Order-6 cubic honeycomb

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation (or honeycomb) in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

Symmetry
A half-symmetry construction of the order-6 cubic honeycomb exists as {4,3[3]}, with two alternating types (colors) of cubic cells. This construction has Coxeter-Dynkin diagram ↔.

Another lower-symmetry construction, [4,3*,6], of index 6, exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram.

This honeycomb contains that tile 2-hypercycle surfaces, similar to the paracompact order-3 apeirogonal tiling, :
 * H2-I-3-dual.svg

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

It has a related alternation honeycomb, represented by ↔. This alternated form has hexagonal tiling and tetrahedron cells.

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including the order-6 cubic honeycomb itself.

The order-6 cubic honeycomb is part of a sequence of regular polychora and honeycombs with cubic cells.

It is also part of a sequence of honeycombs with triangular tiling vertex figures.

Rectified order-6 cubic honeycomb
The rectified order-6 cubic honeycomb, r{4,3,6}, has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.



It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞}, alternating apeirogonal and square faces:
 * H2 tiling 24i-2.png

Truncated order-6 cubic honeycomb
The truncated order-6 cubic honeycomb, t{4,3,6}, has truncated cube and triangular tiling facets, with a hexagonal pyramid vertex figure.



It is similar to the 2D hyperbolic truncated infinite-order square tiling, t{4,∞}, with apeirogonal and octagonal (truncated square) faces:
 * H2 tiling 24i-6.png

Bitruncated order-6 cubic honeycomb
The bitruncated order-6 cubic honeycomb is the same as the bitruncated order-4 hexagonal tiling honeycomb.

Cantellated order-6 cubic honeycomb
The cantellated order-6 cubic honeycomb, rr{4,3,6}, has rhombicuboctahedron, trihexagonal tiling, and hexagonal prism facets, with a wedge vertex figure.



Cantitruncated order-6 cubic honeycomb
The cantitruncated order-6 cubic honeycomb, tr{4,3,6}, has truncated cuboctahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.



Runcinated order-6 cubic honeycomb
The runcinated order-6 cubic honeycomb is the same as the runcinated order-4 hexagonal tiling honeycomb.

Runcitruncated order-6 cubic honeycomb
The runcitruncated order-6 cubic honeycomb, rr{4,3,6}, has truncated cube, rhombitrihexagonal tiling, hexagonal prism, and octagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.



Runcicantellated order-6 cubic honeycomb
The runcicantellated order-6 cubic honeycomb is the same as the runcitruncated order-4 hexagonal tiling honeycomb.

Omnitruncated order-6 cubic honeycomb
The omnitruncated order-6 cubic honeycomb is the same as the omnitruncated order-4 hexagonal tiling honeycomb.

Alternated order-6 cubic honeycomb
In three-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellation (or honeycomb). As an alternation, with Schläfli symbol h{4,3,6} and Coxeter-Dynkin diagram or, it can be considered a quasiregular honeycomb, alternating triangular tilings and tetrahedra around each vertex in a trihexagonal tiling vertex figure.

Symmetry
A half-symmetry construction from the form {4,3[3]} exists, with two alternating types (colors) of triangular tiling cells. This form has Coxeter-Dynkin diagram ↔. Another lower-symmetry form of index 6, [4,3*,6], exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram.

Related honeycombs
The alternated order-6 cubic honeycomb is part of a series of quasiregular polychora and honeycombs.

It also has 3 related forms: the cantic order-6 cubic honeycomb, h2{4,3,6}, ; the runcic order-6 cubic honeycomb, h3{4,3,6}, ; and the runcicantic order-6 cubic honeycomb, h2,3{4,3,6},.

Cantic order-6 cubic honeycomb
The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h2{4,3,6}. It is composed of truncated tetrahedron, trihexagonal tiling, and hexagonal tiling facets, with a rectangular pyramid vertex figure.

Runcic order-6 cubic honeycomb
The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h3{4,3,6}. It is composed of tetrahedron, hexagonal tiling, and rhombitrihexagonal tiling facets, with a triangular cupola vertex figure.

Runcicantic order-6 cubic honeycomb
The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h2,3{4,3,6}. It is composed of truncated hexagonal tiling, truncated trihexagonal tiling, and truncated tetrahedron facets, with a mirrored sphenoid vertex figure.