Order-6 dodecahedral honeycomb

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

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.

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}	t0,3{6,3,5}	tr{6,3,5}	t0,1,3{6,3,5}	t0,1,2,3{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}	t0,1,3{5,3,6}	t0,1,2,3{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	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6] ${\displaystyle {\overline {HP}}_{3}}$, [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} t0,1{5,3,6}
Coxeter diagrams	↔
Cells	t{5,3} {3,6}
Faces	triangle {3} decagon {10}
Vertex figure	hexagonal pyramid
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6] ${\displaystyle {\overline {HP}}_{3}}$, [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.

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	↔
Cells	rr{5,3} rr{6,3} {}x{6}
Faces	triangle {3} square {4} pentagon {5} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6] ${\displaystyle {\overline {HP}}_{3}}$, [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} t0,1,2{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	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6] ${\displaystyle {\overline {HP}}_{3}}$, [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.

Runcitruncated order-6 dodecahedral honeycomb

Runcitruncated order-6 dodecahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{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	${\displaystyle {\overline {HV}}_{3}}$, [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.

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

• 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