Order-5 hexagonal tiling honeycomb

Order-5 hexagonal tiling honeycomb
File:H3 635 FC boundary.png Perspective projection view from center of Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{6,3,5}
Coxeter-Dynkin diagrams	Template:CDD File:CDel 635 index120.png ↔ Template:CDD
Cells	{6,3}
Faces	hexagon {6}
Edge figure	pentagon {5}
Vertex figure	icosahedron
Dual	Order-6 dodecahedral honeycomb
Coxeter group	${\overline{HV}}_3$, [5,3,6]
Properties	Regular

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. 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 order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.

Symmetry

A lower-symmetry construction of index 120, [6,(3,5)^*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.

Images

The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.

File:H2 tiling 25i-1.png

Related polytopes and honeycombs

The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. Template:Regular paracompact H3 honeycombs

There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb. Template:635 family

The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by Template:CDD ↔ Template:CDD, with icosahedron and triangular tiling cells.

It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets: Template:Hexagonal tiling cell tessellations

It is also part of a sequence of regular polychora and honeycombs with icosahedral vertex figures: Template:Icosahedral vertex figure tessellations

Rectified order-5 hexagonal tiling honeycomb

Rectified order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,5} or t1{6,3,5}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,5} File:Uniform polyhedron-53-t2.png r{6,3} or h2{6,3} File:Uniform tiling 63-t1.pngFile:Uniform polyhedron-63-t1-1.svg
Faces	triangle {3} hexagon {6}
Vertex figure	File:Rectified order-5 hexagonal tiling honeycomb verf.png pentagonal prism
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 hexagonal tiling honeycomb, t₁{6,3,5}, Template:CDD has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

Error creating thumbnail:

Truncated order-5 hexagonal tiling honeycomb

Truncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,5} or t0,1{6,3,5}
Coxeter diagram	Template:CDD
Cells	{3,5} File:Uniform polyhedron-53-t2.png t{6,3} Error creating thumbnail:
Faces	triangle {3} dodecagon {12}
Vertex figure	pentagonal pyramid
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The truncated order-5 hexagonal tiling honeycomb, t_0,1{6,3,5}, Template:CDD has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.

File:H3 635-1100.png Template:Clear

Bitruncated order-5 hexagonal tiling honeycomb

Bitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{6,3,5} or t1,2{6,3,5}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{3,6} File:Uniform tiling 63-t12.svg t{3,5} File:Uniform polyhedron-53-t12.png
Faces	pentagon {5} hexagon {6}
Vertex figure	File:Bitruncated order-5 hexagonal tiling honeycomb verf.png digonal disphenoid
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The bitruncated order-5 hexagonal tiling honeycomb, t_1,2{6,3,5}, Template:CDD has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.

File:H3 635-0110.png Template:Clear

Cantellated order-5 hexagonal tiling honeycomb

Cantellated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,5} or t0,2{6,3,5}
Coxeter diagram	Template:CDD
Cells	r{3,5} Error creating thumbnail: rr{6,3} File:Uniform tiling 63-t02.png {}x{5}
Faces	triangle {3} square {4} pentagon {5} hexagon {6}
Vertex figure	File:Cantellated order-5 hexagonal tiling honeycomb verf.png wedge
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The cantellated order-5 hexagonal tiling honeycomb, t_0,2{6,3,5}, Template:CDD has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.

File:H3 635-1010.png Template:Clear

Cantitruncated order-5 hexagonal tiling honeycomb

Cantitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,5} or t0,1,2{6,3,5}
Coxeter diagram	Template:CDD
Cells	t{3,5} File:Uniform polyhedron-53-t12.png tr{6,3} Error creating thumbnail: {}x{5}
Faces	square {4} pentagon {5} hexagon {6} dodecagon {12}
Vertex figure	File:Cantitruncated order-5 hexagonal tiling honeycomb verf.png mirrored sphenoid
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The cantitruncated order-5 hexagonal tiling honeycomb, t_0,1,2{6,3,5}, Template:CDD has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.

