Great rhombated tetrahedral honeycomb
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| Great rhombated tetrahedral honeycomb | |
|---|---|
 | |
| Rank | 4 |
| Type | Uniform, paracompact |
| Space | Hyperbolic |
| Notation | |
| Bowers style acronym | Grath |
| Coxeter diagram | o6x3x3x (    ) |
| Elements | |
| Cells | MN Hexagonal prisms, MN truncated octahedra, 2N hexagonal tilings |
| Faces | 6MN Squares, 2MN+2MN+4MN hexagons |
| Edges | 12MN+12MN+24MN |
| Vertices | 12MN |
| Vertex figure | Sphenoid, edge lengths √2 (2) and √3 (4)  |
| Measures (edge length 1) | |
| Circumradius | |
| Related polytopes | |
| Army | Grath |
| Regiment | Grath |
| Abstract & topological properties | |
| Orientable | Yes |
| Properties | |
| Symmetry | [6,3,3] |
| Convex | Yes |
The great rhombated tetrahedral honeycomb, also called the cantitruncated tetrahedral honeycomb, is a paracompact uniform tiling of 3D hyperbolic space. 1 hexagonal tiling, 2 truncated octahedra, and 1 hexagonal prism meet at each vertex. It is paracompact because it has Euclidean hexagonal tiling cells. As the name suggests, it can be derived by cantitruncation of the tetrahedral honeycomb.
The truncated octahedra are in the form , as great rhombitetratetrahedra, with tetrahedral symmetry; and the hexagonal tilings are in the form , as truncated triangular tilings.
Representations[edit | edit source]
A great rhombated tetrahedral honeycomb has the following Coxeter diagrams:
- o6x3x3x () (full symmetry)
- x3x3x3x3*b (
) (half symmetry)
External links[edit | edit source]
- Wikipedia contributors. "Cantellated order-6 tetrahedral honeycomb".
| truncations | ||||
|---|---|---|---|---|
| Name | OBSA | Schläfli symbol | CD diagram | Image |
| Hexagonal tiling honeycomb | hexah | {6,3,3} | |  |
| Truncated hexagonal tiling honeycomb | thexah | t{6,3,3} | |  |
| Rectified hexagonal tiling honeycomb | rihexah | r{6,3,3} | |  |
| Tetrahedral-hexagonal tiling honeycomb | tehexah | 2t{6,3,3} | |  |
| Rectified tetrahedral honeycomb | rath | r{3,3,6} | |  |
| Truncated tetrahedral honeycomb | tath | t{3,3,6} | |  |
| Tetrahedral honeycomb | thon | {3,3,6} | |  |
| Small rhombated hexagonal tiling honeycomb | srihexah | rr{6,3,3} | |  |
| Great rhombated hexagonal tiling honeycomb | grihexah | tr{6,3,3} | |  |
| Small rhombated tetrahedral honeycomb | srath | rr{3,3,6} | |  |
| Great rhombated tetrahedral honeycomb | grath | tr{3,3,6} | | |
| Small prismated hexagonal tiling honeycomb | sidpithexah | t0,3{6,3,3} | |  |
| Prismatorhombated tetrahedral honeycomb | prath | t0,1,3{6,3,3} | |  |
| Prismatorhombated hexagonal tiling honeycomb | prihexah | t0,1,3{3,3,6} | |  |
| Great prismated hexagonal tiling honeycomb | gidpithexah | t0,1,2,3{6,3,3} | |  |