View source for Square

{{Short description|Shape with four equal sides and angles}}
{{CS1 config|mode=cs1}}
{{About|the shape}}{{Infobox polygon
| name       = Square
| image      = Regular polygon 4 annotated.svg
| caption    =
| type       = {{plainlist|1=
*[[quadrilateral]]
*[[regular polygon]]
*[[hypercube]]
*[[cross-polytope]]
}}
| euler      =
| edges      = 4
| schläfli   = 
| wythoff    = 
| coxeter    = 
| symmetry   = [[Dihedral group of order 8|order-8 dihedral]]
| area       = side<sup>2</sup>
| angle      = {{pi}}/2 (90°)
| perimeter  = 4 · side
| dual       = 
| properties =
}}
In [[geometry]], a '''square''' is a [[regular polygon|regular]] [[quadrilateral]]. It has four straight sides of equal length and four equal [[angle]]s. Squares are special cases of [[rectangle]]s, which have four equal angles, and of [[rhombus]]es, which have four equal sides. As with all rectangles, a square's angles are [[right angle]]s (90 [[degree (angle)|degree]]s, or [[Pi|{{pi}}]]/2 [[radian]]s), making adjacent sides [[perpendicular]]. The [[area]] of a square is the side length multiplied by itself, and so in [[algebra]], multiplying a number by itself is called [[square (algebra)|squaring]].

Equal squares can tile the plane edge-to-edge in the [[square tiling]]. Square tilings are ubiquitous in [[tile]]d floors and walls, [[graph paper]], image [[pixel]]s, and [[game board]]s. Square shapes are also often seen in building [[floor plan]]s, [[origami paper]], food servings, in [[graphic design]] and [[heraldry]], and in instant photos and fine art.

The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for [[squaring the circle]] by [[compass and straightedge]], now known to be impossible. Squares can be inscribed in any smooth or convex curve such as a circle or triangle, but it remains unsolved [[inscribed square problem|whether a square can be inscribed in every simple closed curve]]. Several problems of [[squaring the square]] involve subdividing squares into unequal squares. Mathematicians have also studied packing squares as tightly as possible into other shapes.

Squares can be constructed by [[straightedge and compass]], through their [[Cartesian coordinates]], or by repeated multiplication by <math>i</math> in the [[complex plane]]. They form the [[Ball (mathematics)|metric balls]] for [[taxicab geometry]] and [[Chebyshev distance]], two forms of non-Euclidean geometry. Although [[spherical geometry]] and [[hyperbolic geometry]] both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles, or with right angles and different numbers of sides.

==Definitions and characterizations==
[[File:The square among a family of rectangles or rhombuses.png|thumb|upright=1.25|Among rectangles (top row), the square is the shape with equal sides (blue, middle). Among rhombuses (bottom row), the square is the shape with right angles (blue, middle).]]
Squares can be defined or characterized in many equivalent ways. If a [[polygon]] in the [[Euclidean plane]] satisfies any one of the following criteria, it satisfies all of them:
* A square is a polygon with four equal sides and four [[right angle]]s; that is, it is a quadrilateral that is both a rhombus and a rectangle<ref name=zalgri/>
* A square is a [[rectangle]] with four equal sides.<ref name=zalgri>{{cite book|first1=Zalman|last1=Usiskin|author1-link=Zalman Usiskin|first2=Jennifer|last2=Griffin|title=The Classification of Quadrilaterals: A Study of Definition|publisher=Information Age Publishing|year=2008|page=59|isbn=978-1-59311-695-8}}</ref>
* A square is a [[rhombus]] with a right angle between a pair of adjacent sides.<ref name=zalgri/>
* A square is a rhombus with all angles equal.<ref name=zalgri/>
* A square is a [[parallelogram]] with one right angle and two adjacent equal sides.<ref name=zalgri/>
* A square is a  quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other. That is, it is a rhombus with equal diagonals.<ref>{{Cite web|url=http://jwilson.coe.uga.edu/MATH7200/ProblemSet1.3.html|title=Problem Set 1.3, problem 10|work=Math 5200/7200 Foundations of Geometry I|first=Jim|last=Wilson|publisher=University of Georgia|date=Summer 2010|access-date=2025-02-05}}</ref>
* A square is a  quadrilateral with successive sides <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> whose area is<ref>{{cite book
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger B.
 | contribution = Theorem 9.2.1
 | isbn = 9781470453121
 | page = 186
 | publisher = American Mathematical Society
 | series = Dolciani Mathematical Expositions
 | title = A Cornucopia of Quadrilaterals
 | url = https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA22
 | volume = 55
 | year = 2020}}</ref> <math display=block>A=\frac14(a^2+b^2+c^2+d^2).</math>

Squares are the only [[regular polygon]]s whose [[Internal and external angle|internal angle]], [[central angle#Central angle of a regular polygon|central angle]], and [[Internal and external angle|external angle]] are all equal (they are all right angles).<ref name=rich/>

==Properties==
A square is a special case of a [[rhombus]] (equal sides, opposite equal angles), a [[Kite (geometry)|kite]] (two pairs of adjacent equal sides), a [[trapezoid]] (one pair of opposite sides parallel), a [[parallelogram]] (all opposite sides parallel), a [[quadrilateral]] or tetragon (four-sided polygon), and a [[rectangle]] (opposite sides equal, right-angles),<ref name=zalgri/> and therefore has all the properties of all these shapes, namely:
* All four internal angles of a square are equal (each being 90°, a right angle).<ref name=rich/><ref name=godsid>{{cite book
 | last1 = Godfrey | first1 = Charles
 | last2 = Siddons | first2 = A. W.
 | edition = 3rd
 | page = 40
 | publisher = Cambridge University Press
 | title = Elementary Geometry: Practical and Theoretical
 | url = https://archive.org/details/elementarygeomet00godfuoft/page/40
 | year = 1919}}</ref>
* The central angle of a square is equal to 90°.<ref name=rich>{{cite book
 | last = Rich | first = Barnett
 | page = 132
 | publisher = Schaum
 | title = Principles And Problems Of Plane Geometry
 | url = https://archive.org/details/in.ernet.dli.2015.131938/page/n147
 | year = 1963}}</ref>
* The external angle of a square is equal to 90°.<ref name=rich/>
* The diagonals of a square are equal and [[bisection|bisect]] each other, meeting at 90°.<ref name=godsid/>
* The diagonals of a square bisect its internal angles, forming [[angle#adjacent|adjacent angles]] of 45°.<ref>{{cite book
 | last1 = Schorling | first1 = R.
 | last2 = Clark | first2 = John P.
 | last3 = Carter | first3 = H. W.
 | pages = 124–125
 | publisher = George G. Harrap & Co.
 | title = Modern Mathematics: An Elementary Course
 | url = https://archive.org/details/in.ernet.dli.2015.84435/page/n127
 | year = 1935}}</ref>
* All four sides of a square are equal.{{sfnp|Godfrey|Siddons|1919|p=135}}
* Opposite sides of a square are  [[Parallel (geometry)|parallel]].{{sfnp|Schorling|Clark|Carter|1935|p=101}}

All squares are [[similarity (geometry)|similar]] to each other, meaning they have the same shape.<ref>{{cite book
 | title = Project Mathematics! Program Guide and Workbook: Similarity
 | publisher = California Institute of Technology
 | last = Apostol | first = Tom M. | author-link = Tom M. Apostol
 | year = 1990
 | page = 8–9
 | url = https://archive.org/details/02-the-story-of-pi/01%20Similarity/page/9/mode/1up
}} Workbook accompanying ''[[Project Mathematics!]]'' [https://www.youtube.com/watch?v=vpxWyJg4_1A Ep. 1: "Similarity"] (Video).</ref> One parameter (typically the length of a side or diagonal)<ref>{{cite book
 | last1 = Gellert | first1 = W.
 | last2 = Gottwald | first2 = S.
 | last3 = Hellwich | first3 = M.
 | last4 = Kästner | first4 = H.
 | last5 = Küstner | first5 = H.
 | title = The VNR Concise Encyclopedia of Mathematics | edition = 2nd
 | year = 1989
 | publisher = Van Nostrand Reinhold | place = New York
 | isbn = 0-442-20590-2
 | chapter-url = https://archive.org/details/vnrconciseencycl00gell/page/161/mode/1up?q=%22square+is+given%22
 | chapter = Quadrilaterals
 | at = § 7.5, p. 161
}}</ref> suffices to specify a square's size. Squares of the same size are [[Congruence (geometry)|congruent]].<ref>{{cite book
 | last = Henrici | first = Olaus | author-link = Olaus Henrici
 | page = 134
 | publisher = Longmans, Green
 | title = Elementary Geometry: Congruent Figures
 | url = https://archive.org/details/elementarygeome00henrgoog/page/n164
 | year = 1879}}</ref>

===Measurement===
[[File:YBC-7289-OBV-labeled.jpg|[[YBC 7289]], a [[Babylonian mathematics|Babylonian]] calculation of a square's diagonal from between 1800 and 1600 BCE|thumb]]
[[Image:Five Squared.svg|150px|right|thumb|The area of a square is the product of the lengths of its sides.]]

A square whose four sides have length <math>\ell </math> has [[perimeter]]{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n147 131]}} <math>P=4\ell</math> and [[diagonal]] length <math>d=\sqrt2\ell</math>.{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n135 120]}} The [[square root of 2]], appearing in this formula, is [[irrational number|irrational]], meaning that it is not the ratio of any two [[integer]]s. It is approximately equal to 1.414,<ref>{{cite book |last1=Conway |first1=J. H. |author-link1=John Horton Conway |last2=Guy |first2=R. K. |author-link2=Richard K. Guy |title=The Book of Numbers |title-link=The Book of Numbers (math book) |year=1996 |publisher=Springer-Verlag |location=New York|pages=181–183}}</ref> and its approximate value was already known in [[Babylonian mathematics]].<ref>{{cite journal
 | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
 | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
 | doi = 10.1006/hmat.1998.2209
 | issue = 4
 | journal = [[Historia Mathematica]]
 | mr = 1662496
 | pages = 366–378
 | title = Square root approximations in old Babylonian mathematics: YBC 7289 in context
 | volume = 25
 | year = 1998| doi-access = free
 }}</ref> A square's [[area]] is{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n135 120]}}
<math display="block">A=\ell^2=\tfrac12 d^2.</math>
This formula for the area of a square as the second power of its side length led to the use of the term ''[[Square (algebra)|squaring]]'' to mean raising any number to the second power.<ref>{{cite book|first=James|last=Thomson|author-link=James Thomson (mathematician)|title=An Elementary Treatise on Algebra: Theoretical and Practical|year=1845|location=London|publisher=Longman, Brown, Green, and Longmans|page=4|url=https://archive.org/details/anelementarytre01thomgoog/page/n15}}</ref> Reversing this relation, the side length of a square of a given area is the [[square root]] of the area. Squaring an integer, or taking the area of a square with integer sides, results in a [[square number]]; these are [[figurate number]]s representing the numbers of points that can be arranged into a square grid.{{sfnp|Conway|Guy|1996|pp=30–33,38–40}}

