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<title>Inscribed square problem</title>

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Inscribed square problem

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<div>Unsolved problem in mathematics</div>

<div class="unsolved-body">Does every <a href="Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a> have an inscribed square?</div>

<div class="unsolved-more"><a href="List_of_unsolved_problems_in_mathematics" title="List of unsolved problems in mathematics">More unsolved problems in mathematics</a></div>

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The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an <a href="Open_problem" title="Open problem">unsolved question</a> in <a href="Geometry" title="Geometry">geometry</a>: Does every <a href="Jordan_curve" class="mw-redirect" title="Jordan curve">plane simple closed curve</a> contain all four vertices of some <a href="Square" title="Square">square</a>? This is true if the curve is <a href="Convex_set" title="Convex set">convex</a> or <a href="Piecewise_smooth" class="mw-redirect" title="Piecewise smooth">piecewise smooth</a> and in other special cases. The problem was proposed by <a href="Otto_Toeplitz" title="Otto Toeplitz">Otto Toeplitz</a> in 1911.<a href="#cite_note-1">[1]</a> Some early positive results were obtained by <a href="Arnold_Emch" title="Arnold Emch">Arnold Emch</a><a href="#cite_note-Emch-2">[2]</a> and <a href="Lev_Schnirelmann" title="Lev Schnirelmann">Lev Schnirelmann</a>.<a href="#cite_note-Schnirelmann_1944-3">[3]</a> The general case remains open.<a href="#cite_note-hartnett-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Problem_statement">Problem statement</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> be a <a href="Jordan_curve" class="mw-redirect" title="Jordan curve">Jordan curve</a>. A <a href="Polygon" title="Polygon">polygon</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is inscribed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> if all vertices of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. The inscribed square problem asks:

<dl><dd>Does every Jordan curve admit an inscribed square?</dd></dl>

It is not required that the vertices of the square appear along the curve in any particular order.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

Some figures, such as <a href="Circle" title="Circle">circles</a> and squares, admit infinitely many <a href="Inscribed" class="mw-redirect" title="Inscribed">inscribed</a> squares. There is one <a href="Inscribed_square_in_a_triangle" title="Inscribed square in a triangle">inscribed square in a triangle</a> for any <a href="Obtuse_triangle" class="mw-redirect" title="Obtuse triangle">obtuse triangle</a>, two squares for any <a href="Right_triangle" title="Right triangle">right triangle</a>, and three squares for any <a href="Acute_triangle" class="mw-redirect" title="Acute triangle">acute triangle</a>.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Resolved_cases">Resolved cases</h2></div>

It is tempting to attempt to solve the inscribed square problem by <a href="Mathematical_proof" title="Mathematical proof">proving</a> that a special class of well-behaved curves always contains an inscribed square, and then to approximate an arbitrary curve by a sequence of well-behaved curves and infer that there still exists an inscribed square as a <a href="Limit_(mathematics)" title="Limit (mathematics)">limit</a> of squares inscribed in the curves of the sequence. One reason this argument has not been carried out to completion is that the limit of a sequence of squares may be a single point rather than itself being a square. Nevertheless, many special cases of curves are now known to have an inscribed square.<a href="#cite_note-matschke-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Piecewise_analytic_curves">Piecewise analytic curves</h3></div>

<a href="Arnold_Emch" title="Arnold Emch">Arnold Emch</a> (<a href="#CITEREFEmch1916">1916</a>) showed that <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> <a href="Analytic_curve" class="mw-redirect" title="Analytic curve">analytic curves</a> always have inscribed squares. In particular this is true for polygons. Emch's proof considers the curves traced out by the <a href="Midpoint" title="Midpoint">midpoints</a> of <a href="Secant_line" title="Secant line">secant</a> <a href="Line_segments" class="mw-redirect" title="Line segments">line segments</a> to the curve, <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to a given line. He shows that, when these curves are intersected with the curves generated in the same way for a <a href="Perpendicular" title="Perpendicular">perpendicular</a> family of secants, there are an <a href="Parity_(mathematics)" title="Parity (mathematics)">odd</a> number of crossings. Therefore, there always exists at least one crossing, which forms the center of a <a href="Rhombus" title="Rhombus">rhombus</a> inscribed in the given curve. By rotating the two perpendicular lines <a href="Continuous_(mathematics)" class="mw-redirect" title="Continuous (mathematics)">continuously</a> through a <a href="Right_angle" title="Right angle">right angle</a>, and applying the <a href="Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a>, he shows that at least one of these rhombi is a square.<a href="#cite_note-matschke-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Locally_monotone_curves">Locally monotone curves</h3></div>

