Order-4 hexagonal tiling honeycomb

Order-4 hexagonal tiling honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{6,3,4} {6,3^1,1} t_0,1{(3,6)²}
Coxeter diagrams	↔ ↔ ↔
Cells	{6,3}
Faces	hexagon {6}
Edge figure	square {4}
Vertex figure	octahedron, {3,4}
Dual	Order-6 cubic honeycomb
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1] [(6,3)^[2]]
Properties	Regular, quasiregular honeycomb

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as 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.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, 8 hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.^[1]

Images

Perspective projection
The vertices of a t{(3,∞,3)}, tiling exists as a 2-hypercycle within this honeycomb	It is analogous to the H² order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle

Symmetry

Subgroup relations

It has three reflective simplex symmetry construction. The uniform construction {6,3^1,1} has two types (colors) of hexagonal tilings in the Wythoff construction. ↔ A quarter symmetry construction can have four colors of hexagonal tilings: .

An additional two reflective symmetries exist with nonsimplex fundamental domains:Coxeter notation: [6,3^*,4], index 6, , and [6,(3,4)^*], index 48, with a cube fundamental domain, and octahedral Coxeter diagram with three axial infinite branches: . It can be seen with 8 colors of hexagonal tilings.

This honeycomb contains that tile 2-hypercycle surfaces, similar to this paracompact tilings, :

Related polytopes and honeycombs

Regular 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}

634 honeycombs

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb, {4,3,6}.

[6,3,4] family honeycombs
{6,3,4}	r{6,3,4}	t{6,3,4}	rr{6,3,4}	t_0,3{6,3,4}	tr{6,3,4}	t_0,1,3{6,3,4}	t_0,1,2,3{6,3,4}


{4,3,6}	r{4,3,6}	t{4,3,6}	rr{4,3,6}	2t{4,3,6}	tr{4,3,6}	t_0,1,3{4,3,6}	t_0,1,2,3{4,3,6}

Quasiregular honeycombs

It has a related alternation honeycomb, represented by ↔ , having triangular tiling and octahedron cells.

Regular and Quasiregular honeycombs: {p,3,4} and {p,3^1,1}
Space	Euclidean 4-space	Euclidean 3-space	Hyperbolic 3-space
Name	{3,3,4} {3,3^1,1} = $\left\{3,{3 \atop 3}\right\}$	{4,3,4} {4,3^1,1} = $\left\{4,{3 \atop 3}\right\}$	{5,3,4} {5,3^1,1} = $\left\{5,{3 \atop 3}\right\}$	{6,3,4} {6,3^1,1} = $\left\{6,{3 \atop 3}\right\}$
Coxeter diagram	=	=	=	=
Image
Cells {p,3}

Hexagonal tiling cells

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

{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,∞}

Octahedral vertex figures

This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb all which have octahedral vertex figures.

{p,3,4} regular honeycombs
Space	S³	E³	H³
Form	Finite	Affine	Compact	Paracompact	Noncompact
Name	{3,3,4}	{4,3,4}	{5,3,4}	{6,3,4}	{7,3,4}	{8,3,4}	... {∞,3,4}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

Rectified order-4 hexagonal tiling honeycomb

Rectified order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,4} or t₁{6,3,4}
Coxeter diagrams	↔ ↔ ↔
Cells	{3,4} r{6,3}
Faces	Triangle {3} Hexagon {6}
Vertex figure	Square prism {}×{4}
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1] [4,3^[3]] [3^{[ ]×[3]}]
Properties	Vertex-transitive, edge-transitive

The rectified order-4 hexagonal tiling honeycomb, t₁{6,3,4}, has octahedral and trihexagonal tiling facets, with a square prism vertex figure.

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4}, which alternates apeirogonal and square faces:

Truncated order-4 hexagonal tiling honeycomb

Truncated order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,4} or t_0,1{6,3,4}
Coxeter diagram	↔
Cells	{3,4} t{6,3}
Faces	Triangle {3} Dodecagon {12}
Vertex figure	square pyramid
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1]
Properties	Vertex-transitive

The truncated order-4 hexagonal tiling honeycomb, t_0,1{6,3,4}, has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure.

It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling, t{∞,4}, with apeirogonal and square faces:

Bitruncated order-4 hexagonal tiling honeycomb

Bitruncated order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{6,3,4} or t_1,2{6,3,4}
Coxeter diagram	↔ ↔ ↔
Cells	t{4,3} t{3,6} t{3,6}
Faces	Triangle {3} hexagon {6} octagon {8}
Vertex figure	tetrahedron
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1] [4,3^[3]] [3^{[ ]×[3]}]
Properties	Vertex-transitive

The bitruncated order-4 hexagonal tiling honeycomb, t_1,2{6,3,4}, has Truncated octahedron and hexagonal tiling cells, with a tetrahedral vertex figure.

Cantellated order-4 hexagonal tiling honeycomb

Cantellated order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,4} or t_0,2{6,3,4}
Coxeter diagram	↔
Cells	r{3,4} rr{6,3}
Faces	Triangle {3} square {4} hexagon {6}
Vertex figure	triangular prism
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1]
Properties	Vertex-transitive

The cantellated order-4 hexagonal tiling honeycomb, t_0,2{6,3,4}, has cuboctahedron and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

Runcinated order-4 hexagonal tiling honeycomb

Runcinated order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,3{6,3,4}
Coxeter diagram	↔
Cells	{4,3} {6,3} {}x{6}
Faces	Triangle {3} square {4} hexagon {6}
Vertex figure	triangular antiprism
Coxeter groups	BV₃, [6,3,4] DV₃, [6,3^1,1]
Properties	Vertex-transitive

The runcinated order-4 hexagonal tiling honeycomb, t_0,3{6,3,4}, has cube, hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.

It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6}, with square and hexagonal faces. It also has a half symmetry construction .

	=

Omnitruncated order-4 hexagonal tiling honeycomb

Omnitruncated order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t_0,1,2,3{6,3,4}
Coxeter diagram
Cells	tr{4,3} tr{6,3} {}x{6} {4,3}
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	tetrahedron
Coxeter groups	BV₃, [6,3,4]
Properties	Vertex-transitive

The omnitruncated order-4 hexagonal tiling honeycomb, t_0,1,2,3{6,3,4}, has truncated cuboctahedron, truncated trihexagonal tiling, hexagonal prism, and cube cells, with a tetrahedron vertex figure.

Alternated order-4 hexagonal tiling honeycomb

Alternated order-4 hexagonal tiling honeycomb

Type	Paracompact uniform honeycomb
Schläfli symbols	h{6,3,4}
Coxeter diagrams	↔ ↔
Cells
Faces	Triangle {3} Hexagon {6}
Vertex figure	truncated octahedron
Coxeter groups	BV₃, [6,3,4]
Properties	Vertex-transitive, edge-transitive

Quarter order-4 hexagonal tiling honeycomb

Quarter order-4 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	q{6,3,4}
Coxeter diagram	↔
Cells	{3,6} {3,3} t{3,3} rr{3,6}
Faces	{3}, {6}
Vertex figure	Triangular cupola
Coxeter groups	${\bar {DP}}_{3}$ , [3^{[ ]x[ ]}]
Properties	Vertex-transitive

The quarter order-4 hexagonal tiling honeycomb, q{6,3,4}, or with a triangular cupola vertex figure.

References

↑ 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 (Chapter 16-17: Geometries on Three-manifolds I,II)
Norman Johnson Uniform Polytopes, Manuscript
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- N.W. Johnson: Geometries and Transformations, (2015) Chapter 13: Hyperbolic Coxeter groups

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