Square


In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles, making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring.
Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art.
The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge, now known to be impossible. Squares can be inscribed in any smooth or convex curve, such as a circle or triangle, but it remains unsolved whether a square can be inscribed in every simple closed curve. Several problems of squaring the square involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes.
Squares can be constructed by straightedge and compass, through their Cartesian coordinates, or by repeated multiplication by in the complex plane. They form the metric balls for taxicab geometry and Chebyshev distance, two forms of non-Euclidean geometry. Although spherical geometry and hyperbolic geometry both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles, or with right angles and different numbers of sides.

Definitions and characterizations

Squares can be defined or characterized in many equivalent ways. If a polygon in the Euclidean plane satisfies any one of the following criteria, it satisfies all of them:
  • A square is a polygon with four equal sides and four right angles; that is, it is a quadrilateral that is both a rhombus and a rectangle
  • A square is a rectangle with four equal sides.
  • A square is a rhombus with a right angle between a pair of adjacent sides.
  • A square is a rhombus with all angles equal.
  • A square is a parallelogram with one right angle and two adjacent equal sides.
  • A square is a quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other. That is, it is a rhombus with equal diagonals.
  • A square is a quadrilateral with successive sides,,, whose area is
Squares are the only regular polygons whose internal angle, central angle, and external angle are all equal.

Properties

A square is a special case of a rhombus, a kite, a trapezoid, a parallelogram, a quadrilateral or tetragon, and a rectangle, and therefore has all the properties of all these shapes, namely:
  • All four internal angles of a square are equal.
  • The central angle of a square is equal to 90°.
  • The external angle of a square is equal to 90°.
  • The diagonals of a square are equal and bisect each other, meeting at 90°.
  • The diagonals of a square bisect its internal angles, forming adjacent angles of 45°.
  • All four sides of a square are equal.
  • Opposite sides of a square are parallel.
All squares are similar to each other, meaning they have the same shape. One parameter suffices to specify a square's size. Squares of the same size are congruent.

Measurement

A square whose four sides have length has perimeter and diagonal length. The square root of 2, appearing in this formula, is irrational, meaning that it cannot be written exactly as a fraction. It is approximately equal to 1.414, and its approximate value was already known in Babylonian mathematics. A square's area is
This formula for the area of a square as the second power of its side length led to the use of the term squaring to mean raising any number to the second power. Reversing this relation, the side length of a square of a given area is the square root of the area. Squaring an integer, or taking the area of a square with integer sides, results in a square number; these are figurate numbers representing the numbers of points that can be arranged into a square grid.
Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an equable shape. The only other equable integer rectangle is a three-by-six rectangle.
Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:
with equality if and only if the quadrilateral is a square.

Symmetry

The square is the most symmetrical of the quadrilaterals. Eight rigid transformations of the plane take the square to itself:
For an axis-parallel square centered at the origin, each symmetry acts by a combination of negating and swapping the Cartesian coordinates of points. The symmetries permute the eight isosceles triangles between the half-edges and the square's center ; any of these triangles can be taken as the fundamental region of the transformations. Each two vertices, each two edges, and each two half-edges are mapped one to the other by at least one symmetry. All regular polygons also have these properties, which are expressed by saying that symmetries of a square and, more generally, a regular polygon act transitively on vertices and edges, and simply transitively on half-edges.
Combining any two of these transformations by performing one after the other continues to take the square to itself, and therefore produces another symmetry. Repeated rotation produces another rotation with the summed rotation angle. Two reflections with the same axis return to the identity transformation, while two reflections with different axes rotate the square. A rotation followed by a reflection, or vice versa, produces a different reflection. This composition operation gives the eight symmetries of a square the mathematical structure of a group, called the group of the square or the dihedral group of order eight. Other quadrilaterals, like the rectangle and rhombus, have only a subgroup of these symmetries.
The shape of a square, but not its size, is preserved by similarities of the plane. Other kinds of transformations of the plane can take squares to other kinds of quadrilateral. An affine transformation can take a square to any parallelogram, or vice versa; a projective transformation can take a square to any convex quadrilateral, or vice versa. This implies that, when viewed in perspective, a square can look like any convex quadrilateral, or vice versa. A Möbius transformation can take the vertices of a square to the vertices of a harmonic quadrilateral.
The wallpaper groups are symmetry groups of two-dimensional repeating patterns. For many of these groups the basic unit of repetition can be a square, and for three of these groups, p4, p4m, and p4g, it must be a square.

