Order-6 dodecahedral honeycomb

Order-6 dodecahedral honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol{5,3,6}
{5,3[3]}
Coxeter diagram
↔
Cells{5,3}
Facespentagon {5}
Edge figurehexagon {6}
Vertex figure{3,6}
DualOrder-5 hexagonal tiling honeycomb
Coxeter groupHV3, [5,3,6]
HP3, [5,3[3]]
PropertiesRegular, quasiregular

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

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 in the within Poincaré disk model, with the viewpoint then placed at the origin.

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

Related polytopes and honeycombs

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form and its regular dual, order-5 hexagonal tiling honeycomb, {6,3,5}.

[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}

This honeycomb is a part of a sequence of 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}

This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:

Rectified order-6 dodecahedral honeycomb

Rectified order-6 dodecahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{5,3,6}
t1{5,3,6}
Coxeter diagrams
↔
Cellsr{5,3}
{3,6}
Vertex figure
Hexagonal prism { }×{6}
Coxeter groups, [6,3,5]
[5,3[3]]
PropertiesVertex-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 infinite-order square tiling, r{5,∞} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

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{6,3}
r{6,3}
r{∞,3}
r{∞,3}

Truncated order-6 dodecahedral honeycomb

Truncated order-6 dodecahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolst{5,3,6}
t0,1{5,3,6}
Coxeter diagrams
↔
Cellst{5,3}
{3,6}
Vertex figure
Hexagonal pyramid { }v{6}
Coxeter groups, [6,3,5]
[5,3[3]]
PropertiesVertex-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.

Cantellated order-6 dodecahedral honeycomb

Cantellated order-6 dodecahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsrr{5,3,6}
t0,2{5,3,6}
Coxeter diagrams
↔
Cells
Vertex figure
Coxeter groups, [6,3,5]
[5,3[3]]
PropertiesVertex-transitive

The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}.

Cantitruncated order-6 dodecahedral honeycomb

Cantitruncated order-6 dodecahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolstr{5,3,6}
t0,1,2{5,3,6}
Coxeter diagrams
↔
Cells
Vertex figure
Coxeter groups, [6,3,5]
[5,3[3]]
PropertiesVertex-transitive

The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has a tetrahedral vertex figure.

Runcitruncated order-6 dodecahedral honeycomb

Runcitruncated order-6 dodecahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolst0,1,3{5,3,6}
Coxeter diagrams
Cells
Vertex figure
Coxeter groups, [6,3,5]
[5,3[3]]
PropertiesVertex-transitive

The runcitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has a trapezoidal pyramid vertex figure.

See also

References

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