| Hexagonal tiling-triangular tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | {(3,6,3,6)} or {(6,3,6,3)} |
| Coxeter diagrams |     or or or  |
| Cells | {3,6} {6,3}  r{6,3}  |
| Faces | triangular {3} square {4} hexagon {6} |
| Vertex figure |  rhombitrihexagonal tiling |
| Coxeter group | [(6,3)[2]] |
| Properties | Vertex-uniform, edge-uniform |
In thegeometry ofhyperbolic 3-space, thehexagonal tiling-triangular tiling honeycomb is aparacompact uniform honeycomb, constructed fromtriangular tiling,hexagonal tiling, andtrihexagonal tiling cells, in arhombitrihexagonal tilingvertex figure. It has a single-ring Coxeter diagram,, and is named by its two regular cells.
Ageometric honeycomb is aspace-filling ofpolyhedral or higher-dimensionalcells, so that there are no gaps. It is an example of the more general mathematicaltiling ortessellation in any number of dimensions.
Honeycombs are usually constructed in ordinaryEuclidean ("flat") space, like theconvex uniform honeycombs. They may also be constructed innon-Euclidean spaces, such ashyperbolic uniform honeycombs. Any finiteuniform polytope can be projected to itscircumsphere to form a uniform honeycomb in spherical space.
A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,6,3*)] symmetry, represented by acube fundamental domain, and an octahedralCoxeter diagram.
Thecyclotruncated octahedral-hexagonal tiling honeycomb, has a higher symmetry construction as theorder-4 hexagonal tiling.