Order-6 tetrahedral honeycomb

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.

Symmetry constructions
The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3[3]}. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation, this half symmetry is represented as [3,3,6,1+] ↔ [3,((3,3,3))], or [3,3[3]]: ↔.

Related polytopes and honeycombs
The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling, {3,&infin;}. Both tessellations are regular, and only contain triangles and ideal vertices.
 * Infinite-order triangular tiling.svg

The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

This honeycomb is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the hexagonal tiling honeycomb.

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

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

Rectified order-6 tetrahedral honeycomb
The rectified order-6 tetrahedral honeycomb, t1{3,3,6} has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure.
 * H3 336 CC center 0100.pngHyperbolic rectified order-6 tetrahedral honeycomb.pngPerspective projection view within Poincaré disk model

Truncated order-6 tetrahedral honeycomb
The truncated order-6 tetrahedral honeycomb, t0,1{3,3,6} has truncated tetrahedron and triangular tiling cells arranged in a hexagonal pyramid vertex figure.



Bitruncated order-6 tetrahedral honeycomb
The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb.

Cantellated order-6 tetrahedral honeycomb
The cantellated order-6 tetrahedral honeycomb, t0,2{3,3,6} has cuboctahedron, trihexagonal tiling, and hexagonal prism cells arranged in an isosceles triangular prism vertex figure.



Cantitruncated order-6 tetrahedral honeycomb
The cantitruncated order-6 tetrahedral honeycomb, t0,1,2{3,3,6} has truncated octahedron, hexagonal tiling, and hexagonal prism cells connected in a mirrored sphenoid vertex figure.



Runcinated order-6 tetrahedral honeycomb
The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb.

Runcitruncated order-6 tetrahedral honeycomb
The runcitruncated order-6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb.

Runcicantellated order-6 tetrahedral honeycomb
The runcicantellated order-6 tetrahedral honeycomb is equivalent to the runcitruncated hexagonal tiling honeycomb.

Omnitruncated order-6 tetrahedral honeycomb
The omnitruncated order-6 tetrahedral honeycomb is equivalent to the omnitruncated hexagonal tiling honeycomb.