Inscribed square problem

<h2 id="problem-statement">Problem statement</h2>
<p>Let 
  
    
      
        C
      
    
    {\displaystyle C}
  
 be a <a href="/facts/Jordan_curve/zKpty4QT">Jordan curve</a>. A <a href="/facts/Polygon/yAq7RjPz">polygon</a> 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is inscribed in 
  
    
      
        C
      
    
    {\displaystyle C}
  
  if all vertices of 
  
    
      
        P
      
    
    {\displaystyle P}
  
 belong to 
  
    
      
        C
      
    
    {\displaystyle C}
  
. The inscribed square problem asks:
</p>
<i>Does every Jordan curve admit an inscribed square?</i>
<p>It is <i>not</i> required that the vertices of the square appear along the curve in any particular order.
</p>
<h2 id="examples">Examples</h2>
<p>Some figures, such as <a href="/facts/Circle/9fy5ggKn">circles</a> and <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a>, admit infinitely many <a href="/facts/Inscribed/ltbko1Eh">inscribed</a> squares. There is one <a href="/facts/Inscribed_square_in_a_triangle/e6VE4ynr">inscribed square in a triangle</a> for any <a href="/facts/Obtuse_triangle/YrYTg0Ao">obtuse triangle</a>, two squares for any <a href="/facts/Right_triangle/dD6M9GAn">right triangle</a>, and three squares for any <a href="/facts/Acute_triangle/YrYTg0Ao">acute triangle</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="resolved-cases">Resolved cases</h2>
<p>It is tempting to attempt to solve the inscribed square problem by proving 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="/facts/Limit_(mathematics)/j8JBhj4e">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 class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h3>Piecewise analytic curves</h3>
<p><a href="/facts/Arnold_Emch/djgCgT6Q">Arnold Emch</a> (1916) showed that <a href="/facts/Piecewise/nlQwptpe">piecewise</a> <a href="/facts/Analytic_curve/JhL3z8FP">analytic curves</a> always have inscribed squares. In particular this is true for <a href="/facts/Polygon/yAq7RjPz">polygons</a>. Emch's proof considers the curves traced out by the <a href="/facts/Midpoint/ykRvr0Vk">midpoints</a> of <a href="/facts/Secant_line/gJnaF5g3">secant</a> <a href="/facts/Line_segments/V8E0M4qk">line segments</a> to the curve, parallel to a given line. He shows that, when these curves are intersected with the curves generated in the same way for a perpendicular family of secants, there are an odd number of crossings. Therefore, there always exists at least one crossing, which forms the center of a <a href="/facts/Rhombus/T4pU50GF">rhombus</a> inscribed in the given curve. By rotating the two perpendicular lines continuously through a <a href="/facts/Right_angle/psaslRq7">right angle</a>, and applying the <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a>, he shows that at least one of these rhombi is a square.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h3>Locally monotone curves</h3>
<p>Stromquist has proved that every <i>local monotone</i> plane simple curve admits an inscribed square.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The condition for the admission to happen is that for any point p, the curve C should be locally represented as a graph of a function 
  
    
      
        y
        =
        f
        (
        x
        )
      
    
    {\displaystyle y=f(x)}
  
.
</p><p>In more precise terms, for any given point 
  
    
      
        p
      
    
    {\displaystyle p}
  
 on 
  
    
      
        C
      
    
    {\displaystyle C}
  
, there is a neighborhood 
  
    
      
        U
        (
        p
        )
      
    
    {\displaystyle U(p)}
  
 and a fixed direction 
  
    
      
        n
        (
        p
        )
      
    
    {\displaystyle n(p)}
  
 (the direction of the “
  
    
      
        y
      
    
    {\displaystyle y}
  
-axis”) such that no <a href="/facts/Chord_(geometry)/aAyoEdLd">chord</a> of 
  
    
      
        C
      
    
    {\displaystyle C}
  
 -in this neighborhood- is parallel to 
  
    
      
        n
        (
        p
        )
      
    
    {\displaystyle n(p)}
  
.
</p><p>Locally monotone curves include all types of <a href="/facts/Polygon/yAq7RjPz">polygons</a>, all closed <a href="/facts/Convex_set/vdAuJRJl">convex</a> curves, and all piecewise 
<a href="/facts/Smooth_function/mffL4ch2">
  
    
      
        
          C
          
            1
          
        
      
    
    {\displaystyle C^{1}}
  
</a> curves without any <a href="/facts/Cusp_(singularity)/QH3fEGry">cusps</a>.
</p>
<h3>Curves without special trapezoids</h3>
<p>An even weaker condition on the curve than local monotonicity is that, for some 
  
