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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
, a square is a
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunin ...
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
, which means that it has four equal sides and four equal
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (ge ...
s (90-
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
angles, π/2 radian angles, or
right angles In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
). It can also be defined as a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
in which two adjacent sides have equal length. A square with vertices ''ABCD'' would be denoted .


Characterizations

A
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optical device that focuses or disperses a light beam by means of ...
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
is a square
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondit ...
it is any one of the following: * A
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
with two adjacent equal sides * A
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
with a right vertex angle * A
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
with all angles equal * A
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
with one right vertex angle and two adjacent equal sides * A
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
with four equal sides and four
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or π/2 radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ...

right angle
s * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is A= \tfrac(a^2+c^2)=\tfrac(b^2+d^2).Josefsson, Martin
"Properties of equidiagonal quadrilaterals"
''Forum Geometricorum'', 14 (2014), 129-144.


Properties

A square is a special case of a
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
(equal sides, opposite equal angles), a
kite . This sparless, ram-air inflated kite, has a complex bridle formed of many strings attached to the face of the wing. A kite is a tethered heavier-than-air or lighter-than-air craft with wing A wing is a type of fin that produces lift wh ...
(two pairs of adjacent equal sides), a
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

trapezoid
(one pair of opposite sides parallel), a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
(all opposite sides parallel), a
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
or tetragon (four-sided polygon), and a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
(opposite sides equal, right-angles), and therefore has all the properties of all these shapes, namely: * The
diagonal Image:Cube diagonals.svg, The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length \sqrt 3, while AC (shown in red) is a face diagonal and has length \sqrt 2. In geometry, a diagonal is a line segment joinin ...
s of a square bisect each other and meet at 90°. * The diagonals of a square bisect its angles. * Opposite sides of a square are both
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
and equal in length. * All four angles of a square are equal (each being 360°/4 = 90°, a right angle). * All four sides of a square are equal. * The diagonals of a square are equal. * The square is the n=2 case of the families of n-
hypercubes In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consist ...
and n-
orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular polytope, regular, convex polytope that exists in ''n''-dimensions. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedro ...
es. * A square has
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who ...
. A truncated square, t, is an
octagon In geometry, an octagon (from the Ancient Greek, Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A ''regular polygon, regular octagon'' has Schläfli symbol and can also be constructed as a quasireg ...

octagon
, . An alternated square, h, is a
digon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

digon
, .


Perimeter and area

150px, The area of a square is the product of the length of its sides. The
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length ( one-dimensional). The perimeter of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathema ...

perimeter
of a square whose four sides have length \ell is :P=4\ell and the
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the am ...

area
''A'' is :A=\ell^2. In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term ''
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
'' to mean raising to the second power. The area can also be calculated using the diagonal ''d'' according to :A=\frac. In terms of the
circumradius In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
''R'', the area of a square is :A=2R^2; since the area of the circle is \pi R^2, the square fills 2/\pi \approx 0.6366 of its
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertex (geometry), vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. ...
. In terms of the
inradius (I), excircles, excenters (J_A, J_B, J_C), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt: ...
''r'', the area of the square is :A=4r^2; hence the area of the
inscribed circle (I), excircles, excenters (J_A, J_B, J_C), internal angle bisector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldes ...
is \pi/4 \approx 0.7854 of that of the square. Because it is a
regular polygon In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method cons ...
, 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. Indeed, if ''A'' and ''P'' are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds: :16A\le P^2 with equality if and only if the quadrilateral is a square.


Other facts

* The diagonals of a square are \sqrt (about 1.414) times the length of a side of the square. This value, known as the
square root of 2 The square root of 2, or the one-half power of 2, written in mathematics as \sqrt or 2^, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the principal square root of 2, to di ...
or Pythagoras' constant, was the first number proven to be
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
. * A square can also be defined as a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
with equal diagonals that bisect the angles. * If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square. * A square has a larger area than any other quadrilateral with the same perimeter. * A
square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of , meaning it has ''4'' Square (geometry), squares around every Vertex (geometry), vertex. John Horton Conway ...
is one of three
regular tilings This page lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n'' ...
of the plane (the others are the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular polygon, equiangular; that is, all three internal angles are also con ...

