# Square tiling

Square tiling
Rank3
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymSquat
Coxeter diagramx4o4o ()
Schläfli symbol{4,4}
Elements
FacesN squares
Edges2N
VerticesN
Vertex figureSquare, edge length 2
Measures (edge length 1)
Vertex density${\displaystyle 1}$
Related polytopes
ArmySquat
RegimentSquat
DualSquare tiling
Petrie dualPetrial square tiling
ConjugateNone
Abstract & topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryR3
ConvexYes

The square tiling, or squat, is one of the three regular tilings of the Euclidean plane. 4 squares join at each vertex of this tiling. It is the only one of the three regular tilings to be self-dual. It is also the 2D hypercubic honeycomb.

## Vertex coordinates

Coordinates for the vertices of a square tiling of edge length 1 are given by

• ${\displaystyle \left(i,\,j\right),}$

where i and j range over the integers.

## Representations

A square tiling has the following Coxeter diagrams:

• x4o4o (regular)
• o4x4o (as rectified square tiling)
• x4o4x (as small rhombated square tiling)
• xØx xØx (W2|W2 symmetry, as comb product of two apeirogons)
• s4o4o (as alternated square tiling)
• o4s4o
• s4o4s
• s4x4o (as additional alternated facetings)
• x4s4x
• s4x4s
• x4s4o
• s4s4x
• qo4oo4oq&#zx (as hull of two dual square tilings)
• qo4xx4oq&#zx (as hull of two oopposite variant truncated square tilings)

## In vertex figures

Square tilings in vertex figures
Name Picture Schläfli symbol
Octahedral honeycomb
{3,4,4}
Order-4 square tiling honeycomb
{4,4,4}

## Related tilings

The square tiling is the colonel of a two-member regiment that also includes the square-hemiapeirogonal tiling.

o4o4o truncations
Name OBSA Schläfli symbol CD diagram Picture
Square tiling squat {4,4} x4o4o
Truncated square tiling tosquat t{4,4} x4x4o
Rectified square tiling = Square tiling squat r{4,4} o4x4o
Truncated square tiling tosquat t{4,4} o4x4x
Square tiling squat {4,4} o4o4x
Cantellated square tiling = Square tiling squat rr{4,4} x4o4x
Omnitruncated square tiling = Truncated square tiling tosquat tr{4,4} x4x4x
Snub square tiling snasquat sr{4,4} s4s4s