Does every closed curve contain the vertices of a square?

This is the question on Futility Closet

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See here. –  Ragib Zaman Mar 23 '12 at 9:07
@RagibZaman thanks –  yiyi Mar 23 '12 at 9:12
The question is essentially settled for $C^2$ curves, locally graphical curves if we restrict ourselves to one codimension. There is still more to be done (rectifiable curves would be interesting). –  Glen Wheeler Mar 23 '12 at 9:15
Another place to look would be Jason Cantarella's page on the topic jasoncantarella.com/webpage/index.php?title=Square_Peg_problem –  Louis Mar 23 '12 at 10:46

According to Wikipedia (inscribed square problem), this is an open problem - that is, we don't currently know the answer. It's known to be true for "nice enough" curves (Stromquist's Theorem).

So it is true for e.g. all piecewise smooth curves, which are the kind we tend to imagine. Stromquist more generally implies an affirmative answer for curves that are piecewise graphs (as in $y=f(x)$) of continuous functions. Also see this page for further discussion.

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I TRIED TO FOUND IT . FOR THIS I HAD DONE A SMALL ACTIVITY. THE ACTIVITY IS AS FOLLOWS- 1. TAKE A GRAPH SHEET AND DRAW A CLOSED CURVE THROUGH THE POINTS WHERE TWO LINES MEET. 2. NOW YOU WILL EASILY COME ACROSS THE DIAGONALS (OR) LINES DRAWN FROM OPPOSITE SIDES OF A SQUARE. 3.NOW JOIN ALL SUCH POINTS. 4.THIS IS MORE APPLICABLE FOR SYMMETRIC CURVES ALSO. YES EVERY CLOSED CURVE CONTAINS VERTICES OF A SQUARE. I AM STUDYING 8 CLASS IN SRI LAKSHMI HIGH SCHOOL IN KURNOOL. MY NAME IS K.NAGA HARISH. IF ADDITIONAL INFORMATION PLEASE SEND TO MY E-MAIL ADDRESS.

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This doesn't answer the question. IF you have a question of your own, please use the "Ask Question" link at the upper right of the page. Also, please refrain from writing in all caps; it is quite visually distracting. –  robjohn Jan 19 '13 at 14:00