Square

Honestly, I find Wikipedia articles to be rather… pedestrian. Dry facts, devoid of any real spark. But you want me to rewrite one? And extend it? Fine. Don't expect me to gush over geometry. I'll give you the facts, impeccably arranged, but don't mistake thoroughness for enthusiasm. It’s a square. It’s a shape. Let’s move on.

Square (Geometry)

A square is a fundamental shape in geometry, distinguished by its four equal sides and four equal angles. It stands as a regular polygon, a special kind of quadrilateral where all sides and all interior angles are identical. This inherent regularity places it in a unique position: it is simultaneously a rectangle, possessing four right angles, and a rhombus, boasting four equal sides. The angles of a square are precisely right angles – 90 degrees, or π/2 radians – meaning adjacent sides are always perpendicular to each other. The area calculation is elegantly simple: the length of one side multiplied by itself. This operation, when applied to numbers, is what we call squaring.

The ubiquity of the square is undeniable. In its most basic form, it's the building block for square tilings, a pattern where identical squares fit together edge-to-edge, filling the entire plane without gaps. You see this everywhere: from the floors beneath your feet and the walls around you, to the grid lines on graph paper, the individual pixels that form our digital images, and the playing fields of countless game boards. Beyond these functional applications, the square shape pervades our visual world. It's a common feature in architectural floor plans, the crisp edges of origami paper, the presentation of certain foods, the foundations of graphic design, and the symbolic language of heraldry. Even the snapshots from instant cameras and the canvases of fine art often embrace this simple, balanced form.

The very concept of area in geometry is deeply intertwined with the square. The quest to construct a square with the same area as a given circle, a problem known as squaring the circle, has captivated mathematicians for centuries, though it was eventually proven impossible using only compass and straightedge constructions. This pursuit, however, spurred significant mathematical inquiry. The question of whether every simple closed curve can contain an inscribed square remains an open problem, though it's known to be true for smooth or convex curves. The related problem of squaring the square delves into dissecting a square into smaller, unequal squares, a fascinating combinatorial challenge. Furthermore, mathematicians have explored the intricate puzzles of packing squares as efficiently as possible within other shapes.

The construction of a square is achievable through various geometric means. It can be formed using straightedge and compass, defined by its Cartesian coordinates, or even visualized in the complex plane through repeated multiplication by the imaginary unit $i$ . Intriguingly, the square emerges as the fundamental "circle" – the set of all points equidistant from a center – in certain non-Euclidean geometries, specifically taxicab geometry and Chebyshev distance spaces. While spherical geometry and hyperbolic geometry do not permit squares in the Euclidean sense (four equal sides and right angles), they do feature analogous regular polygons with four sides but different angles, or right angles with varying numbers of sides.

Definitions and Characterizations

A square is more than just a shape; it's a confluence of properties that uniquely define it. Within the realm of Euclidean plane polygons, a shape earns the title of "square" if it fulfills any of the following criteria, which, remarkably, all lead to the same definitive form:

A square is a polygon possessing four sides of equal length and four right angles. This dual characteristic means it is simultaneously a rhombus and a rectangle.[1]
Alternatively, a square can be defined as a rectangle that also has four sides of equal length.[1]
It can also be described as a rhombus that possesses a right angle between any two adjacent sides.[1]
Another perspective defines a square as a rhombus where all interior angles are equal.[1]
From the properties of a parallelogram, a square is identified as one with a single right angle and two adjacent sides of equal length.[1]
In terms of its diagonals, a square is a quadrilateral whose diagonals are not only equal in length but also bisect each other perpendicularly. This makes it a rhombus with equal diagonals.[2]
Mathematically, a square can be characterized by its area. If a quadrilateral has successive sides of lengths $a$ , $b$ , $c$ , and $d$ , its area $A$ is given by the formula $A = \frac{1}{4}(a^2 + b^2 + c^2 + d^2)$ .[3] This peculiar formula highlights that only when $a = b = c = d$ and the angles are right angles will this equation hold true for a square.

