Order-6 dodecahedral honeycomb

Order-6 dodecahedral honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{5,3,6} {5,3[3]}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD
Cells	{5,3}
Faces	pentagon {5}
Edge figure	hexagon {6}
Vertex figure	triangular tiling
Dual	Order-5 hexagonal tiling honeycomb
Coxeter group	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Regular, quasiregular

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

Symmetry

A half symmetry construction exists as Template:CDD with alternately colored dodecahedral cells.

Images

The model is cell-centered within the Poincaré disk model, with the viewpoint then placed at the origin.

The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface.

Related polytopes and honeycombs

The order-6 dodecahedral 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 [5,3,6] Coxeter group family, including this regular form, and its regular dual, the order-5 hexagonal tiling honeycomb. Template:635 family

The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: Template:Triangular tiling vertex figure tessellations small

It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells: Template:Dodecahedral tessellations small

Rectified order-6 dodecahedral honeycomb

Rectified order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{5,3,6} t1{5,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	r{5,3} {3,6}
Faces	triangle {3} pentagon {5}
Vertex figure	hexagonal prism
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-6 dodecahedral honeycomb, t₁{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure.

Perspective projection view within Poincaré disk model

It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces.

Truncated order-6 dodecahedral honeycomb

Truncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{5,3,6} t0,1{5,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{5,3} {3,6}
Faces	triangle {3} decagon {10}
Vertex figure	hexagonal pyramid
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The truncated order-6 dodecahedral honeycomb, t_0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure.

Template:Clear

Bitruncated order-6 dodecahedral honeycomb

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

Cantellated order-6 dodecahedral honeycomb

Cantellated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{5,3,6} t0,2{5,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	rr{5,3} rr{6,3} {}x{6}
Faces	triangle {3} square {4} pentagon {5} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The cantellated order-6 dodecahedral honeycomb, t_0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.

Template:Clear

Cantitruncated order-6 dodecahedral honeycomb

Cantitruncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{5,3,6} t0,1,2{5,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	tr{5,3} t{3,6} {}x{6}
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	mirrored sphenoid
Coxeter groups	${\overline{HV}}_3$, [5,3,6] ${\overline{HP}}_3$, [5,3[3]]
Properties	Vertex-transitive

The cantitruncated order-6 dodecahedral honeycomb, t_0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.

File:H3 635-0111.png Template:Clear

Runcinated order-6 dodecahedral honeycomb

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

Runcitruncated order-6 dodecahedral honeycomb

Runcitruncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{5,3,6}
Coxeter diagrams	Template:CDD
Cells	t{5,3} Error creating thumbnail: rr{6,3} Error creating thumbnail: {}x{10} File:Decagonal prism.png {}x{6} Error creating thumbnail:
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	File:Runcitruncated order-6 dodecahedral honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter groups	${\overline{HV}}_3$, [5,3,6]
Properties	Vertex-transitive

The runcitruncated order-6 dodecahedral honeycomb, t_0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 635-1011.png Template:Clear

Runcicantellated order-6 dodecahedral honeycomb

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

Omnitruncated order-6 dodecahedral honeycomb

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

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