Ingeometry, asquare is aregularquadrilateral. It has four straight sides of equal length and four equalangles. Squares are special cases ofrectangles, which have four equal angles, and ofrhombuses, which have four equal sides. As with all rectangles, a square's angles areright angles (90degrees, orπ/2radians), making adjacent sidesperpendicular. Thearea of a square is the side length multiplied by itself, and so inalgebra, multiplying a number by itself is calledsquaring.
The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods forsquaring the circle bycompass 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 unsolvedwhether a square can be inscribed in every simple closed curve. Several problems ofsquaring the square involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes.
Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).
Squares can be defined or characterized in many equivalent ways. If apolygon in theEuclidean plane satisfies any one of the following criteria, it satisfies all of them:
A square is a polygon with four equal sides and fourright angles; that is, it is a quadrilateral that is both a rhombus and a rectangle[1]
A square is aparallelogram with one right angle and two adjacent equal sides.[1]
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.[2]
A square is a quadrilateral with successive sides,,, whose area is[3]
A square is a special case of arhombus (equal sides, opposite equal angles), akite (two pairs of adjacent equal sides), atrapezoid (one pair of opposite sides parallel), aparallelogram (all opposite sides parallel), aquadrilateral or tetragon (four-sided polygon), and arectangle (opposite sides equal, right-angles),[1] 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).[4][5]
All squares aresimilar to each other, meaning they have the same shape.[9] One parameter (typically the length of a side or diagonal)[10] suffices to specify a square's size. Squares of the same size arecongruent.[11]
YBC 7289, aBabylonian calculation of a square's diagonal from between 1800 and 1600 BCEThe area of a square is the product of the lengths of its sides.
A square whose four sides have length hasperimeter[12] anddiagonal length.[13] Thesquare root of 2, appearing in this formula, isirrational, meaning that it cannot be written exactly as afraction. It is approximately equal to 1.414,[14] and its approximate value was already known inBabylonian mathematics.[15] A square'sarea is[13]This formula for the area of a square as the second power of its side length led to the use of the termsquaring to mean raising any number to the second power.[16] Reversing this relation, the side length of a square of a given area is thesquare root of the area. Squaring aninteger, or taking the area of a square with integer sides, results in asquare number; these arefigurate numbers representing the numbers of points that can be arranged into a square grid.[17]
Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is anequable shape. The only other equable integer rectangle is a three-by-six rectangle.[18]
Because it is aregular 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.[19] Indeed, ifA andP are the area and perimeter enclosed by a quadrilateral, then the followingisoperimetric inequality holds:with equality if and only if the quadrilateral is a square.[20][21]
The axes of reflection symmetry and centers of rotation symmetry of a square (top), rectangle and rhombus (center),isosceles trapezoid, kite, and parallelogram (bottom)
For an axis-parallel square centered at theorigin, each symmetry acts by a combination of negating and swapping theCartesian coordinates of points.[24]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 thefundamental region of the transformations.[25] 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).[22] Allregular polygons also have these properties,[26] which are expressed by saying that symmetries of a square and, more generally, a regular polygon acttransitively on vertices and edges, andsimply transitively on half-edges.[27]
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. Thiscomposition operation gives the eight symmetries of a square the mathematical structure of agroup, called thegroup of the square or thedihedral group of order eight.[23] Other quadrilaterals, like the rectangle and rhombus, have only asubgroup of these symmetries.[28][29]
Three-point perspective of a cube, showing perspective transformations of its six square faces into six different quadrilaterals
Thewallpaper groups are symmetry groups of two-dimensional repeating patterns. For many of these groups the basic unit of repetition (the unit cell of itsperiod lattice) can be a square, and for three of these groups, p4, p4m, and p4g, it must be a square.[35]
Theinscribed 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 (theinradius of the square) is. Because this circle touches all four sides of the square (at their midpoints), the square is atangential quadrilateral. Thecircumscribed circle of a square passes through all four vertices, making the square acyclic quadrilateral. Its radius, thecircumradius, is.[36] If the inscribed circle of a square has tangency points on, on, on, and on, then for any point on the inscribed circle,[37] If is the distance from an arbitrary point in the plane to theth vertex of a square and is thecircumradius of the square, then[38]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.[39]
Architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint. Ancient examples include theEgyptian pyramids,[46]Mesoamerican pyramids such as those atTeotihuacan,[47] theChogha Zanbil ziggurat in Iran,[48] 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 theTaj Mahal in India,[49] the square bases of Buddhiststupas,[50] and East Asianpagodas, buildings that symbolically face to the four points of the compass and reach to the heavens.[51] Normankeeps such as theTower of London often take the form of a low square tower.[52] In modern architecture, a majority ofskyscrapers feature a square plan for pragmatic rather than aesthetic or symbolic reasons.[53]
The squarego board is said to represent the earth, with the 361 crossings of its lines representing days of the year.[65] Thechessboard inherited its square shape from apachisi-like Indian race game and in turn passed it on tocheckers.[66] In two ancient games fromMesopotamia andAncient Egypt, theRoyal Game of Ur andSenet, the game board itself is not square, but rectangular, subdivided into a grid of squares.[67] The ancient GreekOstomachion puzzle (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinesetangram.[68] Another set of puzzle pieces, thepolyominos, are formed from squares glued edge-to-edge.[69] Medieval and Renaissancehoroscopes were arranged in a square format, across Europe, the Middle East, and China.[70] Other recreational uses of squares include the shape oforigami paper,[71] and a common style ofquilting involving the use of square quilt blocks.[72]Scrabble players place square lettered tiles[73] onto a grid of squares on a square board.[74]
