Order-6 dodecahedral honeycomb

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Order-6 dodecahedral honeycomb
H3 536 CC center.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {5,3,6}
{5,3[3]}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png
Cells {5,3} Dodecahedron.png
Faces pentagon {5}
Edge figure hexagon {6}
Vertex figure {3,6}
Uniform tiling 63-t2.png Uniform tiling 333-t1.png
Dual Order-5 hexagonal tiling honeycomb
Coxeter group HV3, [5,3,6]
HP3, [5,3[3]]
Properties Regular, 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[edit]

A half symmetry construction exists as CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch.png with alternately colored dodecahedral cells.

Images[edit]

Order-6 dodecahedral honeycomb.png
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.

H2 tiling 25i-4.png

Related polytopes and honeycombs[edit]

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}
H3 635 FC boundary.png H3 635 boundary 0100.png H3 635-1100.png H3 635-1010.png H3 635-1001.png H3 635-1110.png H3 635-1101.png H3 635-1111.png
H3 536 CC center.png H3 536 CC center 0100.png H3 635-0011.png H3 635-0101.png H3 635-0110.png H3 635-0111.png H3 635-1011.png
{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 H3 336 CC center.png H3 436 CC center.png H3 536 CC center.png H3 636 FC boundary.png
Cells Tetrahedron.png
{3,3}
Hexahedron.png
{4,3}
Dodecahedron.png
{5,3}
Uniform tiling 63-t0.svg
{6,3}
H2 tiling 237-1.png
{7,3}
H2 tiling 238-1.png
{8,3}
H2 tiling 23i-1.png
{∞,3}

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

Rectified order-6 dodecahedral honeycomb[edit]

Rectified order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols r{5,3,6}
t1{5,3,6}
Coxeter diagrams CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch.png
Cells r{5,3} Uniform polyhedron-53-t1.png
{3,6} Uniform tiling 63-t2.png
Vertex figure Rectified order-6 dodecahedral honeycomb verf.png
Hexagonal prism { }×{6}
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel node.png
Coxeter groups , [6,3,5]
[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.

H3 536 CC center 0100.png
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.

H2 tiling 25i-2.png
r{p,3,6}
Space H3
Form Paracompact Noncompact
Name r{3,3,6}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{4,3,6}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{5,3,6}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{6,3,6}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
r{7,3,6}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
... r{∞,3,6}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Image H3 336 CC center 0100.png H3 436 CC center 0100.png H3 536 CC center 0100.png H3 636 boundary 0100.png
Cells
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
Uniform polyhedron-33-t1.png
r{3,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png
r{4,3}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Icosidodecahedron.png
r{6,3}
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
Uniform tiling 63-t1.svg
r{6,3}
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 237-2.png
r{∞,3}
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
H2 tiling 23i-2.png
r{∞,3}
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png

Truncated order-6 dodecahedral honeycomb[edit]

Truncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{5,3,6}
t0,1{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch.png
Cells t{5,3} Uniform polyhedron-53-t01.png
{3,6} Uniform tiling 63-t2.png
Vertex figure Truncated order-6 dodecahedral honeycomb verf.png
Hexagonal pyramid { }v{6}
Coxeter groups , [6,3,5]
[5,3[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.

H3 635-0011.png

Cantellated order-6 dodecahedral honeycomb[edit]

Cantellated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{5,3,6}
t0,2{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel branch 11.png
Cells
Vertex figure Cantellated order-6 dodecahedral honeycomb verf.png
Coxeter groups , [6,3,5]
[5,3[3]]
Properties Vertex-transitive

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

H3 635-0101.png

Cantitruncated order-6 dodecahedral honeycomb[edit]

Cantitruncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{5,3,6}
t0,1,2{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel split1.pngCDel branch 11.png
Cells
Vertex figure Cantitruncated order-6 dodecahedral honeycomb verf.png
Coxeter groups , [6,3,5]
[5,3[3]]
Properties Vertex-transitive

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

H3 635-0111.png

Runcitruncated order-6 dodecahedral honeycomb[edit]

Runcitruncated order-6 dodecahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{5,3,6}
Coxeter diagrams CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.png
Cells
Vertex figure Runcitruncated order-6 dodecahedral honeycomb verf.png
Coxeter groups , [6,3,5]
[5,3[3]]
Properties Vertex-transitive

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

H3 635-1011.png

See also[edit]

References[edit]