Triangular tiling honeycomb
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| Triangular tiling honeycomb | |
|---|---|
 | |
| Rank | 4 |
| Type | Regular, paracompact |
| Space | Hyperbolic |
| Notation | |
| Bowers style acronym | Trah |
| Coxeter diagram | x3o6o3o (    ) |
| Schläfli symbol | {3,6,3} |
| Elements | |
| Cells | NM triangular tilings |
| Faces | NMK triangles |
| Edges | NMK |
| Vertices | NK |
| Vertex figure | Hexagonal tiling, edge length 1 |
| Measures (edge length 1) | |
| Circumradius | 0 |
| Related polytopes | |
| Army | Trah |
| Regiment | Trah |
| Dual | Triangular tiling honeycomb |
| Abstract & topological properties | |
| Orientable | Yes |
| Properties | |
| Symmetry | [3,6,3] |
| Convex | Yes |
The order-3 triangular tiling honeycomb, or just triangular tiling honeycomb, is a paracompact regular tiling of 3D hyperbolic space. 3 ideal triangular tilings meet at each edge. All vertices are ideal points at infinity, with infinitely many triangular tilings meeting at each vertex in a hexagonal tiling arrangement.
Representations[edit | edit source]
The triangular tiling honeycomb has the following Coxeter diagrams:
- x3o6o3o () (full symmetry)
- o6o3x3o3*b (
) (cells of two types, truncated tiangular tiling verf) - x3o3o3o3*a3*c *b3*d (
) (cells of three types)
External links[edit | edit source]
- Klitzing, Richard. "trah".
- Wikipedia Contributors. "Triangular tiling honeycomb".
- lllllllllwith10ls. "Category 1: Regulars" (#12).
| truncations | ||||
|---|---|---|---|---|
| Name | OBSA | Schläfli symbol | CD diagram | Image |
| Triangular tiling honeycomb | trah | {3,6,3} | | |
| Truncated triangular tiling honeycomb = Hexagonal tiling honeycomb | hexah | t{3,6,3} | |  |
| Rectified triangular tiling honeycomb | ritrah | r{3,6,3} | |  |
| Distriangular tiling honeycomb | ditrah | 2t{3,6,3} | |  |
| Rectified triangular tiling honeycomb | ritrah | r{3,6,3} | | |
| Truncated triangular tiling honeycomb = Hexagonal tiling honeycomb | hexah | t{3,6,3} | |  |
| Triangular tiling honeycomb | trah | {3,6,3} | | |
| Small rhombated triangular tiling honeycomb | sritrah | rr{3,6,3} | |  |
| Great rhombated triangular tiling honeycomb | gritrah | tr{3,6,3} | |  |
| Small rhombated triangular tiling honeycomb | sritrah | rr{3,6,3} | |  |
| Great rhombated triangular tiling honeycomb | gritrah | tr{3,6,3} | |  |
| Small prismated triangular tiling honeycomb | spidditrah | t0,3{3,6,3} | |  |
| Prismatorhombated triangular tiling honeycomb | pritrah | t0,1,3{3,6,3} | |  |
| Prismatorhombated triangular tiling honeycomb | pritrah | t0,1,3{3,6,3} | | |
| Great prismated triangular tiling honeycomb | spidditrah | t0,1,2,3{3,6,3} | |  |