Editing Square (section)

==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>

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