The
Schläfli symbol of a square is \{4\} . A
truncated square is an
octagon. The square belongs to a family of
regular polytopes that includes the
cube in three dimensions and the
hypercubes 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, with square holes.
Space-filling curves 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 systems on a square, and the
lemniscate elliptic functions, complex functions periodic on a square grid. Many theorems involve squares. The
Finsler–Hadwiger theorem states that for two squares ABCD and AB'C'D' , the center of both squares and the midpoint of BD' and B'D form a third square. This theorem can be applied repeatedly to prove
van Aubel's theorem, that the centers of four squares constructed on the sides of a quadrilateral form a
midsquare quadrilateral. A square cannot be dissected into an odd number of equal-area triangles, a result of
Monsky's theorem.
Cross's theorem or Vecten's theorem states that, for a triangle formed by the sides of three squares, and connecting the squares' vertices to form another three triangles, all triangles have the same area. of the
Great Britain's map
Minkowski–Bouligand dimension or box-counting dimension involves squares to determine the
fractal dimension of a set in Euclidean space or
metric space, the squares are named "boxes". The fractal dimension of a set is attainable by lying on an evenly spaced grid and counting the number of squares required to cover the set. The method of
box counting is used when the grid is finer. Mathematical puzzles that include squares are square arrays that are filled with numbers in
Sudoku with generalized symbols in
Latin square, and with color or blank in
nonogram, as well as paradoxical optical illusions such as the
missing square puzzle and the
chessboard paradox. The
Langton's ant, in
cellular automaton, is a two-dimensional square lattice consisting of a black-or-white colored grid wherein an ant assigns the combination of those colors and its current direction of motion. A
square trisection is a dissection problem of cutting a square into multiple pieces and then reassembling them into three identical squares.
Tarski's circle-squaring problem challenges the
equidecomposability between a square and a circle; that is, cutting the
disc of a circle into finitely many pieces and reassembling them into a square.
Inscribed squares and the three placements of its largest square. The placement on the long side of the triangle is inscribed; the other two are not. A square is
inscribed 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, 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 side. A square inscribed in a triangle can cover at most half the triangle's area.
Area and quadrature : the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse. Since ancient times, many units for surface
area have been defined from squares, typically with a standard unit of
length as its side, for example, a
square meter or
square inch. 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, a process called
quadrature or
squaring.
Euclid's Elements shows how to do this for rectangles, parallelograms, triangles, and then more generally for
simple polygons by breaking them into triangular pieces. Some shapes with curved sides could also be squared, such as the
lune of Hippocrates and the
parabola. 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 of any
polynomial with
rational coefficients. A construction for squaring the circle could be translated into a polynomial formula for , which does not exist. In philosophy, the concept of a "
square circle" has been used as an example of an
oxymoron since
Aristotle, sparking attempts to find contexts such as
taxicab geometry (
below) in which this phrase is meaningful.
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 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 squares 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.
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 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 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. The surfaces tiled with finitely many squares of the three-dimensional integer lattice are called
polyominoids.
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 rectangles 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 ratios of the rectangles, 3:1, 3:2, and the square of the
plastic ratio, approximately 1.755:1. 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: A
magic 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. For n \times n array, the sum can be formulated as n(n^2 + 1)/2 ; the numbers for n = 1, 2, 3, \dots 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, 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, 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. In spherical geometry, space has uniform positive curvature, and every convex quadrilateral (a
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 the angles of larger squares increase with area. 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, space has uniform negative curvature, and every convex quadrilateral has angles whose sum falls short of 360° by an amount called the
angular 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 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{(x_1-x_2)^2+(y_1-y_2)^2}. 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_{\infty} 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. Relatedly, a square and circle can become an intermediate shape known as
squircle. ==See also==