Square

A square is a special kind of shape in geometry. It is a regular quadrilateral, meaning it has four straight sides that are all the same length and four angles that are all the same size. Each angle in a square is a right angle, which measures 90 degrees or π/2 radians. Because of these perfect angles, the sides of a square are always perpendicular to each other.

Squares are very useful in many areas of life. They appear in tiled floors and walls, graph paper, the tiny squares that make up digital images called pixels, and many game boards. You can also find squares in building floor plans, pieces of paper used for folding like origami paper, servings of food, and even in art and designs.

In math, the area of a square is found by multiplying the length of one side by itself. This idea of multiplying a number by itself is called squaring, which comes from the shape of the square. Squares can be used to solve interesting problems, like trying to draw a square inside a circle using only a compass and straightedge, a challenge that mathematicians now know is impossible.

Definitions and characterizations

A square is a special shape with four equal sides and four right angles. This means it is both a rectangle and a rhombus. There are many ways to describe a square, and if a shape meets any one of these rules, it will meet them all. For example, a square can be a rectangle with all sides equal, or a rhombus with a right angle between two sides. Squares are the only regular shapes where all the angles inside and outside the shape are the same, and they are all right angles.

Properties

A square is a special shape with four equal sides and four equal angles, each being 90 degrees. It is a type of rectangle and rhombus, which means it has all the properties of these shapes.

Squares are all the same shape, but they can be different sizes. If you know the length of one side, you can find out everything about the square. When squares have the same size, they are called congruent.

For a square with side length ℓ, the perimeter is 4ℓ and the diagonal length is √2 ℓ. The area of the square is ℓ², which is why we use "squaring" to mean raising a number to the second power. A square is the shape with the smallest perimeter for a given area and the largest area for a given perimeter.

The square has many symmetries. It can be rotated by 90°, 180°, or 270° and still look the same. It can also be reflected across its diagonals or sides. These symmetries form a group called the dihedral group of order eight.

A square can fit inside a circle, and a circle can pass through all four corners of the square. The circle inside the square touches all four sides, and the circle outside the square touches all four corners.

Applications

Squares are very useful shapes in many areas of life. They are commonly used as the shape of tiles, like in floors and walls. The word for a small tile, "tessera," even comes from an ancient word meaning the number four, because of the four corners of a square tile.

In technology, squares are important too. Graph paper with a grid of squares helps with drawing and measuring. The tiny dots, called pixels, in digital photos and videos are arranged in a square grid. This square pattern helps in storing and showing images on screens. Many modern buildings, from ancient pyramids to modern skyscrapers, have square designs. Artists and designers often use squares in their work for balance and order. Games like chessboard and sports like baseball diamonds also use square shapes. Even some foods, like certain types of waffles, are square. Squares appear in nature and science too, in patterns and molecular structures.

Constructions

Coordinates and equations

A unit square is a square with sides of length one. It can be shown on a grid where the points have coordinates between 0 and 1 on both the x and y axes. The four corners of this square are at points where the x and y coordinates are either 0 or 1.

In the world of complex numbers, multiplying a number by the imaginary unit i turns it 90 degrees. If you take a complex number and multiply it by i four times, you get four new numbers that form the corners of a square centered at the origin. By adding another complex number, you can move this square to any other center point.

Compass and straightedge

It is possible to draw a square with a given side length using just a compass and straightedge, as described in Euclid's Elements. This shows that squares are constructible shapes.

Related topics

The cube and regular octahedron, next steps in sequences of regular polytopes starting with squares.

The Sierpiński carpet is a square fractal with square holes.

The Schläfli symbol of a square is { 4 } {\displaystyle \{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 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.

Many theorems involve squares. 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.

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

Main articles: Inscribed square problem and Inscribed square in a triangle

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

See also: Area, Quadrature (geometry), and Squaring the circle

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.

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.

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.

Tiling and packing

Square tiling

Pythagorean tiling

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.

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.

Counting

A common mathematical puzzle involves counting the squares of all sizes in a square grid of n × n {\displaystyle 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 × 2 {\displaystyle 2\times 2} squares, and one 3 × 3 {\displaystyle 3\times 3} square. The answer to the puzzle is n ( n + 1 ) ( 2 n + 1 ) / 6 {\displaystyle n(n+1)(2n+1)/6} , a square pyramidal number.

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 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.

Other geometries

Concentric squares in the sphere (orthographic projection)

Concentric squares in the hyperbolic plane (conformal disk model)

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.

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.

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.

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 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} is ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 {\displaystyle \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 {\displaystyle L_{1}} distance | x 1 − x 2 | + | y 1 − y 2 | {\displaystyle |x_{1}-x_{2}|+|y_{1}-y_{2}|} . The points with taxicab distance d {\displaystyle d} from any given point form a diagonal square, centered at the given point, with diagonal length 2 d {\displaystyle 2d} . In the same way, axis-parallel squares are the circles for the L ∞ {\displaystyle L_{\infty }} or Chebyshev distance, max ( | x 1 − x 2 | , | y 1 − y 2 | ) {\displaystyle \max(|x_{1}-x_{2}|,|y_{1}-y_{2}|)} . In this metric, the points with distance d {\displaystyle d} from some point form an axis-parallel square, centered at the given point, with side length 2 d {\displaystyle 2d} . Relatedly, a square and circle can become an intermediate shape known as squircle.