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== Problem statement ==
Let ''<math>C''</math> be a [[Jordan curve]]. A [[polygon]] ''<math>P''</math> is '''inscribed in ''<math>C''</math> ''' if all vertices of ''<math>P''</math> belong to ''<math>C''</math>. The '''inscribed square problem''' asks:
: ''Does every Jordan curve admit an inscribed square?''
==Examples==
Some figures, such as [[circle]]s and [[Square (geometry)|square]]s, admit infinitely many [[inscribed]] squares. If ''<math>C''</math> is an [[obtuse triangle]] then it admits exactly one inscribed square; right triangles admit exactly two, and acute triangles admit exactly three.<ref>{{citation | last1 = Bailey | first1 = Herbert | last2 = DeTemple | first2 = Duane | title = Squares inscribed in angles and triangles | journal = [[Mathematics Magazine]] | volume = 71 | issue = 4 | date = 1998 | pages = 278–284 | doi=10.2307/2690699| jstor = 2690699 }}</ref>
== Resolved cases ==
=== Locally monotone curves ===
[[Walter Stromquist|Stromquist]] has proved that every ''local monotone'' plane simple curve admits an inscribed square.<ref name="stromquist">{{citation |last=Stromquist |first=Walter |year=1989 |title=Inscribed squares and square-like quadrilaterals in closed curves |journal = [[Mathematika]]|mr=1045781|doi=10.1112/S0025579300013061|volume=36 |issue=2|pages= 187–197}}</ref> The condition for the admission to happen is that for any point {{mvar|p}}, the curve {{mvar|C}} should be locally represented as a graph of a function {{<math|''>y''{{=}}''f''(''x'')}}</math>.
In more precise terms, for any given point {{mvar|<math>p}}</math> on {{mvar|<math>C}}</math>, there is a neighborhood {{<math|''>U''(''p'')}}</math> and a fixed direction {{<math|''>n''(''p'')}}</math> (the direction of the “{{mvar|<math>y}}</math>-axis”) such that no [[Chord (geometry)|chord]] of {{mvar|<math>C}}</math> -in this neighborhood- is parallel to {{<math|''>n''(''p'')}}</math>.
Locally monotone curves include all types of [[polygon]]s, all closed [[convex set|convex]] curves, and all piecewise
[[Smooth function#Differentiability classes|''C''<supmath>C^1</supmath>]] curves without any [[Cusp (singularity)|cusp]]s.
===Curves without special trapezoids===
An even weaker condition on the curve than local monotonicity is that, for some ε <math>\varepsilon> 0</math>, the curve does not have any inscribed special trapezoids of size ε<math>\varepsilon</math>. A special trapezoid is an [[isosceles trapezoid]] with three equal sides, each longer than the fourth side, inscribed in the curve with a vertex ordering consistent with the clockwise ordering of the curve itself. Its size is the length of the part of the curve that extends around the three equal sides. Here, this length is measured in the ___domain of a fixed [[Parametrization_(geometry)|parametrization]] of <math>C</math>, as <math>C</math> may not be [[rectifiable curve|rectifiable]]. Instead of a limit argument, the proof is based on relative [[obstruction theory]]. This condition is open and dense in the space of all Jordan curves with respect to the [[compact-open topology]]. In this sense, the inscribed square problem is solved for [[Generic property#In topology|generic]] curves.<ref name="matschke"/>
===Curves in annuli===
If a Jordan curve is inscribed in an [[Annulus (mathematics)|annulus]] whose outer radius is at most {{<math|>1 + \sqrt{{sqrt|2}}}}</math> times its inner radius, and it is drawn in such a way that it separates the inner circle of the annulus from the outer circle, then it contains an inscribed square. In this case, if the given curve is approximated by some well-behaved curve, then any large squares that contain the center of the annulus and are inscribed in the approximation are topologically separated from smaller inscribed squares that do not contain the center. The limit of a sequence of large squares must again be a large square, rather than a degenerate point, so the limiting argument may be used.<ref name="matschke"/>
===Symmetric curves===
}}; see also [https://terrytao.wordpress.com/2016/11/22/an-integration-approach-to-the-toeplitz-square-peg-problem/ Tao's blog post on the same set of results]</ref>
=== Jordan curves close to a ''C''<supmath>C^2</supmath> Jordan curve ===
In March 2022, Gregory R. Chambers showed that if γ<math>\gamma</math> is a Jordan curve which is close to a ''C''<supmath>C^2</supmath> Jordan curve β<math>\beta</math> in '''R'''<supmath>\mathbb{R}^2</supmath>, then γ<math>\gamma</math> contains an inscribed square. He showed that, if κ <math>\kappa> 0</math> is the maximum unsigned curvature of β<math>\beta</math> and there is a map ''<math>f''</math> from the image of γ<math>\gamma</math> to the image of β<math>\beta</math> with <math>\||''f''(''x'') − ''-x''\|| < <1/10κ10\kappa</math> and ''<math>f''∘γ\circ\gamma</math> having [[winding number]] <math>1</math>, then γ<math>\gamma</math> has an inscribed square of positive sidelength.<ref>{{Cite arXiv |last=Chambers |first=Gregory |date=March 2022 |title=On the square peg problem |class=math.GT |eprint=2203.02613 }}</ref>
