Square tiling

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
Holes	Apeirogons
Measures (edge length 1)
Vertex density	1
Related polytopes
Army	Squat
Regiment	Squat
Dual	Square tiling
Petrie dual	Petrial square tiling
Halving	Square tiling, edge length ${\sqrt {2}}$
Conjugate	None
Abstract & topological properties
Flag count	8N
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	R3
Flag orbits	1
Convex	Yes
Nature	Tame

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 regular tilings of the plane to be self-dual. It is also the 2D hypercubic honeycomb.

The square tiling is the honeycomb product of two apeirogons. It can also be seen as the result of stacking infinitely many apeirogonal prisms on top of each other.

The square tiling is also notable because it is equivalent to its own rectification and cantellation.

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)
• xØx2xØx () (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 opposite variant truncated square tilings)

Related tilings

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

External links

• Klitzing, Richard. "squat".
• Wikipedia contributors. "Square tiling".