# Apeirogonal tiling

Apeirogonal tiling Rank3
TypeRegular, paracompact
SpaceHyperbolic
Notation
Bowers style acronymAzat
Coxeter diagramx∞o3o (     )
Schläfli symbol{∞,3}
Elements
Faces6N Apeirogons
Edges3NM
Vertices2NM
Vertex figureTriangle, edge length 2
Measures (edge length 1)
Circumradius$\frac{i\sqrt{3}}{2} ≈ 0.86603 i$ Related polytopes
ArmyAzat
RegimentAzat
DualOrder-∞ triangular tiling
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry[∞,3]
ConvexYes

The order-3 apeirogonal tiling, or just apeirogonal tiling or azat, is a paracompact regular tiling of the hyperbolic plane. 3 apeirogons join at each vertex.

It can be formed by truncating the order-∞ apeirogonal tiling.

## Representations

The apeirogonal tiling has the following Coxeter diagrams:

• x∞o3o (full symmetry)
• x∞x∞o (as truncated order-∞ apeirogonal tiling)
• x∞x∞x∞*a (three types of faces) (    )

## 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}     