Order-∞ triangular tiling
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| Order-∞ triangular tiling | |
|---|---|
 | |
| Rank | 3 |
| Type | Regular, paracompact |
| Space | Hyperbolic |
| Notation | |
| Bowers style acronym | Aztrat |
| Coxeter diagram | o∞o3x (    ) |
| Schläfli symbol | {3,∞} |
| Elements | |
| Faces | 2NM triangles |
| Edges | 3NM |
| Vertices | 6N |
| Vertex figure | Apeirogon, edge length 1 |
| Measures (edge length 1) | |
| Circumradius | |
| Related polytopes | |
| Army | Aztrat |
| Regiment | Aztrat |
| Dual | Order-3 apeirogonal tiling |
| Abstract & topological properties | |
| Surface | Sphere |
| Orientable | Yes |
| Genus | 0 |
| Properties | |
| Symmetry | [∞,3] |
| Convex | Yes |
The order-∞ triangular tiling or aztrat, also called the infinite-order triangular tiling is a paracompact regular tiling of the hyperbolic plane. Infinitely many ideal triangles join at each vertex. All vertices are ideal points at infinity.
Representations[edit | edit source]
An order-∞ triangular tiling has the following Coxeter diagrams:
- o∞o3x () (full symmetry)
- x3o∞o3*a (
) (triangles of two types)
External links[edit | edit source]
- Klitzing, Richard. "Aztrat".
- Wikipedia contributors. "Infinite-order triangular tiling".
| truncations | ||||
|---|---|---|---|---|
| Name | OBSA | Schläfli symbol | CD diagram | Image |
| Apeirogonal tiling | azat | {∞,3} | |  |
| Triapeirogonal tiling | tazt | r{∞,3} | |  |
| Order-∞ triangular tiling | aztrat | {3,∞} | | |
| Truncated apeirogonal tiling | tazat | t{∞,3} | |  |
| Small rhombitriapeirogonal tiling | srotazt | rr{∞,3} | |  |
| Truncated order-∞ triangular tiling | taztrat | t{3,∞} | |  |
| Great rhombitriapeirogonal tiling | grotazt | tr{∞,3} | |  |
| Snub triapeirogonal tiling | sr{∞,3} |  |  | |
