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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a square is a regular
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
. It has four straight sides of equal length and four equal
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s. Squares are special cases of
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
s, which have four equal angles, and of
rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
es, which have four equal sides. As with all rectangles, a square's angles are
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s (90 degrees, or /2
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s), making adjacent sides
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
. The
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a square is the side length multiplied by itself, and so in
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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
tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wal ...
d floors and walls, graph paper, image
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
s, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in
graphic design Graphic design is a profession, academic discipline and applied art that involves creating visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdisciplinary branch of ...
and
heraldry Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, Imperial, royal and noble ranks, rank and genealo ...
, 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 Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
by
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
, 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 i in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. They form the metric balls for taxicab geometry and Chebyshev distance, two forms of non-Euclidean geometry. Although spherical geometry and
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
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 geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
in the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
satisfies any one of the following criteria, it satisfies all of them: * A square is a polygon with four equal sides and four
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s; that is, it is a quadrilateral that is both a rhombus and a rectangle * A square is a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
with four equal sides. * A square is a
rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
with a right angle between a pair of adjacent sides. * A square is a rhombus with all angles equal. * A square is a
parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
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 a, b, c, d whose area is A=\frac14(a^2+b^2+c^2+d^2). Squares are the only
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s whose internal angle, central angle, and external angle are all equal (they are all right angles).


Properties

A square is a special case of a
rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
(equal sides, opposite equal angles), a
kite A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
(two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides parallel), a
parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
(all opposite sides parallel), a
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
or tetragon (four-sided polygon), and a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
(opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely: * All four internal angles of a square are equal (each being 90°, a right angle). * 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 (typically the length of a side or diagonal) suffices to specify a square's size. Squares of the same size are congruent.


Measurement

A square whose four sides have length \ell has
perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
P=4\ell and
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
length d=\sqrt2\ell. The
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, appearing in this formula, is irrational, meaning that it is not the ratio of any two
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. It is approximately equal to 1.414, and its approximate value was already known in Babylonian mathematics. A square's
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
is A=\ell^2=\tfrac12 d^2. 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 In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
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 In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
, 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: 16A\le P^2 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 (which stays in place); 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 (exactly one for half-edges). All
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s 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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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 In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
can take a square to any parallelogram, or vice versa; a projective transformation can take a square to any convex
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
, 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 (but not its edges) 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 (the unit cell of its period lattice) 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 (the
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
of the square) is r=\ell/2. Because this circle touches all four sides of the square (at their midpoints), 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 R=\ell/\sqrt2. If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle, 2(PH^2-PE^2) = PD^2-PB^2. If d_i is the distance from an arbitrary point in the plane to the vertex of a square and R is the circumradius of the square, then\frac + 3R^4 = \left(\frac + R^2\right)^2. If L and d_i are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then d_1^2 + d_3^2 = d_2^2 + d_4^2 = 2(R^2+L^2) and d_1^2d_3^2 + d_2^2d_4^2 = 2(R^4+L^4), where R is the circumradius of the square.


