Square (geometry)

Square

A square is a regular quadrilateral.
Edges and vertices 4
Schläfli symbol {4}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D4)
Area t2 (with t = edge length)
Internal angle (degrees) 90°
Dual polygon dual polygon of this shape
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90-degree angles, or right angles)[1]. A square with vertices ABCD would be denoted \square ABCD

The square belong to the families of 2-hypercube and 2-orthoplex.

Contents

Characterizations

A convex quadrilateral is a square if and only if it is any one of the following:[2]

Perimeter and area

The perimeter of a square whose sides have length t is

P=4t \,

and the area is

A=t^2.\,

In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.

Standard coordinates

The coordinates for the vertices of a square centered at the origin and with side length 2 are (±1, ±1), while the interior of the same consists of all points (x0, x1) with −1 < xi < 1.

Equations

The equation max(x^2, y^2) = 1 describes a square of side = 2, centered at the origin. This equation means "x^2 or y^2, whichever is larger, equals 1." The circumradius of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and equals \scriptstyle \sqrt{2}.

Properties

A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a parallelogram (opposite sides parallel), a quadrilateral or tetragon (four-sided polygon), and a rectangle (opposite sides equal, right-angles) and therefore has all the properties of all these shapes, namely:[3]

Other facts

Non-Euclidean geometry

In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles.

In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle.

In hyperbolic geometry, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger squares have smaller angles.

Examples:


Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Squares can tile the Euclidean plane with 4 around each vertex, with each square having an internal angle of 90°. The Schläfli symbol is {4,4}.
Squares can tile the hyperbolic plane with 5 around each vertex, with each square having 72-degree internal angles. The Schläfli symbol is {4,5}.

Graphs

The K4 complete graph is often drawn as a square with all 6 edges connected. This graph also represents an orthographic projection of the 4 vertices and 6 edges of the regular 3-simplex (tetrahedron).


3-simplex (3D)

See also

References

  1. ^ Weisstein, Eric W. "Square." From MathWorld--A Wolfram Web Resource.
  2. ^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 59, ISBN 1593116950.
  3. ^ http://www.mathsisfun.com/quadrilaterals.html/
  4. ^ http://www2.mat.dtu.dk/people/V.L.Hansen/square.html

External links

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