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. That is, it is an [[equable shape]]. The only other equable integer rectangle is a three by six rectangle.<ref>{{cite book |last1=Konhauser |first1=Joseph D. E. | author1-link=Joseph Konhauser |last2=Velleman |first2=Dan |last3=Wagon |first3=Stan |authorlink3=Stan Wagon |date=1997 |title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries |contribution=95. When does the perimeter equal the area? |volume=18 |series=Dolciani Mathematical Expositions |publisher=Cambridge University Press |isbn=9780883853252 |page=29 |url=https://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29}}</ref>

Because it is a [[regular polygon]], 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.<ref>Page 147 of {{cite book
 | last = Chakerian | first = G. D.
 | editor-last = Honsberger | editor-first = Ross | editor-link = Ross Honsberger
 | contribution = A distorted view of geometry
 | isbn = 0-88385-304-3
 | mr = 563059
 | pages = 130–150
 | publisher = Mathematical Association of America | location = Washington, DC
 | series = The Dolciani Mathematical Expositions
 | title = Mathematical Plums
 | volume = 4
 | year = 1979}}</ref>  Indeed, if ''A'' and ''P'' are the area and perimeter enclosed by a quadrilateral, then the following [[isoperimetric inequality]] holds:
<math display="block">16A\le P^2</math>
with equality if and only if the quadrilateral is a square.<ref>{{cite journal
 | last = Fink | first = A. M.
 | date = November 2014
 | doi = 10.1017/S0025557200008275
 | issue = 543
 | journal = [[The Mathematical Gazette]]
 | jstor = 24496543
 | page = 504
 | title = 98.30 The isoperimetric inequality for quadrilaterals
 | volume = 98}}</ref>{{sfnp|Alsina|Nelsen|2020|loc=Theorem 9.2.2|page=187}}

===Symmetry===
{{main|Symmetry group of a square}}
The square is the most symmetrical of the quadrilaterals.<ref name=berger/>  Eight [[rigid transformation]]s of the plane take the square to itself:<ref name=miller>{{cite journal
 | last = Miller | first = G. A. | author-link = George Abram Miller
 | doi = 10.1080/00029890.1903.11997111
 | issue = 10
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2969176
 | mr = 1515975
 | pages = 215–218
 | title = On the groups of the figures of elementary geometry
 | volume = 10
 | year = 1903}}</ref>

{{bi|left=1.3em |{{multiple image
 | width = 180
 | perrow = 4
 | align = none
 | image_style=border:none
 | image1 = Square symmetry – I.png
 | caption1 = The square's initial position <br> (the [[identity transformation]])
 | image2 = Square symmetry – R1.png
 | caption2 = [[Rotation (mathematics)|Rotation]] by 90° anticlockwise
 | image3 = Square symmetry – R2.png
 | caption3 = Rotation by 180°
 | image4 = Square symmetry – R3.png
 | caption4 = Rotation by 270°
 | image5 = Square symmetry – D1.png
 | caption5 = Diagonal NW–SE [[Reflection (mathematics)|reflection]]
 | image6 = Square symmetry – H.png
 | caption6 = Horizontal reflection
 | image7 = Square symmetry – D2.png
 | caption7 = Diagonal NE–SW reflection
 | image8 = Square symmetry – V.png
 | caption8 = Vertical reflection
}} }}

[[File:Quadrilateral symmetries.svg|thumb|The axes of reflection symmetry and centers of rotation symmetry of a square (top), rectangle and rhombus (center), [[isosceles trapezoid]], kite, and parallelogram (bottom)]]

For an axis-parallel square centered at the [[Origin (mathematics)|origin]], each symmetry acts by a combination of negating and swapping the [[Cartesian coordinate]]s of points.<ref name=ers>{{cite conference
 | last1 = Estévez | first1 = Manuel
 | last2 = Roldán | first2 = Érika
 | last3 = Segerman | first3 = Henry | author3-link = Henry Segerman
 | editor1-last = Holdener | editor1-first = Judy | editor1-link = Judy Holdener
 | editor2-last = Torrence | editor2-first = Eve | editor2-link = Eve Torrence
 | editor3-last = Fong | editor3-first = Chamberlain
 | editor4-last = Seaton | editor4-first = Katherine
 | arxiv = 2311.06596
 | contribution = Surfaces in the tesseract
 | contribution-url = https://archive.bridgesmathart.org/2023/bridges2023-441.html
 | isbn = 978-1-938664-45-8
 | location = Phoenix, Arizona
 | pages = 441–444
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2023: Mathematics, Art, Music, Architecture, Culture
 | year = 2023}}</ref>
The symmetries permute the eight isosceles triangles between the half-edges and the square's center (which stays in place); any of these triangles can be taken as the [[fundamental region]] of the transformations.<ref>{{cite book
 | last1 = Grove | first1 = L. C.
 | last2 = Benson | first2 = C. T.
 | doi = 10.1007/978-1-4757-1869-0
 | edition = 2nd
 | isbn = 0-387-96082-1
 | mr = 777684
 | page = 9
 | publisher = Springer-Verlag | location = New York
 | series = Graduate Texts in Mathematics
 | title = Finite Reflection Groups
 | volume = 99
 | year = 1985}}</ref> Each two vertices, each two edges, and each two half-edges are mapped one to the other by at least one symmetry (exactly one for half-edges).<ref name=berger>{{cite book
 | last = Berger | first = Marcel
 | doi = 10.1007/978-3-540-70997-8
 | isbn = 978-3-540-70996-1
 | mr = 2724440
 | page = 509
 | publisher = Springer | location = Heidelberg
 | title = Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry
 | year = 2010}}</ref> All [[regular polygon]]s also have these properties,<ref>{{cite book
 | last = Toth | first = Gabor
 | contribution = Section 9: Symmetries of regular polygons
 | doi = 10.1007/0-387-22455-6_9
 | edition = Second
 | isbn = 0-387-95345-0
 | mr = 1901214
 | pages = 96–106
 | publisher = Springer-Verlag, New York
 | series = Undergraduate Texts in Mathematics
 | title = Glimpses of Algebra and Geometry
 | year = 2002}}</ref> which are expressed by saying that symmetries of a square and, more generally, a regular polygon act [[transitive action|transitively]] on vertices and edges, and [[simply transitively]] on half-edges.<ref>{{cite book
 | last = Davis | first = Michael W.
 | isbn = 978-0-691-13138-2
 | mr = 2360474
 | page = 16
 | publisher = Princeton University Press, Princeton, NJ
 | series = London Mathematical Society Monographs Series
 | title = The Geometry and Topology of Coxeter Groups
 | url = https://books.google.com/books?id=pCDHDgAAQBAJ&pg=PA16
 | volume = 32
 | year = 2008}}</ref>

Combining any two of these transformations by performing one after the other continues to take the square to itself, and therefore produces another symmetry. Repeated rotation produces another rotation with the summed rotation angle. Two reflections with the same axis return to the identity transformation, while two reflections with different axes rotate the square. A rotation followed by a reflection, or vice versa, produces a different reflection. This [[Function composition|composition operation]] gives the eight symmetries of a square the mathematical structure of a [[group (mathematics)|group]], called the ''group of the square'' or the ''[[Dihedral group of order 8|dihedral group of order eight]]''.<ref name=miller/> Other quadrilaterals, like the rectangle and rhombus, have only a [[subgroup]] of these symmetries.<ref>{{cite book|first1=John H.|last1=Conway|author1-link=John Horton Conway|first2=Heidi|last2=Burgiel|first3=Chaim|last3=Goodman-Strauss|author3-link=Chaim Goodman-Strauss|title=The Symmetries of Things|title-link=The Symmetries of Things|year=2008|publisher=AK Peters|isbn=978-1-56881-220-5|contribution=Figure 20.3|page=272}}</ref><ref>{{cite journal |last=Beardon |first=Alan F. |author-link=Alan Frank Beardon |year=2012 |title=What is the most symmetric quadrilateral? |journal=The Mathematical Gazette |volume=96 |number=536 |pages=207–212 |doi=10.1017/S0025557200004435 |jstor=23248552}}</ref>

[[File:Perspective-3point.svg|thumb|upright=0.6|Three-point perspective of a cube, showing perspective transformations of its six square faces into six different quadrilaterals]]
The shape of a square, but not its size, is preserved by [[similarity (geometry)|similarities]] of the plane.<ref>{{cite journal
 | last1 = Frost | first1 = Janet Hart
 | last2 = Dornoo | first2 = Michael D.
 | last3 = Wiest | first3 = Lynda R.
 | date = November 2006
 | issue = 4
 | journal = Mathematics Teaching in the Middle School
 | jstor = 41182391
 | pages = 222–224
 | title = Take time for action: Similar shapes and ratios
 | volume = 12}}</ref> Other kinds of transformations of the plane can take squares to other kinds of quadrilateral. An [[affine transformation]] can take a square to any parallelogram, or vice versa;<ref>{{cite journal
 | last = Gerber | first = Leon
 | doi = 10.1080/00029890.1980.11995110
 | issue = 8
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2320952
 | mr = 600923
 | pages = 644–648
 | title = Napoleon's theorem and the parallelogram inequality for affine-regular polygons
 | volume = 87
 | year = 1980}}</ref> a [[projective transformation]] can take a square to any convex [[quadrilateral]], or vice versa.<ref>{{cite book|first=C. R.|last=Wylie|title=Introduction to Projective Geometry|pages=17–19|publisher=McGraw-Hill|year=1970}} [https://books.google.com/books?id=QoNCAwAAQBAJ&pg=PA17 Reprinted], Dover Books, 2008, {{isbn|9780486468952}}</ref> This implies that, when [[Perspective (graphical)|viewed in perspective]], a square can look like any convex quadrilateral, or vice versa.<ref>{{cite book
 | last = Francis | first = George K.
 | isbn = 0-387-96426-6
 | mr = 880519
 | page = 52
 | publisher = Springer-Verlag | location = New York
 | title = A Topological Picturebook
 | year = 1987}}</ref> A [[Möbius transformation]] can take the vertices of a square (but not its edges) to the vertices of a [[harmonic quadrilateral]].<ref>{{cite book|first=Roger A.|last=Johnson|title=Advanced Euclidean Geometry|publisher=Dover|year=2007|orig-year=1929|isbn=978-0-486-46237-0|page=100|url=https://books.google.com/books?id=559e2AVvrvYC&pg=PA100}}</ref>