Stromquist has proved that every local monotone plane simple curve admits an inscribed square.<a href="#cite_note-stromquist-7">[7]</a> The condition for the admission to happen is that for any point p, the curve C should be locally represented as a <a href="Graph_of_a_function" title="Graph of a function">graph of a function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

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</math><img src="./2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy">.

In more precise terms, for any given point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, there is a <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(p)}">

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<annotation encoding="application/x-tex">{\displaystyle U(p)}</annotation>

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</math><img src="./218ca5a80b1a2b8dd5a21ebae88f55219a5c7f83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.761ex; height:2.843ex;" alt="{\displaystyle U(p)}" loading="lazy"> and a fixed direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n(p)}</annotation>

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</math><img src="./a50fb76fa826b92c207b8d5b7fd43c86a3e48c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.373ex; height:2.843ex;" alt="{\displaystyle n(p)}" loading="lazy"> (the direction of the “<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-axis”) such that no <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> -in this neighborhood- is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(p)}">

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<annotation encoding="application/x-tex">{\displaystyle n(p)}</annotation>

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</math><img src="./a50fb76fa826b92c207b8d5b7fd43c86a3e48c70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.373ex; height:2.843ex;" alt="{\displaystyle n(p)}" loading="lazy">.

Locally monotone curves include all types of polygons, all closed convex curves, and all piecewise

<msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation>

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</math><img src="./bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"></a> curves without any <a href="Cusp_(singularity)" title="Cusp (singularity)">cusps</a>.

<div class="mw-heading mw-heading3"><h3 id="Curves_without_special_trapezoids">Curves without special trapezoids</h3></div>

An even weaker condition on the curve than local monotonicity is that, for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy">, the curve does not have any inscribed special <a href="Trapezoid" title="Trapezoid">trapezoids</a> of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">. A special trapezoid is an <a href="Isosceles_trapezoid" title="Isosceles trapezoid">isosceles trapezoid</a> 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 <a href="Parametrization_(geometry)" title="Parametrization (geometry)">parametrization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> may not be <a href="Rectifiable_curve" class="mw-redirect" title="Rectifiable curve">rectifiable</a>. Instead of a limit argument, the proof is based on relative <a href="Obstruction_theory" title="Obstruction theory">obstruction theory</a>. This condition is <a href="Open_set" title="Open set">open</a> and <a href="Dense_set" title="Dense set">dense</a> in the <a href="Topological_space" title="Topological space">space</a> of all Jordan curves with respect to the <a href="Compact-open_topology" title="Compact-open topology">compact-open topology</a>. In this sense, the inscribed square problem is solved for <a href="Generic_property#In_topology" title="Generic property">generic</a> curves.<a href="#cite_note-matschke-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Curves_in_annuli">Curves in annuli</h3></div>

If a Jordan curve is inscribed in an <a href="Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a> whose outer <a href="Radius" title="Radius">radius</a> is at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\sqrt {2}}}">

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+{\sqrt {2}}}</annotation>

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</math><img src="./0d6647fc0b70302f56dbc87eaf718dc3832ba161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:3.009ex;" alt="{\displaystyle 1+{\sqrt {2}}}" loading="lazy"> 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.<a href="#cite_note-matschke-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Symmetric_curves">Symmetric curves</h3></div>

The affirmative answer is also known for centrally symmetric curves, even <a href="Fractal" title="Fractal">fractals</a> such as the <a href="Koch_snowflake" title="Koch snowflake">Koch snowflake</a>, and curves with reflective symmetry across a line.<a href="#cite_note-nielsen-wright-8">[8]</a>

<div class="mw-heading mw-heading3"><h3 id="Lipschitz_graphs">Lipschitz graphs</h3></div>

