Tetrahedral honeycomb
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| Tetrahedral honeycomb | |
|---|---|
 | |
| Rank | 4 |
| Type | Regular, paracompact |
| Space | Hyperbolic |
| Notation | |
| Bowers style acronym | Thon |
| Coxeter diagram | x4o3o6o (     ) |
| Schläfli symbol | {3,3,6} |
| Elements | |
| Cells | NM tetrahedra |
| Faces | 2NM triangles |
| Edges | NM |
| Vertices | 2N |
| Vertex figure | Triangular tiling, edge length 1 |
| Measures (edge length 1) | |
| Circumradius | 0 |
| Related polytopes | |
| Army | Thon |
| Regiment | Thon |
| Dual | Hexagonal tiling honeycomb |
| Abstract & topological properties | |
| Orientable | Yes |
| Properties | |
| Symmetry | [6,3,3] |
| Convex | Yes |
The order-6 tetrahedral honeycomb, or just tetrahedral honeycomb, is a paracompact regular tiling of 3D hyperbolic space. 6 ideal tetrahedra meet at each edge. All vertices are ideal points at infinity, with infinitely many tetrahedra meeting at each vertex in a triangular tiling arrangement.
Representations[edit | edit source]
A tetrahedral honeycomb has the following Coxeter diagrams:
- x3o3o6o () (full symmetry)
- x3o3o3o3*b (
) (tetrahedra of two alternating types)
External links[edit | edit source]
- Klitzing, Richard. "thon".
- Wikipedia Contributors. "Order-6 tetrahedral honeycomb".
- lllllllllwith10ls. "Category 1: Regulars" (#6).
| 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} | |  |