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 |
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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
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| Dual | Order-5 hexagonal tiling honeycomb |
| Coxeter group |  , [5,3,6]
 , [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.
The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
More information 11 paracompact regular honeycombs ...
| 11 paracompact regular honeycombs |
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{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} |
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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.
More information {6,3,5}, r{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} |
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| {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} |
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The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:
More information Form, Paracompact ...
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 |
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| Cells |
{3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
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It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
More information {5,3,p} polytopes, Space ...
| {5,3,p} polytopes |
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| 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 |
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Vertex
figure |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
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Rectified order-6 dodecahedral honeycomb
More information , ...
| Rectified order-6 dodecahedral honeycomb
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| 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 | , [5,3,6]
, [5,3[3]] |
| Properties | Vertex-transitive, edge-transitive |
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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.
More information Space, H3 ...
r{p,3,6}
| Space |
H3 |
| Form |
Paracompact |
Noncompact |
| Name |
r{3,3,6}
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r{4,3,6}
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r{5,3,6}
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r{6,3,6}
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r{7,3,6}
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... r{∞,3,6}
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| Image |
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Cells
{3,6}
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r{3,3}
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r{4,3}
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r{5,3}
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r{6,3}
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r{7,3}
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r{∞,3}
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Truncated order-6 dodecahedral honeycomb
More information , ...
| Truncated order-6 dodecahedral honeycomb
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| 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 | , [5,3,6]
, [5,3[3]] |
| Properties | Vertex-transitive |
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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
More information , ...
| Cantellated order-6 dodecahedral honeycomb |
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| 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 | , [5,3,6]
, [5,3[3]] |
| Properties | Vertex-transitive |
Close
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
More information , ...
| Cantitruncated order-6 dodecahedral honeycomb |
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| 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 | , [5,3,6]
, [5,3[3]] |
| Properties | Vertex-transitive |
Close
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.
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