Among the regular polygons, the square holds the distinction of being the only one where the internal angle, central angle, and external angle are all identical – each being a right angle.[4]

Properties

The square, by virtue of its perfect symmetry and regularity, inherits all the properties of the shapes it encompasses. It is a special type of rhombus (equal sides, opposite angles equal), a kite (two distinct pairs of equal adjacent sides), a trapezoid (at least one pair of parallel opposite sides), a parallelogram (both pairs of opposite sides parallel), a quadrilateral (a four-sided polygon), and a rectangle (opposite sides equal, all angles right angles).[1] Consequently, it possesses a comprehensive set of characteristics:

All four internal angles of a square are equal, each measuring 90°, a perfect right angle.[4][5]
The central angle of a square, formed by connecting the center to two adjacent vertices, is also 90°.[4]
Similarly, the external angle, formed by extending one side, measures 90°.[4]
The diagonals of a square are not only equal in length but also bisect each other, forming four right angles at their intersection.[5]
Furthermore, these diagonals act as angle bisectors for the square's internal angles, dividing each 90° angle into two 45° angles.[6]
As previously noted, all four sides of a square are of equal length.[7]
The opposite sides of a square are parallel, a defining characteristic of parallelograms.[8]

All squares, irrespective of their size, are similar to one another, meaning they share the same shape.[9] To define a specific square, one only needs a single parameter, typically the length of its side or its diagonal.[10] Squares that are identical in size are considered congruent.[11]

Measurement

The area of a square is straightforwardly calculated by multiplying the length of its side by itself.

For a square with sides of length $\ell$ , its perimeter $P$ is given by the formula $P = 4\ell$ , and the length of its diagonal $d$ is $d = \sqrt{2}\ell$ .[12][13] The value $\sqrt{2}$ , appearing in the diagonal calculation, is an irrational number, meaning it cannot be expressed precisely as a simple fraction. Its approximate value, around 1.414,[14] was already understood in Babylonian mathematics.[15] The area $A$ of a square can also be expressed in terms of its diagonal: $A = \ell^2 = \frac{1}{2}d^2$ .[13]

This connection between area and the square of the side length is the origin of the term squaring in algebra, referring to the operation of multiplying a number by itself. Conversely, finding the side length of a square from its area involves taking the square root. When integers are squared, the results are known as square numbers, which are a type of figurate number representing quantities that can be arranged into a square grid.[17]

A curious property emerges when considering a square with sides of length four. Its area ( $4^2 = 16$ ) is numerically equal to its perimeter ( $4 \times 4 = 16$ ). Such a shape is termed an equable shape. The only other integer rectangle with this property is a 3x6 rectangle.[18]

As a regular polygon, the square possesses a remarkable efficiency: it is the quadrilateral that encloses the largest possible area for a given perimeter, and conversely, it has the least perimeter for a given area.[19] This is encapsulated by the isoperimetric inequality, which states that for any quadrilateral with area $A$ and perimeter $P$ , the relationship $16A \leq P^2$ holds true. Equality is achieved precisely when the quadrilateral is a square.[20][21]

Symmetry

The square is arguably the most symmetrical of all quadrilaterals.[22] A total of eight rigid transformations in the plane map a square perfectly onto itself:[23]

The identity transformation, which leaves the square unchanged.
Rotations by 90°, 180°, and 270° around the square's center.
Reflections across the two diagonals (NW–SE and NE–SW).
Reflections across the horizontal and vertical lines passing through the center.

For a square aligned with the coordinate axes and centered at the origin, these symmetries can be represented by combinations of negating and swapping the Cartesian coordinates of points.[24] These transformations permute the eight isosceles triangles formed by connecting the center to the midpoints of the sides. Any one of these triangles can serve as a fundamental region for understanding the group's action.[25] A key aspect of this symmetry is that any two vertices, any two edges, or any two half-edges of the square can be mapped onto each other by at least one of these symmetry operations.[22] This property is shared by all regular polygons and is mathematically described by saying that the symmetries act transitively on the vertices and edges, and simply transitively on the half-edges.[27]

The composition of any two symmetry operations (performing one after the other) results in another symmetry operation, giving these eight transformations the structure of a mathematical group, known as the dihedral group of order eight.[23] Less symmetrical quadrilaterals, such as rectangles and rhombuses, possess only a subgroup of these symmetries.[28][29]

The visual representation of a square can change dramatically under perspective transformations. A cube, for instance, with its six square faces, will appear as six distinct quadrilaterals when viewed in three-point perspective, demonstrating how a square's apparent shape can be distorted.