Inscribed and circumscribed circles

The inscribed circle of a square is the largest circle that can fit inside that square. Its center is the center point of the square, and its radius is. Because this circle touches all four sides of the square, the square is a tangential quadrilateral. The circumscribed circle of a square passes through all four vertices, making the square a cyclic quadrilateral. Its radius, the circumradius, is. If the inscribed circle of a square has tangency points on, on, on, and on, then for any point on the inscribed circle, If is the distance from an arbitrary point in the plane to the vertex of a square and is the circumradius of the square, then
If and are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then and where is the circumradius of the square.

Applications

Squares are so well-established as the shape of tiles that the Latin word tessera, for a small tile as used in mosaics, comes from an ancient Greek word for the number four, referring to the four corners of a square tile. Graph paper, preprinted with a square tiling, is widely used for data visualization using Cartesian coordinates. The pixels of bitmap images, as recorded by image scanners and digital cameras or displayed on electronic visual displays, conventionally lie at the intersections of a square grid, and are often considered as small squares, arranged in a square tiling. Standard techniques for image compression and video compression, including the JPEG format, are based on the subdivision of images into larger square blocks of pixels. The quadtree data structure used in data compression and computational geometry is based on the recursive subdivision of squares into smaller squares.
Architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint. Ancient examples include the Egyptian pyramids, Mesoamerican pyramids such as those at Teotihuacan, the Chogha Zanbil ziggurat in Iran, the four-fold design of Persian walled gardens, said to model the four rivers of Paradise,
and later structures inspired by their design such as the Taj Mahal in India, the square bases of Buddhist stupas, and East Asian pagodas, buildings that symbolically face to the four points of the compass and reach to the heavens. Norman keeps such as the Tower of London often take the form of a low square tower. In modern architecture, a majority of skyscrapers feature a square plan for pragmatic rather than aesthetic or symbolic reasons.
The stylized nested squares of a Tibetan mandala, like the design of a stupa, function as a miniature model of the cosmos. Some formats for film photography use a square aspect ratio, notably Polaroid cameras, medium format cameras, and Instamatic cameras. Painters known for their frequent and prominent use of square forms and frames include Josef Albers, Kazimir Malevich, Piet Mondrian, and Theo van Doesburg.
Baseball diamonds and boxing rings are square despite being named for other shapes. In the quadrille and square dance, four couples form the sides of a square. In Samuel Beckett's minimalist television play Quad, four actors walk along the sides and diagonals of a square.
The square go board is said to represent the earth, with the 361 crossings of its lines representing days of the year. The chessboard inherited its square shape from a pachisi-like Indian race game and in turn passed it on to checkers. In two ancient games from Mesopotamia and Ancient Egypt, the Royal Game of Ur and Senet, the game board itself is not square, but rectangular, subdivided into a grid of squares. The ancient Greek Ostomachion puzzle involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese tangram. Another set of puzzle pieces, the polyominos, are formed from squares glued edge-to-edge; in this context, a single square is called a monomino. Medieval and Renaissance horoscopes were arranged in a square format, across Europe, the Middle East, and China. Other recreational uses of squares include the shape of origami paper, and a common style of quilting involving the use of square quilt blocks. Scrabble players place square lettered tiles onto a grid of squares on a square board.
Squares are a common element of graphic design, used to give a sense of stability, symmetry, and order. In heraldry, a canton is normally square, and a square flag is called a banner. The flag of Switzerland is square, as are the flags of the Swiss cantons. QR codes are square and feature prominent nested square alignment marks in three corners. Robertson screws have a square drive socket. Crackers and sliced cheese are often square, as are waffles. Square foods named for their square shapes include caramel squares, date squares, lemon squares, square sausage, and Carré de l'Est cheese.
A square can be found in an optical illusion. The Orbison illusion, named after American psychologist William Orbison, consists of a two-dimensional circle and square, superimposed over a background of radial lines or concentric circles.
In stereochemistry, a square planar molecular geometry is a chemical structure with atoms at the corners of a square. An example is xenon tetrafluoride.