Square tiling

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.

Square tiling
Rank	3
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Squat
Coxeter diagram	x4o4o ()
Schläfli symbol	{4,4}
Elements
Faces	N squares
Edges	2N
Vertices	N
Vertex figure	Square, edge length √2
Measures (edge length 1)
Vertex density	$1$
Related polytopes
Army	Squat
Regiment	Squat
Dual	Square tiling
Petrie dual	Petrial square tiling
Conjugate	None
Abstract & topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	R3
Convex	Yes

Vertex coordinates

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

• $\left(i,\,j\right),$

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)

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

External links

• Klitzing, Richard. "squat".

• Wikipedia Contributors. "Square tiling".