== Variants and generalizations ==
One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle ''<math>T''</math> and Jordan curve ''<math>C''</math>, there is a triangle similar to ''<math>T''</math> and inscribed in ''<math>C''</math>.<ref name="meyerson">{{citation |last=Meyerson |first=Mark D. |year=1980 |title=Equilateral triangles and continuous curves |journal=Fundamenta Mathematicae |mr=600575 |volume=110 |issue=1 |pages=1–9 |doi=10.4064/fm-110-1-1-9|doi-access=free }}</ref><ref>{{citation |last1=Kronheimer |first1=E. H. |last2=Kronheimer |first2=P. B. |author2-link=Peter B. Kronheimer |year=1981 |title=The tripos problem |journal=Journal of the London Mathematical Society |series=Second Series |mr=623685 |doi=10.1112/jlms/s2-24.1.182 |volume=24 |issue=1 |pages=182–192}}</ref> Moreover, the set of the vertices of such triangles is [[dense set|dense]] in ''<math>C''</math>.<ref>{{citation |last=Nielsen |first=Mark J. |year=1992 |title=Triangles inscribed in simple closed curves |journal=[[Geometriae Dedicata]] |mr=1181760 |doi=10.1007/BF00151519 | doi-access=free |volume=43 |issue=3 |pages=291–297}}</ref> In particular, there is always an inscribed [[equilateral triangle]].
It is also known that any Jordan curve admits an inscribed [[rectangle]]. This was proved by Vaughan by reducing the problem to the non-embeddability of the [[projective plane]] in ''R''<supmath>\mathbb{R}^3</supmath>; his proof from around 1977 is published in Meyerson.<ref>{{citation |last=Meyerson |first=Mark D. |year=1981 |title=Balancing acts |journal=Topology Proceedings |volume=6 |issue=1 |pages=71 |url=http://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf |access-date=2023-10-06 }}</ref>
In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle.<ref name="morales-villanueva">{{citation |last1=Morales-Fuentes |first1=Ulises |last2=Villanueva-Segovia |first2=Cristina |title=Rectangles Inscribed in Locally Connected Plane Continua |journal=Topology Proceedings |date=2021 |volume=58 |pages=37–43}}</ref> In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve {{mvar|<math>C}}</math> and rectangle ''<math>R''</math> in the Euclidean plane there exists a rectangle similar to ''<math>R''</math> whose vertices lie on ''<math>C''</math>.<ref name=hartnett>{{citation|last=Hartnett|first=Kevin|title=New geometric perspective cracks old problem about rectangles|url=https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/|access-date=2020-06-26|magazine=Quanta Magazine|date=June 25, 2020}}</ref><ref>{{citation |last1=Greene |first1=Joshua Evan |last2=Lobb |first2=Andrew |title=The rectangular peg problem |journal=Annals of Mathematics |date=September 2021 |volume=194 |issue=2 |pages=509–517 |doi=10.4007/annals.2021.194.2.4|arxiv=2005.09193|s2cid=218684701 }}</ref><ref>{{Cite journal|last=Schwartz|first=Richard Evan|date=2021-09-13|title=Rectangles, curves, and Klein bottles|url=https://www.ams.org/bull/2022-59-01/S0273-0979-2021-01755-1/|journal=Bulletin of the American Mathematical Society|language=en|volume=59|issue=1|pages=1–17|doi=10.1090/bull/1755|issn=0273-0979|doi-access=free}}</ref> This generalizes both the existence of rectangles (of arbitrary shape) and the existence of squares on smooth curves, which has been known since the work of {{harvtxt|Šnirel'man|1944}}.<ref name="Schnirelmann 1944"/> In 2021, Green and Lobb extended their 2020 result and proved that every smooth Jordan curve inscribes every cyclic quadrilateral (modulo an orientation-preserving similarity).<ref>{{Cite journal |last=Greene |first=Joshua Evan |last2=Lobb |first2=Andrew |date=2023 |title=Cyclic quadrilaterals and smooth Jordan curves |url=https://link.springer.com/10.1007/s00222-023-01212-6 |journal=Inventiones mathematicae |language=en |volume=234 |issue=3 |pages=931–935 |doi=10.1007/s00222-023-01212-6 |issn=0020-9910}}</ref>
Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general [[continuum (topology)|continua]] in higher dimensional [[Euclidean space]]s. For example, Stromquist proved that every continuous closed curve ''<math>C''</math> in '''R'''<supmath>''\mathbb{R}^n''</supmath> satisfying "Condition A" that no two chords of ''<math>C''</math> in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.<ref name="stromquist"/> This class of curves includes all ''C''<supmath>C^2</supmath> curves. Nielsen and Wright proved that any symmetric continuum ''<math>K''</math> in '''R'''<supmath>''\mathbb{R}^n''</supmath> contains many inscribed rectangles.<ref name="nielsen-wright"/>
== References ==
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