Square Peg Applet
(a.k.a. the Inscribed Square Applet)
If the applet doesn't show, here is a tentative alternative.
Conjecture [Square Peg Problem (a.k.a. Inscribed Square Problem), Toepliz 1911]:
Every continuous simple closed curve in the plane contains the four vertices of a square.|
The Square Peg Problem is only know to hold true for curves that are "smooth enough", e.g. C1-curves do it. Many proofs for different curve classes are known.
Conjecture [Rectangular Peg Problem]: Every smooth simple closed curve in the plane contains the four vertices of a rectangle with a prescribed aspect ratio r:1.
The Rectangular Peg Problem (for smooth curves) is only known to hold true for r=1, that is, for squares.