Inscribed square problem
Does every
The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in
Problem statement
Let be a
- Does every Jordan curve admit an inscribed square?
It is not required that the vertices of the square appear along the curve in any particular order.
Examples
Some figures, such as
Resolved cases
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 limit 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.[6]
Piecewise analytic curves
Locally monotone curves
Stromquist has proved that every local monotone plane simple curve admits an inscribed square.[7] 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 .
In more precise terms, for any given point on , there is a neighborhood and a fixed direction (the direction of the “-axis”) such that no chord of -in this neighborhood- is parallel to .
Locally monotone curves include all types of polygons, all closed convex curves, and all piecewise curves without any cusps.
Curves without special trapezoids
An even weaker condition on the curve than local monotonicity is that, for some , the curve does not have any inscribed special trapezoids of size . 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 of , as may not be
Curves in annuli
If a Jordan curve is inscribed in an annulus whose outer radius is at most 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.[6]
Symmetric curves
The affirmative answer is also known for centrally symmetric curves, even fractals such as the Koch snowflake, and curves with reflective symmetry across a line.[8]
Lipschitz graphs
In 2017, Terence Tao published a proof of the existence of a square in curves formed by the union 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 Lipschitz continuity condition with Lipschitz constant less than one. Tao also formulated several related conjectures.[9]
Jordan curves close to a Jordan curve
In March 2022, Gregory R. Chambers showed that if is a Jordan curve which is close to a Jordan curve in , then contains an inscribed square. He showed that, if is the maximum unsigned curvature of and there is a map from the image of to the image of with and having winding number , then has an inscribed square of positive sidelength.[10]
Variants and generalizations
One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle and Jordan curve , there is a triangle similar to and inscribed in .[11][12] Moreover, the set of the vertices of such triangles is dense in .[13] 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 ; his proof from around 1977 is published in Meyerson.[14] In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle.[15] In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve and rectangle in the Euclidean plane there exists a rectangle similar to whose vertices lie on .[4][16][17] 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).[3] 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).[18]
Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general continua in higher dimensional Euclidean spaces. For example, Stromquist proved that every continuous closed curve in satisfying "Condition A" that no two chords of in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.[7] This class of curves includes all curves. Nielsen and Wright proved that any symmetric continuum in contains many inscribed rectangles.[8]
References
- ^ Toeplitz, O. (1911), "Über einige Aufgaben der Analysis situs", Verhandlungen der Schweizerischen Naturforschenden Gesellschaft (in German), 94: 197
- MR 1506274
- ^ MR 0012531
- ^ a b Hartnett, Kevin (June 25, 2020), "New geometric perspective cracks old problem about rectangles", Quanta Magazine, retrieved 2020-06-26
- JSTOR 2690699
- ^ doi:10.1090/noti1100
- ^ MR 1045781
- ^ MR 1340790
- MR 3731730; see also Tao's blog post on the same set of results
- arXiv:2203.02613 [math.GT].
- MR 0600575
- MR 0623685
- MR 1181760
- ^ Meyerson, Mark D. (1981), "Balancing acts" (PDF), Topology Proceedings, 6 (1): 71, retrieved 2023-10-06
- ^ Morales-Fuentes, Ulises; Villanueva-Segovia, Cristina (2021), "Rectangles Inscribed in Locally Connected Plane Continua", Topology Proceedings, 58: 37–43
- S2CID 218684701
- ISSN 0273-0979.
- ISSN 0020-9910.
Further reading
- ISBN 978-0-88385-315-3
External links
- Mark J. Nielsen, Figures Inscribed in Curves. A short tour of an old problem
- Inscribed squares: Denne speaks at Jordan Ellenberg's blog
- Grant Sanderson, Who cares about topology? (Inscribed rectangle problem), 3Blue1Brown, YouTube a – video showing a topological solution to a simplified version of the problem.