In 2017, <a href="Terence_Tao" title="Terence Tao">Terence Tao</a> published a proof of the existence of a square in curves formed by the <a href="Union_(set_theory)" title="Union (set theory)">union</a> of the graphs of two functions, both of which have the same value at the endpoints of the curves and both of which obey a <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a> condition with Lipschitz constant less than one. Tao also formulated several related <a href="Conjecture" title="Conjecture">conjectures</a>.<a href="#cite_note-9">[9]</a>

In 2024, Joshua Greene and Andrew Lobb published a preprint improving this result to curves with Lipschitz constant less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\sqrt {2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+{\sqrt {2}}}</annotation>

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</math><img src="./0d6647fc0b70302f56dbc87eaf718dc3832ba161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:3.009ex;" alt="{\displaystyle 1+{\sqrt {2}}}" loading="lazy">. <a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Jordan_curves_close_to_a_C2_Jordan_curve">Jordan curves close to a C2 Jordan curve</h3></div>

In March 2022, Gregory R. Chambers showed that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is a Jordan curve which is close to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy"> Jordan curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> contains an inscribed square. He showed that, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa >0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \kappa >0}</annotation>

</semantics>

</math><img src="./f2d8b6e1180bd22cf9e92b0295ede259cb80db64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle \kappa >0}" loading="lazy"> is the maximum unsigned <a href="Curvature" title="Curvature">curvature</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> and there is a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> from the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> to the image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f(x)-x\|<1/10\kappa }</annotation>

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</math><img src="./7195188738a0839f7ad9091083f388a3de8dedc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20ex; height:2.843ex;" alt="{\displaystyle \|f(x)-x\|<1/10\kappa }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle f\circ \gamma }</annotation>

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</math><img src="./7eaed1ad841924bc2c2ed3ecc68a03c97058ff25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.736ex; height:2.676ex;" alt="{\displaystyle f\circ \gamma }" loading="lazy"> having <a href="Winding_number" title="Winding number">winding number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> has an inscribed square of positive sidelength.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Variants_and_generalizations">Variants and generalizations</h2></div>

One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> and Jordan curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, there is a triangle <a href="Similarity_(geometry)" title="Similarity (geometry)">similar</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> and inscribed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.<a href="#cite_note-meyerson-12">[12]</a><a href="#cite_note-13">[13]</a> Moreover, the set of the vertices of such triangles is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.<a href="#cite_note-14">[14]</a> In particular, there is always an inscribed <a href="Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>.

It is also known that any Jordan curve admits an inscribed <a href="Rectangle" title="Rectangle">rectangle</a>. This was proved by Vaughan by reducing the problem to the non-embeddability of the <a href="Projective_plane" title="Projective plane">projective plane</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation>

</semantics>

</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">; his proof from around 1977 is published in Meyerson.<a href="#cite_note-15">[15]</a>

In 2020, Morales and Villanueva characterized locally connected plane <a href="Continuum_(topology)" title="Continuum (topology)">continua</a> that admit at least one inscribed rectangle.<a href="#cite_note-morales-villanueva-16">[16]</a> In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and rectangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> in the Euclidean plane there exists a rectangle similar to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> whose vertices lie on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">.<a href="#cite_note-hartnett-4">[4]</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a> 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 <a href="#CITEREFŠnirel'man1944">Šnirel'man (1944)</a>.<a href="#cite_note-Schnirelmann_1944-3">[3]</a> In 2021, Greene and Lobb extended their 2020 result and proved that every smooth Jordan curve inscribes every <a href="Cyclic_quadrilateral" title="Cyclic quadrilateral">cyclic quadrilateral</a> (modulo an orientation-preserving similarity).<a href="#cite_note-19">[19]</a>

Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general continua in higher dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean spaces</a>. For example, Stromquist proved that every continuous closed curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> satisfying "Condition A", that no two chords of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> in a suitable neighborhood of any point are perpendicular, admits an inscribed quadrilateral with equal sides and equal diagonals.<a href="#cite_note-stromquist-7">[7]</a> This class of curves includes all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math><img src="./4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy"> curves. Nielsen and Wright proved that any symmetric continuum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> contains many inscribed rectangles.<a href="#cite_note-nielsen-wright-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFToeplitz1911" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Otto_Toeplitz" title="Otto Toeplitz">Toeplitz, O.</a> (1911), "Über einige Aufgaben der Analysis situs", Verhandlungen der Schweizerischen Naturforschenden Gesellschaft (in German), 94: 197</cite>

