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Square

A square is a regular quadrilateral in Euclidean geometry, defined as a with four equal-length sides and four right angles of 90 degrees each. This makes it both equilateral (all sides congruent) and equiangular (all angles equal), distinguishing it as the only with an even number of sides that is also a and a . Key properties of the square include its diagonals, which are congruent in length, to each other, and bisect the vertex angles and each other at right angles and at their midpoints. Each diagonal divides the square into two congruent 45-45-90 right triangles. The perimeter of a square with side length s is $4s, while its area is s^2. These attributes contribute to the square's high degree of , belonging to the D_4, with rotational symmetries of 90, 180, and 270 degrees, as well as four lines of reflection. Historically, the square was formalized in mathematics, notably in Euclid's Elements (circa 300 BCE), where it serves as a foundational figure for theorems on , similarity, and area. In modern applications, squares underpin concepts in coordinate , vector spaces, and , and they form the basis for square lattices in and patterns that cover the plane without gaps or overlaps.

Definitions and Characterizations

Basic Definition

A square is a regular quadrilateral in , characterized by four equal-length sides and four right angles, each measuring 90 degrees. This definition establishes the square as both equilateral and equiangular, distinguishing it as a special case among polygons. Unlike a , which has four right angles but opposite sides of equal length without requiring all sides to be equal, a square mandates uniformity in side lengths. Similarly, a features four equal sides but does not necessarily have right angles, allowing for non-perpendicular orientations. These distinctions highlight the square's unique balance of equality in both sides and angles. Visually, a square appears as a flat, four-sided figure with straight edges meeting at precise corners, forming a stable and balanced often exemplified by with side s, where all edges measure s. Squares exhibit as a direct consequence of their equal sides and angles, enabling identical appearance under certain rotations. In terms of geometric transformations, two squares are congruent if their corresponding sides are equal in , preserving both and through rigid motions like or . All squares are similar, sharing identical angles and proportional sides regardless of scale, which underscores their uniform across different .

Etymology and Historical Terminology

The word "square" in the context of geometry derives from the Latin quadratus, meaning "square" or "made square," which is the past participle of quadrare "to square" or "to make square." This Latin term stems from quattuor "four," reflecting the shape's four equal sides and , and entered English in the mid-13th century via esquerre or esquarre, originally denoting a for measuring right before extending to the geometric figure itself. In ancient , the square was referred to by terms emphasizing its four-angled structure, such as tetragōnon (τετράγωνον), meaning "four-angled" or "four-cornered," a of tetra- "four" and gōnia "angle." , in his (c. 300 BCE), formalized the square within Book I, defining it in Definition 22 as "that [quadrilateral figure] which is both equilateral and right-angled," and using tetragōnon throughout to distinguish it from other quadrilaterals like the or oblong. Earlier civilizations also employed square shapes in practical measurements; Babylonian mathematicians around 2000 BCE compiled tables of squares for numbers up to 59 on clay tablets, aiding land surveys and calculations, while Egyptian texts from the (c. 2000 BCE) used squares to compute areas in administrative papyri for and . Over time, the terminology evolved to encompass broader uses beyond the geometric shape, with "square" functioning as a from the 1530s to "to make square" or "to adjust to a ," and as a noun in from the late to denote a "square number" or , such as 16 as the square of 4. Euclid's Elements marked a key milestone by integrating the square into axiomatic , with propositions like I.46 demonstrating its construction on a given , influencing subsequent mathematical across cultures.

Geometric Properties

Sides, Angles, and Diagonals

A is defined by four sides of equal length, with each pair of adjacent sides to one another, forming a closed planar figure. This configuration ensures the boundary consists of straight line segments meeting at right angles, distinguishing the square from other quadrilaterals. The interior of a square each measure exactly 90 degrees, resulting from the perpendicular adjacency of its sides. Consequently, the exterior , formed by extending one side beyond a , also measure 90 degrees at each , as the sum of an interior and adjacent exterior angle is 180 degrees. The diagonals of a square are line segments connecting non-adjacent vertices, opposite corners of the figure. These diagonals are equal in length and bisect each other at a 90-degree angle, intersecting at the center of the square. Additionally, each diagonal bisects the two at the vertices it connects, dividing each 90-degree interior angle into two 45-degree , and they function as axes of for the square's geometric structure.