Applications

Squares are so well-established as the shape of tiles that the
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
word tessera, for a small tile as used in
mosaic A mosaic () is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/Mortar (masonry), mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and ...
s, 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
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, p ...
s of bitmap images, as recorded by
image scanner An image scanner (often abbreviated to just scanner) is a device that optically scans images, printed text, handwriting, or an object and converts it to a digital image. The most common type of scanner used in the home and the office is the flatbe ...
s and
digital camera A digital camera, also called a digicam, is a camera that captures photographs in Digital data storage, digital memory. Most cameras produced today are digital, largely replacing those that capture images on photographic film or film stock. Dig ...
s 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 In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression ...
, including the
JPEG JPEG ( , short for Joint Photographic Experts Group and sometimes retroactively referred to as JPEG 1) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degr ...
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 The Egyptian pyramids are ancient masonry structures located in Egypt. Most were built as tombs for the pharaohs and their consorts during the Old Kingdom of Egypt, Old and Middle Kingdom of Egypt, Middle Kingdom periods. At least 138 identi ...
, Mesoamerican pyramids such as those at
Teotihuacan Teotihuacan (; Spanish language, Spanish: ''Teotihuacán'', ; ) is an ancient Mesoamerican city located in a sub-valley of the Valley of Mexico, which is located in the State of Mexico, northeast of modern-day Mexico City. Teotihuacan is ...
, 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 The Taj Mahal ( ; ; ) is an ivory-white marble mausoleum on the right bank of the river Yamuna in Agra, Uttar Pradesh, India. It was commissioned in 1631 by the fifth Mughal Empire, Mughal emperor, Shah Jahan () to house the tomb of his belo ...
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 The Tower of London, officially His Majesty's Royal Palace and Fortress of the Tower of London, is a historic citadel and castle on the north bank of the River Thames in central London, England. It lies within the London Borough of Tower Hamle ...
often take the form of a low square tower. In modern architecture, a majority of
skyscraper A skyscraper is a tall continuously habitable building having multiple floors. Most modern sources define skyscrapers as being at least or in height, though there is no universally accepted definition, other than being very tall high-rise bui ...
s 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 The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
, notably Polaroid cameras, medium format cameras, and Instamatic cameras. Painters known for their frequent use of square frames and forms include
Josef Albers Josef Albers ( , , ; March 19, 1888March 25, 1976) was a German-born American artist and Visual arts education, educator who is considered one of the most influential 20th-century art teachers in the United States. Born in 1888 in Bottrop, Westp ...
,
Kazimir Malevich Kazimir Severinovich Malevich (
and
Piet Mondrian Pieter Cornelis Mondriaan (; 7 March 1872 – 1 February 1944), known after 1911 as Piet Mondrian (, , ), was a Dutch Painting, painter and Theory of art, art theoretician who is regarded as one of the greatest artists of the 20th century. He w ...
. Baseball diamonds and
boxing ring A boxing ring, often referred to simply as a ring or the squared circle, is the space in which a boxing match occurs. A modern ring consists of a square raised platform with a post at each corner. Four ropes are attached to the posts and pulled p ...
s 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 Samuel Barclay Beckett (; 13 April 1906 – 22 December 1989) was an Irish writer of novels, plays, short stories, and poems. Writing in both English and French, his literary and theatrical work features bleak, impersonal, and Tragicomedy, tra ...
'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 Checkers (American English), also known as draughts (; English in the Commonwealth of Nations, Commonwealth English), is a group of Abstract strategy game, strategy board games for two players which involve forward movements of uniform game ...
. In two ancient games from
Mesopotamia Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...
and
Ancient Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
, 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 (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese tangram. Another set of puzzle pieces, the
polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popu ...
s, are formed from squares glued edge-to-edge. 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. Squares are a common element of
graphic design Graphic design is a profession, academic discipline and applied art that involves creating visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdisciplinary branch of ...
, used to give a sense of stability, symmetry, and order. In
heraldry Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, Imperial, royal and noble ranks, rank and genealo ...
, a canton (a design element in the top left of a shield) 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 Cheese is a type of dairy product produced in a range of flavors, textures, and forms by coagulation of the milk protein casein. It comprises proteins and fat from milk (usually the milk of cows, buffalo, goats or sheep). During prod ...
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.