The [[wallpaper group]]s are symmetry groups of two-dimensional repeating patterns. For many of these groups the basic unit of repetition (the unit cell of its [[period lattice]]) can be a square, and for three of these groups, p4, p4m, and p4g, it must be a square.<ref>{{cite journal
 | last = Schattschneider | first = Doris | author-link = Doris Schattschneider
 | doi = 10.1080/00029890.1978.11994612
 | issue = 6
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2320063
 | mr = 477980
 | pages = 439–450
 | title = The plane symmetry groups: their recognition and notation
 | volume = 85
 | year = 1978}}</ref>
{{bi |left=1.3em |{{multiple image|align=none
|footer=[[Wallpaper group]]s of tilings from ''[[The Grammar of Ornament]]''
|footer_align=center|caption_align=center
|image1=Wallpaper group-p4-1.jpg|caption1=p4, Egyptian tomb ceiling
|image2=Wallpaper group-p4m-1.jpg|caption2=p4m, Nineveh & Persia
|image3=Wallpaper group-p4g-2.jpg|caption3=p4g, China
|total_width=480}} }}

===Inscribed and circumscribed circles===
[[File:Incircle and circumcircle of a square.png|thumb|The [[inscribed circle]] (purple) and [[circumscribed circle]] (red) of a square (black)]]
The [[inscribed circle]] of a square is the largest circle that can fit inside that square. Its center is the center point of the square, and its radius (the [[inradius]] of the square) is <math>r=\ell/2</math>. Because this circle touches all four sides of the square (at their midpoints), the square is a [[tangential quadrilateral]]. The [[circumscribed circle]] of a square passes through all four vertices, making the square a [[cyclic quadrilateral]]. Its radius, the [[circumradius]], is <math>R=\ell/\sqrt2</math>.{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n149 133]}} If the inscribed circle of a square <math>ABCD</math> has tangency points <math>E</math> on <math>AB</math>, <math>F</math> on <math>BC</math>, <math>G</math> on <math>CD</math>, and <math>H</math> on <math>DA</math>, then for any point <math>P</math> on the inscribed circle,<ref>{{Cite web|url=http://gogeometry.com/problem/p331_square_inscribed_circle.htm|title=Problem 331. Discovering the Relationship between Distances from a Point on the Inscribed Circle to Tangency Point and Vertices in a Square|website=Go Geometry from the Land of the Incas|first=Antonio|last=Gutierrez|access-date=2025-02-05}}</ref><math display=block> 2(PH^2-PE^2) = PD^2-PB^2.</math> If <math>d_i</math> is the distance from an arbitrary point in the plane to the {{nowrap|<math>i</math>th}} vertex of a square and <math>R</math> is the [[circumradius]] of the square, then<ref>{{cite journal
 | last = Park | first = Poo-Sung
 | journal = [[Forum Geometricorum]]
 | mr = 3507218
 | pages = 227–232
 | title = Regular polytopic distances
 | url = http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | archive-url = https://web.archive.org/web/20161010184811/http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | archive-date = 2016-10-10
 | url-status = dead
 | volume = 16
 | year = 2016}}</ref><math display=block>\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.</math>
If <math>L</math>  and <math>d_i</math> are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices  respectively, then <math display=block>d_1^2 + d_3^2 = d_2^2 + d_4^2 = 2(R^2+L^2)</math> and <math display=block> d_1^2d_3^2 + d_2^2d_4^2 = 2(R^4+L^4), </math> where <math>R</math> is the circumradius of the square.<ref name=Mamuka >{{cite journal
 | last = Meskhishvili | first = Mamuka
 | issue = 1
 | journal = International Journal of Geometry
 | mr = 4193377
 | pages = 58–65
 | title = Cyclic averages of regular polygonal distances
 | url = https://ijgeometry.com/wp-content/uploads/2020/12/4.-58-65.pdf
 | volume = 10
 | year = 2021}}</ref>

==Applications==
{{multiple image
|image1=Artesanal tile industry (15323096610).jpg|caption1=Square [[tile]]s
|image2=Girl with a Pearl Earring (pixelated).jpg|caption2=[[Pixelated]] ''[[Girl with a Pearl Earring]]''
|total_width=400}}
Squares are so well-established as the shape of [[tiles]] that the [[Latin language|Latin]] word [[tessera]], for a small tile as used in [[mosaic]]s, comes from an ancient Greek word for the number four, referring to the four corners of a square tile.<ref>{{cite web|url=https://wordsmith.org/words/tessera.html|title=Tessera|work=A word a day|first=Anu|last=Garg|author-link=Anu Garg|access-date=2025-02-09}}</ref> [[Graph paper]], preprinted with a [[square tiling]], is widely used for [[data visualization]] using [[Cartesian coordinates]].<ref>{{cite journal
 | last = Cox | first = D. R. | author-link = David Cox (statistician)
 | doi = 10.2307/2346220
 | issue = 1
 | journal = Applied Statistics
 | jstor = 2346220
 | title = Some Remarks on the Role in Statistics of Graphical Methods
 | volume = 27
 | year = 1978| pages = 4–9 }}</ref> The [[pixel]]s of [[bitmap image]]s, as recorded by [[image scanner]]s and [[digital camera]]s or displayed on [[electronic visual display]]s, conventionally lie at the intersections of a square grid, and are often considered as small squares, arranged in a square tiling.<ref>{{cite book
 | last = Salomon | first = David
 | isbn = 9780857298867
 | page = 30
 | publisher = Springer
 | title = The Computer Graphics Manual
 | url = https://books.google.com/books?id=DX4YstV76c4C&pg=PA30
 | year = 2011}}</ref><ref>{{cite tech report
 | last = Smith | first = Alvy Ray | author-link = Alvy Ray Smith
 | year = 1995
 | id = Microsoft Computer Graphics, Technical Memo 6
 | publisher = Microsoft
 | title = A Pixel Is ''Not'' A Little Square, A Pixel Is ''Not'' A Little Square, A Pixel Is ''Not'' A Little Square! (And a Voxel is ''Not'' a Little Cube)
 | url = http://alvyray.com/Memos/CG/Microsoft/6_pixel.pdf }}</ref> Standard techniques for [[image compression]] and [[video compression]], including the [[JPEG]] format, are based on the subdivision of images into larger square blocks of pixels.<ref>{{cite book
 | last = Richardson | first = Iain E.
 | isbn = 9780471485537
 | page = 127
 | publisher = John Wiley & Sons
 | title = Video Codec Design: Developing Image and Video Compression Systems
 | url = https://books.google.com/books?id=8jxbbRKVbkIC&pg=PA127
 | year = 2002}}</ref> The [[quadtree]] data structure used in data compression and [[computational geometry]] is based on the [[recursive]] subdivision of squares into smaller squares.<ref>{{cite book
 | last = Samet | first = Hanan | author-link = Hanan Samet
 | contribution = 1.4 Quadtrees
 | contribution-url = https://books.google.com/books?id=vO-NRRKHG84C&pg=PA28
 | isbn = 9780123694461
 | pages = 28–48
 | publisher = Morgan Kaufmann
 | title = Foundations of Multidimensional and Metric Data Structures
 | year = 2006}}</ref>

[[File:20240815 Site of Luoyang City from Han to Wei Dynasty - Site of the Pagoda of Yongning Temple 04.jpg|thumb|Site of the [[Yongning Pagoda]]]]
Architectural structures from both ancient and modern cultures have featured a square floor plan, base, or footprint. Ancient examples include the [[Egyptian pyramids]],<ref>{{cite journal
 | last = Vafea | first = Flora
 | issue = 1
 | journal = Mediterranean Archaeology and Archaeometry
 | pages = 111–125
 | title = The mathematics of pyramid construction in ancient Egypt
 | url = https://www.maajournal.com/index.php/maa/article/view/36
 | volume = 2
 | year = 2002}}</ref> [[Mesoamerican pyramids]] such as those at [[Teotihuacan]],<ref>{{cite journal
 | last = Sugiyama | first = Saburo
 | date = June 1993
 | doi = 10.2307/971798
 | issue = 2
 | journal = Latin American Antiquity
 | pages = 103–129
 | title = Worldview materialized in Teotihuacan, Mexico
 | volume = 4| jstor = 971798
 }}</ref> the [[Chogha Zanbil]] ziggurat in Iran,<ref>{{cite journal
 | last = Ghirshman | first = Roman | author-link = Roman Ghirshman
 | date = January 1961
 | issue = 1
 | journal = [[Scientific American]]
 | jstor = 24940741
 | pages = 68–77
 | title = The Ziggurat of Tchoga-Zanbil
 | volume = 204| doi = 10.1038/scientificamerican0161-68 | bibcode = 1961SciAm.204a..68G }}</ref> the four-fold design of Persian walled gardens, said to model the four rivers of Paradise,
and later structures inspired by their design such as the [[Taj Mahal]] in India,<ref>{{cite journal
 | last1 = Stiny | first1 = G
 | last2 = Mitchell | first2 = W J
 | doi = 10.1068/b070209
 | issue = 2
 | journal = Environment and Planning B: Planning and Design
 | pages = 209–226
 | title = The grammar of paradise: on the generation of Mughul gardens
 | url = https://www.andrew.cmu.edu/user/ramesh/teaching/course/48-747/subFrames/readings/Stiny&MItchell-1980-EPB7_209-226.TheGrammarOfParadise..pdf
 | volume = 7
 | year = 1980| bibcode = 1980EnPlB...7..209S
 }}</ref> the square bases of Buddhist [[stupa]]s,<ref>{{cite journal
 | last1 = Nakamura | first1 = Yuuka
 | last2 = Okazaki | first2 = Shigeyuki
 | journal = International Understanding
 | pages = 31–43
 | title = The Spatial Composition of Buddhist Temples in Central Asia, Part 1: The Transformation of Stupas
 | url = https://itcs.mukogawa-u.ac.jp/publications/IU_vol6/pdf/IU_vol.6_31-43.pdf
 | volume = 6
 | year = 2016}}</ref> and East Asian [[pagoda]]s, buildings that symbolically face to the four points of the compass and reach to the heavens.<ref>{{cite journal
 | last = Guo | first = Qinghua
 | journal = Construction History
 | jstor = 41613875
 | pages = 3–19
 | title = From tower to pagoda: structural and technological transition
 | volume = 20
 | year = 2004}}</ref> Norman [[keep]]s such as the [[Tower of London]] often take the form of a low square tower.<ref>{{cite journal
 | last = Bruce | first = J. Collingwood
 | date = October 1850
 | doi = 10.1080/00681288.1850.11886925
 | issue = 3
 | journal = Journal of the British Archaeological Association
 | pages = 209–228
 | publisher = Informa UK Limited
 | title = On the structure of the Norman Fortress in England
 | url = https://archive.org/details/journalbritishar06brituoft/page/208
 | volume = 6}} See [https://archive.org/details/journalbritishar06brituoft/page/212 p. 213].</ref> In modern architecture, a majority of [[skyscraper]]s feature a square plan for pragmatic rather than aesthetic or symbolic reasons.<ref>{{cite thesis
 | last = Choi | first = Yongsun
 | id = {{ProQuest|304600838}}
 | publisher = Illinois Institute of Technology
 | title = A Study on Planning and Development of Tall Building: The Exploration of Planning Considerations
 | type = Ph.D. thesis
 | year = 2000}} See in particular pp. 88–90</ref>