The wallpaper groups, which classify repeating patterns in two dimensions, often utilize the square as a fundamental unit. Specifically, three of these groups – p4, p4m, and p4g – are defined by patterns where the basic repeating unit must be a square.[35]

Inscribed and Circumscribed Circles

A square possesses both an inscribed circle and a circumscribed circle.

The inscribed circle, depicted in purple, is the largest circle that can be contained within the square. It touches each of the square's sides at its midpoint. Its center coincides with the square's center, and its radius, known as the inradius, is half the length of the square's side ( $r = \ell/2$ ). Because this circle is tangent to all four sides, the square qualifies as a tangential quadrilateral.

The circumscribed circle, shown in red, passes through all four vertices of the square. This makes the square a cyclic quadrilateral. The radius of this circle, the circumradius, is related to the side length by $R = \ell/\sqrt{2}$ .

For a square $ABCD$ with an inscribed circle touching sides $AB$ , $BC$ , $CD$ , and $DA$ at points $E$ , $F$ , $G$ , and $H$ respectively, a peculiar geometric relationship holds: for any point $P$ on the inscribed circle, the equation $2(PH^2 - PE^2) = PD^2 - PB^2$ is satisfied.[37]

Furthermore, if $d_i$ represents the distance from an arbitrary point in the plane to the $i$ -th vertex of a square, and $R$ is its circumradius, a complex identity emerges: $\frac{d_1^4 + d_2^4 + d_3^4 + d_4^4}{4} + 3R^4 = \left(\frac{d_1^2 + d_2^2 + d_3^2 + d_4^2}{4} + R^2\right)^2$ [38] If $L$ is the distance from the point to the square's centroid and $d_i$ are the distances to the vertices, 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.[39]

Applications

The square's inherent simplicity and stability have made it a cornerstone in numerous applications across various fields.

Tiling and Grids: The most apparent application is in tiling. The word "tessera," referring to a small tile in mosaics, originates from the ancient Greek word for four, alluding to the four corners of a square tile.[40] Graph paper, with its grid of squares, is indispensable for data visualization using Cartesian coordinates.[41] In digital imaging, pixels, the fundamental units of bitmap images, are conventionally arranged in a square grid and often conceptualized as small squares themselves.[42][43] Advanced image and video compression techniques, such as the JPEG format, rely on dividing images into larger square blocks.[44] The quadtree data structure, used in areas like data compression and computational geometry, is built upon the recursive subdivision of squares.[45]

Architecture and Design: Square floor plans and bases have been a recurring motif in architecture since antiquity. The Egyptian pyramids,[46] the Mesoamerican pyramids of Teotihuacan,[47] and the Chogha Zanbil ziggurat all exhibit square foundations. The traditional layout of Persian walled gardens, intended to symbolize the four rivers of Paradise, influenced later designs like the Taj Mahal.[49] Buddhist stupas and East Asian pagodas often feature square bases, symbolizing their cosmic orientation and aspiration towards the heavens.[50][51] Even utilitarian structures like Norman keeps, such as those found in the Tower of London, frequently adopted a square tower form.[52] In contemporary architecture, the prevalence of square floor plans in skyscrapers is often driven by practical considerations of construction and space optimization.[53]

Art and Symbolism: The stylized, nested squares found in Tibetan mandalas serve as microcosmic representations of the universe.[54] Certain photographic formats, notably those used by Polaroid cameras and some medium format and Instamatic cameras, utilize a square aspect ratio.[55][56] Artists such as Josef Albers,[57] Kazimir Malevich,[58] [Piet Mondrian],[59] and Theo van Doesburg are renowned for their consistent and prominent use of square forms in their work.[60]