Constructions


Coordinates and equations

A
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
is a square of side length one. Often it is represented in Cartesian coordinates as the square enclosing the points (x,y) that have 0\le x\le 1 and 0\le y\le 1. Its vertices are the four points that have 0 or 1 in each of their coordinates. An axis-parallel square with its center at the point (x_c,y_c) and sides of length 2r (where r is the inradius, half the side length) has vertices at the four points (x_c\pm r,y_c\pm r). Its interior consists of the points (x,y) with \max(, x-x_c, ,, y-y_c, ) < r, and its boundary consists of the points with \max(, x-x_c, ,, y-y_c, )=r. A diagonal square with its center at the point (x_c,y_c) and diagonal of length 2R (where R is the circumradius, half the diagonal) has vertices at the four points (x_c\pm R,y_c) and (x_c,y_c\pm R). Its interior consists of the points (x,y) with , x-x_c, +, y-y_c, , and its boundary consists of the points with , x-x_c, +, y-y_c, =R. For instance the illustration shows a diagonal square centered at the origin (0,0) with circumradius 2, given by the equation , x, +, y, =2. In the plane of complex numbers, multiplication by the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
i rotates the other term in the product by 90° around the origin (the number zero). Therefore, if any nonzero complex number p is repeatedly multiplied by i, giving the four numbers p, ip, -p, and -ip, these numbers will form the vertices of a square centered at the origin. If one interprets the real part and imaginary part of these four complex numbers as Cartesian coordinates, with p=x+iy, then these four numbers have the coordinates (x,y), (-y,x), (-x,-y), and (-y,-x). This square can be translated to have any other complex number c is center, using the fact that the
translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
from the origin to c is represented in complex number arithmetic as addition with c. The
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
s, complex numbers with integer real and imaginary parts, form a square lattice in the complex plane.


Compass and straightedge

The construction of a square with a given side, using a
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
, is given in Euclid's ''Elements'' I.46. The existence of this construction means that squares are constructible polygons. A regular is constructible exactly when the odd
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s of n are distinct
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: 3, 5, ...
s, and in the case of a square n=4 has no odd prime factors so this condition is vacuously true. ''Elements'' IV.6–7 also give constructions for a square inscribed in a circle and circumscribed about a circle, respectively. Straight Square Inscribed in a Circle 240px.gif, Square with a given circumcircle 01-Quadrat-Seite-gegeben.gif, Square with a given side length, using
Thales' theorem In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as pa ...
01-Quadrat-Diagonale-gegeben.gif, Square with a given diagonal


Related topics

The
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of a square is . A truncated square is an
octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
. The square belongs to a family of
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...
s that includes the
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
in three dimensions and the
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...
s in higher dimensions, and to another family that includes the regular octahedron in three dimensions and the cross-polytopes in higher dimensions. The cube and hypercubes can be given vertex coordinates that are all \pm 1, giving an axis-parallel square in two dimensions, while the octahedron and cross-polytopes have one coordinate \pm 1 and the rest zero, giving a diagonal square in two dimensions. As with squares, the symmetries of these shapes can be obtained by applying a signed permutation to their coordinates. The Sierpiński carpet is a square
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
, with square holes.
Space-filling curve In mathematical analysis, a space-filling curve is a curve whose Range of a function, range reaches every point in a higher dimensional region, typically the unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Pea ...
s including the Hilbert curve, Peano curve, and Sierpiński curve cover a square as the continuous image of a line segment. The Z-order curve is analogous but not continuous. Other mathematical functions associated with squares include Arnold's cat map and the baker's map, which generate chaotic
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s on a square, and the lemniscate elliptic functions, complex functions periodic on a square grid.


Inscribed squares

A square is
inscribed An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same th ...
in a curve when all four vertices of the square lie on the curve. The unsolved inscribed square problem asks whether every simple closed curve has an inscribed square. It is true for every smooth curve, and for any closed convex curve. The only other regular polygon that can always be inscribed in every closed convex curve is the
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, as there exists a convex curve on which no other regular polygon can be inscribed. For an inscribed square in a triangle, at least one side of the square lies on a side of the triangle. Every acute triangle has three inscribed squares, one for each of its three sides. A right triangle has two inscribed squares, one touching its right angle and the other lying on its hypotenuse. An obtuse triangle has only one inscribed square, on its longest. A square inscribed in a triangle can cover at most half the triangle's area.