    
      
        ε
        >
        0
      
    
    {\displaystyle \varepsilon >0}
  
, the curve does not have any inscribed special trapezoids of size 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
. A special trapezoid is an <a href="/facts/Isosceles_trapezoid/Izn5AA4p">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="/facts/Parametrization_(geometry)/SQK3P4sc">parametrization</a> of 
  
    
      
        C
      
    
    {\displaystyle C}
  
, as 
  
    
      
        C
      
    
    {\displaystyle C}
  
 may not be <a href="/facts/Rectifiable_curve/QgTrmLxY">rectifiable</a>. Instead of a limit argument, the proof is based on relative <a href="/facts/Obstruction_theory/h5T62I8z">obstruction theory</a>. This condition is open and dense in the space of all Jordan curves with respect to the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>. In this sense, the inscribed square problem is solved for <a href="/facts/Generic_property/Uz0FcCBW">generic</a> curves.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p>
<h3>Curves in annuli</h3>
<p>If a Jordan curve is inscribed in an <a href="/facts/Annulus_(mathematics)/rhxKuoem">annulus</a> whose outer radius is at most 
  
    
      
        1
        +
        
          
            2
          
        
      
    
    {\displaystyle 1+{\sqrt {2}}}
  
 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 class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>
<h3>Symmetric curves</h3>
<p>The affirmative answer is also known for centrally symmetric curves, even <a href="/facts/Fractal/sXj38JeN">fractals</a> such as the <a href="/facts/Koch_snowflake/JTckxQVK">Koch snowflake</a>, and curves with reflective symmetry across a line.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h3>Lipschitz graphs</h3>
<p>In 2017, <a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a> published a proof of the existence of a square in curves formed by the union of the <a href="/facts/Graph_of_a_function/htYX0BGX">graphs of two functions</a>, both of which have the same value at the endpoints of the curves and both of which obey a <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz continuity</a> condition with Lipschitz constant less than one. Tao also formulated several related conjectures.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
In 2024, Joshua Greene and Andrew Lobb published a preprint improving this result to curves with Lipschitz constant less than 
  
    
      
        1
        +
        
          
            2
          
        
      
    
    {\displaystyle 1+{\sqrt {2}}}
  
. <a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>
<h3>Jordan curves close to a C2 Jordan curve</h3>
<p>In March 2022, Gregory R. Chambers showed that if 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is a Jordan curve which is close to a 
  
    
      
        
          C
          
            2
          
        
      
    
    {\displaystyle C^{2}}
  
 Jordan curve 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 in 
  
    
      
        
          
            R
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} ^{2}}
  
, then 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 contains an inscribed square. He showed that, if 
  
    
      
        κ
        >
        0
      
    
    {\displaystyle \kappa >0}
  
 is the maximum unsigned curvature of 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 and there is a map 
  
    
      
        f
      
    
    {\displaystyle f}
  
 from the image of 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 to the image of 
  
    
      
        β
      
    
    {\displaystyle \beta }
  
 with 
  
    
      
        ‖
        f
        (
        x
        )
        −
        x
        ‖
        <
        1
        
          /
        
        10
        κ
      
    
    {\displaystyle \|f(x)-x\|<1/10\kappa }
  
 and 
  
    
      
        f
        ∘
        γ
      
    
    {\displaystyle f\circ \gamma }
  
 having <a href="/facts/Winding_number/kaquyUL2">winding number</a> 
  
    
      
        1
      
    
    {\displaystyle 1}
  
, then 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 has an inscribed square of positive sidelength.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>
</p>
<h2 id="variants-and-generalizations">Variants and generalizations</h2>
<p>One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle 
  
    
      
        T
      
    
    {\displaystyle T}
  
 and Jordan curve 
  
    
      
        C
      
    
    {\displaystyle C}
  
, there is a triangle similar to 
  
    
      
        T
      
    
    {\displaystyle T}
  
 and inscribed in 
  
    
      
        C
      
    
    {\displaystyle C}
  
.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> Moreover, the set of the vertices of such triangles is <a href="/facts/Dense_set/nPT9Yxao">dense</a> in 
  
    
      
        C
      
    
    {\displaystyle C}
  
.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> In particular, there is always an inscribed <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a>.
</p><p>It is also known that any Jordan curve admits an inscribed <a href="/facts/Rectangle/mdLfIGKA">rectangle</a>. This was proved by Vaughan by reducing the problem to the non-embeddability of the <a href="/facts/Projective_plane/p6ot8oNv">projective plane</a> in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
; his proof from around 1977 is published in Meyerson.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a>
In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and rectangle 
  
