The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an unsolved question in geometry asking whether every plane simple closed curve contains all four vertices of some square. It is known to be true if the curve is convex or piecewise smooth; the problem was proposed by Otto Toeplitz in 1911, with early positive results from Arnold Emch and Lev Schnirelmann, but the general case remains open. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementDoes every plane simple closed curve contain all four vertices of some square? 1 Proposed Year Progress Toward ResolutionTrue for convex and piecewise smooth curves and other special cases; the general case remains open. 1 Classification
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Source Inscribed Square Problem (Wikipedia)
Sources
1. Inscribed Square Problem (Wikipedia)
Wikimedia FoundationLead section
The problem was proposed by Otto Toeplitz in 1911.
Lead section, first sentence
Does every plane simple closed curve contain all four vertices of some square?
Lead section, last sentence
The general case remains open.
In Branch: Geometry, Lead sentence
e Toeplitz conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some
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