Games and Recreation: Despite their names, baseball diamonds and boxing rings are fundamentally square in shape.[61][62] In square dance and quadrille, four couples traditionally form the sides of a square.[63] The minimalist play Quad by Samuel Beckett features four actors tracing the paths of a square.[64] The go board, with its grid of 361 intersections, is said to represent the earth, with its lines symbolizing days of the year.[65] The chessboard, a descendant of Indian race games like pachisi, passed its square form to checkers.[66] Ancient Mesopotamian and Egyptian games like the Royal Game of Ur and Senet utilized rectangular boards divided into smaller squares.[67] Puzzles like the ancient Greek Ostomachion and the Chinese tangram involve rearranging pieces derived from squares,[68] while polyominos are shapes formed by joining squares edge-to-edge.[69] Medieval and Renaissance horoscopes were often presented in a square format across different cultures.[70] Recreational activities like origami frequently employ square paper, and quilting often uses square blocks.[71][72] Even the popular game of Scrabble uses square tiles arranged on a square board.[73][74]

Everyday Objects and Symbols: Squares are a staple in graphic design, conveying stability, symmetry, and order.[75] In heraldry, a canton, a charge in the upper-left corner of a shield, is typically square. A square flag is called a banner.[76] The flag of Switzerland is notably square, as are the flags of many Swiss cantons.[77] QR codes, essential for modern data transfer, are square and feature distinctive square alignment markers.[78] Even mechanical components, like the drive socket of a Robertson screw, are square.[79] Many common food items, including crackers, sliced cheese, and waffles, are presented in square shapes.[80][81][82] Specific food items are even named for their square form, such as caramel squares, date squares, lemon squares,[83] square sausage,[84] and the French cheese Carré de l'Est.[85]

Chemistry: In stereochemistry, the term square planar molecular geometry describes a specific arrangement of atoms at the corners of a square, with an example being xenon tetrafluoride.

Constructions

Coordinates and Equations: A unit square is defined as a square with sides of length one. In the Cartesian coordinate system, it is commonly represented by the set of points $(x, y)$ such that $0 \leq x \leq 1$ and $0 \leq y \leq 1$ . Its vertices are the four points where the coordinates are either 0 or 1.[87]

An axis-parallel square centered at $(x_c, y_c)$ with sides of length $2r$ (where $r$ is the inradius, or half the side length) has vertices at $(x_c \pm r, y_c \pm r)$ . The interior of this square is defined by the inequality $\max(|x - x_c|, |y - y_c|) < r$ , while its boundary consists of points where $\max(|x - x_c|, |y - y_c|) = r$ .[88]

A diagonal square, also centered at $(x_c, y_c)$ and with a diagonal of length $2R$ (where $R$ is the circumradius, or half the diagonal), has vertices at $(x_c \pm R, y_c)$ and $(x_c, y_c \pm R)$ . The interior of this square is described by the inequality $|x - x_c| + |y - y_c| < R$ , and its boundary by $|x - x_c| + |y - y_c| = R$ .[88] The accompanying illustration shows such a diagonal square centered at the origin $(0, 0)$ with a circumradius of 2, defined by the equation $|x| + |y| = 2$ .

In the complex plane, multiplication by the imaginary unit $i$ results in a 90° rotation around the origin. Consequently, if a non-zero complex number $p$ is repeatedly multiplied by $i$ , yielding $p$ , $ip$ , $-p$ , and $-ip$ , these four numbers form the vertices of a square centered at the origin.[89] Interpreting the real and imaginary parts as Cartesian coordinates, with $p = x + iy$ , these vertices correspond to $(x, y)$ , $(-y, x)$ , $(-x, -y)$ , and $(-y, -x)$ .[90] This square can be translated to any other location in the complex plane by adding a complex number $c$ to each vertex.[91] The Gaussian integers – complex numbers with integer real and imaginary parts – form a regular square lattice in the complex plane.[91]