File:H3 635-1110.png

Runcinated order-5 hexagonal tiling honeycomb

Runcinated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,5}
Coxeter diagram	Template:CDD
Cells	{6,3} Error creating thumbnail: {5,3} File:Uniform polyhedron-53-t0.png {}x{6} Error creating thumbnail: {}x{5} Error creating thumbnail:
Faces	square {4} pentagon {5} hexagon {6}
Vertex figure	Error creating thumbnail: irregular triangular antiprism
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The runcinated order-5 hexagonal tiling honeycomb, t_0,3{6,3,5}, Template:CDD has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure.

File:H3 635-1001.png Template:Clear

Runcitruncated order-5 hexagonal tiling honeycomb

Runcitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,3{6,3,5}
Coxeter diagram	Template:CDD
Cells	t{6,3} rr{5,3} Error creating thumbnail: {}x{5} Error creating thumbnail: {}x{12} File:Dodecagonal prism.png
Faces	triangle {3} square {4} pentagon {5} dodecagon {12}
Vertex figure	File:Runcitruncated order-5 hexagonal tiling honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The runcitruncated order-5 hexagonal tiling honeycomb, t_0,1,3{6,3,5}, Template:CDD has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Error creating thumbnail: Template:Clear

Runcicantellated order-5 hexagonal tiling honeycomb

The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.

Omnitruncated order-5 hexagonal tiling honeycomb

Omnitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{6,3,5}
Coxeter diagram	Template:CDD
Cells	tr{6,3} File:Uniform tiling 63-t012.png tr{5,3} File:Uniform polyhedron-53-t012.png {}x{10} File:Decagonal prism.png {}x{12} File:Dodecagonal prism.png
Faces	square {4} hexagon {6} decagon {10} dodecagon {12}
Vertex figure	File:Omnitruncated order-5 hexagonal tiling honeycomb verf.png irregular tetrahedron
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The omnitruncated order-5 hexagonal tiling honeycomb, t_0,1,2,3{6,3,5}, Template:CDD has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure.

File:H3 635-1111.png Template:Clear

Alternated order-5 hexagonal tiling honeycomb

Alternated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	h{6,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	{3[3]} Error creating thumbnail: {3,5} File:Uniform polyhedron-53-t2.png
Faces	triangle {3}
Vertex figure	File:Uniform polyhedron-53-t12.png truncated icosahedron
Coxeter groups	${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive, edge-transitive, quasiregular

The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, Template:CDD ↔ Template:CDD, has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.

Cantic order-5 hexagonal tiling honeycomb

Cantic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2{6,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	h2{6,3} File:Uniform polyhedron-63-t1-1.svg t{3,5} File:Uniform polyhedron-53-t12.png r{5,3} Error creating thumbnail:
Faces	triangle {3} pentagon {5} hexagon {6}
Vertex figure	File:Cantic order-5 hexagonal tiling honeycomb verf.png triangular prism
Coxeter groups	${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The cantic order-5 hexagonal tiling honeycomb, h₂{6,3,5}, Template:CDD ↔ Template:CDD, has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.

Runcic order-5 hexagonal tiling honeycomb

Runcic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h3{6,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	{3[3]} Error creating thumbnail: rr{5,3} Error creating thumbnail: {5,3} File:Uniform polyhedron-53-t0.png {}x{3} File:Triangular prism.png
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	Error creating thumbnail: triangular cupola
Coxeter groups	${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The runcic order-5 hexagonal tiling honeycomb, h₃{6,3,5}, Template:CDD ↔ Template:CDD, has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.

Runcicantic order-5 hexagonal tiling honeycomb

Runcicantic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2,3{6,3,5}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	h2{6,3} File:Uniform polyhedron-63-t1-1.svg tr{5,3} File:Uniform polyhedron-53-t012.png t{5,3} Error creating thumbnail: {}x{3} File:Triangular prism.png
Faces	triangle {3} square {4} hexagon {6} decagon {10}
Vertex figure	File:Runcicantic order-5 hexagonal tiling honeycomb verf.png rectangular pyramid
Coxeter groups	${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The runcicantic order-5 hexagonal tiling honeycomb, h_2,3{6,3,5}, Template:CDD ↔ Template:CDD, has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, Template:Isbn (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition Template:Isbn (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, (2018) Chapter 13: Hyperbolic Coxeter groups