Editing Square (section)

===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}} }}