 6
Vertex figure	tetrahedron
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3] P¯3displaystyle bar P_3, [3,3[3]]
Properties	Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t_1,26,3,3, 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	rr6,3,3 or t0,26,3,3
Coxeter diagram
Cells	r3,3 rr6,3 ×3
Faces	Triangle 3 Square 4 Hexagon 6
Vertex figure	Irreg. triangular prism
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The cantellated hexagonal tiling honeycomb, t_0,26,3,3, 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	tr6,3,3 or t0,1,26,3,3
Coxeter diagram
Cells	t3,3 tr6,3 ×3
Faces	Triangle 3 Square 4 Hexagon 6
Vertex figure	Irreg. tetrahedron
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t_0,1,26,3,3, 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	t0,36,3,3
Coxeter diagram
Cells	3,3 t0,26,3 ×6 ×3
Faces	Triangle 3 Square 4 Hexagon 6
Vertex figure	Octahedron
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The runcinated hexagonal tiling honeycomb, t_0,36,3,3, 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	t0,1,36,3,3
Coxeter diagram
Cells	rr3,3 x3 x12 t6,3
Faces	Triangle 3 Square 4 Hexagon 6 Dodecagon 12
Vertex figure	quad-pyramid
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t_0,1,36,3,3, 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	t0,2,36,3,3
Coxeter diagram
Cells	t3,3 x6 rr6,3
Faces	Triangle 3 Square 4 Hexagon 6
Vertex figure	quad-pyramid
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t_0,2,36,3,3, 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	t0,1,2,36,3,3
Coxeter diagram
Cells	tr3,3 x6 x12 tr6,3
Faces	Square 4 Hexagon 6 Dodecagon 12
Vertex figure	tetrahedron
Coxeter groups	V¯3displaystyle bar V_3, [6,3,3]
Properties	Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t_0,1,2,36,3,3, 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. .mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  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]
  

External links

• 
  John Baez, Visual Insight: 6,3,3 Honeycomb (2014/03/15)


• 
  John Baez, Visual Insight: 6,3,3 Honeycomb in Upper Half Space (2013/09/15)


• 
  John Baez, Visual Insight: Truncated 6,3,3 Honeycomb (2016/12/01)