equilateral triangle
and the
regular hexagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

regular hexagon
). * The square is in two families of polytopes in two dimensions:
hypercube In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

hypercube
and the
cross-polytope In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
. The
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who ...
for the square is . * The square is a highly symmetric object. There are four lines of
reflectional symmetry 250px, Figures with the axes of asymmetric.">asymmetry.html" ;"title="symmetry drawn in. The figure with no axes is asymmetry">asymmetric. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to Re ...

reflectional symmetry
and it has
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...
of order 4 (through 90°, 180° and 270°). Its
symmetry group In group theory, the symmetry group of a geometric object is the group (mathematics), group of all Transformation (geometry), transformations under which the object is invariant (mathematics), invariant, endowed with the group operation of Fun ...
is the
dihedral group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
 D4. * If the inscribed circle of a square ''ABCD'' has tangency points ''E'' on ''AB'', ''F'' on ''BC'', ''G'' on ''CD'', and ''H'' on ''DA'', then for any point ''P'' on the inscribed circle, :: 2(PH^2-PE^2) = PD^2-PB^2. * If d_i is the distance from an arbitrary point in the plane to the ''i''-th vertex of a square and R is the
circumradius In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
of the square, then ::\frac + 3R^4 = \left(\frac + R^2\right)^2. * If L and d_i are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, 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 of the square.


Coordinates and equations

The coordinates for the vertices of a square with vertical and horizontal sides, centered at the origin and with side length 2 are (±1, ±1), while the interior of this square consists of all points (''x''i, ''y''i) with and . The equation :\max(x^2, y^2) = 1 specifies the boundary of this square. This equation means "''x''2 or ''y''2, whichever is larger, equals 1." The
circumradius In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
of this square (the radius of a circle drawn through the square's vertices) is half the square's diagonal, and is equal to \sqrt. Then the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertex (geometry), vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. ...
has the equation :x^2 + y^2 = 2. Alternatively the equation :\left, x - a\ + \left, y - b\ = r. can also be used to describe the boundary of a square with center
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
(''a'', ''b''), and a horizontal or vertical radius of ''r''. The square is therefore the shape of a topological ball according to the L1 distance metric.


Construction

The following animations show how to construct a square using a
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ...
. This is possible as 4 = 22, a
power of two Visualization of powers of two from 1 to 1024 (20 to 210) A power of two is a number of the form where is an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can b ...
.


Symmetry

The ''square'' has Dih4 symmetry,
order Order or ORDER or Orders may refer to: * Orderliness Orderliness is associated with other qualities such as cleanliness Cleanliness is both the abstract state of being clean and free from germs, dirt, trash, or waste, and the habit of achieving a ...
8. There are 2 dihedral subgroups: Dih2, Dih1, and 3
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

cyclic
subgroups: Z4, Z2, and Z1. A square is a special case of many lower symmetry quadrilaterals: * A rectangle with two adjacent equal sides * A quadrilateral with four equal sides and four
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or π/2 radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ...

right angle
s * A parallelogram with one right angle and two adjacent equal sides * A rhombus with a right angle * A rhombus with all angles equal * A rhombus with equal diagonals These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order. Each subgroup symmetry allows one or more degrees of freedom for irregular
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

quadrilateral
s. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
, and p4 is the symmetry of a
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

rhombus
. These two forms are
duals ''Duals'' is a compilation album by the Irish Rock music, rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose ...
of each other, and have half the symmetry order of the square. d2 is the symmetry of an
isosceles trapezoid In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...

isosceles trapezoid
, and p2 is the symmetry of a
kite . This sparless, ram-air inflated kite, has a complex bridle formed of many strings attached to the face of the wing. A kite is a tethered heavier-than-air or lighter-than-air craft with wing A wing is a type of fin that produces lift wh ...
. g2 defines the geometry of a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

parallelogram
. Only the g4 subgroup has no degrees of freedom, but can seen as a square with
directed edge In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

directed edge
s.


Squares inscribed in triangles

Every
acute triangleAn acute triangle (or acute-angled triangle) is a triangle A triangle is a polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmet ...

acute triangle
has three
inscribed{{unreferenced, date=August 2012 Image:Inscribed circles.svg, frame, Inscribed circles of various polygons image:Circumcentre.svg, An inscribed triangle of a circle In geometry, an inscribed plane (geometry), planar shape or solid (geometry), solid ...
squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a
right triangle A right triangle (American English) or right-angled triangle (British English, British ) is a triangle in which one angle is a right angle (that is, a 90-Degree (angle), degree angle). The relation between the sides and angles of a right triangle ...

right triangle
two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two ''distinct'' inscribed squares. An
obtuse triangleAn acute triangle (or acute-angled triangle) is a triangle A triangle is a polygon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmet ...
has only one inscribed square, with a side coinciding with part of the triangle's longest side. The fraction of the triangle's area that is filled by the square is no more than 1/2.