Compass and Straightedge: Euclid's Elements, Book I, Proposition 46, provides a method for constructing a square with a given side length using only a compass and straightedge.[92] This construction confirms that squares are constructible polygons. The general condition for a regular $n$ -gon to be constructible involves its odd prime factors being distinct Fermat primes.[93] For $n=4$ , there are no odd prime factors, so the condition is vacuously true.[94] Euclid's Elements also details constructions for squares inscribed within and circumscribed about a given circle (Book IV, Propositions 6 and 7, respectively).[95]

Inscribed Squares

A square is said to be inscribed in a curve if all four of its vertices lie on that curve. The inscribed square problem poses the question of whether every simple closed curve can accommodate an inscribed square. This has been proven true for any smooth curve[114] and for any closed convex curve. Interestingly, the equilateral triangle is the only other regular polygon that can always be inscribed in every closed convex curve; for other regular polygons, there exist convex curves that cannot contain them.[115]

When considering an inscribed square in a triangle, at least one side of the square must lie along a side of the triangle. An acute triangle can contain three such inscribed squares, one associated with each side. A right triangle can have two inscribed squares: one touching the right-angle vertex and another lying on the hypotenuse. An obtuse triangle, however, can only accommodate one inscribed square, situated on its longest side. It's also worth noting that an inscribed square can cover at most half the area of the triangle it resides in.[116]

Area and Quadrature

Since antiquity, units of surface area have been derived from squares, typically by defining a standard unit of length as the side of the square – examples include the square meter and square inch.[117]

In ancient Greek deductive geometry, the process of quadrature, or "squaring," involved constructing a square with the same area as a given planar shape using only a finite number of compass and straightedge operations. Euclid's Elements demonstrates this for various shapes, starting with rectangles, parallelograms, and triangles, and extending to more complex simple polygons by decomposing them into triangles.[118] Notably, certain shapes with curved boundaries, such as the lune of Hippocrates[119] and the parabola,[120] were also found to be squarable.

The use of squares as the standard for area measurement is evident in the Greek formulation of the Pythagorean theorem. This formulation states that the sum of the areas of the squares constructed on the two shorter sides of a right triangle equals the area of the square constructed on the hypotenuse.[121] While this theorem could be expressed using other shapes (like equilateral triangles or semicircles) instead of squares, the Greeks specifically chose squares.[122] In modern mathematics, this geometric interpretation has largely been superseded by the algebraic formulation: $a^2 + b^2 = c^2$ , where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse.[123]

The ambition to "square the circle" – constructing a square with the same area as a given circle using only compass and straightedge – occupied mathematicians for centuries. However, in 1882, this task was definitively proven impossible. The proof relies on the Lindemann–Weierstrass theorem, which establishes that pi (π) is a transcendental number, meaning it cannot be the root of any polynomial with rational coefficients. A successful construction of a squarable circle would imply such a polynomial for π, which does not exist.[124] Philosophically, the concept of a "square circle" has been a classic example of an oxymoron since the time of Aristotle, prompting explorations into contexts, such as taxicab geometry, where the phrase might hold a peculiar meaning.[125]

Tiling and Packing

Tiling: The square tiling, a familiar pattern seen in floors and game boards, is one of only three regular tilings of the plane, alongside the equilateral triangle and regular hexagon tilings.[126] The points where the corners of the squares meet in a square tiling form a square lattice.[127] It is also possible to tile the plane using squares of different sizes,[128][129] as exemplified by the Pythagorean tiling, named for its connection to certain proofs of the Pythagorean theorem.[130]

Packing: Problems related to square packing investigate the smallest square or circle capable of containing a specified number of unit squares. While a chessboard perfectly packs a square number of unit squares within a larger square, finding optimal solutions for packing multiple unit squares into various containers remains a challenge, with many such problems unsolved.[131][132][133] The complexity increases when packing circles into a square (circle packing in a square).[134] Packing squares into other shapes can become computationally formidable; for instance, determining if a given number of unit squares can fit into a rectilinear polygon with specific coordinate properties is an NP-complete problem.[135]