Area, Perimeter, and Diagonal Formulas

The perimeter of a square, which measures the total length around its boundary, is calculated as P = 4s, where s is the length of each side. This formula arises directly from the square's four equal sides. The area of a square, representing the space enclosed within its boundaries, is given by A = s^2. To derive this, divide the square into two congruent right-angled triangles by drawing one diagonal; each triangle has legs of length s serving as base and height, yielding an area of \frac{1}{2} s \cdot s = \frac{1}{2} s^2 per triangle, for a total area of $2 \times \frac{1}{2} s^2 = s^2. The length of a diagonal d, connecting two opposite vertices, is d = s \sqrt{2}. This is obtained by applying the to the formed by two adjacent sides and the diagonal: d^2 = s^2 + s^2 = 2s^2 \implies d = s \sqrt{2}. The is established as Proposition I.47 in Euclid's Elements. For the unit square with s = 1, the perimeter is 4, the area is 1, and the diagonal is \sqrt{2}. Under uniform scaling by a factor k > 0, linear measures such as the perimeter and diagonal scale proportionally by k (yielding P' = 4ks and d' = k \sqrt{2}), while the area scales by k^2 (yielding A' = k^2 s^2).

Symmetry Group

The symmetry group of the square is the D_4, a of order 8 that captures all transformations leaving the square invariant. This group consists of four rotations—by $0^\circ, $90^\circ, $180^\circ, and $270^\circ about the square's —and four reflections across its axes of symmetry: the two diagonals and the two midlines passing through the midpoints of opposite sides. These elements can be represented as permutations of the square's vertices or as $2 \times 2 orthogonal matrices preserving the figure.) All elements of D_4 are isometries of the , meaning they preserve distances between points and angles between lines, ensuring that the square maps rigidly onto itself under each transformation. maintain , while reflections reverse it, but both types rigidly superimpose the square on its original position. In contrast, the of a non-square is the smaller D_2 (isomorphic to the ), which includes only the , a $180^\circ , and reflections over the horizontal and vertical midlines, lacking the $90^\circ rotations and diagonal reflections due to the rectangle's reduced regularity. The structure of D_4 highlights the square's high degree of invariance, with relations such as rotations composing additively modulo $360^\circ and reflections satisfying specific conjugation rules with rotations. In group-theoretic terms, a fundamental domain for the action of D_4 on the square is a that intersects each exactly once; for instance, a right-angled formed by the center and one vertex serves as such a domain, with area one-eighth of the square, and its images under D_4 cover the square without overlap or gap, illustrating how symmetries generate the full object from a minimal representative. This concept underscores the role of D_4 in modular constructions and periodic extensions.

Relation to Inscribed and Circumscribed Circles

A square admits both an inscribed circle, known as the incircle, and a circumscribed circle, known as the . The incircle is to all four sides of the square, with its center at the square's center of symmetry. For a square with side length s, the r of the incircle is r = \frac{s}{2}, derived directly from the where the equals the side length. The area of the incircle is \pi r^2 = \pi \left( \frac{s}{2} \right)^2 = \frac{\pi s^2}{4}, which is \frac{\pi}{4} times the area of the square s^2. The circumcircle passes through all four vertices of the square, also centered at the square's center. Its radius R is R = \frac{s \sqrt{2}}{2}, obtained by halving the diagonal length d = s \sqrt{2}, as the diagonal spans the diameter of the circumcircle. Thus, R = \frac{d}{2}. The area of the circumcircle is \pi R^2 = \pi \left( \frac{s \sqrt{2}}{2} \right)^2 = \frac{\pi s^2}{2}, or \frac{\pi}{2} times the square's area. Among all rectangles inscribed in a fixed (sharing the same circumradius), the square achieves the maximum area. This follows from the l^2 + w^2 = 4R^2 for l and width w, where the area A = l w is maximized when l = w = s = 2R / \sqrt{2}, yielding A = 2 R^2.

Constructions

Coordinate and Equation Representations

In , a square of side length s is often placed on the coordinate plane with vertices at (0,0), (s,0), (s,s), and (0,s), aligning its sides parallel to the coordinate axes. This standard positioning facilitates calculations in Cartesian coordinates and serves as a basis for translations and rotations. For a square centered at (h, k) with sides parallel to the axes, the boundary equation is given by \max(|x - h|, |y - k|) = \frac{s}{2}, corresponding to the unit ball in the L^\infty norm scaled appropriately. The interior region satisfies \max(|x - h|, |y - k|) \leq \frac{s}{2}. This representation highlights the square's equivalence to a rotated and scaled in other norms, but emphasizes its axis-aligned form here. Squares rotated by an \theta relative to the axes can be described using transformed coordinates or polar forms centered at (h, k). In polar coordinates (r, \phi), where \phi is the from the positive x-axis, the boundary satisfies r = \frac{s/2}{\max(|\cos(\phi - \theta)|, |\sin(\phi - \theta)|)}. The interior consists of points where the from the center r \leq \frac{s\sqrt{2}}{2} along the diagonals, with tighter angle-dependent constraints along the sides to containment within the square's boundaries. This maximum \frac{s\sqrt{2}}{2} represents half the diagonal , providing for rotated configurations. Parametric equations offer a versatile way to describe the sides and diagonals. For the axis-aligned square with vertices as above, the four sides can be parametrized piecewise over t \in [0, s] as follows: bottom side x = t, y = 0; right side x = s, y = t; top side x = s - t, y = s; left side x = 0, y = s - t. The is given by x = t, y = t for t \in [0, s], while the other diagonal uses x = s - t, y = t. These linear parametrizations extend naturally to centered or rotated squares via and matrices.