{{multiple image
|image1=Mandalatibet (cropped).jpg|caption1=A Tibetan [[mandala]]
|image2=Piet Mondrian, 1942 - Broadway Boogie Woogie.jpg|caption2=''[[Broadway Boogie Woogie]]'', [[Piet Mondrian]]
|total_width=360}}
The stylized nested squares of a Tibetan [[mandala]], like the design of a stupa, function as a miniature model of the cosmos.<ref>{{cite journal
 | last = Xu | first = Ping
 | date = Fall 2010
 | issue = 3
 | journal = Journal of Architectural and Planning Research
 | jstor = 43030905
 | pages = 181–203
 | title = The mandala as a cosmic model used to systematically structure the Tibetan Buddhist landscape
 | volume = 27}}</ref> Some formats for film photography use a square [[Aspect ratio (image)|aspect ratio]], notably [[Polaroid camera]]s, [[medium format]] cameras, and [[Instamatic]] cameras.<ref>{{cite journal
 | last = Chester | first = Alicia
 | date = September 2018
 | doi = 10.1525/aft.2018.45.5.10
 | issue = 5
 | journal = Afterimage
 | pages = 10–15
 | publisher = University of California Press
 | title = The outmoded instant: From Instagram to Polaroid
 | volume = 45}}</ref><ref>{{cite book
 | last = Adams | first = Ansel |author-link = Ansel Adams
 | year = 1980
 | title = The Camera
 | place = Boston | publisher = New York Graphic Society
 | chapter = Medium-Format Cameras |at=Ch. 3, {{pgs|21–28}}
}}</ref> Painters known for their frequent use of square frames and forms include [[Josef Albers]],<ref>{{cite conference
 | last = Mai | first = James
 | editor1-last = Torrence | editor1-first = Eve | editor1-link = Eve Torrence
 | editor2-last = Torrence | editor2-first = Bruce
 | editor3-last = Séquin | editor3-first = Carlo | editor3-link = Carlo Séquin
 | editor4-last = McKenna | editor4-first = Douglas
 | editor5-last = Fenyvesi | editor5-first = Kristóf
 | editor6-last = Sarhangi | editor6-first = Reza
 | contribution = Planes and frames: spatial layering in Josef Albers' ''Homage to the Square'' paintings
 | contribution-url = https://archive.bridgesmathart.org/2016/bridges2016-233.html
 | isbn = 978-1-938664-19-9
 | location = Phoenix, Arizona
 | pages = 233–240
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture
 | year = 2016}}</ref> [[Kazimir Malevich]]<ref>{{cite journal
 | last = Luecking | first = Stephen
 | date = June 2010
 | doi = 10.1080/17513471003744395
 | issue = 2
 | journal = Journal of Mathematics and the Arts
 | pages = 87–100
 | title = A man and his square: Kasimir Malevich and the visualization of the fourth dimension
 | volume = 4}}</ref> and [[Piet Mondrian]].<ref>{{cite journal
 | last = Millard | first = Charles W.
 | date = Summer 1972
 | issue = 2
 | journal = The Hudson Review
 | jstor = 3849001
 | pages = 270–274
 | title = Mondrian
 | volume = 25| doi = 10.2307/3849001
 }}</ref>

[[Baseball diamond]]s<ref>{{cite journal
 | last = Battista | first = Michael T.
 | date = April 1993
 | doi = 10.5951/mt.86.4.0336
 | issue = 4
 | journal = The Mathematics Teacher
 | jstor = 27968332
 | pages = 336–342
 | title = Mathematics in Baseball
 | volume = 86}} See p. 339.</ref> and [[boxing ring]]s are square despite being named for other shapes.<ref>{{cite book
 | last = Chetwynd | first = Josh
 | isbn = 9781607748113
 | quote = The decision to go oxymoron with a squared "ring" had taken place by the late 1830s ... Despite the geometric shift, the language was set.
 | page = 122
 | publisher = Ten Speed Press
 | title = The Field Guide to Sports Metaphors: A Compendium of Competitive Words and Idioms
 | url = https://books.google.com/books?id=xmPhCwAAQBAJ&pg=PA122
 | year = 2016}}</ref> In the [[quadrille]] and [[square dance]], four couples form the sides of a square.<ref>{{cite journal
 | last1 = Sciarappa | first1 = Luke
 | last2 = Henle | first2= Jim
 | year = 2022
 | title = Square Dance from a Mathematical Perspective
 | journal = The Mathematical Intelligencer
 | volume = 44 | number = 1
 | pages = 58–64
 | doi = 10.1007/s00283-021-10151-0
| pmid = 35250151
 | pmc = 8889875
 }}</ref> In [[Samuel Beckett]]'s minimalist television play ''[[Quad (play)|Quad]]'', four actors walk along the sides and diagonals of a square.<ref>{{cite book
 | last = Worthen | first = William B. 
 | year = 2010
 | title = Drama: Between Poetry and Performance
 | chapter = Quad: Euclidean Dramaturgies
 | at = Ch. 4.i, {{pgs|196–204}}
 | publisher = Wiley
 | isbn = 978-1-405-15342-3
 | chapter-url = https://www.textures-archiv.geisteswissenschaften.fu-berlin.de/wp-content/uploads/2010/08/worthen_bill_2010_08.pdf
 }}</ref>

{{multiple image
|image1=Chess and goose game board MET ES4614.jpg|caption1=16th-century Indian [[chessboard]]
|image2=Stomachion.JPG|caption2=[[Ostomachion]]
|image3=Horoskop Johannette Maria zu Wied 1615 img01.jpg|caption3=1615 [[horoscope]]
|total_width=480}}
The square [[go board]] is said to represent the earth, with the 361 crossings of its lines representing days of the year.<ref>{{cite book
 | last1 = Lang | first1 = Ye
 | last2 = Liangzhi | first2 = Zhu
 | contribution = ''Weiqi'': A Game of Wits
 | doi = 10.1007/978-981-97-4511-1_38
 | isbn = 9789819745111
 | pages = 469–476
 | publisher = Springer Nature Singapore
 | title = Insights into Chinese Culture
 | year = 2024}} See page 472.</ref> The [[chessboard]] inherited its square shape from a [[pachisi]]-like Indian race game and in turn passed it on to [[checkers]].<ref>{{cite journal
 | last = Newman | first = James R.
 | date = August 1961
 | issue = 2
 | journal = [[Scientific American]]
 | jstor = 24937045
 | pages = 155–161
 | title = About the rich lore of games played on boards and tables (review of ''Board and Table Games From Many Civilizations'' by R. C. Bell)
 | volume = 205| doi = 10.1038/scientificamerican0861-155
 }}</ref> In two ancient games from [[Mesopotamia]] and [[Ancient Egypt]], the [[Royal Game of Ur]] and [[Senet]], the game board itself is not square, but rectangular, subdivided into a grid of squares.<ref>{{cite book
 | last = Donovan | first = Tristan
 | isbn = 9781250082725
 | pages = 10–14
 | publisher = St. Martin's
 | title = It's All a Game: The History of Board Games from Monopoly to Settlers of Catan
 | url = https://books.google.com/books?id=PAyrDgAAQBAJ&pg=PA10
 | year = 2017}}</ref> The ancient Greek [[Ostomachion]] puzzle (according to some interpretations) involves rearranging the pieces of a square cut into smaller polygons, as does the Chinese [[tangram]].<ref>{{cite journal
 | last = Klarreich | first = Erica | author-link = Erica Klarreich
 | date = May 15, 2004
 | journal = Science News
 | pages = 314–315
 | title = Glimpses of Genius: mathematicians and historians piece together a puzzle that Archimedes pondered
 | doi = 10.2307/4015223 | jstor = 4015223 | url = https://www.sciencenews.org/article/glimpses-genius}}</ref> Another set of puzzle pieces, the [[polyomino]]s, are formed from squares glued edge-to-edge.<ref>{{cite book
 | last = Golomb | first = Solomon W.
 | edition = 2nd
 | isbn = 0-691-08573-0
 | mr = 1291821
 | publisher = Princeton University Press
 | title = Polyominoes: Puzzles, Patterns, Problems, and Packings
 | year = 1994}}</ref> Medieval and Renaissance [[horoscope]]s were arranged in a square format, across Europe, the Middle East, and China.<ref>{{cite book
 | last = Thomann | first = Johannes
 | editor1-last = Forêt | editor1-first = Philippe
 | editor2-last = Kaplony | editor2-first = Andreas
 | contribution = Chapter Five: Square Horoscope Diagrams In Middle Eastern Astrology And Chinese Cosmological Diagrams: Were These Designs Transmitted Through The Silk Road?
 | doi = 10.1163/ej.9789004171657.i-248.45
 | isbn = 9789004171657
 | pages = 97–118
 | publisher = BRILL
 | series = Brill's Inner Asian Library
 | title = The Journey of Maps and Images on the Silk Road
 | volume = 21
 | year = 2008}}</ref> Other recreational uses of squares include the shape of [[origami]] paper,<ref>{{cite journal
 | last = Cipra | first = Barry A.
 | issue = 8
 | journal = SIAM News
 | title = In the Fold: Origami Meets Mathematics
 | url = https://mpadocuments.s3.amazonaws.com/origami/InTheFold.pdf
 | volume = 34}}</ref> and a common style of [[quilting]] involving the use of square quilt blocks.<ref>{{cite journal
 | last = Wickstrom | first = Megan H.
 | date = November 2014
 | doi = 10.5951/teacchilmath.21.4.0220
 | issue = 4
 | journal = Teaching Children Mathematics
 | jstor = 10.5951/teacchilmath.21.4.0220
 | pages = 220–227
 | title = Piecing it together
 | volume = 21}}</ref>

