Order-6 dodecahedral honeycomb

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.

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	↔
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

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.

Symmetry

A half symmetry construction exists as 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.

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}

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.

[6,3,5] family honeycombs
{6,3,5}	r{6,3,5}	t{6,3,5}	rr{6,3,5}	t_0,3{6,3,5}	tr{6,3,5}	t_0,1,3{6,3,5}	t_0,1,2,3{6,3,5}


{5,3,6}	r{5,3,6}	t{5,3,6}	rr{5,3,6}	2t{5,3,6}	tr{5,3,6}	t_0,1,3{5,3,6}	t_0,1,2,3{5,3,6}

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

Hyperbolic uniform honeycombs: {p,3,6}
Form	Paracompact				Noncompact
Name	{3,3,6}	{4,3,6}	{5,3,6}	{6,3,6}	{7,3,6}	{8,3,6}	... {∞,3,6}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:

{5,3,p} polytopes
Space	S³	H³
Form	Finite	Compact		Paracompact	Noncompact
Name	{5,3,3}	{5,3,4}	{5,3,5}	{5,3,6}	{5,3,7}	{5,3,8}	... {5,3,∞}
Image
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

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	↔
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.

r{p,3,6}
Space	H³
Form	Paracompact				Noncompact
Name	r{3,3,6}	r{4,3,6}	r{5,3,6}	r{6,3,6}	r{7,3,6}	... r{∞,3,6}
Image
Cells {3,6}	r{3,3}	r{4,3}	r{5,3}	r{6,3}	r{7,3}	r{∞,3}

Truncated order-6 dodecahedral honeycomb

Truncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{5,3,6} t_0,1{5,3,6}
Coxeter diagrams	↔
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.

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} t_0,2{5,3,6}
Coxeter diagrams	↔
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.

Cantitruncated order-6 dodecahedral honeycomb

Cantitruncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{5,3,6} t_0,1,2{5,3,6}
Coxeter diagrams	↔
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.

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	t_0,1,3{5,3,6}
Coxeter diagrams
Cells	t{5,3} rr{6,3} {}x{10} {}x{6}
Faces	square {4} hexagon {6} decagon {10}
Vertex figure	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.

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.

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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 (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

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