Compass and Straightedge Construction

In classical , the construction of a square using only a and , given a as its side, is a fundamental operation that demonstrates the existence of such a figure on any finite straight line. This method, which requires erecting perpendiculars, transferring lengths, and drawing parallels, ensures the resulting has four equal sides and four right angles. The procedure is rigorously outlined in Euclid's Elements, Book I, Proposition 46, where it serves as a building block for further geometric developments, such as applications in Books II, VI, and XIII. To construct the square on a given straight AB, proceed as follows: The validity of this construction is proven by verifying that ADEB is equilateral (all sides AB = BE = ED = DA by direct equality and parallelogram properties) and rectangular (angle BAD is right by construction, angle ADE is right via the properties of parallels and transversals per Proposition I.29, and opposite angles are equal per Proposition I.34). Thus, ADEB satisfies the definition of a square as a regular quadrilateral with equal sides and right angles. This approach requires approximately six circle drawings and four straight lines, though alternative methods using fewer steps exist while adhering to the same Euclidean postulates. A key limitation in compass and straightedge constructions arises when the side length is not pre-given: starting from a unit length (rational), only squares with constructible side lengths can be formed, where constructible numbers are those generated by finite iterations of addition, subtraction, multiplication, division, and square roots over the rationals, yielding quadratic field extensions. Irrational lengths like √2 (the diagonal of a unit square) are constructible, but arbitrary irrationals, such as π, cannot be achieved without incorporating the desired length as an initial element, as they do not lie in any such tower of quadratic extensions. This restriction underscores the algebraic foundations of Euclidean constructions, limiting them to numbers algebraic of degree a power of 2 over the rationals.

Applications

In Architecture, Art, and Design

In , squares form the foundational elements of intricate geometric patterns used for floors, walls, and windows, often interlaced with circles and stars to create repeating motifs that symbolize unity and . These designs, prominent in mosques and madrasas from the 8th century onward, emphasize for aesthetic harmony and are evident in structures like the in , where facilitate complex star patterns. The square base has been a staple in monumental architecture for stability and symbolic representation of the earthly plane. The Great Pyramid of Giza, constructed around 2580–2560 BCE, features a nearly square base measuring approximately 230 meters on each side, providing a level foundation on uneven terrain and aligning with cardinal directions. In modern city planning, grid layouts based on square blocks promote efficient land use and navigation; William Penn's 1682 plan for Philadelphia divided the city into a rectangular grid of squares to foster orderly growth and communal spaces. Similarly, James Oglethorpe's 1733 design for Savannah, Georgia, incorporated a grid of wards centered around public squares, blending functionality with green areas for social interaction. In , squares underpin perspective techniques to simulate on a two-dimensional surface. During the , artists like employed square as underdrawings to ensure proportional accuracy in linear , as detailed in his 1525 Underweysung der Messung, where a of squares helps objects receding to a . Piet Mondrian's Neoplasticist works from the 1920s, such as Composition with Red, Blue, and Yellow (1930), reduce forms to intersecting black lines forming squares and rectangles filled with primary colors, embodying universal harmony through . Square-based modularity enhances adaptability in furniture design by allowing interchangeable components that maximize space efficiency. In mid-20th-century Scandinavian modernism, designers like used square modules in shelving systems, such as the 1933 Artek collection, to create customizable units that stack and reconfigure without waste. Early digital art relied on pixels—fundamental square units of color on raster displays—to build images, originating in the 1960s with NASA's color imaging systems and evolving into by the 1970s through limited-resolution . This square grid structure, seen in pioneering works like those from the of Technology's computer art lab, constrained yet inspired minimalist compositions in video games and algorithmic drawings.