{{multiple image
|image1=CHE Vuadens Flag.svg
|caption1=Square flag of the municipality of [[Vuadens]], based on the Swiss flag
|image2=QR code for mobile English Wikipedia.svg
|caption2=[[QR code]] for the mobile English Wikipedia
|image3=Waffles (1).jpg
|caption3=Square [[waffle]]s
|total_width=480}}
Squares are a common element of [[graphic design]], used to give a sense of stability, symmetry, and order.<ref>{{cite book|title=Everything Graphic Design: A Comprehensive Understanding of Visual Communications for Beginners & Creatives|first=Jeff|last=Nyamweya|publisher=Bogano|year=2024|isbn=9789914371413|page=78|url=https://books.google.com/books?id=3404EQAAQBAJ&pg=PA76}}</ref> In [[heraldry]], a [[Canton (heraldry)|canton]] (a design element in the top left of a shield) is normally square, and a square flag is called a banner.<ref>{{cite book
 | last = Boutell | first = Charles
 | edition = 2nd
 | location = London
 | page = [https://archive.org/details/heraldryhistori00boutgoog/page/n58 31], [https://archive.org/details/heraldryhistori00boutgoog/page/88 89]
 | publisher = Bentley
 | title = Heraldry, Historical and Popular
 | year = 1864}}</ref> The [[flag of Switzerland]] is square, as are the [[Flags and arms of cantons of Switzerland|flags of the Swiss cantons]].<ref>{{cite book
 | edition = 7th
 | pages = 200–206
 | publisher = DK Penguin Random House
 | title = Complete Flags of the World: The Ultimate Pocket Guide
 | url = https://books.google.com/books?id=aIEvEQAAQBAJ&pg=PA200
 | year = 2021| isbn = 978-0-7440-6001-0
 }}</ref> [[QR code]]s are square and feature prominent nested square alignment marks in three corners.<ref>{{cite book
 | last1 = Kan | first1 = Tai-Wei
 | last2 = Teng | first2 = Chin-Hung
 | last3 = Chen | first3 = Mike Y.
 | editor-last = Furht | editor-first = Borko
 | contribution = QR code based augmented reality applications
 | doi = 10.1007/978-1-4614-0064-6_16
 | isbn = 9781461400646
 | pages = 339–354
 | publisher = Springer
 | title = Handbook of Augmented Reality
 | year = 2011}} See especially [https://books.google.com/books?id=fG8JUdrScsYC&pg=PA341 Section 2.1, Appearance, pp. 341–342].</ref> [[Robertson screw]]s have a square drive socket.<ref>{{cite book
 | last = Rybczynski | first = Witold
 | pages = 80–83
 | publisher = Scribner
 | title = One Good Turn: A Natural History of the Screwdriver and the Screw
 | url = https://books.google.com/books?id=nv9L_FxyhuIC&pg=PA80
 | year = 2000| isbn = 978-0-684-86730-4
 }}</ref> [[Cracker (food)|Crackers]] and sliced [[cheese]] are often square,<ref>{{cite book|title=Math and Science for Young Children|first1=Rosalind|last1=Charlesworth|first2=Karen|last2=Lind|publisher=Delmar Publishers|year=1990|isbn=9780827334021|page=195}}</ref> as are [[waffle]]s.<ref>{{cite book|title=Waffles: Sweet, Savory, Simple|first=Dawn|last=Yanagihara|publisher=Chronicle Books|year=2014|isbn=9781452138411|page=11|url=https://books.google.com/books?id=vkr0AgAAQBAJ&pg=PA11}}</ref><ref>{{cite book|title=Street Food around the World: An Encyclopedia of Food and Culture|first1=Bruce|last1=Kraig|first2=Colleen Taylor|last2=Sen|publisher=Bloomsbury Publishing USA|year=2013|isbn=9781598849554|page=50|url=https://books.google.com/books?id=VhXHEAAAQBAJ&pg=PA50}}</ref> Square foods named for their square shapes include [[caramel square]]s, [[date square]]s, [[lemon square]]s,<ref>{{cite book|title=Fat-Back and Molasses|first=Ivan F.|last=Jesperson|publisher=Breakwater Books|year=1989|isbn=9780920502044}} Caramel squares and date squares, [https://books.google.com/books?id=xWEjlBCsSaMC&pg=PA134 p. 134]; lemon squares, [https://books.google.com/books?id=xWEjlBCsSaMC&pg=PA104 p. 104].</ref> [[square sausage]],<ref>{{cite book|title=Sausage: A Global History|first=Gary|last=Allen|publisher=Reaktion Books|year=2015|isbn=9781780235554|page=57|url=https://books.google.com/books?id=nz0pCgAAQBAJ&pg=PA57}}</ref> and [[Carré de l'Est]] cheese.<ref>{{cite book|title=World Cheese Book|first=Juliet|last=Harbutt|publisher=Penguin|year=2015|isbn=9781465443724|page=45|url=https://books.google.com/books?id=kMvlBwAAQBAJ&pg=PA45}}</ref>

{{-}}

==Constructions==
===Coordinates and equations===
[[File:Square equation plot.svg|thumb|right|<math>|x| + |y| = 2</math> plotted on ''[[Cartesian coordinates]]''.]]
A [[unit square]] is a square of side length one. Often it is represented in [[Cartesian coordinate]]s as the square enclosing the points <math>(x,y)</math> that have <math>0\le x\le 1</math> and <math>0\le y\le 1</math>. Its vertices are the four points that have 0 or 1 in each of their coordinates.<ref>{{cite book
 | last1 = Rosenthal | first1 = Daniel
 | last2 = Rosenthal | first2 = David
 | last3 = Rosenthal | first3 = Peter
 | doi = 10.1007/978-3-030-00632-7
 | edition = 2nd
 | isbn = 9783030006327
 | series = Undergraduate Texts in Mathematics
 | page = 108
 | publisher = Springer International Publishing
 | title = A Readable Introduction to Real Mathematics
 | url = https://books.google.com/books?id=JQGQDwAAQBAJ&pg=PA108
 | year = 2018}}</ref>

An axis-parallel square with its center at the point <math>(x_c,y_c)</math> and sides of length <math>2r</math> (where <math>r</math> is the inradius, half the side length) has vertices at the four points <math>(x_c\pm r,y_c\pm r)</math>. Its interior consists of the points <math>(x,y)</math> with <math>\max(|x-x_c|,|y-y_c|) < r</math>, and its boundary consists of the points with <math>\max(|x-x_c|,|y-y_c|)=r</math>.<ref name=iobst>{{cite journal
 | last = Iobst | first = Christopher Simon
 | issue = 1
 | journal = Ohio Journal of School Mathematics
 | pages = 27–31
 | title = Shapes and Their Equations: Experimentation with Desmos
 | date = 14 June 2018
 | url = https://ojs.library.osu.edu/index.php/OJSM/article/view/6367
 | volume = 79}}</ref>

A diagonal square with its center at the point <math>(x_c,y_c)</math> and diagonal of length <math>2R</math> (where <math>R</math> is the circumradius, half the diagonal) has vertices at the four points <math>(x_c\pm R,y_c)</math> and <math>(x_c,y_c\pm R)</math>. Its interior consists of the points <math>(x,y)</math> with <math>|x-x_c|+|y-y_c|<R</math>, and its boundary consists of the points with <math>|x-x_c|+|y-y_c|=R</math>.<ref name=iobst/> For instance the illustration shows a diagonal square centered at the origin <math>(0,0)</math> with circumradius 2, given by the equation <math>|x|+|y|=2</math>.

[[File:A square with Gaussian integer vertices.png|thumb|A square formed by multiplying the complex number {{mvar|p}} by powers of {{mvar|i}}, and its translation obtained by adding another complex number {{mvar|c}}. The background grid shows the [[Gaussian integer]]s.]]
In the [[Complex plane|plane of complex numbers]], multiplication by the [[imaginary unit]] <math>i</math> rotates the other term in the product by 90° around the origin (the number zero). Therefore, if any nonzero complex number <math>p</math> is repeatedly multiplied by <math>i</math>, giving the four numbers <math>p</math>, <math>ip</math>, <math>-p</math>, and <math>-ip</math>, these numbers will form the vertices of a square centered at the origin.<ref>{{citation
 | last = Vince | first = John
 | doi = 10.1007/978-0-85729-154-7
 | isbn = 9780857291547
 | page = 11
 | publisher = Springer | location = London
 | title = Rotation Transforms for Computer Graphics
 | year = 2011| bibcode = 2011rtfc.book.....V
 }}</ref> If one interprets the [[real part]] and [[imaginary part]] of these four complex numbers as Cartesian coordinates, with <math>p=x+iy</math>, then these four numbers have the coordinates <math>(x,y)</math>, <math>(-y,x)</math>, <math>(-x,-y)</math>, and <math>(-y,-x)</math>.<ref>{{cite book
 | last = Nahin | first = Paul
 | isbn = 9781400833894
 | page = [https://books.google.com/books?id=OPyPwaElDvUC&pg=PA54 54]
 | publisher = Princeton University Press
 | title = An Imaginary Tale: The Story of <math>\sqrt{-1}</math>
 | year = 2010}}</ref> This square can be translated to have any other complex number <math>c</math> is center, using the fact that the [[translation (geometry)|translation]] from the origin to <math>c</math> is represented in complex number arithmetic as addition with <math>c</math>.<ref name=numberverse/> The [[Gaussian integer]]s, complex numbers with integer real and imaginary parts, form a [[square lattice]] in the complex plane.<ref name=numberverse>{{cite book
 | last1 = McLeman | first1 = Cam
 | last2 = McNicholas | first2 = Erin
 | last3 = Starr | first3 = Colin
 | doi = 10.1007/978-3-030-98931-6
 | isbn = 9783030989316
 | series = Undergraduate Texts in Mathematics
 | page = [https://books.google.com/books?id=G7OiEAAAQBAJ&pg=PA7 7]
 | publisher = Springer International Publishing
 | title = Explorations in Number Theory: Commuting through the Numberverse
 | year = 2022}}</ref>

===Compass and straightedge===
The construction of a square with a given side, using a [[Compass and straightedge constructions|compass and straightedge]], is given in [[Euclid's Elements|Euclid's ''Elements'']] I.46.<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book I, Proposition 46. [http://aleph0.clarku.edu/~djoyce/elements/bookI/propI46.html Online English version] by [[David E. Joyce]].</ref> The existence of this construction means that squares are [[constructible polygon]]s. A regular {{nowrap|<math>n</math>-gon}} is constructible exactly when the odd [[prime factor]]s of <math>n</math> are distinct [[Fermat prime]]s,<ref>{{cite book |last= Martin |first= George E. |date= 1998 |title= Geometric Constructions |title-link= Geometric Constructions |publisher= Springer-Verlag, New York |series= Undergraduate Texts in Mathematics |isbn= 0-387-98276-0|page=46}}</ref> and in the case of a square <math>n=4</math> has no odd prime factors so this condition is [[vacuously true]].