Squaring the circle

Squaring the circle Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "meas ...

Squaring the circle
, proposed by
ancient Ancient history is the aggregate of past eventsWordNet Search – 3.0
"History"
from ...
geometers A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BC to 1 BC * Baudhayana (fl. c. 800 BC) - Euclidean geometry, geometric algebra * Manava ...
, is the problem of constructing a square with the same area as a given
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a ...
, by using only a finite number of steps with
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ...
. In 1882, the task was proven to be impossible as a consequence of the
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following. In other words the extension field has transcendence degree In abst ...
, which proves that () is a
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
rather than an algebraic irrational number; that is, it is not the
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They most often lie bel ...
of any
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtra ...

polynomial
with
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
coefficients.


Non-Euclidean geometry

In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles. In
spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...
, a square is a polygon whose edges are
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
arcs of equal distance, which meet at equal angles. Unlike the square of plane geometry, the angles of such a square are larger than a right angle. Larger spherical squares have larger angles. In
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any ...
, squares with right angles do not exist. Rather, squares in hyperbolic geometry have angles of less than right angles. Larger hyperbolic squares have smaller angles. Examples:


Crossed square

A crossed square is a
faceting Image:CubeAndStel.svg Stella octangula as a faceting of the cube In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest ...
of the square, a self-intersecting polygon created by removing two opposite edges of a square and reconnecting by its two diagonals. It has half the symmetry of the square, Dih2, order 4. It has the same
vertex arrangement In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
as the square, and is
vertex-transitive In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertex (geometry), vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the sam ...
. It appears as two 45-45-90 triangle with a common vertex, but the geometric intersection is not considered a vertex. A crossed square is sometimes likened to a
bow tie#REDIRECT Bow tie The bow tie is a type of necktie. A modern bow tie is tied using a common shoelace knot, which is also called the bow knot for that reason. It consists of a ribbon of fabric tied around the collar (clothing), collar of a shir ...
or
butterfly Butterflies are insect Insects or Insecta (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known ...
. the
crossed rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...
is related, as a faceting of the rectangle, both special cases of
crossed quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle), tetra ...
s. The interior of a crossed square can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise. A square and a crossed square have the following properties in common: * Opposite sides are equal in length. * The two diagonals are equal in length. * It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°). It exists in the
vertex figure In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
of a uniform star polyhedra, the .


Graphs

The K4
complete graph In the mathematics, mathematical field of graph theory, a complete graph is a simple graph, simple undirected graph in which every pair of distinct vertex (graph theory), vertices is connected by a unique edge (graph theory), edge. A complete digr ...
is often drawn as a square with all 6 possible edges connected, hence appearing as a square with both diagonals drawn. This graph also represents an
orthographic projection Orthographic projection (sometimes referred to as orthogonal projection, used to be called analemma) is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines ...
of the 4 vertices and 6 edges of the regular 3-
simplex In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

simplex
(
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (g ...

tetrahedron
).


See also

*
Cube In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
*
Pythagorean theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

Pythagorean theorem
*
Square lattice Upright square tiling. The vertices of all squares together with their centers form an upright square lattice. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale √2 times as large as t ...
*
Square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
*
Square root In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

Square root
*
Squaring the square Squaring the square is the problem of tessellation, tiling an integral square using only other integral squares. (An integral square is a square (geometry), square whose sides have integer length.) The name was coined in a humorous analogy with squa ...

Squaring the square
*
Squircle A squircle is a shape A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color Color ( American English), or colour ( Commonwealth English), is the characte ...
*
Unit square 300px, The unit square in the Euclidean geometry, real plane In mathematics, a unit square is a square (geometry), square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian coordinate system#Ca ...


References


External links


Animated course (Construction, Circumference, Area)


With interactive applet

{{Authority control Elementary shapes Types of quadrilaterals 4 (number) Constructible polygons