Editing Square (section)

==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>