''Elements'' IV.6–7 also give constructions for a square inscribed in a circle and circumscribed about a circle, respectively.<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book IV, [http://aleph0.clarku.edu/~djoyce/elements/bookIV/propIV6.html Proposition 6], [http://aleph0.clarku.edu/~djoyce/elements/bookIV/propIV7.html Proposition 7]. Online English version by [[David E. Joyce]].</ref>
<gallery widths=180 heights=180>
Straight Square Inscribed in a Circle 240px.gif|Square with a given circumcircle
01-Quadrat-Seite-gegeben.gif|Square with a given side length, using [[Thales' theorem]]
01-Quadrat-Diagonale-gegeben.gif|Square with a given diagonal
</gallery>

==Related topics==
{{multiple image
|image1=Dual Cube-Octahedron.svg|caption1=The [[cube]] and [[regular octahedron]], next steps in sequences of [[regular polytope]]s starting with squares
|image2=Sierpinski carpet 6, white on black.svg|caption2=The [[Sierpiński carpet]], a square [[fractal]] with square holes
|image3=Ising-tartan.png|caption3=An invariant measure for the [[baker's map]]
|total_width=480}}
The [[Schläfli symbol]] of a square is {4}.<ref>{{cite book 
 | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter
 | title = Regular Polytopes | title-link = Regular Polytopes (book)
 | publisher = Methuen and Co.
 | year = 1948
 | page = 2}}</ref> A [[truncation (geometry)|truncated]] square is an [[octagon]].{{sfnp|Coxeter|1948|p=148}} The square belongs to a family of [[regular polytope]]s that includes the [[cube]] in three dimensions and the [[hypercube]]s in higher dimensions,{{sfnp|Coxeter|1948|pp=122–123}} and to another family that includes the [[regular octahedron]] in three dimensions and the [[cross-polytope]]s in higher dimensions.{{sfnp|Coxeter|1948|pp=121–122}} The cube and hypercubes can be given vertex coordinates that are all <math>\pm 1</math>, giving an axis-parallel square in two dimensions, while the octahedron and cross-polytopes have one coordinate <math>\pm 1</math> and the rest zero, giving a diagonal square in two dimensions.{{sfnp|Coxeter|1948|pp=122, 126}} As with squares, the [[hyperoctahedral group|symmetries of these shapes]] can be obtained by applying a [[signed permutation]] to their coordinates.<ref name=ers/>

The [[Sierpiński carpet]] is a square [[fractal]], with square holes.<ref>{{cite book
 | last1 = Barker | first1 = William
 | last2 = Howe | first2 = Roger | author2-link = Roger Evans Howe
 | doi = 10.1090/mbk/047
 | isbn = 978-0-8218-3900-3
 | mr = 2362745
 | page = [https://books.google.com/books?id=88EjDwAAQBAJ&pg=PA528 528]
 | publisher = American Mathematical Society | location = Providence, Rhode Island
 | title = Continuous Symmetry: From Euclid to Klein
 | year = 2007}}</ref> [[Space-filling curve]]s including the [[Hilbert curve]], [[Peano curve]], and [[Sierpiński curve]] cover a square as the continuous image of a line segment.<ref>{{cite book
 | last = Sagan | first = Hans
 | doi = 10.1007/978-1-4612-0871-6
 | isbn = 0-387-94265-3
 | mr = 1299533
 | publisher = Springer-Verlag | location = New York
 | series = Universitext
 | title = Space-Filling Curves
 | year = 1994}} For the Hilbert curve, see p. 10; for the Peano curve, see p. 35; for the Sierpiński curve, see p. 51.</ref> The [[Z-order curve]] is analogous but not continuous.<ref>{{cite journal
 | last1 = Burstedde | first1 = Carsten
 | last2 = Holke | first2 = Johannes
 | last3 = Isaac | first3 = Tobin
 | arxiv = 1505.05055
 | doi = 10.1007/s10208-018-9400-5
 | issue = 4
 | journal = [[Foundations of Computational Mathematics]]
 | mr = 3989715
 | pages = 843–868
 | title = On the number of face-connected components of Morton-type space-filling curves
 | volume = 19
 | year = 2019}}</ref> Other mathematical functions associated with squares include [[Arnold's cat map]] and the [[baker's map]], which generate chaotic [[dynamical system]]s on a square,<ref>{{cite book
 | last = Ott | first = Edward | author-link = Edward Ott
 | contribution = 7.5 Strongly chaotic systems
 | contribution-url = https://books.google.com/books?id=PfXoAwAAQBAJ&pg=PA296
 | edition = 2nd
 | isbn = 9781139936576
 | page = 296
 | publisher = Cambridge University Press
 | title = Chaos in Dynamical Systems
 | year = 2002}}
</ref> and the [[lemniscate elliptic functions]], complex functions periodic on a square grid.<ref>{{cite book
 | last = Vlăduț | first = Serge G.
 | contribution = 2.2 Elliptic functions
 | contribution-url = https://books.google.com/books?id=YLcPxfZW47EC&pg=PA20
 | isbn = 2-88124-754-7
 | location = New York
 | mr = 1121266
 | page = 20
 | publisher = Gordon and Breach Science Publishers
 | series = Studies in the Development of Modern Mathematics
 | title = Kronecker's Jugendtraum and Modular Functions
 | volume = 2
 | year = 1991}}</ref>
{{-}}

===Inscribed squares===
[[File:Calabi triangle.svg|thumb|The [[Calabi triangle]] and the three placements of its largest square.{{sfnp|Conway|Guy|1996|p=[https://books.google.com/books?id=0--3rcO7dMYC&pg=PA206 206]}} The placement on the long side of the triangle is inscribed; the other two are not.]]
{{Main|Inscribed square problem|Inscribed square in a triangle}}
A square is [[inscribed figure|inscribed]] in a curve when all four vertices of the square lie on the curve. The unsolved [[inscribed square problem]] asks whether every [[simple closed curve]] has an inscribed square. It is true for every [[smooth curve]],<ref>{{cite journal |last=Matschke |first=Benjamin |year=2014 |title=A survey on the square peg problem |journal=[[Notices of the American Mathematical Society]] |doi=10.1090/noti1100 |volume=61 |issue=4 |pages=346–352|doi-access=free |hdl=21.11116/0000-0004-15B8-5 |hdl-access=free }}</ref> and for any closed [[convex curve]]. The only other regular polygon that can always be inscribed in every closed convex curve is the [[equilateral triangle]], as there exists a convex curve on which no other regular polygon can be inscribed.<ref>{{cite journal
 | last = Eggleston | first = H. G.
 | doi = 10.1080/00029890.1958.11989144
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2308878
 | mr = 97768
 | pages = 76–80
 | title = Figures inscribed in convex sets
 | volume = 65
 | year = 1958| issue = 2
 }}</ref>

For an [[inscribed square in a triangle]], at least one side of the square lies on a side of the triangle. Every [[acute triangle]] has three inscribed squares, one for each of its three sides. A [[right triangle]] has two inscribed squares, one touching its right angle and the other lying on its hypotenuse. An [[obtuse triangle]] has only one inscribed square, on its longest. A square inscribed in a triangle can cover at most half the triangle's area.<ref name=gardner>{{cite journal
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | date = September 1997
 | doi = 10.1080/10724117.1997.11975023
 | issue = 1
 | journal = [[Math Horizons]]
 | pages = 18–22
 | title = Some surprising theorems about rectangles in triangles
 | volume = 5}}</ref>
{{-}}

===Area and quadrature <span class="anchor" id="Squaring the circle"></span>===
{{also|Area|Quadrature (geometry)|Squaring the circle}}
[[File:Pythagorean.svg|thumb|upright=0.8|The [[Pythagorean theorem]]: the two smaller squares on the sides of a right triangle have equal total area to the larger square on the hypotenuse.]]
[[File:Squaring the Circle J.svg|thumb|A circle and square with the same area|upright=0.6]]
Since ancient times, many units for surface [[area]] have been defined from squares, typically with a standard unit of [[length]] as its side, for example a [[square meter]] or [[square inch]]. The area of an arbitrary rectangle can then be simply computed as the product of its length and its width, and more complicated shapes can be measured by conceptually breaking them up into unit squares or into arbitrary triangles.

In [[Euclidean geometry|ancient Greek deductive geometry]], the area of a planar shape was measured and compared by constructing a square with the same area by using only a finite number of steps with [[compass and straightedge]], a process called ''[[Quadrature (geometry)|quadrature]]'' or ''squaring''. [[Euclid's Elements|Euclid's ''Elements'']] shows how to do this for rectangles, parallelograms, triangles, and then more generally for [[simple polygon]]s by breaking them into triangular pieces.<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book II, Proposition 14. [http://aleph0.clarku.edu/~djoyce/elements/bookII/propII14.html Online English version] by [[David E. Joyce]].</ref> Some shapes with curved sides could also be squared, such as the [[lune of Hippocrates]]<ref>{{cite journal|title=The problem of squarable lunes|journal=[[The American Mathematical Monthly]]|volume=107|issue=7|year=2000|pages=645–651|jstor=2589121|first=M. M.|last=Postnikov|author-link=Mikhail Postnikov|doi=10.2307/2589121}}</ref> and the [[Quadrature of the Parabola|parabola]].<ref>{{cite journal
 | last = Berendonk | first = Stephan
 | doi = 10.1007/s00591-016-0173-0
 | issue = 1
 | journal = Mathematische Semesterberichte
 | mr = 3629442
 | pages = 1–13
 | title = Ways to square the parabola—a commented picture gallery
 | volume = 64
 | year = 2017}}</ref>

This use of a square as the defining shape for area measurement also occurs in the Greek formulation of the [[Pythagorean theorem]]: squares constructed on the two sides of a [[right triangle]] have equal total area to a square constructed on the [[hypotenuse]].<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book I, Proposition 47. [http://aleph0.clarku.edu/~djoyce/elements/bookI/propI47.html Online English version] by [[David E. Joyce]].</ref> Stated in this form, the theorem would be equally valid for other shapes on the sides of the triangle, such as equilateral triangles or semicircles,<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book VI, Proposition 31. [http://aleph0.clarku.edu/~djoyce/elements/bookVI/propVI31.html Online English version] by [[David E. Joyce]].</ref> but the Greeks used squares. In modern mathematics, this formulation of the theorem using areas of squares has been replaced by an algebraic formulation involving [[Square (algebra)|squaring numbers]]: the lengths of the sides and hypotenuse of the right triangle obey the equation <math>a^2+b^2=c^2</math>.<ref>{{cite book|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|page=xi|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|year=2019}}</ref>

Because of this focus on quadrature as a measure of area, the Greeks and later mathematicians sought unsuccessfully to [[squaring the circle|square the circle]], constructing a square with the same area as a given circle, again using finitely many steps with a compass and straightedge. In 1882, the task was proven to be impossible as a consequence of the [[Lindemann–Weierstrass theorem]]. This theorem proves that [[pi]] ({{pi}}) is a [[transcendental number]] rather than an [[algebraic number|algebraic irrational number]]; that is, it is not the [[root of a function|root]] of any [[polynomial]] with [[rational number|rational]] coefficients. A construction for squaring the circle could be translated into a polynomial formula for {{pi}}, which does not exist.<ref>{{cite journal
 | last = Kasner | first = Edward | author-link = Edward Kasner
 | date = July 1933
 | issue = 1
 | journal = [[The Scientific Monthly]]
 | jstor = 15685
 | pages = 67–71
 | title = Squaring the circle
 | volume = 37}}</ref>
{{-}}