</li>

<li id="cite_note-Emch-2"><a href="#cite_ref-Emch_2-0">^</a> <cite id="CITEREFEmch1916" class="citation cs2"><a href="Arnold_Emch" title="Arnold Emch">Emch, Arnold</a> (1916), "On some properties of the medians of closed continuous curves formed by analytic arcs", American Journal of Mathematics, 38 (1): 6–18, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2370541">10.2307/2370541</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2370541">2370541</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1506274">1506274</a></cite>

</li>

<li id="cite_note-Schnirelmann_1944-3">^ <a href="#cite_ref-Schnirelmann_1944_3-0">a</a> <a href="#cite_ref-Schnirelmann_1944_3-1">b</a> <cite id="CITEREFŠnirel'man1944" class="citation cs2"><a href="Lev_Schnirelmann" title="Lev Schnirelmann">Šnirel'man, L. G.</a> (1944), "On certain geometrical properties of closed curves", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 10: 34–44, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0012531">0012531</a></cite>

</li>

<li id="cite_note-hartnett-4">^ <a href="#cite_ref-hartnett_4-0">a</a> <a href="#cite_ref-hartnett_4-1">b</a> <cite id="CITEREFHartnett2020" class="citation cs2">Hartnett, Kevin (June 25, 2020), <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/">"New geometric perspective cracks old problem about rectangles"</a>, Quanta Magazine, retrieved 2020-06-26</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBaileyDeTemple1998" class="citation cs2">Bailey, Herbert; DeTemple, Duane (1998), "Squares inscribed in angles and triangles", <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>, 71 (4): 278–284, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690699">10.2307/2690699</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690699">2690699</a></cite>

</li>

<li id="cite_note-matschke-6">^ <a href="#cite_ref-matschke_6-0">a</a> <a href="#cite_ref-matschke_6-1">b</a> <a href="#cite_ref-matschke_6-2">c</a> <a href="#cite_ref-matschke_6-3">d</a> <cite id="CITEREFMatschke2014" class="citation cs2">Matschke, Benjamin (2014), "A survey on the square peg problem", <a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>, 61 (4): 346–352, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1100">10.1090/noti1100</a>, <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/21.11116%2F0000-0004-15B8-5">21.11116/0000-0004-15B8-5</a></cite>

</li>

<li id="cite_note-stromquist-7">^ <a href="#cite_ref-stromquist_7-0">a</a> <a href="#cite_ref-stromquist_7-1">b</a> <cite id="CITEREFStromquist1989" class="citation cs2">Stromquist, Walter (1989), "Inscribed squares and square-like quadrilaterals in closed curves", <a href="Mathematika" title="Mathematika">Mathematika</a>, 36 (2): 187–197, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FS0025579300013061">10.1112/S0025579300013061</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1045781">1045781</a></cite>

</li>

<li id="cite_note-nielsen-wright-8">^ <a href="#cite_ref-nielsen-wright_8-0">a</a> <a href="#cite_ref-nielsen-wright_8-1">b</a> <cite id="CITEREFNielsenWright1995" class="citation cs2">Nielsen, Mark J.; Wright, S. E. (1995), "Rectangles inscribed in symmetric continua", <a href="Geometriae_Dedicata" title="Geometriae Dedicata">Geometriae Dedicata</a>, 56 (3): 285–297, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01263570">10.1007/BF01263570</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1340790">1340790</a></cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFTao2017" class="citation cs2">Tao, Terence (2017), "An integration approach to the Toeplitz square peg problem", Forum of Mathematics, 5: e30, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Ffms.2017.23">10.1017/fms.2017.23</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3731730">3731730</a></cite>; see also <a rel="nofollow" class="external text" href="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</a>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFGreeneLobb2024" class="citation arxiv cs2">Greene, Joshua; Lobb, Andrew (2024), "Square pegs between two graphs", <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2407.07798">2407.07798</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.SG">math.SG</a>]</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFChambers2022" class="citation arxiv cs2">Chambers, Gregory (March 2022), "On the square peg problem", <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2203.02613">2203.02613</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GT">math.GT</a>]</cite>

