Editing Square (section)

==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 }}</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]]
Conventionally, since ancient times, most units of [[area]] have been defined in terms of various squares, typically a square with a standard unit of [[length]] as its side, for example a [[square meter]] or [[square inch]].<ref name=Treese>{{cite book |last=Treese |first=Steven A. |year=2018 |chapter=Historical Area |title=History and Measurement of the Base and Derived Units |publisher=Springer |doi=10.1007/978-3-319-77577-7_5 |pages=301–390 |isbn=978-3-319-77576-0 }}</ref> 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.{{r|Treese}}

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}}</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>