# 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 | |

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 | R_{3} |

Convex | Yes |

## Vertex coordinatesEdit

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

where *i* and *j* range over the integers.

## RepresentationsEdit

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 figuresEdit

Name | Picture | Schläfli symbol |
---|---|---|

Octahedral honeycomb | {3,4,4} | |

Order-4 square tiling honeycomb | {4,4,4} |

## Related tilingsEdit

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

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 linksEdit

- Klitzing, Richard. "squat".

- Wikipedia Contributors. "Square tiling".