Square

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 combining prefix is squ-, as in squipdip.

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.

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

Naming
The name square is descended from Old French esquarre, which itself is descended from the latin quadro (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.

Vertex coordinates
The vertices of a square of edge length 1 centered at the origin are:
 * (±1/2, ±1/2).

Representations
A square can be represented by the following Coxeter diagrams:


 * x4o (full symmetry) 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)


 * x x (digonal symmetry, generally

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

Other kinds of quadrilaterals
Besides the regular square, there are various other kinds of quadrilaterals:

Rectangle
A rectangle has two pairs of opposite parallel sides, with all angles still being equal at 90º.

Bowtie
A bowtie, also known as a crossed rectangle, is a type of crossed quadrilateral where the vertices form a rectangle. It has one pair of opposite parallel sides.

Trapezoid
A trapezoid is a quadrilateral with one pair of parallel sides. The two vertices on either end of one of the parallel sides are identical.

Butterfly
A butterfly, also known as an antiparallelogram, is a type of crossed quadrilateral where the vertices form a trapezoid. Angles and edges come in two types.

Crossed trapezoid
A crossed trapezoid is a type of crossed quadrilateral where the vertices form a trapezoid. Angles come in two types, but unlike the butterfly, edges come in three types.