# Order-∞ square tiling

Order-∞ square tiling Rank3
TypeRegular, paracompact
SpaceHyperbolic
Notation
Bowers style acronymAsquat
Coxeter diagramo∞o4x (     )
Schläfli symbol{4,∞}
Elements
FacesNM squares
Edges2NM
Vertices4N
Vertex figureApeirogon, edge length 2
Measures (edge length 1)
Circumradius$0$ Related polytopes
ArmyAsquat
RegimentAsquat
DualOrder-4 apeirogonal tiling
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry[∞,4]
ConvexYes

The order-∞ square tiling or infinite-order square tiling is a paracompact regular tiling of the hyperbolic plane. Infinitely many ideal squares join at each vertex. All vertices are ideal points at infinity.

The tiling can be alternated to produce the order-∞ apeirogonal tiling.

## Representations

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

• o∞o4x (full symmetry)
• o4x4o∞*a (squares of two types)

## Related polytopes

o∞o4o truncations
Name OBSA Schläfli symbol CD diagram Picture
Order-4 apeirogonal tiling squazat {∞,4}     Truncated order-4 apeirogonal tiling tosquazat t{∞,4}     Tetraapeirogonal tiling tezt r{∞,4}     Truncated order-∞ square tiling tazsquat t{4,∞}     Order-∞ square tiling azsquat {4,∞}     Small rhombitetraapeirogonal tiling srotezt rr{∞,4}     Great rhombitetraapeirogonal tiling grotezt tr{∞,4}     Snub tetraapeirogonal tiling sr{∞,4}     