===Tiling and packing===
{{main|Square tiling|Square packing|Circle packing in a square|Squaring the square}}
{{multiple image
|image1=Tiling Regular 4-4 Square.svg|caption1=[[Square tiling]]
|image2=Academ Periodic tiling by squares of two different sizes.svg|caption2=[[Pythagorean tiling]]
|total-width=240}}
The [[square tiling]], familiar from flooring and game boards, is one of three [[Tiling by regular polygons|regular tilings]] of the plane. The other two use the [[equilateral triangle]] and the [[regular hexagon]].<ref>{{cite book
 | last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum
 | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard
 | publisher = W. H. Freeman
 | title = Tilings and Patterns
 | title-link = Tilings and patterns
 | year = 1987
 | contribution = Figure 1.2.1
 | page = 21}}</ref> The vertices of a square tiling form a [[square lattice]].{{sfnp|Grünbaum|Shephard|1987|p=29}} Squares of more than one size can also tile the plane,{{sfnp|Grünbaum|Shephard|1987|pp=76–78}}<ref>{{cite book
 | last = Fisher | first = Gwen L.
 | year = 2003
 | contribution = Quilt Designs Using Non-Edge-to-Edge Tilings by Squares
 | pages = 265–272
 | title = Meeting Alhambra: ISAMA-BRIDGES Conference Proceedings
 | contribution-url = https://archive.bridgesmathart.org/2003/bridges2003-265.html
}}</ref> for instance in the [[Pythagorean tiling]], named for its connection to proofs of the [[Pythagorean theorem]].<ref>{{cite journal|title=Paintings, plane tilings, and proofs|first=Roger B.|last=Nelsen|journal=[[Math Horizons]]|date=November 2003|volume=11|issue=2|pages=5–8|doi=10.1080/10724117.2003.12021741|s2cid=126000048|url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/New%20Problems/paintings.pdf}}</ref>

[[File:Packing 11 unit squares in a square with side length 3.87708359....svg|thumb|upright=0.8|The smallest known square that can contain 11 unit squares has side length approximately 3.877084.<ref name=friedman/>]]
[[Square packing]] problems seek the smallest square or circle into which a given number of [[unit square]]s can fit. A chessboard optimally packs a square number of unit squares into a larger square, but beyond a few special cases such as this, the optimal solutions to these problems remain unsolved;<ref name=friedman>{{cite journal
 | last = Friedman
 | first = Erich
 | year = 2009
 | title = Packing unit squares in squares: a survey and new results
 | journal = [[Electronic Journal of Combinatorics]]
 | volume = 1000
 | doi = 10.37236/28
 | at = Dynamic Survey 7
 | mr = 1668055
 | url = http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS7
 | doi-access = free
 | access-date = 2018-02-23
 | archive-date = 2018-02-24
 | archive-url = https://web.archive.org/web/20180224053022/http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS7
 | url-status = live
 }}</ref><ref>{{cite journal
 | last1 = Chung | first1 = Fan | author1-link = Fan Chung
 | last2 = Graham | first2 = Ron | author2-link = Ronald Graham
 | year = 2020
 | title = Efficient packings of unit squares in a large square
 | journal = [[Discrete & Computational Geometry]]
 | doi = 10.1007/s00454-019-00088-9
 | volume = 64
 | issue = 3
 | pages = 690–699
 | url = https://www.math.ucsd.edu/~fan/wp/spacking.pdf
}}</ref><ref>{{cite journal
 | last1 = Montanher | first1 = Tiago
 | last2 = Neumaier | first2 = Arnold
 | last3 = Markót | first3 = Mihály Csaba
 | last4 = Domes | first4 = Ferenc
 | last5 = Schichl | first5 = Hermann
 | doi = 10.1007/s10898-018-0711-5
 | issue = 3
 | journal = Journal of Global Optimization
 | mr = 3916193
 | pages = 547–565
 | title = Rigorous packing of unit squares into a circle
 | volume = 73
 | year = 2019| pmid = 30880874
 }}</ref> the same is true for [[circle packing in a square]].<ref>{{cite book |title=Unsolved Problems in Geometry |last=Croft |first=Hallard T. |author2=Falconer, Kenneth J. |author3=Guy, Richard K. |year=1991 |publisher=Springer-Verlag |location=New York |isbn=0-387-97506-3 |pages=[https://archive.org/details/unsolvedproblems0000crof/page/108 108–110] |url=https://archive.org/details/unsolvedproblems0000crof/page/108|contribution=D.1 Packing circles or spreading points in a square }}</ref> Packing squares into other shapes can have high [[computational complexity]]: testing whether a given number of unit squares can fit into an [[Orthogonal convexity|orthogonally convex]] [[rectilinear polygon]] with [[half-integer]] vertex coordinates is [[NP-complete]].<ref>{{cite conference
 | last1 = Abrahamsen | first1 = Mikkel
 | last2 = Stade | first2 = Jack
 | arxiv = 2404.09835
 | contribution = Hardness of packing, covering and partitioning simple polygons with unit squares
 | doi = 10.1109/FOCS61266.2024.00087
 | pages = 1355–1371
 | publisher = IEEE
 | title = 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024, Chicago, IL, USA, October 27–30, 2024
 | year = 2024| isbn = 979-8-3315-1674-1
 }}</ref>

[[Squaring the square]] involves subdividing a given square into smaller squares, all having integer side lengths. A subdivision with distinct smaller squares is called a perfect squared square.<ref>{{cite journal
 | last = Duijvestijn | first = A. J. W.
 | doi = 10.1016/0095-8956(78)90041-2
 | issue = 2
 | journal = [[Journal of Combinatorial Theory, Series B]]
 | mr = 511994
 | pages = 240–243
 | title = Simple perfect squared square of lowest order
 | volume = 25
 | year = 1978| doi-access = free
 }}</ref> Another variant of squaring the square called "Mrs. Perkins's quilt" allows repetitions, but uses as few smaller squares as possible in order to make the [[greatest common divisor]] of the side lengths be 1.<ref>{{cite journal
 | last = Trustrum | first = G. B.
 | doi = 10.1017/s0305004100038573
 | journal = [[Proceedings of the Cambridge Philosophical Society]]
 | mr = 170831
 | pages = 7–11
 | title = Mrs Perkins's quilt
 | volume = 61
 | issue = 1
 | year = 1965| bibcode = 1965PCPS...61....7T
 }}</ref> The entire plane can be tiled by squares, with exactly one square of each integer side length.<ref>{{cite journal
 | last1 = Henle | first1 = Frederick V.
 | last2 = Henle | first2 = James M.
 | doi = 10.1080/00029890.2008.11920491
 | issue = 1
 | journal = [[The American Mathematical Monthly]]
 | jstor = 27642387
 | pages = 3–12
 | s2cid = 26663945
 | title = Squaring the plane
 | url = http://www.fredhenle.net/stp/monthly003-012.pdf
 | volume = 115
 | year = 2008}}</ref>

{{multiple image
|image1=Clifford-torus.gif|caption1=[[Stereographic projection]] into 3d of a rotating [[Clifford torus]]
|image2=Petrie-1.gif|caption2=[[Regular skew apeirohedron]] with six squares per vertex
|image3=Teabag.jpg
|caption3=Numerical simulation of an inflated square pillow
|total_width=450}}
In higher dimensions, other surfaces than the plane can be tiled by equal squares, meeting edge-to-edge. One of these surfaces is the [[Clifford torus]], the four-dimensional [[Cartesian product]] of two congruent circles; it has the same intrinsic geometry as a single square with each pair of opposite edges glued together.<ref>{{cite book
 | last = Thorpe | first = John A.
 | contribution = Chapter 14: Parameterized surfaces, Example 9
 | doi = 10.1007/978-1-4612-6153-7
 | isbn = 0-387-90357-7
 | mr = 528129
 | page = 113
 | publisher = Springer-Verlag | location = New York & Heidelberg
 | series = Undergraduate Texts in Mathematics
 | title = Elementary Topics in Differential Geometry
 | year = 1979}}</ref> Another square-tiled surface, a [[regular skew apeirohedron]] in three dimensions, has six squares meeting at each vertex.<ref>{{cite journal
 | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter
 | doi = 10.1112/plms/s2-43.1.33
 | issue = 1
 | journal = [[Proceedings of the London Mathematical Society]]
 | mr = 1575418
 | pages = 33–62
 | series = Second Series
 | title = Regular skew polyhedra in three and four dimension, and their topological analogues
 | volume = 43
 | year = 1937}} Reprinted in ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, pp. 75–105.</ref> The [[paper bag problem]] seeks the maximum volume that can be enclosed by a surface tiled with two squares glued edge to edge; its exact answer is unknown.<ref>{{cite journal
 | last1 = Pak | first1 = Igor | author1-link = Igor Pak
 | last2 = Schlenker | first2 = Jean-Marc
 | doi = 10.1142/S140292511000057X
 | issue = 2
 | journal = [[Journal of Nonlinear Mathematical Physics]]
 | mr = 2679444
 | pages = 145–157
 | title = Profiles of inflated surfaces
 | volume = 17
 | year = 2010| arxiv = 0907.5057
 }}</ref> Gluing two squares in a different pattern, with the vertex of each square attached to the midpoint of an edge of the other square (or alternatively subdividing these two squares into eight squares glued edge-to-edge) produces a pincushion shape called a [[biscornu]].<ref>{{Cite journal |last=Seaton |first=Katherine A. |date=2021-10-02 |title=Textile D-forms and D 4d |journal=[[Journal of Mathematics and the Arts]] |volume=15 |issue=3–4 |pages=207–217 |doi=10.1080/17513472.2021.1991134|doi-access=free |arxiv=2103.09649 }}</ref>

===Counting===
{{main|Square pyramidal number|Dividing a square into similar rectangles}}
[[File:Two square counting puzzles.svg|thumb|upright=1.2|Two square-counting puzzles: There are 14 squares in a {{math|3 × 3}} grid of squares (top), but as a {{math|4 × 4}} grid of points it has six more off-axis squares (bottom) for a total of 20.]]
A common [[mathematical puzzle]] involves counting the squares of all sizes in a square grid of <math>n\times n</math> squares. For instance, a square grid of nine squares has 14 squares: the nine squares that form the grid, four more <math>2\times 2</math> squares, and one <math>3\times 3</math> square. The answer to the puzzle is <math>n(n+1)(2n+1)/6</math>, a [[square pyramidal number]].<ref>{{cite journal
 | last1 = Duffin | first1 = Janet
 | last2 = Patchett | first2 = Mary
 | last3 = Adamson | first3 = Ann
 | last4 = Simmons | first4 = Neil
 | date = November 1984
 | issue = 5
 | journal = Mathematics in School
 | jstor = 30216270
 | pages = 2–4
 | title = Old squares new faces
 | volume = 13}}</ref> For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A000330|Square pyramidal numbers}}</ref>

{{block indent|left=1.6|1, 5, 14, 30, 55, 91, 140, 204, 285, ...}}

A variant of the same puzzle asks for the number of squares formed by a grid of <math>n\times n</math> points, allowing squares that are not axis-parallel. For instance, a grid of nine points has five axis-parallel squares as described above, but it also contains one more diagonal square for a total of six.<ref>{{cite journal
 | last = Bright | first = George W.
 | date = May 1978
 | doi = 10.5951/at.25.8.0039
 | issue = 8
 | journal = The Arithmetic Teacher
 | jstor = 41190469
 | pages = 39–43
 | publisher = National Council of Teachers of Mathematics
 | title = Using Tables to Solve Some Geometry Problems
 | volume = 25}}</ref> In this case, the answer is given by the ''4-dimensional pyramidal numbers'' <math>n^2(n^2-1)/12</math>. For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A002415|4-dimensional pyramidal numbers}}</ref>

