Order-6 dodecahedral honeycomb

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.

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

Images
The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,&infin;}, with pentagonal faces, and with vertices on the ideal surface.
 * H2 tiling 25i-4.png

Related polytopes and honeycombs
The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

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.

The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:

It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells:

Rectified order-6 dodecahedral honeycomb
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.pngPerspective projection view within Poincaré disk model

It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,&infin;} with pentagon and apeirogonal faces.
 * H2 tiling 25i-2.png

Truncated order-6 dodecahedral honeycomb
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
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
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
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.