Editing Square (section)

===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>