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<li id="cite_note-meyerson-12"><a href="#cite_ref-meyerson_12-0">^</a> <cite id="CITEREFMeyerson1980" class="citation cs2">Meyerson, Mark D. (1980), "Equilateral triangles and continuous curves", Fundamenta Mathematicae, 110 (1): 1–9, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-110-1-1-9">10.4064/fm-110-1-1-9</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0600575">0600575</a></cite>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFKronheimerKronheimer1981" class="citation cs2">Kronheimer, E. H.; <a href="Peter_B._Kronheimer" title="Peter B. Kronheimer">Kronheimer, P. B.</a> (1981), "The tripos problem", Journal of the London Mathematical Society, Second Series, 24 (1): 182–192, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs2-24.1.182">10.1112/jlms/s2-24.1.182</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0623685">0623685</a></cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFNielsen1992" class="citation cs2">Nielsen, Mark J. (1992), "Triangles inscribed in simple closed curves", <a href="Geometriae_Dedicata" title="Geometriae Dedicata">Geometriae Dedicata</a>, 43 (3): 291–297, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00151519">10.1007/BF00151519</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1181760">1181760</a></cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFMeyerson1981" class="citation cs2">Meyerson, Mark D. (1981), <a rel="nofollow" class="external text" href="http://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf">"Balancing acts"</a> (PDF), Topology Proceedings, 6 (1): 71, retrieved 2023-10-06</cite>

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<li id="cite_note-morales-villanueva-16"><a href="#cite_ref-morales-villanueva_16-0">^</a> <cite id="CITEREFMorales-FuentesVillanueva-Segovia2021" class="citation cs2">Morales-Fuentes, Ulises; Villanueva-Segovia, Cristina (2021), "Rectangles Inscribed in Locally Connected Plane Continua", Topology Proceedings, 58: 37–43</cite>

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFGreeneLobb2021" class="citation cs2">Greene, Joshua Evan; Lobb, Andrew (September 2021), "The rectangular peg problem", Annals of Mathematics, 194 (2): 509–517, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2005.09193">2005.09193</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2021.194.2.4">10.4007/annals.2021.194.2.4</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:218684701">218684701</a></cite>

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFSchwartz2021" class="citation journal cs2">Schwartz, Richard Evan (2021-09-13), <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2022-59-01/S0273-0979-2021-01755-1/">"Rectangles, curves, and Klein bottles"</a>, Bulletin of the American Mathematical Society, 59 (1): 1–17, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fbull%2F1755">10.1090/bull/1755</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0273-0979">0273-0979</a></cite>

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFGreeneLobb2023" class="citation journal cs2">Greene, Joshua Evan; Lobb, Andrew (2023), <a rel="nofollow" class="external text" href="https://link.springer.com/10.1007/s00222-023-01212-6">"Cyclic quadrilaterals and smooth Jordan curves"</a>, Inventiones Mathematicae, 234 (3): 931–935, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2011.05216">2011.05216</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023InMat.234..931G">2023InMat.234..931G</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00222-023-01212-6">10.1007/s00222-023-01212-6</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</a></cite>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFKleeWagon1991" class="citation cs2"><a href="Victor_Klee" title="Victor Klee">Klee, Victor</a>; <a href="Stan_Wagon" title="Stan Wagon">Wagon, Stan</a> (1991), "Inscribed squares", Old and New Unsolved Problems in Plane Geometry and Number Theory, The Dolciani Mathematical Expositions, vol. 11, Cambridge University Press, pp. 58–65, 137–144, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-315-3</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Mark J. Nielsen, <a rel="nofollow" class="external text" href="http://www.webpages.uidaho.edu/~markn/squares/">Figures Inscribed in Curves. A short tour of an old problem</a></li>

<li><a rel="nofollow" class="external text" href="http://quomodocumque.wordpress.com/2007/08/31/inscribed-squares-denne-speaks/">Inscribed squares: Denne speaks</a> at Jordan Ellenberg's blog</li>

<li>Grant Sanderson, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=IQqtsm-bBRU">This open problem taught me what topology is</a>, <a href="3Blue1Brown" title="3Blue1Brown">3Blue1Brown</a>, YouTube – a video showing a topological solution to a simplified version of the problem.</li></ul></div><div><div class="zim-footer">

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