Area and quadrature

Conventionally, since ancient times, most units of
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
have been defined in terms of various squares, typically a square with a standard unit of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
as its side, for example a square meter or square inch. The area of an arbitrary rectangle can then be simply computed as the product of its length and its width, and more complicated shapes can be measured by conceptually breaking them up into unit squares or into arbitrary triangles. In ancient Greek deductive geometry, the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
, a process called '' quadrature'' or ''squaring''. Euclid's ''Elements'' shows how to do this for rectangles, parallelograms, triangles, and then more generally for
simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
s by breaking them into triangular pieces. Some shapes with curved sides could also be squared, such as the lune of Hippocrates and the
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
. This use of a square as the defining shape for area measurement also occurs in the Greek formulation of the Pythagorean theorem: squares constructed on the two sides of a right triangle have equal total area to a square constructed on the hypotenuse. Stated in this form, the theorem would be equally valid for other shapes on the sides of the triangle, such as equilateral triangles or semicircles, but the Greeks used squares. In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involving squaring numbers: the lengths of the sides and hypotenuse of the right triangle obey the equation a^2+b^2=c^2. Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to square the circle, constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the Lindemann–Weierstrass theorem. This theorem proves that pi () is a transcendental number rather than an algebraic irrational number; that is, it is not the
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
of any
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with rational coefficients. A construction for squaring the circle could be translated into a polynomial formula for , which does not exist.


Tiling and packing

The square tiling, familiar from flooring and game boards, is one of three regular tilings of the plane. The other two use the
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
and the regular hexagon. The vertices of a square tiling form a square lattice. Squares of more than one size can also tile the plane, for instance in the Pythagorean tiling, named for its connection to proofs of the Pythagorean theorem. Square packing problems seek the smallest square or circle into which a given number of
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
s can fit. A chessboard optimally packs a square number of unit squares into a larger square, but beyond a few special cases such as this, the optimal solutions to these problems remain unsolved; the same is true for circle packing in a square. Packing squares into other shapes can have high computational complexity: testing whether a given number of unit squares can fit into an orthogonally convex rectilinear polygon with half-integer vertex coordinates is
NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...
. Squaring the square involves subdividing a given square into smaller squares, all having integer side lengths. A subdivision with distinct smaller squares is called a perfect squared square. Another variant of squaring the square called "Mrs. Perkins's quilt" allows repetitions, but uses as few smaller squares as possible in order to make the
greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...
of the side lengths be 1. The entire plane can be tiled by squares, with exactly one square of each integer side length. In higher dimensions, other surfaces than the plane can be tiled by equal squares, meeting edge-to-edge. One of these surfaces is the Clifford torus, the four-dimensional
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
of two congruent circles; it has the same intrinsic geometry as a single square with each pair of opposite edges glued together. Another square-tiled surface, a regular skew apeirohedron in three dimensions, has six squares meeting at each vertex. The paper bag problem seeks the maximum volume that can be enclosed by a surface tiled with two squares glued edge to edge; its exact answer is unknown. Gluing two squares in a different pattern, with the vertex of each square attached to the midpoint of an edge of the other square (or alternatively subdividing these two squares into eight squares glued edge-to-edge) produces a pincushion shape called a biscornu.


Counting

A common mathematical puzzle involves counting the squares of all sizes in a square grid of n\times n squares. For instance, a square grid of nine squares has 14 squares: the nine squares that form the grid, four more 2\times 2 squares, and one 3\times 3 square. The answer to the puzzle is n(n+1)(2n+1)/6, a square pyramidal number. For n=1,2,3,\dots these numbers are: A variant of the same puzzle asks for the number of squares formed by a grid of n\times n points, allowing squares that are not axis-parallel. For instance, a grid of nine points has five axis-parallel squares as described above, but it also contains one more diagonal square for a total of six. In this case, the answer is given by the ''4-dimensional pyramidal numbers'' n^2(n^2-1)/12. For n=1,2,3,\dots these numbers are: Another counting problem involving squares asks for the number of different shapes of rectangle that can be used when dividing a square into similar rectangles. A square can be divided into two similar rectangles only in one way, by bisecting it, but when dividing a square into three similar rectangles there are three possible
aspect ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
s of the rectangles, 3:1, 3:2, and the square of the plastic ratio. The number of proportions that are possible when dividing into n rectangles is known for small values of n, but not as a general formula. For n=1,2,3,\dots these numbers are:


