# Square

Square | |
---|---|

Rank | 2 |

Type | Regular |

Notation | |

Bowers style acronym | Square |

Coxeter diagram | x4o () |

Schläfli symbol | {4} |

Tapertopic notation | 11 |

Toratopic notation | II |

Bracket notation | [II] |

Elements | |

Edges | 4 |

Vertices | 4 |

Vertex figure | Dyad, length √2 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | 1 |

Angle | 90° |

Height | 1 |

Central density | 1 |

Number of external pieces | 4 |

Level of complexity | 1 |

Related polytopes | |

Army | Square |

Dual | Square |

Conjugate | None |

Abstract & topological properties | |

Flag count | 8 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | B_{2}, order 8 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

The **square**, **regular quadrilateral**, or **regular tetragon** is the 4-sided regular polygon. It has 4 equal sides and 4 equal angles. It is the 2-dimensional hypercube, as well as the 2-dimensional orthoplex.

The square is one of three regular polygons that can tile the plane, the others being the equilateral triangle and regular hexagon. This tiling is called the square tiling, and it has 4 squares joining at a vertex.

This is one of two polygons without a stellation, the other being the triangle, and one of three without a non-compound stellation, the third being the hexagon.

The square is one of two possible segmentogons, being a dyad atop a dyad. The other is the triangle.

A square can be seen as a dyadic prism, a dyadic tegum, or a dyadic antiprism.

Squares are the faces of one of the Platonic solids, the cube.

## Naming[edit | edit source]

The name *square* is descended from Old French *esquarre*, which itself is descended from the Latin *quadrare* (to make square) and *quadrus* (square). Alternate names include:

**Quadrilateral**, from Latin*quadri-*(four) and*lateralis*(sided), referring to the number of sides.**Tetragon**, from Ancient Greek*τετράς*(four) and*γωνία*(angle), referring to the number of angles. More consistent with other polygons.

The combining prefix is **squ-**, as in **squ**ipdip or **s-**, as in ti**s**dip.

## Vertex coordinates[edit | edit source]

The vertices of a square of edge length 1 centered at the origin are:

- .

## Representations[edit | edit source]

A square can be represented by the following Coxeter diagrams:

- x4o () (full symmetry)
- x x () (digonal symmetry, generally a rectangle)
- qo oq&#zx (digonal symmetry, generally a rhombus)
- xx&#x (axial edge-first, generally a trapezoid)
- oqo&#xt (axial vertex-first, generally a kite)
- oooo&#xr (no symmetry, generally an irregular tetragon)

## In vertex figures[edit | edit source]

The square appears as the vertex figure of one uniform polyhedron, namely the regular octahedron. This vertex figure has an edge length of 1.

It is also the vertex figure of the square tiling (with edge length √2) and an infinite family of regular hyperbolic tilings starting with the pentagonal tiling.

## Other kinds of quadrilaterals[edit | edit source]

Besides the regular square, there are various other kinds of quadrilaterals. All of these have angles that add to 360°.

- Rectangle - digonal symmetry, all equal angles, two alternating side lengths
- Rhombus - digonal symmetry, equal sides, two types of alternating angles
- Parallelogram - two pairs of parallel sides, central inversion symmetry only
- Isosceles trapezoid - bilateral symmetry, one pair of parallel sides, other two sides the same length
- Kite - bilateral symmetry, two side lengths, only 2 of 4 angles are identical
- Dart - concave version of the kite
- Trapezoid - no symmetry, one pair of parallel sides
- Irregular tetragon - convex quadrilateral that does not fit in the above categories
- Bowtie or crossed rectangle - crossed, with vertices forming a rectangle
- Crossed trapezoid - crossed, with vertices forming a trapezoid
- Butterfly - similar to above

## External links[edit | edit source]

- Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

- Klitzing, Richard. "Polygons"
- Wikipedia contributors. "Square".
- Hi.gher.Space Wiki Contributors. "Square".

- Hartley, Michael. "{4}*8".