Square flag of the municipality ofVuadens, based on the Swiss flag
Aunit square is a square of side length one. Often it is represented inCartesian coordinates as the square enclosing the points that have and. Its vertices are the four points that have 0 or 1 in each of their coordinates.[87]
An axis-parallel square with its center at the point and sides of length (where is the inradius, half the side length) has vertices at the four points. Its interior consists of the points with, and its boundary consists of the points with.[88]
A diagonal square with its center at the point and diagonal of length (where is the circumradius, half the diagonal) has vertices at the four points and. Its interior consists of the points with, and its boundary consists of the points with.[88] For instance the illustration shows a diagonal square centered at the origin with circumradius 2, given by the equation.
A square formed by multiplying the complex numberp by powers ofi, and its translation obtained by adding another complex numberc. The background grid shows theGaussian integers.
In theplane of complex numbers, multiplication by theimaginary unit rotates the other term in the product by 90° around the origin (the number zero). Therefore, if any nonzero complex number is repeatedly multiplied by, giving the four numbers,,, and, these numbers will form the vertices of a square centered at the origin.[89] If one interprets thereal part andimaginary part of these four complex numbers as Cartesian coordinates, with, then these four numbers have the coordinates,,, and.[90] This square can be translated to have any other complex number is center, using the fact that thetranslation from the origin to is represented in complex number arithmetic as addition with.[91] TheGaussian integers, complex numbers with integer real and imaginary parts, form asquare lattice in the complex plane.[91]
TheSchläfli symbol of a square is {4}.[96] Atruncated square is anoctagon.[97] The square belongs to a family ofregular polytopes that includes thecube in three dimensions and thehypercubes in higher dimensions,[98] and to another family that includes theregular octahedron in three dimensions and thecross-polytopes in higher dimensions.[99] The cube and hypercubes can be given vertex coordinates that are all, giving an axis-parallel square in two dimensions, while the octahedron and cross-polytopes have one coordinate and the rest zero, giving a diagonal square in two dimensions.[100] As with squares, thesymmetries of these shapes can be obtained by applying asigned permutation to their coordinates.[24]
TheFinsler–Hadwiger theorem states that for two squares and, the center of both squares and the midpoint of and form a third square. This theorem can be applied repeatedly to provevan Aubel's theorem, that the centers of four squares constructed on the sides of a quadrilateral form amidsquare quadrilateral.[106] A square cannot be dissected into an odd number of equal-area triangles, a result ofMonsky's theorem.[107]
TheCalabi triangle and the three placements of its largest square.[113] The placement on the long side of the triangle is inscribed; the other two are not.
A square isinscribed in a curve when all four vertices of the square lie on the curve. The unsolvedinscribed square problem asks whether everysimple closed curve has an inscribed square. It is true for everysmooth curve,[114] and for any closedconvex curve. The only other regular polygon that can always be inscribed in every closed convex curve is theequilateral triangle, as there exists a convex curve on which no other regular polygon can be inscribed.[115]
For aninscribed square in a triangle, at least one side of the square lies on a side of the triangle. Everyacute triangle has three inscribed squares, one for each of its three sides. Aright triangle has two inscribed squares, one touching its right angle and the other lying on its hypotenuse. Anobtuse triangle has only one inscribed square, on its longest side. A square inscribed in a triangle can cover at most half the triangle's area.[116]
ThePythagorean theorem: the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.A circle and square with the same area
Since ancient times, many units for surfacearea have been defined from squares, typically with a standard unit oflength as its side, for example asquare meter orsquare inch.[117]
This use of a square as the defining shape for area measurement also occurs in the Greek formulation of thePythagorean theorem: squares constructed on the two sides of aright triangle have equal total area to a square constructed on thehypotenuse.[121] 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,[122] but the Greeks used squares. In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involvingsquaring numbers: the lengths of the sides and hypotenuse of the right triangle obey the equation.[123]
Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully tosquare 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 theLindemann–Weierstrass theorem. This theorem proves thatpi (π) is atranscendental number rather than analgebraic irrational number; that is, it is not theroot of anypolynomial withrational coefficients. A construction for squaring the circle could be translated into a polynomial formula forπ, which does not exist.[124] In philosophy, the concept of a "square circle" has been used as an example of anoxymoron sinceAristotle, sparking attempts to find contexts such astaxicab geometry (below) in which this phrase is meaningful.[125]
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.[136] 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 thegreatest common divisor of the side lengths be 1.[137] The entire plane can be tiled by squares, with exactly one square of each integer side length.[138]
In higher dimensions, other surfaces than the plane can be tiled by equal squares, meeting edge-to-edge. One of these surfaces is theClifford torus, the four-dimensionalCartesian product of two congruent circles; it has the same intrinsic geometry as a single square with each pair of opposite edges glued together.[139] Another square-tiled surface, aregular skew apeirohedron in three dimensions, has six squares meeting at each vertex.[140] Thepaper 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.[141] 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 abiscornu.[142] The surfaces tiled with finitely many squares of the three-dimensional integer lattice are calledpolyominoids.[143]
Two square-counting puzzles: There are 14 squares in a3 × 3 grid of squares (top), but as a4 × 4 grid of points it has six more off-axis squares (bottom) for a total of 20.