Squaring the Square: This involves dissecting a large square into smaller squares, each with integer side lengths. A "perfect squared square" is one where all the smaller squares are of distinct sizes.[136] A variation, known as "Mrs. Perkins's quilt," allows for repeated square sizes but aims to minimize the number of squares used while ensuring their side lengths have a greatest common divisor of 1.[137] It's also possible to tile the entire plane using squares such that there is exactly one square of each positive integer side length.[138]

Higher Dimensions and Surfaces: Beyond the flat plane, squares can tile more complex surfaces. The Clifford torus, a four-dimensional object formed by the Cartesian product of two circles, possesses the same intrinsic geometry as a square with opposite edges identified.[139] In three dimensions, a regular skew apeirohedron features six squares meeting at each vertex.[140] The paper bag problem explores the maximum volume enclosed by a surface formed by two squares joined edge-to-edge, a problem whose exact solution remains elusive.[141] Different joining patterns of squares can create other shapes, such as the biscornu. Surfaces composed of finitely many squares within the three-dimensional integer lattice are termed [polyominoids].[143]

Counting

Squares in a Grid: A common mathematical puzzle involves counting all possible squares within a larger square grid. For an $n \times n$ grid of squares, the total number of squares is given by the sum of the first $n$ square pyramidal numbers: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ .[144] For example, a $3 \times 3$ grid contains 14 squares: nine $1 \times 1$ squares, four $2 \times 2$ squares, and one $3 \times 3$ square. The sequence for $n=1, 2, 3, \dots$ begins: 1, 5, 14, 30, 55, 91, ...[145]

A variation considers squares formed by points, allowing for tilted squares. For an $n \times n$ grid of points, the number of squares (including tilted ones) is given by $\frac{n^2(n^2-1)}{12}$ .[147] For $n=1, 2, 3, \dots$ , this sequence starts: 0, 1, 6, 20, 50, 105, ...

Dividing a Square into Similar Rectangles: Another counting problem concerns the number of distinct aspect ratios possible when dividing a square into $n$ similar rectangles. While a square can be divided into two similar rectangles in only one way (by bisecting it), three distinct aspect ratios are possible for three similar rectangles.[148] The number of possible proportions for $n=1, 2, 3, \dots$ is 1, 1, 3, 11, 51, 245, 1372, ...[150]

Magic Squares: A magic square is an $n \times n$ grid filled with numbers such that the sum of the numbers in each row, each column, and both main diagonals is the same. This magic sum is given by $\frac{n(n^2+1)}{2}$ .[151][152] The sequence of magic sums for $n=1, 2, 3, \dots$ is: 1, 5, 15, 34, 65, 111, ...

Other Geometries

In the familiar Euclidean geometry, space is flat, leading to the definition of a square as a regular quadrilateral with four equal sides and four 90° angles. However, in curved spaces like spherical geometry and hyperbolic geometry, this definition shifts.

Spherical Geometry: In positively curved spherical space, the sum of internal angles in a convex quadrilateral exceeds 360°, with the excess proportional to the area. Small spherical squares approximate Euclidean squares, but larger ones have angles that increase with area.[153] A hemisphere, for instance, can be considered a spherical square with four straight angles.[156] An octant of a sphere forms a regular spherical triangle with three right angles.[157]

Hyperbolic Geometry: In negatively curved hyperbolic space, the sum of internal angles falls short of 360°, with the defect proportional to the area. Hyperbolic squares are smaller than their Euclidean counterparts for a given side length, and their angles decrease with increasing area.[154] Special cases include squares with angles of $360^\circ/n$ for $n>4$ , which can tile the hyperbolic plane. An "ideal square" has vertices at infinity and 0° angles, yet possesses a finite area.[159][160]

Metric Geometries: The Euclidean distance defines our standard notion of distance. However, other distance functions create different geometric worlds. In taxicab geometry, based on the $L_1$ distance, diagonal squares represent "circles" (points equidistant from a center).[161][162] Conversely, in Chebyshev distance (the $L_\infty$ distance), axis-parallel squares function as circles.[163]