Other geometries

In the familiar Euclidean geometry, space is flat, and every convex quadrilateral has internal angles summing to 360°, so a square (a regular quadrilateral) has four equal sides and four right angles (each 90°). By contrast, in spherical geometry and
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
, space is curved and the internal angles of a convex quadrilateral never sum to 360°, so quadrilaterals with four right angles do not exist. Both of these geometries have regular quadrilaterals, with four equal sides and four equal angles, often called squares, but some authors avoid that name because they lack right angles. These geometries also have regular polygons with right angles, but with numbers of sides different from four. In spherical geometry, space has uniform positive curvature, and every convex quadrilateral (a
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
with four great-circle arc edges) has angles whose sum exceeds 360° by an amount called the angular excess, proportional to its surface area. Small spherical squares are approximately Euclidean, and larger squares' angles increase with area. See paragraph about spherical squares, p. 48. One special case is the face of a spherical cube with four 120° angles, covering one sixth of the sphere's surface. Another is a hemisphere, the face of a spherical square dihedron, with four straight angles; the Peirce quincuncial projection for world maps conformally maps two such faces to Euclidean squares. An octant of a sphere is a regular spherical triangle, with three equal sides and three right angles; eight of them tile the sphere, with four meeting at each vertex, to form a spherical octahedron. A spherical lune is a regular digon, with two semicircular sides and two equal angles at antipodal vertices; a right-angled lune covers one quarter of the sphere, one face of a four-lune hosohedron. In
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
, space has uniform negative curvature, and every convex quadrilateral has angles whose sum falls short of 360° by an amount called the
angular defect In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the ''exces ...
, proportional to its surface area. Small hyperbolic squares are approximately Euclidean, and larger squares' angles decrease with increasing area. Special cases include the squares with angles of for every value of larger than , each of which can tile the hyperbolic plane. In the infinite limit, an ideal square has four sides of infinite length and four vertices at ideal points outside the hyperbolic plane, with internal angles; an ideal square, like every ideal quadrilateral, has finite area proportional to its angular defect of . It is also possible to make a regular hyperbolic polygon with right angles at every vertex and any number of sides greater than four; such polygons can uniformly tile the hyperbolic plane, dual to the tiling with squares about each vertex. The Euclidean plane can be defined in terms of the real coordinate plane by adoption of the Euclidean distance function, according to which the distance between any two points (x_1,y_1) and (x_2,y_2) is \textstyle \sqrt. Other metric geometries are formed when a different distance function is adopted instead, and in some of these geometries shapes that would be Euclidean squares become the " circles" (set of points of equal distance from a center point). Squares tilted at 45° to the coordinate axes are the circles in taxicab geometry, based on the L_1 distance , x_1-x_2, +, y_1-y_2, . The points with taxicab distance d from any given point form a diagonal square, centered at the given point, with diagonal length 2d. In the same way, axis-parallel squares are the circles for the L_ or Chebyshev distance, \max(, x_1-x_2, ,, y_1-y_2, ). In this metric, the points with distance d from some point form an axis-parallel square, centered at the given point, with side length 2d.


See also

* Finsler–Hadwiger theorem on a square derived from two squares sharing a vertex * Midsquare quadrilateral, a polygon whose edge midpoints form a square * Monsky's theorem, on subdividing a square into an odd number of equal-area triangles * Square planar molecular geometry, chemical structure with atoms at the corners of a square * Square trisection, a problem of cutting and reassembling one square into three squares * Squircle, a shape intermediate between a square and a circle * Tarski's circle-squaring problem, dividing a disk into sets that can be rearranged into a square * Van Aubel's theorem and Thébault's theorem, on squares placed on the sides of a quadrilateral


References

{{Authority control Elementary shapes Types of quadrilaterals 4 (number) Constructible polygons