    
      
        R
      
    
    {\displaystyle R}
  
 in the Euclidean plane there exists a rectangle similar to 
  
    
      
        R
      
    
    {\displaystyle R}
  
 whose vertices lie on 
  
    
      
        C
      
    
    {\displaystyle C}
  
.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></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 Šnirel'man (1944).<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> In 2021, Greene and Lobb extended their 2020 result and proved that every smooth Jordan curve inscribes every cyclic quadrilateral (modulo an orientation-preserving similarity).<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a>
</p><p>Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general <a href="/facts/Continuum_(topology)/MBAh8aro">continua</a> in higher dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean spaces</a>. For example, Stromquist proved that every continuous closed curve 
  
    
      
        C
      
    
    {\displaystyle C}
  
 in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 satisfying "Condition A" that no two chords of 
  
    
      
        C
      
    
    {\displaystyle C}
  
 in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> This class of curves includes all 
  
    
      
        
          C
          
            2
          
        
      
    
    {\displaystyle C^{2}}
  
 curves. Nielsen and Wright proved that any symmetric continuum 
  
    
      
        K
      
    
    {\displaystyle K}
  
 in 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 contains many inscribed rectangles.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a>
</p>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Victor_Klee/NMRFBGm6">Klee, Victor</a>; <a href="/facts/Stan_Wagon/7LGzJEN6">Wagon, Stan</a> (1991), "Inscribed squares", <i>Old and New Unsolved Problems in Plane Geometry and Number Theory</i>, The Dolciani Mathematical Expositions, vol. 11, Cambridge University Press, pp. 58–65, 137–144, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-88385-315-3</li></ul>
<h2 id="external-links">External links</h2>
<ul><li>Mark J. Nielsen, <a href="http://www.webpages.uidaho.edu/~markn/squares/">Figures Inscribed in Curves. A short tour of an old problem</a></li>
<li><a 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 href="https://www.youtube.com/watch?v=IQqtsm-bBRU">This open problem taught me what topology is</a>, <a href="/facts/3Blue1Brown/EMrw99Kq">3Blue1Brown</a>, YouTube – a video showing a topological solution to a simplified version of the problem.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Toeplitz, O. (1911), "Über einige Aufgaben der Analysis situs", Verhandlungen der Schweizerischen Naturforschenden Gesellschaft (in German), 94: 197 <a href="/wiki/Otto_Toeplitz" target="_blank">/wiki/Otto_Toeplitz</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Emch, Arnold (1916), "On some properties of the medians of closed continuous curves formed by analytic arcs", American Journal of Mathematics, 38 (1): 6–18, doi:10.2307/2370541, JSTOR 2370541, MR 1506274 <a href="/wiki/Arnold_Emch" target="_blank">/wiki/Arnold_Emch</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Šnirel'man, L. G. (1944), "On certain geometrical properties of closed curves", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 10: 34–44, MR 0012531 <a href="/wiki/Lev_Schnirelmann" target="_blank">/wiki/Lev_Schnirelmann</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Hartnett, Kevin (June 25, 2020), "New geometric perspective cracks old problem about rectangles", Quanta Magazine, retrieved 2020-06-26 <a href="https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/" target="_blank">https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Bailey, Herbert; DeTemple, Duane (1998), "Squares inscribed in angles and triangles", Mathematics Magazine, 71 (4): 278–284, doi:10.2307/2690699, JSTOR 2690699 <a href="/wiki/Mathematics_Magazine" target="_blank">/wiki/Mathematics_Magazine</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 <a href="/wiki/Notices_of_the_American_Mathematical_Society" target="_blank">/wiki/Notices_of_the_American_Mathematical_Society</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 <a href="/wiki/Notices_of_the_American_Mathematical_Society" target="_blank">/wiki/Notices_of_the_American_Mathematical_Society</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Stromquist, Walter (1989), "Inscribed squares and square-like quadrilaterals in closed curves", Mathematika, 36 (2): 187–197, doi:10.1112/S0025579300013061, MR 1045781 <a href="/wiki/Mathematika" target="_blank">/wiki/Mathematika</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 <a href="/wiki/Notices_of_the_American_Mathematical_Society" target="_blank">/wiki/Notices_of_the_American_Mathematical_Society</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 <a href="/wiki/Notices_of_the_American_Mathematical_Society" target="_blank">/wiki/Notices_of_the_American_Mathematical_Society</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Nielsen, Mark J.; Wright, S. E. (1995), "Rectangles inscribed in symmetric continua", Geometriae Dedicata, 56 (3): 285–297, doi:10.1007/BF01263570, MR 1340790 <a href="/wiki/Geometriae_Dedicata" target="_blank">/wiki/Geometriae_Dedicata</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Tao, Terence (2017), "An integration approach to the Toeplitz square peg problem", Forum of Mathematics, 5: e30, doi:10.1017/fms.2017.23, MR 3731730; see also Tao's blog post on the same set of results <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Greene, Joshua; Lobb, Andrew (2024), "Square pegs between two graphs", arXiv:2407.07798 [math.SG] <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Chambers, Gregory (March 2022), "On the square peg problem", arXiv:2203.02613 [math.GT] <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Meyerson, Mark D. (1980), "Equilateral triangles and continuous curves", Fundamenta Mathematicae, 110 (1): 1–9, doi:10.4064/fm-110-1-1-9, MR 0600575 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Kronheimer, E. H.; Kronheimer, P. B. (1981), "The tripos problem", Journal of the London Mathematical Society, Second Series, 24 (1): 182–192, doi:10.1112/jlms/s2-24.1.182, MR 0623685 <a href="/wiki/Peter_B._Kronheimer" target="_blank">/wiki/Peter_B._Kronheimer</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>Nielsen, Mark J. (1992), "Triangles inscribed in simple closed curves", Geometriae Dedicata, 43 (3): 291–297, doi:10.1007/BF00151519, MR 1181760 <a href="/wiki/Geometriae_Dedicata" target="_blank">/wiki/Geometriae_Dedicata</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
<li id="fn:18"><p>Meyerson, Mark D. (1981), "Balancing acts" (PDF), Topology Proceedings, 6 (1): 71, retrieved 2023-10-06 <a href="http://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf" target="_blank">http://topology.nipissingu.ca/tp/reprints/v06/tp06107.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li>
<li id="fn:19"><p>Morales-Fuentes, Ulises; Villanueva-Segovia, Cristina (2021), "Rectangles Inscribed in Locally Connected Plane Continua", Topology Proceedings, 58: 37–43 <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li>
<li id="fn:20"><p>Hartnett, Kevin (June 25, 2020), "New geometric perspective cracks old problem about rectangles", Quanta Magazine, retrieved 2020-06-26 <a href="https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/" target="_blank">https://www.quantamagazine.org/new-geometric-perspective-cracks-old-problem-about-rectangles-20200625/</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li>
<li id="fn:21"><p>Greene, Joshua Evan; Lobb, Andrew (September 2021), "The rectangular peg problem", Annals of Mathematics, 194 (2): 509–517, arXiv:2005.09193, doi:10.4007/annals.2021.194.2.4, S2CID 218684701 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li>
<li id="fn:22"><p>Schwartz, Richard Evan (2021-09-13), "Rectangles, curves, and Klein bottles", Bulletin of the American Mathematical Society, 59 (1): 1–17, doi:10.1090/bull/1755, ISSN 0273-0979 <a href="https://www.ams.org/bull/2022-59-01/S0273-0979-2021-01755-1/" target="_blank">https://www.ams.org/bull/2022-59-01/S0273-0979-2021-01755-1/</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li>
<li id="fn:23"><p>Šnirel'man, L. G. (1944), "On certain geometrical properties of closed curves", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 10: 34–44, MR 0012531 <a href="/wiki/Lev_Schnirelmann" target="_blank">/wiki/Lev_Schnirelmann</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li>
<li id="fn:24"><p>Greene, Joshua Evan; Lobb, Andrew (2023), "Cyclic quadrilaterals and smooth Jordan curves", Inventiones Mathematicae, 234 (3): 931–935, arXiv:2011.05216, Bibcode:2023InMat.234..931G, doi:10.1007/s00222-023-01212-6, ISSN 0020-9910 <a href="https://link.springer.com/10.1007/s00222-023-01212-6" target="_blank">https://link.springer.com/10.1007/s00222-023-01212-6</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li>
<li id="fn:25"><p>Stromquist, Walter (1989), "Inscribed squares and square-like quadrilaterals in closed curves", Mathematika, 36 (2): 187–197, doi:10.1112/S0025579300013061, MR 1045781 <a href="/wiki/Mathematika" target="_blank">/wiki/Mathematika</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li>
<li id="fn:26"><p>Nielsen, Mark J.; Wright, S. E. (1995), "Rectangles inscribed in symmetric continua", Geometriae Dedicata, 56 (3): 285–297, doi:10.1007/BF01263570, MR 1340790 <a href="/wiki/Geometriae_Dedicata" target="_blank">/wiki/Geometriae_Dedicata</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li>
</ol>

Inscribed square problem open-in-new

Inscribed square problem