# Order-∞ triangular tiling

Order-∞ triangular tiling Rank3
TypeRegular, paracompact
SpaceHyperbolic
Notation
Bowers style acronymAztrat
Coxeter diagramo∞o3x (     )
Schläfli symbol{3,∞}
Elements
Faces2NM Triangles
Edges3NM
Vertices6N
Vertex figureApeirogon, edge length 1
Measures (edge length 1)
Circumradius$0$ Related polytopes
ArmyAztrat
RegimentAztrat
DualOrder-3 apeirogonal tiling
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry[∞,3]
ConvexYes

The order-∞ triangular tiling or trazat, 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

An order–∞ triangular tiling has the following Coxeter diagrams:

• o∞o3x (full symmetry)
• o3x3o∞*a (triangles of two types)

## Related polytopes

o∞o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Apeirogonal tiling azat {∞,3}     Truncated apeirogonal tiling tazat t{∞,3}     Triapeirogonal tiling tazt r{∞,3}     Truncated order-∞ triangular tiling taztrat t{3,∞}     Order-∞ triangular tiling aztrat {3,∞}     Small rhombitriapeirogonal tiling srotazt rr{∞,3}     Great rhombitriapeirogonal tiling grotazt tr{∞,3}     Snub triapeirogonal tiling sr{∞,3}     o∞o3o3*a truncations
Name OBSA CD diagram Picture
Ditrigonary triapeirogonal tiling    Order-∞ triangular tiling    Triapeirogonal tiling    Rectified ditrigonary triapeirogonal tiling    Truncated order-∞ triangular tiling    Snub ditrigonary triapeirogonal tiling    