In Mathematics, Physics, and Computing

In mathematics, the square serves as a fundamental domain for various analytical techniques. The unit square, defined as the set [0,1] \times [0,1] in the Cartesian plane, is commonly used in methods, such as , where random points are sampled uniformly within its boundaries to approximate multidimensional integrals by estimating probabilities or volumes. For example, integrating a f(x,y) over the unit square involves averaging function values at sampled points and multiplying by the area, which is 1, providing a probabilistic estimate of the integral's value. In , the square domain facilitates in partial differential equations, where double expand solutions on rectangular regions, including squares, to solve problems like heat conduction on a square plate. Additionally, the geometric square relates to algebraic structures through square matrices in linear algebra, which are n \times n arrays representing linear transformations on n-dimensional vector spaces, with the square shape ensuring compatibility for operations like inversion and computation when the matrix is non-singular. This linkage underscores the square's role as a basis for conceptualizing finite-dimensional spaces, where the matrix's square form mirrors the balanced dimensions of the geometric figure. In physics, squares underpin coordinate systems and quantum models. Cartesian grids, composed of square cells aligned with orthogonal axes, are essential for discretizing space in simulations of physical phenomena, such as fluid dynamics or electromagnetism, enabling efficient numerical solutions to governing equations on uniform meshes. A prominent example is the infinite square well potential in quantum mechanics, a one-dimensional model where a particle is confined between impenetrable walls at x=0 and x=a, leading to quantized energy levels E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2} and stationary wave functions \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n \pi x}{a}\right) for n = 1, 2, \dots, illustrating boundary-induced quantization. In computing, squares form the basis of and algorithms. Raster images represent visuals as grids of square , where each pixel is a small square holding color data, allowing scalable rendering in displays and image processing by mapping continuous scenes onto discrete square arrays. For in video games, axis-aligned bounding boxes—rectangular approximations often simplified to squares for symmetric objects—enable efficient real-time checks by comparing coordinate overlaps, such as verifying if the projections on x- and y-axes intersect, reducing in dynamic environments.

Related Topics

Quadrature and Squaring the Circle

In , quadrature refers to the historical problem of constructing a square with the same area as a given or curved figure using geometric methods. The specific challenge of involves constructing, with compass and straightedge, a square whose area equals that of a given of r, or \pi r^2. This task originated in as one of the three classical problems of , alongside and cube duplication. Early attempts date to the fifth century BCE, when the Sophist proposed inscribing polygons within a and iteratively doubling the number of sides to approximate the 's area, though his method incorrectly assumed the process could exhaust the 's curvature in finitely many steps. Around 440 BCE, advanced the problem by demonstrating the of certain lunes—crescent-shaped regions bounded by circular arcs—such as those formed by a and segments of circles with diameters equal to the 's sides, showing these areas equal that of a . In the third century BCE, provided a practical by inscribing and circumscribing regular 96-gons around a , yielding bounds $3 \frac{10}{71} < \pi < 3 \frac{1}{7}, which allowed construction of a near-equivalent square via polygonal areas. The impossibility of exact compass-and-straightedge was established in 1882 by , who proved \pi is transcendental—meaning it is not the root of any non-zero with rational coefficients—implying no such algebraic construction exists, as constructible lengths are algebraic. In modern , while exact geometric remains impossible, numerical methods approximate the circle's area for computational purposes, such as , which estimates \pi by randomly sampling points in a square enclosing a and computing the ratio of points inside the circle.

Tiling, Packing, and Dissection

Squares form one of the three regular tessellations of the , known as the or square grid, where congruent squares are arranged such that four meet at each with 90-degree . This monohedral uses a single prototile shape—the square—and covers the plane without gaps or overlaps, exhibiting translational, rotational, and reflectional symmetries. The tiling's is itself a square, and it corresponds to the {4,4}, highlighting its uniform polyhedral properties in two dimensions. In , squares achieve the maximum possible of 1 when arranged in a square lattice to fill the entire without voids, as they perfectly tessellate the . For finite containers, such as packing equal smaller squares into a larger square, the optimal arrangements minimize wasted space, with known configurations for small numbers like n=1 (trivial) or n=4 (2x2 grid). Packing equal squares into a is more challenging due to , often requiring rotated placements to approach higher densities, though exact remain open for many n. Dissection of squares involves cutting them into pieces that reassemble into other shapes, governed by the Bolyai–Gerwien theorem, which states that any two polygons of equal area, including squares, can be dissected into each other using finitely many polygonal pieces. A notable case is "," the problem of a square with smaller squares of unequal sizes and integer sides (perfect squared square); the simplest such tiling uses 21 distinct squares and was discovered by A. J. W. Duijvestijn in via computer search, proving it the lowest-order simple perfect example. For three-dimensional extensions involving cubic dissections, the Dehn invariant provides an obstruction, determining when polyhedra of equal volume, such as cubes derived from squares, can be dissected into each other, as resolved in .