{{block indent|left=1.6|0, 1, 6, 20, 50, 105, 196, 336, 540, ...}}

[[File:Plastic square partitions.svg|thumb|Partitions of a square into three similar rectangles]]
Another counting problem involving squares asks for the number of different shapes of rectangle that can be used when [[dividing a square into similar rectangles]].<ref>{{cite news |last=Roberts |first=Siobhan | author-link = Siobhan Roberts |date=February 7, 2023 |title=The quest to find rectangles in a square|newspaper=[[The New York Times]] |url=https://www.nytimes.com/2023/02/07/science/puzzles-rectangles-mathematics.html}}</ref> A square can be divided into two similar rectangles only in one way, by bisecting it, but when dividing a square into three similar rectangles there are three possible [[aspect ratio]]s of the rectangles, 3:1, 3:2, and the square of the [[plastic ratio]]. The number of proportions that are possible when dividing into <math>n</math> rectangles is known for small values of <math>n</math>, but not as a general formula. For <math>n=1,2,3,\dots</math> these numbers are:<ref>{{cite OEIS|A359146|Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible}}</ref>

{{block indent|left=1.6|1, 1, 3, 11, 51, 245, 1372, ...}}
{{-}}

===Other geometries===
{{multiple image
| image1 = Concentric squares on the sphere.png | caption1 = Concentric squares in the [[sphere]] ([[orthographic projection]])
| image2 = Concentric squares in the hyperbolic plane.png | caption2 = Concentric squares in the [[hyperbolic plane]] ([[conformal disk model]])
|total_width=400}}
{{multiple image
| image1 = Octant_of_a_sphere.png | caption1= An octant is a regular spherical triangle with right angles.
| image2 = H2 tiling 246-1.png | caption2 = Regular hexagons with right angles [[order-4 hexagonal tiling|can tile the hyperbolic plane]] with four hexagons meeting at each vertex.
|total_width=400}}

In the familiar Euclidean geometry, space is flat, and every convex quadrilateral has internal angles summing to 360°, so a square (a regular quadrilateral) has four equal sides and four right angles (each 90°). By contrast, in [[spherical geometry]] and [[hyperbolic geometry]], space is curved and the internal angles of a convex quadrilateral never sum to 360°, so quadrilaterals with four right angles do not exist. Both of these geometries have regular quadrilaterals, with four equal sides and four equal angles, often called squares,<ref name=maraner/> but some authors avoid that name because they lack right angles. These geometries also have regular polygons with right angles, but with numbers of sides different from four.<ref name=singer/>

In spherical geometry, space has uniform positive curvature, and every convex quadrilateral (a [[spherical polygon|polygon]] with four [[great-circle arc]] edges) has angles whose sum exceeds 360° by an amount called the [[angular excess]], proportional to its surface area. Small spherical squares are approximately Euclidean, and larger squares' angles increase with area.<ref name=maraner>{{cite journal
 | last = Maraner | first = Paolo
 | doi = 10.1007/s00283-010-9152-9
 | issue = 3
 | journal = [[The Mathematical Intelligencer]]
 | mr = 2721310
 | pages = 46–50
 | title = A spherical Pythagorean theorem
 | volume = 32
 | year = 2010}} See paragraph about spherical squares, p. 48.</ref> One special case is the face of a [[spherical cube]] with four 120° angles, covering one sixth of the sphere's surface.<ref>{{cite book |title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere |last=Popko |first=Edward S. |publisher=CRC Press |year=2012 |isbn=9781466504295 |pages=100–10 1|url=https://books.google.com/books?id=WLAFlr1_2S4C&pg=PA100 }}</ref> Another is a [[hemisphere (geometry)|hemisphere]], the face of a spherical square [[dihedron]], with four [[straight angle]]s; the [[Peirce quincuncial projection]] for [[world map]]s [[Conformal mapping|conformally maps]] two such faces to Euclidean squares.<ref>{{cite journal|last=Lambers|first=Martin|issue=2|journal=[[Journal of Computer Graphics Techniques]]|pages=1–21|title=Mappings between sphere, disc, and square|url=https://jcgt.org/published/0005/02/01/|volume=5|year=2016}}</ref> An [[octant of a sphere]] is a regular [[spherical triangle]], with three equal sides and three right angles; eight of them tile the sphere, with four meeting at each vertex, to form a [[spherical octahedron]].<ref>{{cite book
 | last = Stillwell | first = John | author-link = John Stillwell
 | doi = 10.1007/978-1-4612-0929-4
 | isbn = 0-387-97743-0
 | mr = 1171453
 | page = 68
 | publisher = Springer-Verlag | location = New York
 | series = Universitext
 | title = Geometry of Surfaces
 | year = 1992}}</ref> A [[spherical lune]] is a regular [[digon]], with two semicircular sides and two equal angles at [[antipodal points|antipodal]] vertices; a right-angled lune covers one quarter of the sphere, one face of a four-lune [[hosohedron]].<ref>{{cite journal
 | last1 = Coxeter | first1 = H. S. M. | author1-link = H. S. M. Coxeter
 | last2 = Tóth | first2 = László F. | author2-link = László Fejes Tóth
 | title = The Total Length of the Edges of a Non-Euclidean Polyhedron with Triangular Faces
 | journal = The Quarterly Journal of Mathematics
 | volume = 14 | number = 1
 | pages = 273–284
 | doi = 10.1093/qmath/14.1.273
}}</ref>

In [[hyperbolic geometry]], space has uniform negative curvature, and every convex quadrilateral  has angles whose sum falls short of 360° by an amount called the [[angular defect]], proportional to its surface area. Small hyperbolic squares are approximately Euclidean, and larger squares' angles decrease with increasing area. Special cases include the squares with angles of {{math|360°/''n''}} for every value of {{mvar|n}} larger than {{math|4}}, each of which can tile the [[hyperbolic plane]].<ref name=singer/> In the infinite limit, an [[Ideal point#Polygons with ideal vertices|ideal square]] has four sides of infinite length and four vertices at [[ideal point]]s outside the hyperbolic plane, with {{math|0°}} internal angles;<ref>{{cite book |last=Bonahon |first=Francis |title=Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots |pages=115–116 |publisher=American Mathematical Society |year=2009 |isbn=978-0-8218-4816-6 |url=https://books.google.com/books?id=F5qIAwAAQBAJ&pg=PA116&dq=%22ideal+square%22 |url-access=limited }}</ref> an ideal square, like every ideal quadrilateral, has finite area proportional to its angular defect of {{math|360°}}.<ref>{{cite journal |last=Martin |first=Gaven J. |title=Random ideal hyperbolic quadrilaterals, the cross ratio distribution and punctured tori |journal=Journal of the London Mathematical Society |volume=100 |number=3 |year=2019 |pages=851–870 |doi=10.1112/jlms.12249 |arxiv=1807.06202 }}</ref> It is also possible to make a regular hyperbolic polygon with right angles at every vertex and any number of sides greater than four; such polygons can [[Uniform tilings in hyperbolic plane|uniformly tile the hyperbolic plane]], [[Dual polyhedron|dual]] to the tiling with {{mvar|n}} squares about each vertex.<ref name=singer>{{cite book
 | last = Singer | first = David A.
 | contribution = 3.2 Tessellations of the Hyperbolic Plane
 | doi = 10.1007/978-1-4612-0607-1
 | isbn = 0-387-98306-6
 | mr = 1490036
 | pages = 57–64
 | publisher = Springer-Verlag, New York
 | series = Undergraduate Texts in Mathematics
 | title = Geometry: Plane and Fancy
 | year = 1998}}</ref>

[[File:Metric circles.png|thumb|upright=0.9|Metric circles using Chebyshev, Euclidean, and taxicab distance functions]]
The Euclidean plane can be defined in terms of the [[real coordinate plane]] by adoption of the [[Euclidean distance]] function, according to which the distance between any two points <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math> is <math>\textstyle \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}</math>. Other metric geometries are formed when a different [[Metric space|distance function]] is adopted instead, and in some of these geometries shapes that would be Euclidean squares become the "[[Ball (mathematics)#In normed vector spaces|circles]]" (set of points of equal distance from a center point). Squares tilted at 45° to the coordinate axes are the circles in [[taxicab geometry]], based on the <math>L_1</math> distance <math>|x_1-x_2|+|y_1-y_2|</math>. The points with taxicab distance <math>d</math> from any given point form a diagonal square, centered at the given point, with diagonal length <math>2d</math>. In the same way, axis-parallel squares are the circles for the <math>L_{\infty}</math> or [[Chebyshev distance]], <math>\max(|x_1-x_2|,|y_1-y_2|)</math>. In this metric, the points with distance <math>d</math> from some point form an axis-parallel square, centered at the given point, with side length <math>2d</math>.<ref>{{cite journal
 | last = Scheid | first = Francis | author-link = Francis Scheid
 | date = May 1961
 | doi = 10.5951/mt.54.5.0307
 | issue = 5
 | journal = [[The Mathematics Teacher]]
 | jstor = 27956386
 | pages = 307–312
 | title = Square Circles
 | volume = 54}}</ref><ref>{{cite journal
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | date = November 1980
 | issue = 5
 | journal = [[Scientific American]]
 | jstor = 24966450
 | pages = 18–34
 | title = Mathematical Games: Taxicab geometry offers a free ride to a non-Euclidean locale
 | volume = 243| doi = 10.1038/scientificamerican1280-18 }}</ref><ref>{{cite book
 | last = Tao | first = Terence | author-link = Terence Tao
 | doi = 10.1007/978-981-10-1804-6
 | isbn = 978-981-10-1804-6
 | mr = 3728290
 | pages = 3–4
 | publisher = Springer
 | series = Texts and Readings in Mathematics
 | title = Analysis II
 | volume = 38
 | year = 2016}}</ref>

==See also==
{{Portal|Mathematics}}
* [[Finsler–Hadwiger theorem]] on a square derived from two squares sharing a vertex
* [[Midsquare quadrilateral]], a polygon whose edge midpoints form a square
* [[Monsky's theorem]], on subdividing a square into an odd number of equal-area triangles
* [[Square planar molecular geometry]], chemical structure with atoms at the corners of a square
* [[Square trisection]], a problem of cutting and reassembling one square into three squares
* [[Squircle]], a shape intermediate between a square and a circle
* [[Tarski's circle-squaring problem]], dividing a disk into sets that can be rearranged into a square
* [[Van Aubel's theorem]] and [[Thébault's theorem]], on squares placed on the sides of a quadrilateral

==References==
{{Reflist|30em}}

{{Commons category|Square (geometry)}}
{{Polytopes}}
{{Polygons}}

{{Authority control}}

[[Category:Elementary shapes]]
[[Category:Types of quadrilaterals]]
[[Category:4 (number)]]
[[Category:Constructible polygons]]