# Square

(Redirected from Squares)
Square
Rank2
TypeRegular
Notation
Bowers style acronymSquare
Coxeter diagramx4o ()
Schläfli symbol{4}
Tapertopic notation11
Toratopic notationII
Bracket notation[II]
Elements
Edges4
Vertices4
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Area1
Angle90°
Height1
Central density1
Number of external pieces4
Level of complexity1
Related polytopes
ArmySquare
DualSquare
ConjugateNone
Abstract & topological properties
Flag count8
Euler characteristic0
OrientableYes
Properties
SymmetryB2, order 8
ConvexYes
NatureTame

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.

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

## Naming

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 squipdip or s-, as in tisdip.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A square can be represented by the following Coxeter diagrams:

• x4o () (full symmetry) rectangle)
• x x () (digonal symmetry, generally
• 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

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 and an infinite family of regular hyperbolic tilings starting with the pentagonal tiling.

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