Squares in Algebra and Number Theory

In , a is the square of an , denoted as n^2 where n is an , forming a such as 0, , 4, 9, , and so on. These numbers exhibit notable properties in ; for instance, every equals the sum of the first n odd positive integers, a recognition attributed to early Pythagorean mathematicians. Additionally, modulo a prime p, the correspond to the residues, which are the distinct values of x^2 \mod p for x = 0, [1](/page/1), \dots, p-1, numbering exactly (p+1)/2 including 0. The square root function extracts the principal (non-negative) root of a non-negative real number, but in the context of integers, \sqrt{k} is irrational unless k is a perfect square. A classic example is \sqrt{2}, proven irrational by contradiction in Euclid's Elements: assume \sqrt{2} = p/q in lowest terms with p, q positive integers; then p^2 = 2q^2, implying p is even (so p = 2r), substituting yields $2r^2 = q^2, so q is even, contradicting the fraction's reduced form. Key algebraic identities involving squares include the binomial expansion (a + b)^2 = a^2 + 2ab + b^2, which algebraically decomposes the square of a sum and admits a geometric interpretation: the area of a square with side length a + b equals the areas of squares on sides a and b plus twice the area of a rectangle with sides a and b. In , square-free integers are those whose prime contains no squared prime factors (i.e., all exponents are ), such as 1, 2, 3, 5, 6, 7, 10; the number 1 is conventionally square-free, and their asymptotic density among positive integers is $6/\pi^2 \approx 0.6079. A profound result linking squares to all s is , which states that every can be expressed as the sum of four integer squares, first proved by in 1770 using identities like Euler's four-square product formula and properties of primes.

Squares in Other Geometries

In hyperbolic geometry, squares are regular quadrilaterals with four equal-length sides and four congruent interior angles, each measuring less than 90 degrees, as the total angle sum for any quadrilateral falls below 360 degrees due to the negative curvature of the space. This contrasts with Euclidean squares, where angles are exactly 90 degrees, and arises from the hyperbolic parallel postulate allowing more parallel lines through a point not on a given line. In the Poincaré disk model, which represents the hyperbolic plane as the interior of a unit disk with hyperbolic lines as circular arcs orthogonal to the boundary circle, such squares can be constructed using geodesic segments of equal length, and their vertices lie within the disk. Tilings of the hyperbolic plane with these squares are feasible, permitting five or more squares to meet at a vertex, satisfying the condition that the angle defect allows infinite tessellations without gaps or overlaps, as governed by the inequality \frac{2}{k} + \frac{2}{n} < 1 for n-gons with k meeting at each vertex. Spherical geometry, characterized by positive on the surface of a , defines squares as quadrilaterals bounded by four great circle arcs of equal length, with interior angles exceeding 90 degrees and a total angle sum greater than 360 degrees, reflecting the spherical excess proportional to the enclosed area. For a small spherical square near the 's surface, the angles approach 90 degrees, but larger ones exhibit increased angular measure. These properties stem from the geometry's , where great circles (the "lines") always intersect, leading to no true parallels and forcing angles to bulge outward compared to planar figures. The symmetries of squares adapt here to rotations and reflections preserving great circles. In , also known as geometry, the d((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2| measures along axis-aligned paths, transforming familiar shapes: the unit appears as a (a square rotated 45 degrees relative to the coordinate axes) with vertices at (\pm 1, 0) and (0, \pm 1). Consequently, "squares" in this geometry are equilateral and equiangular figures under the taxicab , manifesting as diamonds in the embedding, where sides follow the metric's grid-like paths and right angles are preserved in the L1 norm sense. This redefinition highlights how the geometry prioritizes horizontal and vertical movements, altering perimeter and area calculations—for example, the taxicab perimeter of such a square doubles the diagonal length. The concept of a square extends to higher dimensions as the n-dimensional , with the 4-dimensional case known as the serving as the direct analog. A consists of 8 cubic cells ( faces), 24 square faces (2D faces), 32 edges, and 16 vertices, each vertex connecting to 4 edges, mirroring the connectivity progression from square (4 edges per vertex in 2D) to (3 edges per vertex in ). Its volume is given by side length s raised to the , V = s^4, and it exhibits hyperoctahedral , generalizing the square's D_4. Projections of the into space reveal nested cubes connected by edges, illustrating its extra-dimensional structure without violating lower-dimensional embedding constraints.