Inscribed square problem

Unsolved problem in mathematics:
Does every Jordan curve have an inscribed square?
(more unsolved problems in mathematics)
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 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] The general case remains open.[4]

  1. ^ Toeplitz, O. (1911), "Über einige Aufgaben der Analysis situs", Verhandlungen der Schweizerischen Naturforschenden Gesellschaft (in German), 94: 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, JSTOR 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. ^ Cite error: The named reference hartnett was invoked but never defined (see the help page).