A commonmathematical puzzle involves counting the squares of all sizes in a square grid of squares. For instance, a square grid of nine squares has 14 squares: the nine squares that form the grid, four more squares, and one square. The answer to the puzzle is, asquare pyramidal number.[144] For these numbers are:[145]
1, 5, 14, 30, 55, 91, 140, 204, 285, ...
A variant of the same puzzle asks for the number of squares formed by a grid of 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.[146] In this case, the answer is given by the4-dimensional pyramidal numbers. For these numbers are:[147]
0, 1, 6, 20, 50, 105, 196, 336, 540, ...
Partitions of a square into three similar rectangles
Another counting problem involving squares asks for the number of different shapes of rectangles that can be used whendividing 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 possibleaspect ratios of the rectangles,[148] 3:1, 3:2, and the square of theplastic ratio, approximately 1.755:1.[149] The number of proportions that are possible when dividing into rectangles is known for small values of, but not as a general formula. For these numbers are:[150]
1, 1, 3, 11, 51, 245, 1372, ...
Amagic square is a square array of numbers, where the sums of the positive numbers in each row, each column, and both main diagonals are the same.[151] For array, the sum can be formulated as; the numbers for are:[152]
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, inspherical geometry andhyperbolic geometry, 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 feature regular quadrilaterals, characterized by four equal sides and four equal angles, often referred to as squares,[153] although some authors prefer to avoid this name because they lack right angles. These geometries also feature regular polygons with right angles, but with the number of sides differing from four.[154]
In spherical geometry, space has uniform positive curvature, and every convex quadrilateral (apolygon with fourgreat-circle arc edges) has angles whose sum exceeds 360° by an amount called theangular excess, proportional to its surface area. Small spherical squares are approximately Euclidean, and the angles of larger squares increase with area.[153] One special case is the face of aspherical cube with four 120° angles, covering one sixth of the sphere's surface.[155] Another is ahemisphere, the face of a spherical squaredihedron, with fourstraight angles; thePeirce quincuncial projection forworld mapsconformally maps two such faces to Euclidean squares.[156] Anoctant of a sphere is a regularspherical triangle, with three equal sides and three right angles; eight of them tile the sphere, with four meeting at each vertex, to form aspherical octahedron.[157] Aspherical lune is a regulardigon, with two semicircular sides and two equal angles atantipodal vertices; a right-angled lune covers one quarter of the sphere, one face of a four-lunehosohedron.[158]
Inhyperbolic geometry, space has uniform negative curvature, and every convex quadrilateral has angles whose sum falls short of 360° by an amount called theangular defect, proportional to its surface area. Small hyperbolic squares are approximately Euclidean, and larger squares decrease with increasing area. Special cases include the squares with angles of360°/n for every value ofn larger than4, each of which can tile thehyperbolic plane.[154] In the infinite limit, anideal square has four sides of infinite length and four vertices atideal points outside the hyperbolic plane, with0° internal angles;[159] an ideal square, like every ideal quadrilateral, has finite area proportional to its angular defect of360°.[160] 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 canuniformly tile the hyperbolic plane,dual to the tiling withn squares about each vertex.[154]
Metric circles using Chebyshev, Euclidean, and taxicab distance functions
The Euclidean plane can be defined in terms of thereal coordinate plane by adoption of theEuclidean distance function, according to which the distance between any two points and is. Other metric geometries are formed when a differentdistance 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 intaxicab geometry, based on the distance. The points with taxicab distance from any given point form a diagonal square, centered at the given point, with diagonal length. In the same way, axis-parallel squares are the circles for the orChebyshev distance,. In this metric, the points with distance from some point form an axis-parallel square, centered at the given point, with side length.[161][162][163]
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