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.[1]
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 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]]: 
↔ 
.
The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling, {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices.
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.
The rectified order-6 tetrahedral honeycomb, t1{3,3,6} has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure.
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.
The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling 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.
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.
The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb.
The runcitruncated order-6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb.
The runcicantellated order-6 tetrahedral honeycomb is equivalent to the runcitruncated hexagonal tiling honeycomb.
The omnitruncated order-6 tetrahedral honeycomb is equivalent to the omnitruncated hexagonal tiling honeycomb.