Inscribed square problem

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Example: The black dashed curve goes through all corners of several blue squares.

The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? This is known to be true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911.[1] Some early positive results were obtained by Arnold Emch[2] and Lev Schnirelmann.[3] As of 2012, the general case remains open.

Problem statement

Let C be a Jordan curve. A polygon P is inscribed in C if all vertices of P belong to C. The inscribed square problem asks:

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.

Resolved cases

Some figures, such circles and squares, admit infinitely many inscribed squares. If C is an obtuse triangle then it admits exactly one inscribed square.

The most encompassing result to date is due to Stromquist, who proved that every local monotone plane simple curve admits an inscribed square.[4] The condition is that for any point p, the curve C can be locally represented as a graph of a function y = f(x). More precisely, for any point p on C there is a neighborhood U(p) such that no chord of C in this neighborhood is parallel to a fixed direction n(p) (the direction of the "y-axis"). Locally monotone curves include all closed convex curves and all piecewise-C1 curves without cusps.

The affirmative answer is also known for centrally symmetric curves.[5]

Variants and generalizations

One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C.[6][7] Moreover, the set of the vertices of such triangles is dense in C.[8] In particular, there is always an inscribed equilateral triangle. It is also known that any Jordan curve admits an inscribed rectangle.

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 C in Rn satisfying "Condition A" that no two chords of C in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals.[4] This class of curves includes all C2 curves. Nielsen and Wright proved that any symmetric continuum K in Rn contains many inscribed rectangles.[5] H.W. Guggenheimer proved that every hypersurface C3-diffeomorphic to the sphere Sn−1 contains 2n vertices of a regular Euclidean n-cube.[9]

References

  1. ↑ Toeplitz, O. : Ueber einige aufgaben der analysis situs Verhandlungen der Schweizerischen Naturforschenden Gesellschaft in Solothurn, 94 (1911), p. 197.
  2. ↑ 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, MR 1506274 .
  3. ↑ 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 .
  4. ↑ 4.0 4.1 Stromquist, Walter (1989), "Inscribed squares and square-like quadrilaterals in closed curves", Mathematika 36 (2): 187–197, doi:10.1112/S0025579300013061, MR 1045781 .
  5. ↑ 5.0 5.1 Nielsen, Mark J.; Wright, S. E. (1995), "Rectangles inscribed in symmetric continua", Geometriae Dedicata 56 (3): 285–297, doi:10.1007/BF01263570, MR 1340790 .
  6. ↑ Meyerson, Mark D. (1980), "Equilateral triangles and continuous curves", Fundamenta Mathematicae 110 (1): 1–9, MR 600575 .
  7. ↑ 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 623685 .
  8. ↑ Nielsen, Mark J. (1992), "Triangles inscribed in simple closed curves", Geometriae Dedicata 43 (3): 291–297, doi:10.1007/BF00151519, MR 1181760 .
  9. ↑ Guggenheimer, H. (1965), "Finite sets on curves and surfaces", Israel Journal of Mathematics 3: 104–112, doi:10.1007/BF02760036, MR 0188898 .

Additional reading

External links

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