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Order-6 dodecahedral honeycomb

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(Redirected from Order-3-6 pentagonal honeycomb) Regular geometrical object in hyperbolic space
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}
Coxeter diagram
↔
Cells {5,3}
Faces pentagon {5}
Edge figure hexagon {6}
Vertex figure
triangular tiling
Dual Order-5 hexagonal tiling honeycomb
Coxeter group H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
H P ¯ 3 {\displaystyle {\overline {HP}}_{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.

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 Coxeter group family, including this regular form, and its regular dual, the order-5 hexagonal tiling honeycomb.

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 S H
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 H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
H P ¯ 3 {\displaystyle {\overline {HP}}_{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 H
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 H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} ,
Properties Vertex-transitive

The truncated order-6 dodecahedral honeycomb, t0,1{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}
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 H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} ,
Properties Vertex-transitive

The cantellated order-6 dodecahedral honeycomb, t0,2{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 H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
H P ¯ 3 {\displaystyle {\overline {HP}}_{3}} ,
Properties Vertex-transitive

The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{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 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 H V ¯ 3 {\displaystyle {\overline {HV}}_{3}} ,
Properties Vertex-transitive

The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{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.

See also

References

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